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Physics 117 Lab Manual

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1. Physics 117 Uncertainty Analysis Fall 2011 91 Note that our new measured result is perfectly consistent with our previous result with its somewhat larger uncertainty We cannot tell which is right or which is wrong without further investigation and comparison F 4 2 x Statistical Technique The x technique produces a number which tells you how well your data match the theory you are trying to test Let s imagine that we have a theory which suggests that two measured variables x and y are related y y x As a test of this theory let s further imagine that we do an experiment which generates x and y 0 We are assuming here that the error in the z is negligibly small The x for this data set is given by al x 2 y yi 1 i F 25 il 2 oO where N is the number of data points The reduced x written x is defined by 2 X vat F 26 V where v N n and n is the number of free parameters in the fit A straight line fit has two free parameters the slope and the intercept Despite its appearance x is fairly easy to interpret To see this let s consider a large set of data with only a small number of free parameters so that v N If the data fit well to the theory then we would expect on average that y y a 0 Thus we would also expect that x N and therefore x 1 If on the other hand our theory doesn t fit the data well we would expect y y a gt o on averag
2. The coils are connected to the digital oscilloscope which records the signal as a function of time and then displays that signal on the oscilloscope screen Make a careful sketch of the pattern you observe in your lab notebook Each of the observed wiggles indicates that the magnet has passed through one of the coils Note that the size and separation of the wiggles changes with time Explain why this happens in your lab notebook Repeat the experiment with the magnet reversed Again draw and explain what you see in your lab notebook 5 3 2 Objects Dropped through an Aluminum Tube There are two identical blue cylindrical objects in the box near the apparatus Drop them one at a time through the long aluminum tube Write down your observations in your lab notebook along with the answers to the following two questions Can you explain why the two objects behave differently Why does the spring scale which supports the metal tube read differently in the two cases 5 3 3 Magnetically Damped Pendulum 1 Move the big magnet far away from the pendulum then start the pendulum swinging with a small angular displacement about 10 209 Observe that the pendulum swings Physics 117 Experiment 5 Fall 2011 34 many times before coming to a halt 2 Stop the pendulum Place the magnet so the pendulum can pass easily between the pole faces 3 Now release the pendulum from roughly the same starting displacement as y
3. it Figure 6 1 6 1 2 Inductors Now consider the potential difference for an inductor The voltage across an inductor is proportional to the rate of change of current through the inductor di t dt uL t L 6 4 Eqs 6 3 and 6 4 suggest that capacitances and inductances are in some sense complements of one another e For a capacitor the current is proportional to the time derivative of the voltage e For an inductor the voltage is proportional to the time derivative of the current This situation might remind you of the relationship between velocity and position for the motion of a mass on a spring or the motion of a simple pendulum In fact these systems show very similar behavior simple harmonic motion and resonance in particular to that of the circuit you will study today This is due to the fact these mechanical systems are also described by Eqs 6 3 and 6 4 with the symbols changed as we ll see below 6 1 3 LC Oscillations a simple case Fig 6 2 shows the simplest imaginable circuit containing both an inductor and a capacitor Suppose that at t 0 the capacitor is somehow charged as shown As in an RC circuit current will start to flow but unlike the case of the RC circuit q t will not simply decay monotonically to zero it will overshoot The capacitor will become oppositely charged then current will flow clockwise and so on The system continues to oscillate spontaneous
4. accepted value for the speed of light 3 Wu s result is worse because it disagrees with the accepted value for the speed of light 4 Wu made a mistake in measuring the speed of light Without knowing the uncertainties in these measurements however it turns out that we cannot assess the results at all Physics 117 Uncertainty Analysis Fall 2011 83 F 1 Expressing Experimental Uncertainties Suppose that we have measured the distance between two points on a piece of paper There are two common ways of expressing the uncertainty associated with this measurement ab solute uncertainty and relative uncertainty In both ways the measured quantity is expressed in the form Ccasuned e TOT F 1 Here Test is the best measured value usually from an average of a set of measurements and x is the uncertainty in the best measured value The measurement is always a range of values not just the best value F 1 1 Absolute Uncertainty We might express the result of the measurement as 5 1 cm 0 1 cm F 2 By this we mean that the result usually an average result of the set of measurements is 5 1 cm but given the conditions under which the measurements were made the fuzziness of the points and the refinement of our distance measuring equipment it is our best judgment that the actual distance might lie between 5 0 cm and 5 2 cm Incidentally an alternative shorthand way of expressing this uncertainty looks
5. The solution of the differential equation with the initial condition that Q t 0 0 since the capacitor is initially completely discharged is Q t VoC 1 E e 4 7 Exercise 1 Verify that Eq 4 7 provides a solution to Eq 4 6 when V t Vo You can do this by evaluating the derivative of Eq 4 7 and substituting this result and V t into the differential equation Note that the product RC determines the time scale required for the capacitor to charge This product is called the time constant for the circuit Te RC Unfortunately we cannot measure the charge stored in the capacitor directly but we can measure the potential difference across the capacitor say with the oscilloscope The electrical potential across the capacitor is given by Volt 20 y 1 zs gee l 4 8 Physics 117 Experiment 4 Fall 2011 25 Exercise 2 Using Eq 4 8 draw a graph of Vo t as a function of t Now suppose that the capacitor is completely charged The potential difference across it is equal to Vo Then suddenly the voltage drops back to 0 Let s reset our time axis so that this new drop occurs at a new t 0 In this case the charge stored in the capacitor and the potential across the capacitor begin to decrease with time Q t VCe EC 4 9 Volt We EC 4 10 Note once again that the product RC sets the time scale for the capacitor to discharge Exercise 3 Using Eq 4 10 sketch a graph of the electrical potential across
6. inevitable experimental uncertainty in reading their locations from the graph paper The two points should also be located at easy to read crossings on the graph paper Mark each of those points with a heavy but not too large dot and draw a circle around the dot Read the coordinates of each point off the graph The slope of the line is defined as the change in y the vertical coordinate divided by the corresponding change in x the horizontal coordinate You may know this in some other Physics 117 Graphical Presentation of Data Fall 2011 62 form such as rise over run To calculate the slope use slope m LZ B 2 Ta Ly substituting your values for the points 21 y1 and 2 y2 For example if your points are 1 0 sec 8 8 m sec and 6 0 sec 46 3 m sec then m 7 5 m sec Notice that the units of m are the units of rise over run Now that you have the slope find the intercept from intercept b y mz B 3 That is you can read b directly off off the graph or you can use the slope and one point to determine b Use either point for x1 y1 Both lie on the line so either will work In the example above we get that b 1 3 m sec Notice that the units of b are those of the y variable Once you have determined the values of m and b from the graph you can quote the equation for your straight line For example if m 7 5 m sec and b 1 3 m sec then the equation of your straight line i
7. with each other in person or electronically once the writing process has begun Specific questions concerning the writing of reports should be directed to the instructor or teaching fellow In addition laboratory partners are expected to share equally in the collection of data The sharing of drafts of reports use of any data or calculations other than one s own or the modeling of discussion or analysis after that found in another student s report is considered a violation of the statement of intellectual responsibility We wish to emphasize that intellectual responsibility in lab work extends beyond simply not copying someone else s work to include the notion of scientific integrity i e respect for the data By this we mean you should not alter fudge or make up data just to have your results agree with some predetermined notions Analysis of the data may occasionally cause you to question the validity of those data It is always best to admit that your results do not turn out the way you had anticipated and to try to understand what went wrong You should NEVER ERASE data that appear to be wrong It is perfectly legitimate to state that you are going to ignore some data in your final analysis if you have a justifiable reason to suspect a particular observation or calculation Experiment 1 Kirchhoff s Rules For the first half of the semester you will study various topics of electromagnetism One of the most successful
8. you may wish to reconsider your capacitor geometry to make the capacitance big ger For the purposes of prediction in this lab take the dielectric constant for wax paper to be x 3 7 1 0 4 4 RC Response to a Step Change in Voltage We wish to verify that the equations developed in Section 4 1 describe the response of an RC circuit to step changes in the applied voltage We will then use that response to determine the capacitance of the capacitor you have built from measurements of the time constant T RC of the circuit For a circuit with a large time constant i e a minute or more it would be possible to study the charge and discharge processes by placing an ordinary high resistance voltmeter across the capacitor and reading the potential difference every few seconds Alternatively the vertical input to an oscilloscope might be connected across the capacitor The time constants of the circuits you are provided with however are only a fraction of a second and the methods described above are inadequate The technique we shall employ consists of arranging to have the capacitor go through a series of identical charges and discharges in a periodic fashion The scope will be triggered so that similar charging and discharging cycles are superimposed on the screen We will use the function generator to create the switching on and off of the voltage When set for a square wave output the function generator acts like a battery with a
9. 2011 12 2 1 4 Resistors in Series and Parallel Optional Suitably alter your circuit to test the hypothesis that resistors in series add according to the equation R Ri Ro 2 1 and that resistors in parallel add according to 1 1 1 RR 2 2 R To ee Again five to ten data points are all that are required for each case Note you should still be measuring current through the resistors in the same way you did in Sec 2 1 2 2 2 The Laboratory Report This experiment will be written as your first formal lab report Please follow the guidelines for writing a formal lab report in appendix E Your report should include e diagrams of the circuits used e all data in neat and organized tables e graphs of the data The graphs should be titled and axes should be labeled e answers to the questions raised e and a conclusion of the results and findings Experiment 3 Introduction to the Oscilloscope The oscilloscope is one of the most powerful and versatile devices in science and technical fields from physics to medicine e g heart monitors In today s experiment you will be come familiar with some of the common operations of an oscilloscope Ask your instructors to tell you more details about the history and inner workings of your scope 3 1 Comments e It is impossible to damage one of these scopes by twiddling the knobs So do not be inhibited try things out and see what happens It is possible however to da
10. College Writing Center also has links on its webpage to several online resources The style of scientific writing is definitive concrete and fact based It is not poetic literary sarcastic or opinionated Specific styles of writing the should not be used in your report Editorial expressions of opinion or commentary The language should be simple and substantive based on the evidence presented in the report It should not be an expression of how the writer feels about the experiment Bombastic inflated or grandiose language Reports are not exercises in creative writing The purpose of the report is to educate the reader not impress the reader Keep the language simple Verbiage excessively wordy but conveying little or no information Do not ramble Abstract void of concrete real world meaning Use physical terminology and use it correctly Circumlocution excessive use of words to explain a concept or idea Do not over explain the experiment from first principles Be specific to the subject matter Standard conventions include Write in narrative prose not outline form This is especially true when writing sections of the report like the experimental technique Do not recite the procedure outlined in the lab manual Use simple complete sentences Sentences are expressions of one complete thought fact or idea The simpler the sentence the better Avoid excessive use of qualifiers modifier and subordinat
11. F 2 Determining Experimental Uncertainties o F 3 Propagation of Uncertainties ee F 4 Assessing Uncertainties and Deviations from Expected Results F 5 The User s Guide to Uncertainties Bibliography 95 General Instructions The laboratory sessions of Physics 117 are designed to help you become more familiar with fundamental physical concepts by actually carrying out quantitative measurements of physical phenomena The labs are designed to help you develop several basic skills and several higher level skills The basic skills include the following 1 Being able to relate abstract concepts to observable quantities For example know ing how one determines the electrical resistance of a device from easily measured quantities This skill includes the ability to estimate and measure important physical quantities at various levels of precision 2 Knowing and applying some generally useful measurement techniques for improving the reliability and precision of measurements such as using repeated measurements 3 Being able to estimate the experimental uncertainties in quantities obtained from measurements The higher level skills include the following 1 Planning and preparing for measurements 2 Executing and checking measurements intelligently 3 Analyzing the results of measurements both numerically and wherever applicable graphically This skill includes assessing experimental u
12. There are special BNC cables that can be plugged into these connectors you should try this out now to get the sense of how these connectors work Note that they push on and then turn to make the solid connection The voltage difference on a BNC cable is measured between the central pin and the outer shield There are also adapters that convert from BNC to banana in which case there is a red connector and a black connector These are very similar to the connectors on the handheld multimeter with one important exception that we will discuss in a little while 3 gt 300 v AN 300V CAT II Ea Figure 3 3 The three inputs to the oscilloscope Figure from the Tektronix user manual 3 3 Procedure 1 Initial Settings a Push the button to activate the CH 1 menu If you press the button again the trace will disappear You can make it reappear if you press it once more You should now see some menu items on the right of the screen each associated with a soft key button just to its right b Using the appropriate soft key to the right of the screen set the CH 1 Coupling to DC c Adjust the vertical Scale knob for CH 1 to 1 VOLT DIV The value of the scale appears in the bottom left corner of the screen Fig 3 2 d Push the button to activate the CH 2 menu Set up this channel to also have 1 VOLT DIV and DC Coupling The letters BNC stand for Bayonet Neill Concelman after the style of the connector and it
13. Yi C 4 i 1 i 1 and N N gt bai mz y Ei C 5 i 1 i 1 The constants m and b although unknown can be factored out of the summations This factorization leaves behind a slightly mysterious looking term int he first equation since N N S bby 1 bN C 6 i l 4 1 Physics 117 Linear Regression Analysis Fall 2011 68 since the sum of 1 over N observations is just N We may now write Eqs C 4 and C 5 as Nb md zi gt y yx m Sox a C 8 where we have left off the limits of the sums for the sake of typographical simplicity PES Q xN The sums in Eqs C 7 and C 8 may be calculated from your data and should be treated as known quantities Hence you have two equations in two unknowns m and b and the procedure for finding them is straightforward Solving the equations above for m and b gives N iYi i i p Naw ALA as NO x7 X zi and 01034 037 D ziy b A TAN RN NALA C 10 NQ_27 X zi Equations C 9 and C 10 give you an expression for the slope m and intercept b of the best fit straight line for your data in terms of your measurements Remember here that best fit means the line that minimizes the squares of the differences between the observed y values and the calculated y values If you have a reasonably standard scientific calculator it probably has a canned routine for performing this kind of fit EXERCISE I Performing a linear regression by hand with the assist
14. You might try the analogous experiment with a pendulum constructed from an object tied to a string Hold one end of the string in your hand and shake it at various frequencies Here is another way of coming to the same conclusion If we consider Eq 6 3 and assume that the potential across the capacitor is given by volt Vo sin wt 6 10 then Eq 6 3 gives ilt wC Vo cos wt 6 11 Equation 6 11 tells us two things First the amplitude of the current is related to the amplitude of the oscillating EMF I wCVcg This looks something like Ohm s Law with 1 wC playing the role of resistance In the jargon of electronics 1 wC is called the reactance or impedance of the capacitor The crucial point is that the impedance of the capacitor varies with frequency For high frequencies for which the current is rapidly Physics 117 Experiment 6 Fall 2011 39 oscillating in time the capacitor does not have a chance to charge or discharge very much and its effect impedance in the circuit is small For very low frequencies the capacitor has a very high impedance and prevents very slowly changing currents from flowing in the circuit Secondly there is a phase difference of 90 between the current and the potential difference for a capacitor Exercise 1 e Show that 1 wC has units of ohms e Sketch a graph of v t and i t from Eqs 6 10 and 6 11 Now let s look at an inductor using Eq 6 4 Suppose that the current varies sinusoi
15. a straight line filling a spherically symmetric volume If the total energy per unit time produced by the source is constant the intensity energy per unit area per unit time must fall off with distance from the source In fact since the area of a spherical shell surrounding the source increases as the square of the distance r from the source the intensity is expected to exhibit inverse square law behavior when measured as a function of r Exercise Write down a mathematical expression that relates J to r as described in words in the previous sentence Draw a picture if this helps Experiment and Analysis On an optical bench align a light bulb 10 cm in front of a photo sensor light detector as shown in Fig 7 3 The detector is designed so that it will give an output voltage proportional to the intensity of the light hitting the sensor so long as the voltage is less than 300 mV That is for low intensities the detector responds linearly to the intensity We will want to conduct this experiment in the linear region of the detector To make sure this happens place a holder that contains a collection of light absorbers these are actually just partially darkened overhead transparencies cut into 2 inch squares between the bulb and the detector so that the output reading is close to but not more than 300 mV The absorber filter allows only a fraction of the light to penetrate through it It should be positioned very close to the bu
16. almost total decay of one of the exponential curves but sufficiently spread out so that you can measure it We recommend using the vertical Position knob on the scope to put the level that the exponential decay is approaching on the bottom line of the screen Measure Vc t as a function of t for about 10 values of t Remember your raw data for both Vo and t are in ticks or boxes do not forget to note oscilloscope VOLTS DIV and SEC DIV settings You can also use the Cursor function of the scope to measure voltages at several values of t Alternatively you can save the scope trace to a USB thumb drive and then import that data into a computer Now change the 10 kQ resistor shown above Fig 4 5 to a 100 kQ resistor and repeat your measurement 4 5 Analysis Analyze and graph at least one of the decays of the sort just described before leaving the lab You will show your graph and your results to one of the instructors The theory developed in Section 4 1 says that the voltage across the capacitor is of the form Vo t Ae RO 4 12 where A is some constant Whence it follows that t 1 t ln A 4 1 n Vo t ln RC 4 13 Physics 117 Experiment 4 Fall 2011 28 So a graph of ln Vo vs t should be a straight line Use a spreadsheet program to plot the data with error bars and to find the slope of the best fit straight line Then find C You have two values of C from the two sets of measurements with di
17. amp A 1 ampere 1 C s 1 A To get water to flow through a pipe there must be a pressure difference between one end and the other The analogous concept with electrical circuits is electrical potential difference The symbol for electric potential difference is V and its units are volts V 1 volt 1 J C 1 V Hooking both ends of a hose up to the inlet and outlet of a water pump will cause water to flow through the hose A battery plays a similar role for electric circuits Attaching one end of a wire to the positive end of a battery and the other to the negative end will cause charge to flow through the wire Please take our word for this and refrain from discharging the batteries in this way Batteries are special kinds of electrical pumps because they pretty much always maintain the same potential difference between the positive and negative ends no matter what the current The ratio between the potential difference between say the ends of a wire and the electric current flowing through the wire is called the resistance R of the wire _ AV R I 1 1 The units of resistance are ohms Q 1 ohm 1 V A 1Q Because electrical circuits can get enormously complicated an elaborate system of symbols has been developed to help us draw the circuits we study A battery for instance is represented by the symbols in Fig 1 2 The longer vertical line indicates the position of the positive end of the battery A resistor is repre
18. applications of electromagnetism is the development of electric circuits You will construct simple circuits and study their electrical properties To have some initial understanding of how circuits work you will employ two simple rules of electric circuits Kirchhoff s Rules At the heart of these rules are laws of conservation specifically the law of conservation of energy which you learn about in P116 or P123 and the conservation of electric charge Before we can go further we will need to introduce two more electrical quantities namely electrical current and electrical potential difference or voltage 1 1 Background In this lab you will be causing electric charge to flow through a variety of materials This flow of charge will not be covered in lecture for several more weeks In lab therefore we will be taking a less formal more phenomenological point of view The motion of charge is called an electric current in analogy to the flow of fluid Consider the flow of a fluid like water through a pipe Fig 1 1 P ee A Figure 1 1 Flow of fluid across a plane P Physics 117 Experiment 1 Fall 2011 2 The current through this pipe is characterized by the amount of fluid measured by volume flowing past a plane P per unit time Similarly if charge is flowing down a wire the electrical current is characterized by the amount of charge flowing through P per unit time The symbol for electric current is J and its units are amperes
19. compute them numerically For example of 2 EFI Y a 59 2 Ae 2 F 15 Connection to the traditional simple rules To see where the usual rules for combining uncertainties come from let s look at a simple functional form R xz y F 16 Using our procedure developed above we find that r R x 6yR dy F 17 and combining uncertainties yields OR y da dy F 18 Physics 117 Uncertainty Analysis Fall 2011 89 The traditional rule for handling an additive relationship says that we should add the two absolute uncertainty contributions We see that the traditional method overestimates the uncertainty to some extent Exercise Work out the result for a multiplicative functional relationship R f x y ty Compare our method with the traditional method of adding relative uncertainties Example Suppose we have made some measurements of a mass m a distance r and a frequency f with the following results for the means and standard deviations of the measured quantities m 150 2 0 1 r 5 80 0 02 f 52 3 0 4 Note that we have omitted the units and hence lose 5 points on our lab report From these measured values we want to determine the best value and uncertainty for the following computed quantity F mrf F 19 The best value is computed by simply using the best values of m r and f F 2382875 2 we ll tidy up the number of significant figures later on Let s
20. digital multimeter The real power of an oscilloscope comes when we allow the oscilloscope to control the horizontal voltage for us Change the DISPLAY Format to YT Adjust the horizontal Scale knob to 500ms div The value of the scale appears at center bottom of the screen You should now see a line slowly scanning horizontally across the screen Use a stop watch to time how long it takes to go all the way across the screen Is your result consistent with the 500ms div switch setting 5 A Rapidly Varying Voltage We will now use a function generator also called a frequency generator or a signal generator to produce a rapidly varying signal for the oscilloscope Use a BNC to banana adapter on the signal generator and connect the black terminal to the ground also black connection on the scope and its red terminal to the CH 1 input Other settings of the function generator are VOLTS OUT push in SWEEP out position Physics 117 Experiment 3 Fall 2011 18 RANGE Hz push in 1k button AMPLITUDE fully clockwise FUNCTION push in sinewave button FREQUENCY set to 1 0 DC OFFSET push in Turn on the generator s power switch the red button at the left of the front panel Set the oscilloscope s horizontal Scale knob to 1 ms div Does the scope correctly tell you that the sinusoid you are observing has a frequency of about 1 kHz The frequency of the signal appears on the bottom right of the screen Note
21. of data points that are presented on a graph however the overall uncertainties in the slope and intercept must result from some composite consideration of the uncertainties in all of the data a t once Some graphing programs such as Excel do allow for such a calculation But what does it mean Let us first consider a graph containing only the raw data points for which no uncertainty bars are included You can refer to Appendix B Graphical Presentation of Data for a discussion of uncertainty bars You can get a feel for the effect of the uncertainties in the slope and intercept due solely to the scatter in the data by identifying two worst cases that are still consistent with the data One case is the line that is systematically a little too low at the left hand side of the graph and too high and the right hand side but fits the data well near the middle This line combines the largest acceptable slope with the smallest acceptable intercept that is its slope is m Am and its intercept is b Ab where Am and Ab are the uncertainties associated with the slope and intercept respectively The other worst case combines the smallest acceptable slope with the largest acceptable intercept this line is a little too high at the left end a little too low at the right end and a fairly good fit in the middle Exercise II gives an example that allows you to visualize the extremes for a fit line EXERCISE 2 Assume that for the data abov
22. record all of your data so that you will not fall prey to the temptation to erase As mentioned above Appendix A gives instructions on how to keep a good lab note book You will be expected to adhere to these guidelines throughout the semester In fact we feel that keeping a good laboratory notebook is so important that we have decided to base part of your laboratory grade equivalent to one formal lab Physics 117 General Instructions Fall 2011 vii report on the quality of your lab notes Your notebook will be evaluated at the end of the semester Note You should have your lab notebook initialed by one of the instructors before you leave each lab session 2 Naturally you will need to pay attention to your data taking technique Throughout the semester you will be learning how to use various types of measurement equip ment sometimes crude and sometimes sophisticated In all cases the quality of your data will depend on your understanding when and how to use the equipment most effectively It is always more important to put care and thought into the setup for a measurement than it is to attain a sometimes deceptively high level of accuracy from a meter Of course you will want to optimize the accuracy but only when you re sure you re making the right measurement For example if you build a circuit incorrectly it doesn t matter how many digits you get out of a voltage reading If the circuit is wired wrong the results will be
23. related through y mx b On the other hand if x and y are completely unrelated to each other r 0 Real experimental data points always include some experimental error so real correlation coefficients always have absolute magnitudes less than 1 If the correlation coefficient for your data is close to 1 you are justified in concluding that x and y are really related in some way The correlation coefficient is therefore most useful when you re trying to decide if your data are good enough for you to claim that a variable y does in fact depend on another variable x The correlation coefficient is not very useful however when you re pretty sure for say theoretical reasons that y is determined by x In that case you re assuming that the correlation is valid and you want to know the slope and the intercept of the line The correlation coefficient will tell you in a general way how good your data are but it doesn t tell you anything about the experimental uncertainty in the calculated values of the slope and the intercept The correlation coefficient is also not very good for deciding if a linear fit is good enough or if any of your measurements are suspect It s surprisingly insensitive to curvature in the data or the presence of a single dubious data point For that reason we do not report the correlation coefficient with the results of the fit Appendix D Power Law Curve Fitting Although many processes in nature are linear m
24. resonant frequency fo 6 2 The Experiment LCR Series Resonance We will want to use the oscilloscope to examine the potential difference across the inductor and the capacitor as well as across the resistor In order to carry out these measurements we need to set up the oscilloscope in its so called differential mode in which the display is proportional to the potential difference between the connection to the CH1 input and the connection to the CH2 input We need to use this mode because we can only have one ground point in a series circuit First connect a wire of one color to the CH1 input red terminal and a wire of a different color to the CH2 input red terminal Use the following oscilloscope settings e Press the yellow button to activate CH 1 menu Make sure Probe is set to 1x Voltage e Press the blue button to activate CH 2 menu Make sure Probe is set to 1x Voltage e Adjust both CH 1 and CH 2 vertical Scale knobs to 200mV div It is important that both channels be on the same scale e Adjust the horizontal Scale knob to 5us div Remember us 107 s e Press the red button to activate the MATH menu e Set MATH Operation to e Set MATH Sources to CH1 CH2 e Press the button to activate TRIGGER menu e Set TRIGGER Source to Ext e Set TRIGGER Mode to Auto Use the following function generator settings Physics 117 Experiment 6 Fall 2011 41 e SINE WAVE output e VOLTS OUT OUT position
25. shown on the left not as shown on the right you construct the circuit e Use the screwdriver to turn the variable resistors or pots all the way clockwise Be careful not to twist too much e Record the current as measured by the ammeter The uncertainty in the measurement is 1 unit of the last digit e Use the analog voltmeter to measure the voltage across the 470 Q resistor Fig 1 9 Estimate the uncertainty e Now use Eq 1 1 to calculate the value of the resistance of the resistor Does the calculated value agree with the given value What is the uncertainty in the calculated value Use Eq F 38 to propagate the uncertainty e Now calculate the power dissipated by the resistor In electrical terms the power is defined P IAV 1 2 Power is measured in units of J sec or watts Using the units of voltage and current can you show the product of voltage and current gives power in watts e In calculating resistance and power you are multiplying or dividing two measured quantities each with its own uncertainty Use Eq F 37 to propagate the uncertainty in the power e Now replace the analog voltmeter with the second digital multimeter in the voltage mode and measure the voltage across the resistor again What is the uncertainty in the new measurement of voltage Use the new measured voltage and calculate the resistance and power Physics 117 Experiment 1 Fall 2011 6 TRIPLETT 1101 8 Figure 1 10 Triplett 110
26. simply How much greater or smaller than the stated value could the measured quantity have been be fore you could tell the difference with your measuring instruments If for instance you measure the distance between two marks as 2 85 cm and judge that you can estimate halves of mm the finest gradations on your meter stick you should report your results as 2 85 0 05 cm More details on uncertainties are given in Appendix F An important if not the most important part of the analysis of an experiment is an assessment of the agreement between the actual results of the experiment and Physics 117 General Instructions Fall 2011 viii the expected results of the experiment The expected results might be based on theoretical calculations or the results obtained by other experiments If you have correctly determined the experimental uncertainty for your results you should expect your results to agree with the theoretical or previously determined results within the combined uncertainties If your results do not agree with the expected results you must determine why Several common possibilities are the following a You underestimated the experimental uncertainties b There is an undetected systematic error in your measurement c The theoretical calculation is in error d d e Some combination of the above The previous measurements are in error or Sometimes these deviations are real and in
27. solutions are oscillatory in time with frequency w 1 VLC Physics 117 Experiment 6 Fall 2011 38 6 1 4 LCR circuit with a sinusoidal EMF What has been omitted from the preceding theory is the role of the resistance that is inevitably present Whereas capacitors store energy in electric fields and inductors store energy in the magnetic fields resistors dissipate energy producing heat If a small amount of resistance is present we will observe a damped sinusoidal oscillation a sinusoid of steadily diminishing amplitude If a large amount of resistance is present the overshoot characteristic of oscillation may not even be seen Let us add some resistance to the circuit shown in Figure 6 2 and to compensate for the loss of energy we will add a sinusoidal EMF of adjustable frequency Now our circuit looks like that shown in Fig 6 3 L R Figure 6 3 The EMF can drive an oscillatory current in this circuit alternately clockwise and coun terclockwise maintaining the amplitude of such a current at a steady value in spite of the energy being dissipated in the resistance The frequency of this current will be the same as the frequency of the oscillatory EMF but one might well expect correctly that the size of the resulting current will be greatest when the frequency of the driving EMF matches the natural resonant frequency of the LC circuit the frequency of natural oscillations that you would predict from Eq 6 8
28. t need to be too fussy about taking up the whole page or making the divisions nice You should label the axes and title the graph though Flaky data points show up almost immediately in a graph which is one reason to graph your data in the lab Skipping this low level graphing step can allow problems in the data collection to propagate undetected and require you to perform the experiment again from the beginning Graphing each point as you take it is probably not the best idea though Doing so can be inefficient and may prejudice you about the value of the next data point So your best bet is to take five or six data points and graph them all at once Physics 117 Graphical Presentation of Data Fall 2011 61 Graphing your data right away also flags regions in your data range where you should take more data Typically people take approximately evenly spaced data points over the entire range of the independent variable which is certainly a good way to start A graph of that survey data will tell you if there are regions where you should look more closely regions where you graph is changing rapidly going through a minimum or maximum or changing curvature for example The graph helps you identify interesting sections where you should get more data and saves you from taking lots of data in regions where nothing much is happening B 2 Analyzing your Graph Graphical data analysis is usually a euphemism for find the slope and in
29. the capacitor as a function of time If you are curious your instructors can show you how to derive these two solutions to the differential equation Eq 4 6 4 2 2 Sinusoidal Voltage For the second type of circuit behavior we let the function generator produce a voltage that varies sinusoidally in time V t Vosin wt Vo sin 27 ft 4 11 where w 27 f is called the angular frequency and f is the frequency in Hz or cycles per second In this case the same differential equation 4 6 describes the behavior of the circuit but V t is replaced by the sinusoidal expression given in Eq 4 11 The general solution to the differential equation in this case is fairly complicated If we carry out measurements on this circuit however we find that if we wait a time equal to a few times RC after turning on the circuit then all the voltages in the circuit vary sinusoidally with the same frequency f but may be shifted in phase relative to the voltage source and may have different amplitudes We ll be looking qualitatively at how the amplitude changes as a function of frequency today 4 3 Build Your Own Capacitor Using the materials available in the laboratory build your own parallel plate capacitor Predict what capacitance your capacitor will have using Eq 4 4 If your predicted capac itance is significantly smaller than say 1 nF you may have some trouble in the following Physics 117 Experiment 4 Fall 2011 26 sections
30. the circular wall of the gelatin and exits at the midpoint of the straight side see Fig 7 1 below The light will enter the gelatin normal to its surface You will need to set the height of the laser by propping it up on a book or some such object so that you can see the entrance and exit of the laser light 2 Rotate the ray table until you discover the angle 0 for total internal reflection for water Make sure that the beam is exiting at the midpoint of the straight side Physics 117 Experiment 7 Fall 2011 45 Figure 7 1 Figure 7 2 Basic Optics Ray Table What is it Use it to determine the index of refraction of gelatin from the relation sin0c 1 nc 3 Record the incident and refracted beam angles 0 and 62 respectively starting with 01 0 and increasing 6 in 5 increments for as many data points as you can Make sure to read the 02 values relative to the normal i e relative to the 180 line Plot your data to see if Snell s Law describes your results If so use it to get a value for na How does this value compare with the known value ny 1 333 the index of refraction for water Physics 117 Experiment 7 Fall 2011 46 7 2 Inverse Square Law Behavior Here you will explore the phenomenon that light from a point source propagates outward uniformly in all directions about the source In the absence of any lenses mirrors interfaces etc light will propagate outward from a point source in
31. to get the overall uncertainty in the result The usual argument is the following If we assume that the variables are independent so that variations in one do not affect the variations in the others then we argue that the net uncertainty is calculated as the square root of the sum of the squares of the individual contributions R 4 6 R 2 8 R 2 8 R F 14 The formal justification of this statement comes from the theory of statistical distributions and assumes that the distribution of successive measurement values is described by the so called Gaussian or equivalently normal distribution In rough terms we can think of the fluctuations in the results as given by a kind of motion in a space of variables z y and z If the motion is independent in the xz y and z directions then the net speed is given as the square root of the sum of the squares of the velocity components In most cases we simply assume that the fluctuations due to the various variables are independent and use Eq F 14 to calculate the net effect of combining the contributions to the uncertainties Note that our general method applies no matter what the functional relationship between R and the various measured quantities It is not restricted to additive and multiplicative relationships as are the usual simple rules for handling uncertainties In most cases we do not need extremely precise values for the partial derivatives and we may
32. uncertainty is the same for a certain set of data you can simply indicate that uncertainty at the top of the column of that data You will need at least two columns one for the independent variable and one for each dependent variable It s also good to have an additional column usually at the right hand edge of the page labeled Remarks That way if you make a measurement and decide that you didn t quite carry out your procedure correctly you can make a note to that effect in the Remarks column For example suppose that you realize in looking at your pendulum data that one of your measurements must have timed only nine swings instead of ten If you indicate that with say 9 swings you could justify to a suspicious reader your decision to omit that point from your analysis Do not Erase It is always a good idea to record data comments and calculations in ink rather than in pencil That way you avoid the temptation to erase data that you think are incorrect You never should erase calculations data comments etc because the original data calculation and so on may turn out to be correct after all and in any case you want to keep a complete record of your work even the false starts If you believe that a calculation for example is Physics 117 Keeping a Lab Notebook Fall 2011 58 wrong it is better to draw a line through it and make a note in the margin than to erase the calculation You can always make th
33. use our partial derivative method to find the uncertainty First let s determine the effect to to m F ink 5 m rf 8m 1586 F 20 om Next we look at the effect of r F r mf 8r 8217 F 21 r And finally the effect of f is given by F 5 f 2mrfof 36449 F 22 We see immediately that the measurement of f has the largest effect on the uncertainty of F If we wanted to decrease the uncertainty of our results we ought to work hardest at decreasing the uncertainty in f Physics 117 Uncertainty Analysis Fall 2011 90 Finally let s combine the uncertainties using the square root of the sum of the squares method From that computation we find that we ought to give F in the following form F 2 383 0 037 x 10 F 23 or F 2 38 0 04 x 10 F 24 in the appropriate units Note that we have adjusted the number of significant figures to conform to the stated uncertainty As mentioned above for most purposes citing the uncer tainty itself to one significant figure is adequate For certain high precision measurements we might cite the uncertainty to two significant figures F 4 Assessing Uncertainties and Deviations from Expected Results The primary reason for keeping track of measurement uncertainties is that the uncertainties tell us how much confidence we should have in the results of the measurements If the results of our measurements are compared to the results expect
34. well organized partially so you can find things and partially so that if anyone questions your results not only will they be able to find things but the layout of your notebook will suggest that you investigated the problem carefully and systematically You should use a bound lab notebook that is not a loose leaf notebook So called quadrille Physics 117 Keeping a Lab Notebook Fall 2011 56 notebooks with rectangular grids on each page are particularly handy for making graphs and tables We strongly recommend that you leave every other sheet in your lab notebook free so that you can jot down additional comments and or add graphs onto those blank sheets after the fact If you wish to add a graph done on a computer or a graph done on regular graph paper to the notebook you may simply tape or glue the graph into your notebook Next we will discuss some of the information that goes into your lab notebook Introduction You should begin each new experiment on a fresh page in your notebook Leave some room for pre lab lecture notes Start with the date and brief title for the experiment just enough to remind you what that section of your notebook is about Then give a list of the equipment identifying large pieces of equipment with manufacturer s name and the model For large pieces of equipment record the serial number too With this information you can repeat the experiment with the identical equipment if for some reason you ar
35. what happens to the display when you turn the Scale knob either direction Explain your observations From these exercises we learn that the oscilloscope can give us a visual display of a voltage that varies rapidly in time 6 Trigger Now let s see how the trigger controls work Push the Trig Menu button to activate the TRIGGER menu You should see the TRIGGER Slope set to Rising Look at the screen at the top of the grid and you should see a white arrow point down at the trigger point of the trace If the arrow is pointing to the left or right then press the button that will bring the zero of time back to the center of the display Is the slope of the trace at this point positive or negative Now switch the TRIGGER Slope to Falling What is the slope of the signal at the trigger point now You can also adjust the Trigger Level knob that controls the arrow on the right side of the grid The values of the trigger slope and level appear on the bottom right of the screen How does the signal change as you adjust the trigger level Why 7 Square Waves and Triangle Waves Replace the frequency generators s sinusoidal output with its square wave output and see what you get on the oscilloscope screen Also vary the frequency setting of the frequency generator Then try the triangle wave output 8 Yet another time varying signal Connect the output voltage of one of the small aluminum boxes to the CH 1 input Adjust the horizontal Scale knob
36. wrong too So take care to think first and always critically assess your measurements as you go along to see if they make sense In fact this is really a preliminary part of the analysis process Analysis 1 In addition to the ongoing analysis you conduct during data taking you will be ex pected to perform a more thorough analysis for each experiment Most importantly you will be asked to obtain meaningful physical results from the measurements Of ten though not always this will be done in the context of a graphical analysis That is usually you will use the method of straight line graphing see Appendix C that you have come to know and love from Physics 16 23 or equivalent to create plots that theoretically should be linear From the slopes of these graphs you will often be able to determine a result that is effectively an average measurement from all of your data Of secondary though not insignificant importance is the need for you to specify some limits of accuracy about your results This phase of the operation is often mistakenly referred to as Error Analysis In fact the expression for determining the range of uncertainty in a particular measurement is appropriately Uncertainty Analysis Below are some guidelines to help with this process 2 Uncertainties The stated results of any measurement is incomplete unless accompanied by the un certainty in the measured quantity By the uncertainty we mean
37. 1 Set vertical Scale knob to 1V div 2 Set horizontal Scale knob to 25ms div 3 Press the button to activate the MEASURE menu 4 Press the top menu button and set Measure 1 Source to CH1 5 Set Measure 1 Type to Pk Pk Physics 117 Experiment 5 Fall 2011 32 6 Press the bottom menu button to go Back to the main MEASURE menu 7 Press the second menu and set Measure 2 Source to CH1 8 Set Measure 2 Type to Freq 9 Press the bottom menu button to go Back to the main MEASURE menu Detailed Procedures 1 Mount the coil on the rotator Make sure the coil will be able to rotate freely in the magnet gap 2 Record the frequency of rotation from MEASURE CH 1 Freq N B Don t forget that there is a difference between angular frequency radians per second and ordinary frequency cycles per second That is w 27f Also measure the frequency of rotation using a stopwatch and the mechanical counter on the rotator Do the two values of frequency agree 3 Record the peak to peak height of the signal from MEASURE CH 1 Pk Pk The amplitude Emax Pk Pk 2 4 Use at least 5 different rotation frequencies ranging from very slow to very fast and record Emax from the oscilloscope trace as a function of rotation frequency 5 Make a graph of max versus angular frequency From these data and the N and A values for your coil given in Table 5 1 obtain a value for B 6 Estimate an uncertainty to be ass
38. 1 B digital multimeter 1 3 1 Series and Parallel Circuits Now let s see what happens when more than one device is in the circuit Take the red device and connect it in series with the 470 2 resistor A series circuit is one in which there is only one path for the current to flow If this is true then right away we know that the current through one device is the same as the current through other device e Measure the voltage across each device using the digital voltmeter What is the sum of the voltages What is the uncertainty of the sum of voltages e Now measure the voltage across the terminals of your power supply How does this compare to the sum of the voltages across the two devices This exercise illustrates one of two principles that underlie much of our understanding of circuitry The principle is call Kirchhoff s Voltage Rule The voltage rule states The algebraic sum of voltages around a closed circuit loop is zero Y AV 0 1 3 How do we interpret our measurements in the context of Kirchhoff s Voltage Rules The power supply is providing energy to establish a current in the circuit remember the units of voltage is Joules Coulomb So between the negative and positive terminals of the power supply the potential increases As the current flows the energy is consumed to move the current through each device i e resistor so there is a potential drop across each device Physics 117 Experiment 1 Fall 2011 7 ey co
39. 2 653 0 5 cm F 28 The value Alice should quote for the perimeter is therefore P 12 0 0 5 cm F 5 2 Multiplication and Division For multiplication and division uncertainties propagate in a slightly different manner One must first calculate the fractional uncertainty of a quantity If some value q has an associated uncertainty q then fractional uncertainty F 29 q Once we know the fractional uncertainties for each measured quantity in the product or quotient we can add them in quadrature to get the fractional uncertainty of the result To get the absolute uncertainty of the result simply multiply the fractional uncertainty by the result Example 2 Bob wants to find the area of a triangle He knows the length of the base b 4 2 0 2 cm and the height h 5 8 0 1 cm What is the area of Bob s triangle Physics 117 Uncertainty Analysis Fall 2011 93 Answer The equation for the area is A 30h 12 2 cm The final uncertainty in the result is found by summing the fractional errors in quadrature and then multiplying by the 2 2 JA A 5 0 6 cm F 30 Bob should quote his total area as A 12 2 0 6 cm result F 5 3 Multiple Operations For combinations of operations the best approach is to break the problem up into pieces that can be solved by using the rules given above and then combine the uncertainties of each of these pieces appropriately The following example sho
40. 7 3 Also note that it would be silly to give six significant figures for X Common sense suggests reporting the value of X as say X 339 5 7 3 N m s or X 339 7N m s F 3 2 General Method The general treatment of the propagation of uncertainties is given in detail in texts on the statistical analysis of experimental data A particularly good reference at this level is Taylor 2 Here we will develop a very simple but general method for finding the effects of uncertainties Suppose we want to calculate some result R which depends on the values of several mea sured quantities x y and z R f 0 y z F 12 Let us also suppose that we know the mean values and standard deviations for each of these quantities Then the uncertainty in R due to the uncertainty in x for example is calculated from R s72 dx F 13 where the subscript on 6 reminds us that we are calculating the effect due to x alone Note that the partial derivative is evaluated with the mean values of the measured quantities In a similar fashion we may calculate the effects due to dy and 6z Physics 117 Uncertainty Analysis Fall 2011 88 N B By calculating each of these contributions to the uncertainty individually we can find out which of the variables has the largest effect on the uncertainty of our final result If we want to improve the experiment we then know how to direct our efforts We now need to combine the individual contributions
41. A series RC circuit attached to a time varying voltage source Applying Kirchhoff s Voltage Rule to the preceding circuit gives us the following equation V t I t R Volt Er de A t 4 5 Physics 117 Experiment 4 Fall 2011 24 where we have made use of the relationship between the potential difference across the capacitor Vo t and the charge Q t stored on one of the capacitor plates Note that the current through the circuit I t is simply the rate of change of the charge Q t piling up on the capacitor plate Dividing through by R gives a differential equation for Q t dQ V t 1 oS Fe ot 4 dt R RC S We will be studying two different types of behavior described by this equation 4 2 1 Step Changes in the Voltage V t For the first type of behavior we will have an voltage source that jumps very quickly between two voltage values one we call Vo the other value will be taken to be 0 Let s consider the following scenario Suppose that the voltage supplied by the function generator has been 0 for a long time What long means will become apparent in a moment Then we know that the voltage across the capacitor will be zero the capacitor is completely discharged Next let s assume that the function generator voltage suddenly jumps to the value Vo and stays at that value Let s call the time at which that jump occurs t 0 In the differential equation for Q t Eq 4 6 we treat the voltage value as a constant
42. Figure 1 The caption should be a brief description of the graph and the quantities plotted Adjust scale of axes so data points fill the whole graph Empty space is a waste For more details refer to Appendix B Graphical Presentation of Data Tables Data tables should be organized in columns The head of each column should be labeled and include units If the quantities in the column all have the same uncertainty then the uncertainty can be indicated at the head of the column as well for example Time 0 001 sec Each table should have a descriptive title starting with a number Table 1 for easy reference Do not split tables across pages of the report Do not include long tables like the data tables from motion sensor Keep those tables in a spreadsheet E 2 Composition A formal lab report is paper similar to other papers written in other courses and should follow the accepted conventions of composition The report should be written in narrative style Correct spelling grammar punctuation and syntax are essential Physics 117 Formal Reports Fall 2011 79 Always make use of resources when writing dictionary thesausus handbook Probably the best handbook for college writing is Strunk and White The Elements of Style 4th ed It is particularly good in its no nonsense approach to writing and it has been a standard handbook for collegiate writing for almost 100 years and it is the standard handbook for this course Amherst
43. Physics 117 Lab Manual source Amherst College Fall 2011 Contents General Instructions v 1 Kirchhoff s Rules 1 Iel Background io nda eis ig og de ee p kw Ed sde dhe p a A y 1 1 2 Digital Multimeter a a S a a a a a a a E 4 1 3 The Experiment a so e toeris aoura sa a ka ESA ee 4 1 4 Supplement Power Supply Circuit Diagram 9 2 Ohm s Law Formal 10 2 17 Ohms LaW ale 8s the ya a a eS a Re Ge ew Se el ds 10 2 2 The Laboratory Report e 12 3 Introduction to the Oscilloscope 13 3A Comments a riea iia ara a A SR eee ee ee So 13 3 2 SOSCUIGSCOPE se a od ip ea Be od Sg aetna ee dee es dal Se E Rod Sepa 13 323 Proc diires s la lips o A dd Bak eee ara wok ea ele 15 3A Checking Outs ar ao asa doses ee pee eel eee eee a GLA eee a 20 4 Capacitors 21 Aril Introduction siae de a a ta do 4 2 Charging and Discharging 0 2 000000 ee eee 4 3 Build Your Own Capacitor 00 00 0002 4 4 RC Response to a Step Change in Voltage 2 A57 AMAS Va ir ALR oP th tale Wa de nes Be hee AE Ged Ae tal Ed ys 4 6 Sinusoidal Response optional 0 o oe o Faraday s Law and Induction 5 1 Faraday s Law Introduction 2 0 0 0 0 0 eee ee 5 2 Determining B field strength from the induced EMF 5 3 Minilabs on Induction sse idou 0000000000 eee eee RLC Circuits Formal 6l Antroductionics 3 aac ee e Rob ae Oe NA She 6 2 The Exper
44. a set of measurements of s and s to determine the focal length of the composite lens Once you have found the combined focal length solve for the focal length of the negative lens using e fi fo fomi where one of the focal lengths f on the left hand side of Eq 8 2 is the focal length of the converging lens you used 8 2 Experiment 9 Interference Diffraction and Polarization In our last laboratory on geometric optics we treated light as if it were composed of rays While this approximation is usually adequate for objects which are large compared to the wavelength of light it is not adequate for describing the interaction of light with small objects In this case it is necessary to consider light as an electromagnetic wave The electric and magnetic fields associated with a light wave are always perpendicular to the light propagation direction The magnitude of the electric field associated with a beam of light propagating in the x direction may be written as E Fo cos 27 G ft 9 1 This describes a travelling wave with an amplitude Ey and a velocity equal to fA where f is the frequency of the light and A is the light s wavelength The intensity of an electromagnetic wave is proportional to the total electric field squared E Because light is a wave it may exhibit the property of interference Consider what would happen if two electromagnetic waves were travelling in the same direction but with their pha
45. al way Physics 117 Power Law Curve Fitting Fall 2011 73 Beam Deflection vs Length 0 8 H 0 6 H 0 4 L Deflection mm 0 2 H 0 0 e AAA 0 2 0 4 0 6 0 8 1 0 Length m Figure D 1 Linear plot of beam deflection as a function of beam length Length m Deflection mm 0 2 0 007 0 4 0 053 0 6 0 165 0 8 0 393 1 0 0 822 Table D 1 Beam deflection data That is y AU _ 2701 _ logy logy D 4 Au uu logz log z As an example Fig D 1 below shows data for an experiment studying the deflection of a loaded beam as a function of the length of the beam The data are listed in Table D 1 A linear plot of the data is clearly not a straight line but the curve might be a power law because it passes through the point 0 0 To test this hypothesis plot log D vs log L sure enough the new graph shown in Fig D 2 looks pretty straight The log log graph passes through the points 0 824 2 538 and 0 0 0 084 These points are the logarithms of the points 0 15 0 0029 and 1 0 0 825 The slope of this line is therefore log 0 825 log 0 0029 0 084 2 538 log 1 0 log 0 15 0 0 0 824 Calculating the logarithms if you have more than five or ten data points even with a calculator gets tedious after a while To reduce this tedium special graph paper called Physics 117 Power Law Curve Fitting Fall 2011 74 Beam Deflection vs Length log log plot
46. ance of a calculator Our experience suggests that people find the derivation above somewhat bewildering ap parently as a result of all the summation signs To make the process of linear regression seem more familiar in this exercise you will carry out one linear regression calculation with a small number of points by hand following the worksheet below You will also check that the calculated slope and intercept do indeed give a smaller value of the squared differences than other similar values We have filled in the first row and a few other spaces so that you can check your work measurement X X Y XY 1 os os 0 45 0 27 a a A 3231 135 1 fo e wis ra Physics 117 Linear Regression Analysis Fall 2011 69 slope NOTA OL A 0 Y C 11 NO XJ LX AAA C 12 N XJ XP intercept After carrying out the calculations above plot the four data points on a sheet of graph paper Then add your best fit line and see how well the line and the observed data seem to agree Then compare the data and your best fit line to two other similar lines y 0 5x 0 25 and y 0 45x 0 3 That is plot those lines on the graph as well Now compare the squared differences using your calculated slope and intercept to squared differences using the other two trial lines above In the tables below Yop is the observed value of Y from the first table and is filled in for you Ya is the value of Y you calculate fr
47. and a stopwatch Use one of the coils for which data are given see Table 5 1 and record its identification number Also record the identification number the ACPL number of the magnet you use Coil N A cm Coil N A cm 1 50 1 51 10 100 3 15 2 50 1 51 11 200 3 40 3 100 1 60 12 200 3 40 4 100 1 60 13 50 5 96 5 200 1 78 14 50 5 96 6 200 1 78 15 100 6 12 7 50 3 02 16 100 6 12 8 50 3 02 17 200 6 47 9 100 3 15 18 200 6 47 Table 5 1 Coil data Oscilloscope Initial Settings 1 Press the Run Stop button until the word Stop appears at the top center of the screen 2 Set the vertical Scale to 200mV div 3 Set the horizontal Scale to 100ms div 4 Press the button until the word Ready appears at the top of the screen Connect the coil directly to CH 1 of the oscilloscope Set the Coupling for that channel to DC You will want to experiment with the horizontal and vertical scale settings Carefully insert the coil between the poles of the magnet and orient the coil so that the plane of the loop is perpendicular to the magnetic field Quickly pull the magnet away from the coil Observe the sign of the induced EMF and its approximate magnitude Do this a few Physics 117 Experiment 5 Fall 2011 31 times and write down your observations Remember to press the Single button to rearm the oscilloscope each time you make a measurement Estimate how long it took you to withdraw the magnet from the width
48. any others are not Power law dependen cies of the form y kr D 1 are particularly common You have in fact already encountered one such relation in the simple pendulum experiment the relation between the length of a pendulum and its period In that experiment you could use dimensional analysis to argue that the value of the exponent should be 0 5 but often you re not so lucky And you don t know the value of the multiplying factor k either What do you do One strategy assuming you do know the exponent amounts to organized trial and error to find k You could for example try different values of k calculate a y x curve for each value and see which value best matches your experimental data This approach is clearly pretty tedious if you have only a calculator And if you start with the wrong value of n your value of k isn t going to be worth much Sometimes you don t even have a value for n Al is not lost Suppose you take the logarithm with respect to any base of both sides of Eq D 1 The result is log y log k n log z D 2 Now let log x u and log y v With this substitution and a slight rearrangement Eq D 2 becomes v nu log k D 3 Now you have the equation of a straight line That is if you graph logy vs log x you should wind up with a straight line Furthermore the slope of this line is n the value of the exponent in Eq D 1 You find the value of n by calculating the slope in the usu
49. ar regression indicate the two points you used on the graph Draw the line through the two points label it Best fit line or something similar and give its slope and intercept on the graph in some large clear space Mass vs Temperature 307 ons Mass kg 5 gt 300 310 320 330 340 350 360 370 Temperature K Figure B 2 Mass vs Temperature this is a sample graph illustrating all the features of a high level graph The solid line represents the best fit to the sample data Graphing Checklist e Axes scaled correctly with divisions equal to nice intervals 1 2 5 or 10 Physics 117 Graphical Presentation of Data Fall 2011 65 Graph drawn to as large a scale as possible Scales on axes labeled for entire length Axes labeled including units Graph titled and numbered and Points used to calculate slope and intercept clearly marked if that method is used Appendix C Linear Regression Analysis C 1 Introduction The purpose of this appendix is to introduce you to a method for finding the slope and the intercept of the straight line that best represents your data in the presence of the inevitable experimental uncertainty of your measurements This method is known as linear least squares or linear regression Because the entire procedure is somewhat involved we are breaking up the background material into two parts We do not expect you to be able to reproduc
50. asured results along with any other relevant comments The notebook is an informal record of your work but it must be sufficiently neat and well organized so that both you and the instructor can understand exactly what you have done It is also advantageous for your own professional development that you form the habit of keeping notes on your experimental work notes of sufficient clarity that you can understand them at a later time Developing good lab notebook technique requires consistent effort and discipline skills that will be of great value in any professional career Tf you become a research scientist you will often while writing reports or planning a new experiment find yourself referring back to work you have done months or even years before it is essential that your notes be sufficiently complete and unambiguous that you can understand exactly what you did then In keeping a laboratory notebook it is better to err on the side of verbosity and redundancy than to leave out possibly important details NEVER ERASE data or calculations from your notebook If you have a good reason to suspect some data for example you forgot to turn on a power supply in the system or a calculation you entered the wrong numbers in your calculator simply draw a line through the data or calculation you wish to ignore and write a comment in the margin It is surprising how often wrong data sets turn out to be useful after all In fact use a pen to
51. cilloscope for each of the devices in the circuit including the function generator Note the difference in times when each device reaches a peak The resolution of the second puzzle the large amplitude of the potential difference across the capacitor or the inductor relative to the emf value is a little harder to explain Can you speculate what might be going on using an analogy with resonance in a pendulum Experiment 7 Properties of Light 7 1 Snell s Law In this part of the lab you will investigate the fundamental relationship between incident and refracted light beams This is the foundation for building more complicated refractive optics such as lenses and optical instruments consisting of combinations of lenses Snell s Law states that for light incident at an angle 61 on a smooth interface between two materials of indices of refraction n and no the angle 92 of the refracted light is related to 01 by n sin 01 na sin bo 7 1 where the angles are measured between the light rays and the normal to the interface see Fig 7 1 7 1 1 Procedure 1 Place the D shaped semicircle of gelatin stiff water within its plastic holder on top of the ray table Fig 7 2 Place the gelatin in such a way that the straight side is aligned with the COMPONENT line on the paper and so that the midpoint of the straight side sits at the center of the ray table as shown in Fig 7 1 Set the laser so that the light beam shines through
52. dally with time i t Isinwt 6 12 Then Eq 6 4 tells us that v t wLI coswt 6 13 Again we see that there is a 90 phase difference between the current and the potential across the inductor We also see that the amplitude of vzr is proportional to the amplitude of i t Vr wLI The product wL gives us the impedance of the inductor This impedance is high at high frequencies because the inductor strongly opposes the rapidly changing current For low frequencies the inductor has almost no effect Look again at Fig 6 3 A capacitor acts like a resistance of size 1 wC and an inductor like a resistance of size wL and there is a 180 phase difference between the potential across the inductor and the potential across the capacitor At some intermediate frequency where those two resistances are of equal size then the blocking effects of the L and C cancel each other out then the maximum amplitude of current can flow This frequency dependent response is called RESONANCE The frequency where the can celation occurs is called the resonant frequency denoted fo and the corresponding angular frequency is denoted by wo From our arguments we predict that the resonant frequency satisfies the following condition woL 6 14 WoC CAUTION wo fo because 27 1 Physics 117 Experiment 6 Fall 2011 40 Exercise 2 Suppose L 10 mH 107 H and C 0 001 pF Find the numerical value of the
53. dicate that something interesting has been discovered In most cases unfortunately the explanation of the deviation is rather mundane but nevertheless important Remember that small deviations from expected results have led to several Nobel prizes Lab Reports You will prepare a report for each laboratory session We will have two types 1 short informal reports with an exit interview conducted by one of the laboratory instructors and 2 longer written formal reports Both types depend on your having kept a careful record of your work in the lab notebook Informal reports will in general focus on your in class record of the experiment during the lab time along with your answers to the questions posed in the handouts for each lab These short reports need not describe the entire experiment however they should be complete and self contained and distinct from any pre lab lecture notes which are also written in your lab notebook Your notebooks will be collected at the end of informal labs and graded Formal reports will be required for three of the labs see the Laboratory Syllabus For formal reports you are to prepare a somewhat longer written account of your experimental work These reports should include a complete description of the experiment and its results They should be typed you will likely prefer to use a word processor on separate sheets of paper not in your lab notebook and are to be turned in within one week after
54. e grooves on the CD N B In this case the angles are not small so you need to use As dsin nA for constructive interference 9 2 Malus s Law for Polarization Certain materials such as the polymers in our polarizers have the ability to respond to an incident electric field by absorbing all of the light that has an electric field aligned with the polymers and re radiating the light that does not We can think of light incident on a polarizer as having two components of electric field one aligned with the polymers and one perpendicular to that direction The third dimension perpendicular to the plane of the polarizer is unaffected and need not be considered here The light that makes it through the polarizer is the perpendicular component We say that the transmitted light is polarized because it comes out having an E field in a single well defined direction that we might mark on the polarizer with an arrow regardless of what its E field orientation was to begin with This is good news for those of us who wear polarized sunglasses since the polarizer acts somewhat as an E field filter and cuts the transmitted light intensity way down To get polarized light in the first place we can pass unpolarized light light with many random orientations of the E field vector such as the light from our bulb through one Physics 117 Experiment 9 Fall 2011 54 polarizer It turns out that in that case half of the incident light inte
55. e the uncertainty in the slope is Am 0 064 and the uncertainty in the intercept is Ab 0 13 Plot a second graph showing the original data the best fit line and the two worst acceptable lines determined from the uncertainties given above Notice that the two worst acceptable lines seem to be heavily influenced by the ends of the graph C 3 The Correlation Coefficient and Other Loose Ends Since now you have a nice numerical procedure for getting the best line you may wonder if you still need to graph your data Yes Although the least squares method will calculate the slope and the intercept of your line and the uncertainties in those quantities it will not tell you if you have a bad data point Nor can it decide if your data are close enough to being linear to justify fitting a straight line to them You can only make these decisions Physics 117 Linear Regression Analysis Fall 2011 71 intelligently by plotting your data and examining them particularly in cases where you don t have a good theory to go on Many calculators with built in linear fitting routines include a calculation of a quantity called the correlation coefficient usually called r The correlation coefficient tells you how well correlated your data points are A perfect fit no experimental error will have a correlation coefficient of 1 indicating perfect agreement between the observed and calculated values of y and suggesting that x and y are in fact
56. e and therefore x2 gt 1 If we have overestimated our errors or chosen too many free parameters then y y xi lt 0 on average and x2 lt 1 A full analysis of this technique is given in Bevington 3 In practice this is a difficult technique to apply with any rigor because estimating errors is so difficult For the purposes of this course we will call any fit with 0 5 lt x2 lt 2 a good fit Physics 117 Uncertainty Analysis Fall 2011 92 F 5 The User s Guide to Uncertainties The rules can be derived using the results of Sec F 3 2 F 5 1 Addition and Subtraction For addition and subtraction one should combine the absolute uncertainties in the measured quantities Typically one calculates the final uncertainty by adding the uncertainties in quadrature which means taking the square root of the sums of the squares For example the quadrature sum of the three uncertainties 6x1 0x2 and 6x3 is OXtotal y 011 912 6x3 F 27 The following simple example shows how to propagate uncertainties for the case of a simple sum Example 1 Alice measures the lengths of the sides of a triangle finding s 2 9 0 2 cm s2 4 2 0 4 cm and s3 4 9 0 1 cm What is the perimeter of the triangle and Alice s uncertainty in its value Answer The equation for the perimeter is P s 52 83 12 0 cm The final uncertainty in its value is found by summing the individual errors in quadrature 951 68
57. e clauses and phrases Get to the point and stay on topic Write in paragraph style The paragraph is the building block of the report Each paragraph should address one topic of the report Indent Use correct terminology spelling and grammar There are some words in everyday language that have specific meaning in the context of physics Make sure the terminology is consistent with the subject matter of the report Always use the standard spelling of words Physics 117 Formal Reports Fall 2011 80 Write formally do not use slang or colloquialisms Do not write in a casual manner for example the word plug as in I plugged the numbers into the equa tion This is a sloppy lazy style of writing Formal reports should be written in a formal style E 3 Content The content of the report addresses the subject matter the principles ideas and concepts the report is about Since we believe physics is a logical self consistent science the content of the report should be logical and self consistent as well Clarity the underlying principles are clearly articulated all relevant terminology is defined You should be specific in the language used Avoid vague or ambiguous statements Completeness all elements of the report are present Missing or omitted content will mislead or confuse the reader Conciseness specific and to the point The writer should avoid redundant irrelevant or circuitous statements S
58. e form of an equation Define the quantities to be determined and how they are related to the directly measured quantities Experimental technique The experiment technique should be a detailed narrative of the experimental procedure What was measured and how was it measured Include a simple diagram of the apparatus whenever possible Indicate the primary sources of measurement uncertainty Give numerical estimates of uncertainties associated with each directly measured quantity Data analysis and results Display the data in one or more appropriate forms tables graphs etc Discuss how the final results are obtained Give estimates of the uncertainty of the results based upon measurements uncertainties Be sure to include some discussion of experimental uncertainties and how those uncertainties affect the evaluation of your results Discussion of results and Conclusion The conclusion should reflect your overall understanding of the experiment i e what have you learned about the particular subject of physics studied in the experiment It should consist of a logical sequence of statements substantiated by the evidence presented in the report Was the goal of the experiment accomplished Were the experimental results consistent with theoretical expectations That is do they agree within the range of uncertainty What are the Physics 117 Formal Reports Fall 2011 78 implications of your results It is good practice to resta
59. e interrupted and have to the return to the experiment much later Or if you are suspicious of some piece of equipment having this information will let you avoid that particular item Sketch of the Setup Also make a quick sketch of the setup or a schematic diagram for electronics Schematics will be especially helpful when you will be connecting various pieces of electronic equipment together in mildly complicated ways Also in optics experiments ray diagrams are useful to keep track of the paths of various light rays Outline of Methods Next give a short paragraph noting the main goal of the experiment and outlining how you expect to carry out your measurements This should not be too detailed since you will probably modify your procedure as you go along But this opening paragraph will help you settle in your own mind what you do to get started Particularly as the semester goes on and you develop more and more of the experimental procedure yourself you will find yourself modifying your initial procedure discovering additional variables that should be recorded and revising your approach So you don t want to get too locked in to one format But you also should avoid writing down data or procedures in the nearest blank space or you ll be cursing yourself when you look for those pieces of information later So the cardinal rule of keeping a lab notebook is this give yourself plenty of space Doing so makes extending tables or descripti
60. e s from the lens see Fig 8 1 Here s and s are related to the focal length f of the lens by the Gaussian lens equation 1 1 1 Al 8 1 s F s f where f is a length characteristic of the particular lens used Its value depends upon the radii of curvature of the spherical surfaces of the lens and on the material of which the lens is made i e its index of refraction Note in the drawing we assume s gt f source Figure 8 1 If a screen were placed at s then a bright spot would appear on the screen If instead of a point source we had an extended source a distance s from the lens then a focused image of this extended source would appear on the screen at a position s Indeed if we placed a photographic plate instead of a screen at this position we would have the makings of a film camera In today s lab you will test the validity of equation 8 1 Physics 117 Experiment 8 Fall 2011 49 Part I If an object is very far from the lens s gt f then 1 s lt 1 f In this case we expect from the equation that the distant object will be focused at a distance s f To get a crude idea of the focal length of the lens you are using find some very distant bright object e g a mountain or a tree and measure the distance from the lens to a point where the light is focused Be sure to include with your measurement an estimate of the uncertainty associated with f Part II You have been provided with a ligh
61. e this derivation but we feel that it is important for you to know at least in outline form how the method of least squares works We assume that we have accumulated a set of N observed values yg y a y each with an associated value of the independent variable 1 2 Ty To develop a procedure for determining the line that best fits a set of experimental data points we first must agree on a quantitative criterion for the best fit Several criteria are possible but the one most commonly used is called a least squares best fit This expression means that the square of some quantity will be minimized by the choice of slope and intercept of the line The specific quantity that is minimized is the sum of the squares of all the discrepancies between the observed data points and the values calculated from the slope and intercept That is suppose that the true equation describing your data has the form y ma b where as usual m is the slope and b is the y intercept of the straight line The least squares best fit line is the line with the slope and intercept that minimizes the quantity N N s Dtos ye Yue mai b C 1 i 1 i 1 At the moment you don t know the values of m and b but those values can be found using Physics 117 Linear Regression Analysis Fall 2011 67 the procedure below For each of your N values of the independent variable x you have a measured value y of t
62. ectronic circuit elements 4 1 Introduction Any set of conductors which are not electrically connected to one another can be considered a capacitor In its simplest and most practical form a capacitor is made of just two conductors each of which has a wire attached the capacitor is thus a two terminal device just like a resistor Unlike a resistor however no steady state current can flow through a capacitor since the conductors within are not connected to one another For simplicity let s consider a capacitor that is made of two flat parallel conducting plates each of area A and both separated by some distance d If we put a charge Q on one of the sheets and Q on the other an electric field develops in the region between the two plates as shown in Fig 4 1 below The charge resides on the sides of the plates that face one another since this is the only configuration that ensures that the electric field within the conductors themselves is zero Using Gauss s law we can find the electric field between the two conductors E 0 0 where Q A is the surface charge density on one of the plates We are assuming that the plate area A is quite large so that we can neglect what happens near the edge of the plates This is a uniform electric field We can also find the potential difference between the two plates by doing a line integral over the field from one plate to the other With the plates oriented as shown in Fig 4 1 P
63. ed on the basis of theoretical calculations or on the basis of previous experiments we expect that if no mistakes have been made the results should agree with each other within the combined uncertainties Note that even a theoretical calculation may have an uncertainty associated with it because there may be uncertainties in some of the numerical quantities used in the calculation or various mathematical approximations may have been used in reaching the result There are several ways to assess whether our data support the theory we are trying to test F 4 1 Rule of Thumb As a rule of thumb if the measured results agree with the expected results within a factor of about two times the combined uncertainties we usually can view the agreement as sat isfactory If the results disagree by more than about two times the combined uncertainties something interesting is going on and further examination is necessary Example Suppose a theorist from Harvard predicts that the value of X in the previous example should be 333 1 N m s Since our result 339 7 N m s overlaps the theoretical pre diction within the combined uncertainties we conclude that there is satisfactory agreement between the measured value and the predicted value given the experimental and theoretical uncertainties However suppose that we refine our measurement technique and get a new result 340 1 0 1 N m s Now the measured result and the theoretical result to not agree
64. er Please note that you should NOT set up your circuit to require current to flow through a voltmeter Fig 1 9 In E Figure 1 4 A real circuit with battery and resistor Physics 117 Experiment 1 Fall 2011 4 Figure 1 5 A simple circuit diagram Figure 1 6 The ammeter R R O do this don t do this Figure 1 7 Use the setup on the left not the right for an ammeter 1 2 Digital Multimeter Today we will use a digital multimeter to make measurements of current and voltage As the name implies a digital multimeter is a multi function device It can be set to make measurements like a voltmeter ammeter and more You will typically have two multimeters on hand so you can make simultaneous measurements of current and voltage Fig 1 10 is a picture of the multimeter we will use in lab 1 3 The Experiment In this experiment you will construct a simple circuit and utilize an ammeter and voltmeter to make measurements of the current and voltage respectively Begin by constructing a circuit using your breadboard power supply Fig 1 11 ammeter and 470 Q resistor provided you can identify the resistor by the yellow purple brown bands around it Remember in order to measure current the ammeter must be a part of the circuit Use Fig 1 7 to help Physics 117 Experiment 1 Fall 2011 5 Figure 1 8 The voltmeter R v R do this don t do this Figure 1 9 Hook up a voltmeter as
65. evice Your laboratory notebook should contain a current voltage graph for each of the three unknown devices IMPORTANT Be certain to check whether or not these 1 V values depend upon the direction in which the current flows through the unknown you ll have to figure out how to reverse the direction of the current at some point so that you can get data for current going both directions BEWARE reversing the meter leads does NOT reverse the direction of current through the device you are testing but it does multiply your subsequent measurements by the factor 1 Your graphs should have the origin somewhere in the middle of the sheet so that an J V graph would look something like that shown in Fig 2 1 Which of these devices obey Ohm s Law Determine the resistances of those that do What can you say about the electrical properties of those unknowns if any that do not obey Ohm s Law What limits the precision of your measurement How could you get around this limitation 2 1 3 Role of Ammeter and Voltmeter The resistance of your ammeter is nominally very small and that of your voltmeter is approximately 10 MQ Does the fact that these resistances do not have the ideal values of zero for the ammeter and infinity for the voltmeter affect your interpretation of the results of Sec 2 1 2 Be prepared to explain qualitatively how your results would be affected by these particular meter resistances Physics 117 Experiment 2 Fall
66. ew Often we will abbreviate the circuit diagram above like this red vw S lt black 52 75 kQ Figure 1 15 Abbreviated power supply circuit diagram NOTE The board on which the power supply is mounted also holds four independent auxiliary binding posts which may be convenient for mounting resistors etc Experiment 2 Ohm s Law Formal 2 1 Ohm s Law Electronic devices are often classified in terms of various classes Devices for which R is a constant are said to be ohmic Ohm s Law AV IR R constant Devices that obey Ohm s Law are called resistors 2 1 1 Equipment You will be given the following 1 A power supply consisting of two nominal 1 5 V cells in series with a set of variable resistors a diagram of this circuit is given in the appendix to these notes The word nominal is a code word in physics that means don t trust that this is actually so until you have measured it i e caveat emptor 2 Two digital multi meters one used as a voltmeter and the other as an ammeter 3 Three unknown devices one red one white and one blue 10 Physics 117 Experiment 2 Fall 2011 11 Figure 2 1 How your I V graph might look 2 1 2 The Experiment Construct a circuit which will enable you to determine whether your three unknown devices have current voltage characteristics described by Ohm s Law Use your circuit to obtain 5 to 10 1 V values for each d
67. fferent values of R If the two values are not in agreement that is within the combined experimental uncertainty then something is wrong Discuss the situation with your lab partner and or consult with the instructors What is your conclusion as to what C is Does it agree to within an order of magnitude with your predicted value based on the geometry of your capacitor 4 6 Sinusoidal Response optional Now use the same RC circuit with your capacitor and the 10 kQ resistor but with the function generator set to produce sine waves Turn the amplitude control to its highest setting and leave it there We will measure the amplitude of the potential across the capacitor as a function of fre quency with the following circuit Fig 4 6 R Scope Ch 1 Scope Ground Figure 4 6 Measuring the frequency dependence of an RC circuit First make a few quick observations at high and low frequencies to see the general trend of the behavior Then judiciously select 10 frequencies between 100 Hz and 100 kHz for more careful measurements Explain qualitatively why you see the behavior you do Plot your results for the amplitude as a function of frequency on a log log plot What is the amplitude as a fraction of its maximum value at the frequency f 1 27RC From this and your graph do you see why this is called the corner frequency Experiment 5 Faraday s Law and Induction CAUTION The strong magnets we use ca
68. for 0 20 V setting e all other push buttons in the OUT position e AMPLITUDE fully clockwise Connect the the SYNC output of the function generator to the Ext Trig of the oscilloscope using a coaxial cable This allows the the function generator to tell the scope when to trigger Set up the following circuit this is really just like Fig 6 3 L C CH1 v t 10 Q To Oscilloscope CH2 Figure 6 4 For L use the inductor provided For C use a capacitance substitution box set initially at 0 001 uF Using the value of L you found earlier make a rough estimate of the expected resonant frequency You should have done this in Exercise 2 Set the Function Generator to produce a sinusoidal EMF and look for the resonant fre quency Vary the Function Generator s frequency until you find the resonance Once you have it admire the resonant character of the circuit s response by varying the frequency back and forth through fo Measure fo with the oscilloscope Now repeat the experiment for about eight other C values in the range 0 001 uF to 0 22 uF finding fo for each capacitance Be sure to record the uncertainties in your frequency measurements which will not necessarily be the accuracy with which the counter counts Ask an instructor if you re uncertain about your uncertainty Measure the C values in the substitution box using a digital capacitance meter It is more accurate than the nominal values printed on the substitut
69. from the two slits will combine to illuminate the screen see Fig 9 1 x As dsin 0 HET ai rte dir Figure 9 1 The distance traveled by the wave from slit 2 to the screen is longer than that traveled by the wave from slit 1 The difference in the path lengths is approximately equal to As dsin0 9 6 where 0 is the angle shown in the figure Now if y lt L then is small and sin y L The difference in travel distances is then As yd L If this path difference is equal to nA where n is an integer then there will be a constructive interference and we will see a bright spot on the screen Thus the positions of the bright Physics 117 Experiment 9 Fall 2011 52 spots will be at positions y given by Ynd nLA n or m 9 7 The separation between two successive bright spots will then be given by LA Ay Yn 1 Yn T 9 8 Similarly for points where the path difference As creates a phase shift of 180 there will be destructive interference and no light will be observed These points of destructive in terference will be halfway between the bright maxima They will also be separated by a distance LA Ay 9 9 ua 9 9 If we measure Ay d and L we can in principle measure the wavelength of the incident light In order to perform this experiment a helium neon He Ne laser will be used Lasers create light that propagates only in a particular direction and is monochromatic i e it has onl
70. he dependent variable If you knew the values of m and b you could also predict a value for y x given the measured value of x In general your predicted and observed values of yi for a given x would not agree As stated above the least squares procedure minimizes the sum of the squares of the amounts by which the observed and calculated values disagree with each other But you may argue you still don t know the values of the slope and intercept to make these calculations It turns out that m and b can be calculated from your experimental results Here s how to do it call the sum of the squared differences S and write an expression for S in terms of the unknown coefficients m and b and your collection of N measured data points x and y as we did in the second part of Eq C 1 S may be minimized by the usual method of setting the derivative to zero but since S depends on two unknown quantities m and b we must calculate two derivatives called partial derivatives We calculate each partial derivative by treating the other unknown as a constant for the purposes of that derivative obs Using just y for y ers we write the two partial derivatives for S as follows N A Y 2 yi maj b C 2 i 1 and N 2 X_ 2z yi ma b C 3 1 Setting these derivatives equal to zero for a minimum dividing through by the factor of 2 and rearranging terms for purely aesthetic reasons yields N N Sob mri y
71. hysics 117 Experiment 4 Fall 2011 22 plate area A dl E AV dee Jal ple 0 Figure 4 1 A parallel plate capacitor we have b b Qd AV E dl Edz 4 1 f 2d 4 1 where d is the separation between the two plates Observe that the potential difference AV is proportional to the charge we have placed on the two plates Q on one Q on the other AV Q If we call the proportionality constant 1 C then we have AV Q C or Q CAV 4 2 The quantity C is the capacitance of the two conductors In our example we have considered what is known as a parallel plate capacitor and its capacitance is eg A C 4 3 4 3 where A is the plate area and d is the plate separation Capacitance is therefore a purely geometric factor and can also be calculated without knowing the details of the charge on and potential difference between the plates Different conductor configurations lead to different values for C In our water pipe language a capacitor behaves like a tub of water with the area of the bottom of the tub playing the role of capacitance The amount of charge on a capacitor is analogous to the volume of water in the tub and the height of water in the tub is analogous Physics 117 Experiment 4 Fall 2011 23 the potential difference between the two conductors As you fill the tub with water the height of the water goes up and the more capacity area the tub has the less the water height ri
72. icant Ex 230504 6 significant figures 3 Leading zeros to left of nonzero digit are not significant Such zeros only indicate position of decimal point Ex 0 002 1 significant digit 4 Trailing zeros to right of decimal point are significant Ex 0 0340 3 significant digits 5 Trailing zeros to the left of the decimal point may or may not be significant Ex 50 600 3 4 or 5 significant figures 6 When adding or subtracting numbers the final answer is round off to the decimal place equal to the number with the fewest decimals 7 When multiplying or dividing numbers the final answer is round off the same number of significant figures equal to the number with the fewest significant figures F 2 Determining Experimental Uncertainties There are several methods for determining experimental uncertainties Here we mention three methods which can be used easily in most of the laboratory measurements in this course F 2 1 Estimate Technique In this method we estimate the precision with which we can measure the quantity of interest based on an examination of the measurement equipment scales balances meters Physics 117 Uncertainty Analysis Fall 2011 85 etc being used and the quantity being measured which may be fuzzy changing in time etc For example if we were using a scale with 0 1 cm marks to measure the distance between two points on a piece of paper we might estimate the uncertainty in the mea
73. igure 3 1 The Tektronix 2001c Figure from the Tektronix user manual 3 2 1 Display Once you have plugged in the oscilloscope you can turn it on by pressing the power button on the top of the device After a few moments of self testing the main display will show a grid and possibly yellow and or blue lines similar to Fig 3 2 Tek Es Trig d M Pos 0 000s CH1 m rs POS ss PO T E E O eee E E a E e E TA ji ji ji d i qt 2 A eae ai SOMH2 Volts Div Fine dea vice DD CH1 200m CH2 500mY M S00us CH1 Z 241m 25 Jan 11 11 47 1 00000kHz Figure 3 2 A typical oscilloscope display with channel 1 in yellow and channel 2 in blue You will have only the square wave trace at this point The softmenu selections for channel 1 appear to the right of the voltage graph Physics 117 Experiment 3 Fall 2011 15 The main display graphs the voltage appearing on one or both of its input channels as a function of time This is a major improvement over the voltage averaging provided by the multimeter but with this great power comes great responsibility as attested by the many controls on the front panel In this lab we will spend time on the most essential oscilloscope controls 3 2 2 Inputs There are three inputs to the oscilloscope see Fig 3 3 The two signal inputs are the first two on the left marked 1 and 2 we may talk about the third input marked Ext Trig later These inputs are so called BNC connectors
74. iment LCR Series Resonance 0 00000 eae 6 3 Lwo Puzzles ee sn ee Ath ee AE ge Oe ee BAe oes Properties of Light TE Spell SLEW a oe amp SR Geek OS 4 4d Sad Gee God 7 2 Inverse Square Law Behavior 0 e Geometric Optics Formal Interference Diffraction and Polarization 9 1 Young s Double Slit Experiment 0 2 2 2 0200 9 2 Malus s Law for Polarization 0 0 0 e 11 21 23 25 26 27 28 29 29 30 32 35 35 40 42 44 44 46 48 50 Keeping a Lab Notebook Graphical Presentation of Data BL Introduction Lolo A ook a ad A A Be B 2 Analyzing your Graph yea acid a A ee es B 3 Uncertainty Bars oia fei ot eae ek oo Ses ed ed B 4 Graphical Presentation Guidelines e e e Linear Regression Analysis Gel Imtroductio a a Ad ee teh as a C 2 Uncertainty in the Slope and the Intercept C 3 The Correlation Coefficient and Other Loose Ends Power Law Curve Fitting Guidelines for Formal Laboratory Reports By Dee Formata oie a SN WEA hye en eas Ree ke E R E 2 Composition te 6 iaa a de a Be ee a i ee ee as E3 Contente sso a age dees dee Orap Bode aS Se Be Raa Woe Heys A GTIN E 4 Questions and Exercises 2 ee ee E 5 Some general writing guidelines o a 20200004 Uncertainty Analysis F 1 Expressing Experimental Uncertainties o e
75. ing E 1 Format The report should be formatted in a way that clearly presents all the relevant information to the reader text equations figures etc Some standard report formatting include Typeset using a word processor 76 Physics 117 Formal Reports Fall 2011 77 Lines double spaced 12 point Times New Roman font 1 left and right margins Text is full justification Reports are no more than 6 pages long The organization of the report is crucial The reader anticipates a particular order for the report to be presented Deviation from that order will mislead or confuse the reader The report should be organized as follows Title The title should be a simple descriptive phrase centered at the top of the first page of the report Also include your name the date the lab section and the name s of your partner s Introduction The introduction is a short single paragraph statement of the exper iment What is the purpose the main goal of experiment and why is the experiment a worthwhile means of exploring a particular physical concept Theoretical background The theoretical background should state what the un derlying physics of the experiment is What the theory predicts what assumptions have been made and how the experiment relates to the theory of the physics being studied Terminology specific to the experiment should be defined Often the theory can be best expressed analytically in th
76. ings neat in your report Sequences of Measurements You will often be performing experiments in which you have two independent variables Usually in such experiments you fix the value of one independent variable and make a series of measurements working through several values of the other variable Then you change the value of the first variable and run through the measurements with the other variable again then you change the first independent variable again make another set of measurements and so on It s usually easier to set up this sort of sequence in your notebook as a series of two column tables or three columns with Remarks rather than a big rectangular grid Title each table with the value of the independent variable that you re holding fixed and keep the format of all of the tables the same Comment on Results Once you have completed the experiment and performed any necessary calculations in the notebook you should look back to the main goal and write down to what extent it was achieved If for example you were making a measurement of g you should include a clear statement of the value of g along with its uncertainty Be aware that there are often secondary goals as well to become familiar with a particular physical system or measurement technique for example Comment on your success in attaining these goals as well This serves as a statement of conclusion and gives you the chance to make sure the lab was comple
77. ion box the principles on which this capacitance meter operates are the similar to those used in Lab 4 Capacitors From Eq 6 14 it follows that a graph of C vs 1 f should be a straight line from whose slope you can find L Make such a graph and find L Physics 117 Experiment 6 Fall 2011 42 6 3 Two Puzzles 6 3 1 Initial Measurements Pick one of the eight capacitance values you used in Sec 6 2 that has a capacitance lt 0 015 uF Replace the 100 resistor with a 1 kQ one With the CH1 and CH2 wires connected to opposite ends of the 1 kQ resistor set the function generator frequency so that it is at the resonant frequency for this capacitor Be sure to use the 1 kQ resistor for this part Then move the CH1 and CH2 wires to determine successively the amplitude of e the emf from the function generator e the potential drop across the capacitor e the potential drop across the inductor e and the potential drop across the resistor There should be two puzzling aspects of your results 1 Kirchhoff s Voltage Rule appears to be violated the emf amplitude from the function generator is not equal to the sum of the amplitudes of the potential drops around the circuit 2 The amplitude of the potential difference across the capacitor or the inductor is larger than the amplitude of the emf from the function generator we are getting out more voltage than we are putting in 6 3 2 Resolution of the Puzzles T
78. issajous figures produced by the oscillator and the 60 Hz z axis voltage for oscillator frequencies of 20 30 45 60 80 90 120 150 240 300 360 Hz and higher if you can We suggest you start with 60 Hz Then do 20 30 120 150 240 300 and 360 Hz Then return to do 45 90 and 150 Hz which are trickier Make a sketch in your notebook of the patterns you observe for fy 60 Hz and fy 30 60 120 240 Hz By the way look at the Romer Art Machine in the hallway and the operating instructions that are posted nearby How can you make a fish on the Art Machine On the scope More fun with the oscilloscope optional If you wish to play some more with the oscilloscope you can for instance measure frequencies of notes from a musical instrument or see what your own voice looks like singing or talking as picked up by a microphone and displayed on the scope Physics 117 Experiment 3 Fall 2011 20 3 4 Checking Out For this informal lab no further write up is required Please write answers to the embedded questions in this handout in your lab notebook and a brief conclusion and see one of the instructors for an exit interview before you leave Experiment 4 Capacitors In today s lab we apply the techniques we have learned in the previous labs to a new object the capacitor Capacitors are to be found in almost every electronic device and together with resistors are the most important passive el
79. lb filter G i q light detector bulb Figure 7 3 Note distance r is measured from light source filament to sensor Examine qualitatively what happens to the intensity when the source detector spacing r is gradually increased Take quantitative measurements of the detector output voltage V vs Physics 117 Experiment 7 Fall 2011 47 r for at least 10 readings between 10 cm and 80 cm Take more data points closer to the bulb since that s where the change in V is the greatest You will have to subtract off any nonzero background voltage i e the voltage when your bulb is off from the V readings so make sure to record the background in your notebook too First make a plot of V corrected vs r Then employ the commonly used method of straight line graphing to plot your data in such a way that you observe a straight line A log log plot will help with this Why How linear are your results What is the slope and how does it compare to the expected value of slope from the expression you determined in the Exercise Can you identify reasons for any discrepancies by looking carefully at your plot For your exit interview We will ask you to show all calculations and graphs Please make sure to answer all questions posed above Experiment 8 Geometric Optics Formal If a point source is a distance s from a thin lens then light diverging from the source will be refracted by the lens and will converge at a distanc
80. lent of log x 0 and reading the value off the vertical axis You still have to find the slope by calculating A log y A log x as shown above One further variation some data may be described by a power law added to a constant term For example y kr B D 6 where B is independent of x How could you apply a log log plot to test this hypothesis Hint we can estimate B by looking at what happens for small x if n is positive What would you do if n were negative Appendix E Guidelines for Formal Laboratory Reports The formal lab report should be a complete presentation of your work on the experiment It should be written for someone who has a physics background equivalent to Physics 117 but who does not know anything about the experiment and the measurements you carried out There are three principal components to every formal report Format the organization and presentation of the report Composition the style in which the report is written Content the subject matter of the report All three are essential for writing a complete and self consistent report The purpose of the reports is to test both your analytical skills and your writing skills at communicating phys ical concepts It must be emphasized formal reports are short papers not questionnaires or fill in the blank The reports will be graded with same importance as papers in other course All components of the report will critiqued in the grad
81. like this 5 1 1 cm F 3 where the number in parentheses represents the uncertainty in the last digit Feel free to use this form in your lab work F 1 2 Relative or Percent Uncertainty The relative uncertainty is defined Ox Fa F 4 Zbestl We might express the same measurement result as Tmeasured Chest fz F 5 For example 5 1 cm 2 F 6 Here the uncertainty is expressed as a percentage of the measured value Both means of expressing uncertainties are in common use and of course express the same uncertainty Physics 117 Uncertainty Analysis Fall 2011 84 F 1 3 An aside on significant figures The number of significant figures quoted for a given result should be consistent with the uncertainty in the measurement In the example it would be inappropriate to quote the results as 5 cm 0 1 cm too few significant figures in the result or as 5 132 cm 0 1 cm too many significant figures in the result Some scientists prefer to give the best estimate of the next significant figure after the one limited by the uncertainty for example 5 13 cm 0 1 cm The uncertainties since they are estimates are usually quoted with only one significant figure in some cases e g for very high precision measurements the uncertainties may be quoted with two significant figures F 1 4 Rules for significant figures 1 All nonzero digits are significant Ex 1 9 2 Zeros between nonzero digits are signif
82. log Deflection log Length Figure D 2 Log log plot of beam deflection as a function of beam length The axes are labeled as they would appear on log log graph paper log log paper was invented A sample is given at the end of this appendix The logarithms of numbers are spaced uniformly along both the horizontal and vertical axes as you can confirm by comparing the spacing of 2 and 4 and the spacing of 4 and 8 or 8 and 16 and son In effect the graph paper calculates the logarithms for you Log log paper is most useful when you suspect your data has a power law dependence and you want to test your suspicion Sometimes your suspicion is based on a theoretical prediction sometimes on a previous linear plot Figure D 3 below is a typical linear graph that could be a power law with a negative exponent What happens as the independent variable goes to zero 25 N o a e Power per Unit Area W m a 0 0 0 2 0 4 0 6 0 8 1 0 Distance m Figure D 3 Possible power law behavior with a negative exponent For either positive or negative exponents your next step is to plot your data on log log paper If the graph on log log paper is a pretty good straight line within your experimental Physics 117 Power Law Curve Fitting Fall 2011 75 uncertainty you can conclude that your data does indeed have a power law dependence You can find the constant k in Eq D 1 by extrapolating the line back to x 1 the equiva
83. ly much like the undamped oscillations of a mass on a spring Physics 117 Experiment 6 Fall 2011 37 Figure 6 2 Here is the mathematical argument Use Kirchhoff s Voltage Loop Rule and add the voltages around the loop equating the sum to 0 di t alt a 6 5 L The signs are correct though the proof of that claim is omitted the sign conventions for i t and q t are those adopted in Fig 6 2 We want to focus our attention on the charge q t so we note that ilt 6 6 and taking a derivative yields de dt 8 Using Eq 6 7 in Eq 6 5 gives us Palt 1 LLO a 6 8 But this is an old friend the Simple Harmonic Oscillator equation describing a mass on a spring d x t a kalt 6 9 m Comparing Eqs 6 8 and 6 9 you might well choose to describe L as an inertia mass term for a circuit Just as a mass on a spring overshoots the equilibrium position because of its inertia Newton s First Law the charge through a coil tends to keep going Faraday s law inhibits sudden current changes just as Newton s First Law inhibits sudden velocity changes Similarly the reciprocal of the capacitance plays the role of a spring constant Since you know the solution to Eq 6 9 simple harmonic oscillation with angular frequency w yk m sometimes called for reasons that will become clear in this lab the resonant frequency you can immediately predict the solution to Eq 6 8 Those
84. ly you used the meter stick or how quickly you were able to react when starting and stopping a stopwatch You will not always be expected to put error bars on all of your plotted points but you should know how it is done and be able to apply it to the first lab Physics 117 Graphical Presentation of Data Fall 2011 63 Pendulum Period T vs Initial Angle 2 00 Period seconds 1 80 A 1 A I A 1 A i 1 A 1 A L A 0 5 10 15 20 25 30 35 40 Initial Angle degrees Figure B 1 A graph of one data point of the pendulum period as a function of angle showing uncertainty bars for both variables An example of such an uncertainty bar is shown in Fig B 1 The single data point plotted corresponds to a measured pendulum period T of 1 93 sec 0 03 s for an initial release angle 0 of 20 2 The dotted lines are not part of the graph but are included to show you how the point and the uncertainty bars are related to the axes Notice also that the T axis does not begin at T 0 B 4 Graphical Presentation Guidelines Use these guidelines for higher level graphs 1 Draw your hand plotted graphs in pencil mistakes are easy to make If you wish go back later and touch them up in ink Computer drawn graphs are fine as long as they comply with the remaining guidelines 2 Scale your axes to take the best possible advantage of the graph paper That is draw as large a graph as possible but the divisions of the g
85. mage them by dropping them on the floor hitting them with a hammer or spilling water on them Please resist any temptation to do these things e These scopes may seem at first to have a bewildering array of knobs and switches That is the price we pay for versatility We will suggest some initial settings As you learn to know and love these scopes you will become more adventurous If you get too adventurous too quickly and lose the signal altogether first try by thinking about the functions of the various controls to get it back if you fail go back to the suggested initial settings If that fails an instructor may be able to help 3 2 Oscilloscope The oscilloscopes we will use in this class are Tektronix 2001c which can measure signals up to about 50 MHz on two input channels They are general purpose digital oscilloscopes of a kind that has become nearly ubiquitous in research laboratories around the country Unlike the digital multimeter you must plug these oscilloscopes in although there are now Physics 117 Experiment 3 Fall 2011 14 battery powered versions that are liberated from the grid c Tektronix TDS y el AutoRange ry Measure Acquire O O O Display a 0 L Ojo e se f y Tri O Position Xx E g HO E CE Ment 200 000 5 ee m 300 v 300v cat am En 1 2 2 USB 5 nline 2 2722 001 F
86. n For example if a calculation is solving for a velocity m sec but the solution has units in kilograms then there may be an error in the calculation Do not mix together discussion of theory procedure and analysis Use subheadings for example Theory Analysis Conclusion to keep the report organized and the reader s attention focused Do not over explain the experimental setup A well drawn diagram of the setup is better than a lot of prose Keep the topic of the report on the physics Do not over explain the use of instruments like computers calculators or software like Excel Computers and softwares are only tools Keep to the physics Always present a result in the form tmeas Chest Ax It is nonsense to present the best value separate from the uncertainty Do not wrap text around figures tables or equations Do not include long tables of data in report especially data from motion sensors Long tables are boring to the reader and the data are better presented in a graph Do not include summary Output Page from Regression Analysis in report Keep Output Page in your notebook Only extract the necessary values for you report Do not include long calculations and algebra derivations Reference the equation that is evaluated and state result Only give derivations if it is asked for in a specific exercise Keep a record of any detail calculations in your lab notebook Use drafting tools like rules p
87. n destroy a watch We suggest removing watches to begin with and placing them in a safe location Ignore this suggestion at your own risk 5 1 Faraday s Law Introduction It is possible to induce an EMF e in a coil of wire by changing the magnetic flux passing through the coil that bounds an area A This laboratory provides a test both qualitative and quantitative of some of the ideas inherent in Faraday s law and Lenz s law sues dt 5 1 where a B A BAcos0 5 2 for a uniform magnetic field B Here 0 is the angle between B and the normal to the plane of the coil Note that 4 is equal to BA if B is parallel to or BA if in the opposite direction we have to denote arbitrarily one direction or the other as the direction of or equal to zero if B is perpendicular to If the coil has N turns then the net EMF will be N times the EMF for a single loop of wire d 4 Physics 117 Experiment 5 Fall 2011 30 5 2 Determining B field strength from the induced EMF 5 2 1 One Shot Measurements For this part only a very rough estimate of B need be obtained if you obtain a result you can trust to within a factor of two you re doing OK For the experiment you will have available an oscilloscope a permanent magnet which has a reasonably uniform magnetic field between its pole faces a coil of N turns and cross sectional area A both given a motor drive for rotating the coil in the field
88. n in this measurement measure the interval over several successive minima and divide by the number of intervals From your measurement determine the wavelength of the He Ne laser Be sure to include an estimate Physics 117 Experiment 9 Fall 2011 53 of the uncertainty in your measurement CAUTION You will see successive brightening and dimming of the maxima as you move away from the center This is an interference effect associated with the finite slit widths Do not confuse it with the double slit interference pattern Part III Repeat Part II for your second slit pair Obtain a second value for the He Ne wavelength In which measurement do you have more confidence Why Part IV A remarkable prediction of our treatment of light as a wave is that the intensity at the points of destructive interference will be zero when both slits are open but non zero when light is arriving from either slit alone That is when we open the second slit we decrease the light intensity at these points See if you can observe this effect by covering and uncovering one of your slits Part V Now that you know the laser s wavelength you can reverse the process and use the observed interference to determine the spacing between the slits Shine your laser beam on a compact disk Observe the bright spots due to constructive interference You will probably see only a few From the geometry of the interference pattern determine the spacing between th
89. n quadrature i e SA BN f8C If A BxC then ee 5 5 For a ratio Add the relative uncertainties in quadrature i e B SA IBN 6C LAS then ea 5 5 For multiplication by a constant Multiple uncertainty by the constant i e If A kB then A k B For a square root Divide the relative uncertainty by 2 JA 16B For powers Multiple relative uncertainty by power i e JA B If A B then a In For functions Differentiate the function i e A If A A x then sa 2 x dx 94 F 35 F 36 F 37 F 38 F 39 F 40 F 41 F 42 Bibliography 1 W Strunk and E B White The Elements of Style Allyn amp Bacon Needham Heights MA fourth edition 2000 2 J R Taylor An Introduction to Error Analysis University Science Books Sausalito California second edition 1997 3 P R Bevington and D K Robinson Data Reduction and Error Analysis for the Physical Sciences McGraw Hill New York NY second edition 1992 95
90. n squared deviation also called the standard deviation usually denoted as dx read delta x Note that here x does not mean the change in x but rather is a measure of the spread in x values in the set of measurements Formally the standard deviation is computed as F 8 Although determining the standard deviation may be tedious for a large array of data it is generally accepted as the best estimate of the measurement uncertainty Physics 117 Uncertainty Analysis Fall 2011 86 N B In general we cannot expect exact agreement among the various methods of de termining experimental uncertainties As a rule of thumb we usually expect the different methods of determining the uncertainty to agree within a factor of two or three EXAMPLE Suppose that five independent observers measure the distance between two rather fuzzy marks on a piece of paper and obtain the following results d 5 05 cm d 5 10 cm d3 5 15 cm d4 5 20 cm ds 5 10 cm If the observers were using a scale with 0 1 cm markings method 1 would suggest an uncertainty estimate of about 0 05 cm Method 3 yields a mean value d 5 12 cm and for the standard deviation 0 057 cm 0 06 cm We see that in this case we have reasonable agreement between the two methods of determining the uncertainties We should quote the result of this measurement as 5 12 cm 0 06 cm or 5 12 cm 1 F 3 Propagation of Uncertainties In mo
91. ncertainties and deviations from expected results 4 Being able to describe talk about and write about physical measurements The laboratory work can be divided into three parts preparation execution and anal ysis The preparation of course must be done before you come to your laboratory session The execution and analysis for the most part will be done during the three hour laboratory sessions Some suggestions for performing these three parts successfully are given below Please also refer to Appendix A Keeping a Laboratory Notebook since good note taking will be essential in all phases of the lab Physics 117 General Instructions Fall 2011 vi Preparation 1 Read the laboratory instruction carefully Make sure that you understand what the ultimate goal of the experiment is 2 Review relevant concepts in the text and in your lecture notes 3 Outline the measurements to be made 4 Understand how one goes from the measured quantities to the desired results 5 Organize tables for recording data and the equations needed to relate measured quan tities to the desired results Execution 1 One of the most important elements of executing the experiment will be keeping a step by step record of what you ve done how you ve done it and in what order You will use an inexpensive permanently bound notebook provided to you for recording your laboratory data your analysis of the data and the conclusions you draw from the me
92. nnectors O Y O O O O power supply terminals Figure 1 11 The breadboard power supply 470 Q Figure 1 12 Series circuit Therefore the total potential provided by the battery is matched by the total potential that drops across the devices For your circuit that means Vsupply Vazon Vrea 0 1 4 Now construct a circuit in which the 470 Q resistor and the red device are in parallel with each other Two devices are in parallel with each other when they share two junctions In this case the voltage across each device is the same e Measure the current flowing out of the battery i e before the first junction e Measure the current through each device How does the sum of the individual currents compare to the current before the first junction Physics 117 Experiment 1 Fall 2011 8 470 Q Figure 1 13 Parallel circuit This illustrates a second principle called Kirchhoff s Current Rule The sum of currents flowing into a junction is equal to the sum of currents flowing out of a junction ho da 1 5 Let s consider the case of the 470 Q resistor and the red device Lin Ig700 Irea 1 6 Physics 117 Experiment 1 Fall 2011 9 1 4 Supplement Power Supply Circuit Diagram black Figure 1 14 Power supply circuit diagram This symbol of a squiggly line with an arrow through it denotes a variable resistor whose value can be varied by turning a knob or in our case a scr
93. nsity makes it through If you shine that polarized light onto a second polarizer the polarization of the output light is determined by the second polarizer as discussed above But the intensity of that light depends on the relative polarization angles of the polarizers Malus s Law describes the overall output intensity as a function of the angle 0 between the alignment axes of the two polarizers In fact Malus s law says that the output intensity out of the second polarizer and the input intensity out of the first polarizer are related by the factor cos 0 Exercise I Write down the mathematical expression that is described in the previous sentence Exercise II Get a qualitative feel for the effect of crossed polarizers by looking through a stack of two polarizers and rotating one relative to the other You ll probably need to do this in a well lit room What do you notice about the light intensity Do your observations make sense in light of the above discussion Appendix A Keeping a Lab Notebook Keeping a good lab notebook seems like a simple and obvious task but it requires more care and thought than most people realize It is a skill that requires consistent effort and discipline and is worth the effort to develop Your lab notebook is your written record of everything you did in the lab Hence it includes not only your tables of data but notes on your procedure and your data analysis as well With practice you will bec
94. o resolve the first puzzle adjust the oscilloscope to display two or three cycles of the potential across the inductor Adjust the VOLTS DIV on both CH1 and CH2 so that the display nearly fills the screen vertically Draw a careful graph of the signal you observe Now without changing any of the oscilloscope or function generator settings move your CH1 and CH2 wires to observe the other three signals listed in Sec 6 3 1 In carrying out these measurements it is important to keep the order of the CH1 and CH2 wires the same as you move around the circuit For example if you use the CH1 wire connected to point D in Fig 6 5 below and the CH2 wire connected to point E to measure the potential difference across the resistor then when you measure the potential difference across the Physics 117 Experiment 6 Fall 2011 43 capacitor CH1 should be connected to point B and CH2 to point D Record these other three signals on your graph L A B V C R E D Figure 6 5 Does Kirchhoff s Voltage Law describe the behavior of the circuit at each instant of time The lesson here is that your amplitude measurements did not take into account the phase differences in the circuit the various potential differences do not reach their maximum values at the same time Using the function generator emf as your time reference what are the phase differences of the other potential differences in the circuit Sketch the V vs t you observe on the os
95. ociated with this value of B Do this by folding in results from two methods First consider the scatter in the data and thereby the uncertainty in the slope Second take one representative data point in the middle of your data range and use the Propagation of Errors technique in the equation Emax NBAw 7 As a means of comparison use the commercial Hall effect gaussmeter to measure B for your magnet and compare it to your EMF determination of B You may assume that the uncertainty in the gaussmeter is 2 in the last digit of the reading 5 3 Minilabs on Induction We will have only one setup of each of the following demonstrations You may do them at any time during the lab period In fact it will be helpful and more efficient if you do at least one or two of them early in the lab session e Magnet dropped through a series of coils Physics 117 Experiment 5 Fall 2011 33 e Objects dropped through an aluminum tube e A magnetically damped pendulum e Hand driven generator with light bulb 5 3 1 Falling Magnet In this demonstration you drop a magnet through a glass tube that has six coils of wire wound around it The coils are spaced approximately 20 cm apart Procedure The apparatus here is tricky since the scope has to be programmed to make a single sweep record of the trace Please consult the Instructor before doing this minilab As the magnet passes each coil it induces an EMF in that coil Why
96. of the peak and measure the height of the peak for the size of the EMF Now using the given values of N and A make a rough estimate for B the magnitude of the magnetic field Remember that SI units must be used your result for B will come out correctly in teslas only if voltages are in volts areas in square meters and times in seconds Now vary the orientation of the coil in its initial position between the pole faces and again withdraw the coil Observe qualitatively how the size of the EMF varies with the angle of orientation and with the speed of withdrawal of the coil Also notice what happens when you quickly move the coil into instead of out of the region between the pole faces Write down all of your observations 5 2 2 A Rotating Coil A Simple Electrical Generator and a Measure ment of Magnetic Field Strength If the coil is rotated with an angular frequency w radians second while located in a steady magnetic field the flux will vary sinusoidally with time a BAsinut 5 4 For simplicity s sake we will call t 0 the time when 4 happens to be zero because the plane of the coil is at that time parallel to B Then the induced EMF is by Eq 5 3 N times the time derivative of this magnetic flux e NB Aw cos wt 5 5 That is the induced EMF is predicted to vary sinusoidally with angular frequency w and with an amplitude center to peak of Emax NBAw 5 6 Oscilloscope Settings
97. om the trial values of m and b and the value of X Trial 1 Use m 0 5 b 0 25 X Yobs Yeale Yate Vesta Cos 045 055 om asmo oo a ho al C ee a Trial 2 Use m 0 45 b 0 3 Trial 3 Use your values of m and b from the least squares procedure C 2 Uncertainty in the Slope and the Intercept The slope m and intercept b of the best fit line calculated above are of course based on experimental data Like the data they must be uncertain to some degree or other In practice we can calculated the standard deviations to be associated with the slope and intercept values using the same data we used to find the slope and intercept However the calculations and their derivations are beyond the scope of this course The computer programs Excel and Graphical Analysis available on the computers in the lab can do the Physics 117 Linear Regression Analysis Fall 2011 70 linear regression calculations for you But what can such a program tell you about the uncertainty in your slope resulting from the data Ideally you would like a computer program to accept information about the experimental uncertainties in your measured values and to combine them to give you an uncertainty in the results for the slope and intercept You already know how to do a calculation by hand for the uncertainty in for example the product of two measured values For a review of the method refer to Appendix F For the collection
98. ome adept at sharing your time fairly between conducting the experiment and recording relevant information in your notebook as you go along You want all this information in one place for three main reasons and these reasons continue to be valid even after you leave the introductory physics laboratory That is even or rather especially practicing scientists keep lab notebooks First your lab notebook contains the information you will need to write a convincing report on your work whether that report is for a grade in a course or a journal article Second you may need to return to your work months or even years after you have finished an experiment It is surprising how often some early experiment or calculation is important in your later work Hence you need a reasonably complete account of what you have done Third your lab notebook is also the source to which you turn in case someone questions the validity of your results You may have heard about the famous David Baltimore case of alleged scientific fraud in which the lab notebook of one of Baltimore s collaborators was the subject of careful scrutiny Your notebook therefore serves two purposes that may not be completely compatible with each other On one hand you should write things down pretty much as they occur and before you have a chance to forget them so that you have a complete record of your work in the lab On the other hand your notebook should be reasonably neat and
99. ons of procedure easy and typically also makes your notebook easier to Physics 117 Keeping a Lab Notebook Fall 2011 57 read If you find that you haven t allotted enough space for a table feel free to start it over on the next page labeling the new table of course and making a note at the old table directing you or someone else to the new one Using only the odd numbered or even numbered if you re left handed pages also works well the blank facing page can be used later to reduce data where you can see both the raw data and the reduced data or for graphs Or you can use all the pages but start out using only the top half of each page Procedure In recording your procedure write in complete sentences and complete paragraphs This is part of the discipline required for keeping a good lab notebook Single words or phrases rapidly become mysterious and only with a sentence or two about what you re measuring such as the period of the pendulum as a function of length will you be able to understand later on what you did Give more details where necessary if for example the lab manual does not give a more detailed procedure or if you depart from the procedure in the manual Numerical Data When recording numerical data keep your results in an orderly table You should label the columns and indicate the units in which quantities are measured You should also indicate the uncertainty to be associated with each measurement If the
100. ou did in part 5 3 3 2 Write down your observations in your lab notebook Can you explain what happens 4 Hold the magnet with two hands and pull it quickly away from the pendulum Again write down and explain what happens 5 3 4 Hand Driven Generator with Light Bulb Turn the crank on the generator to get the bulb to illuminate Repeat the experiment with the bulb disconnected Is it now easier or harder to turn the crank Why Experiment 6 RLC Circuits Formal In this lab we will put together the three linear passive devices with which we have worked in the past few weeks the resistor the capacitor and the inductor Remarkably whenever capacitors and inductors appear together in a circuit Kirchhoff s voltage rule yields the equation for a simple harmonic oscillator One important consequence is that the circuit will display the phenomenon of resonance 6 1 Introduction 6 1 1 Capacitors Recall that the voltage across a capacitor is proportional to the charge on one of the plates v t alt 6 1 We ll use lower case letters to represent quantities that change as a function of time If we indicate the current t that flows as the capacitor is discharging we have the following situation Fig 6 1 Since the current results in the charge q t changing with time we have with the signs used in the figure i at 6 2 zga 6 3 dt Physics 117 Experiment 6 Fall 2011 36 it a q 1 q
101. power supply How does the oscilloscope trace deflect Now move the banana to BNC adapter to the CH 2 input and the same connections as above What happens Why 3 Comparing Voltage Measuring Devices The scope can act as a voltmeter similarly to a hand held digital voltmeter Set up the circuit you used to verify that resistors in series add like R R Ro Your circuit might have looked like that in Fig 3 4 Now connect the digital multimeter in parallel with the input of the oscilloscope as shown in Fig 3 5 Turn the multimeter into a voltmeter by adjusting its mode switch to DCV Use the pair of leads to the oscilloscope input to measure the voltage difference across various elements of the circuit Pay attention to polarity Record your data in your lab notebook and organize it as follows Circuit Element Scope Reading CH1 Voltage Voltage read of divisions VOLTS DIV determined by the DMM of deflection by the scope Physics 117 Experiment 3 Fall 2011 17 l white red l l l l 52 75 KQ l l 10 Q l l Figure 3 4 Circuit for measuring resistances in series scope black Figure 3 5 Hooking up the scope and digital multimeter in parallel Do the voltages measured by the multimeter and oscilloscope agree What is the sum of the voltage differences across each of the circuit elements 4 Horizontal Time Sweep At this stage the oscilloscope is no more useful to us than a
102. raph paper should correspond to some nice interval like 1 2 or 5 times some power of 10 If you have to make the graph smaller to get a nice interval make it smaller but check that you ve picked the nice interval that gives you the largest graph Making the graph large will display your data in as much detail as possible When using log log or semi log paper choose paper with the number of cycles that gives the largest possible graph 3 The lower left hand corner need not be the point 0 0 Choose the range of values for each axis to be just wide enough to display all the data you want If 0 0 does not appear on the graph it s customary but not necessary to mark the break in the axis with two wavy lines Physics 117 Graphical Presentation of Data Fall 2011 64 4 Mark the scale of each axis the number of units corresponding to each division for the entire length of the axis 5 Label both axes identifying the quantity being plotted on each axis and the units being used 6 Give each graph a title or provide a figure caption The title should summarize the information contained in the axes and also gives any additional information needed to distinguish this graph from other graphs in the report 7 Give each graph a number e g Figure 2 which you can use in the body of the report to refer quickly to the graph 8 If you calculate the slope and intercept of the graph from two points rather than using line
103. rotractors compasses or or applications like MS Paint to draw diagrams No free hand diagrams DO NOT DOWNLOAD FIGURES FROM INTERNET Produce your own figures Ask for help Don t be afraid to ask questions if you are unclear on something There is no need to guess what we want in the report Appendix F Uncertainty Analysis An intrinsic feature of every measurement is the uncertainty associated with the result of that measurement No measurement is ever exact Being able to determine and assess measurement uncertainties intelligently is an important skill in any type of scientific work The measurement or experimental uncertainty should be considered an essential part of every measurement Why make such a fuss over measurement uncertainties Indeed in many cases the uncer tainties are so small that for some purposes we needn t worry about them On the other hand there are many situations in which small changes might be very significant A clear statement of measurement uncertainties helps us assess deviations from expected results For example suppose that two scientists report measurements of the speed of light in vac uum Scientist Curie reports 2 99 x 10 m s Scientist Wu reports 2 98 x 10 m s There are several possible conclusions we could draw from these reported results 1 These scientists have discovered that the speed of light is not a universal constant 2 Curie s result is better because it agrees with the
104. s y 7 5 m sec x 1 3 m sec B 4 This equation gives a complete description of the line and the job is done B 3 Uncertainty Bars Individual data points plotted on any graph should include uncertainty bars sometimes misleadingly called error bars showing the uncertainty range associated with each data point You should show both vertical and horizontal uncertainty bars if the uncertainties are large enough to be visible on the graph If they aren t large enough you should mention this in your report so we don t think you ve forgotten them You can draw uncertainty bars by indicating the best guess value typically the measured value or average of several measurements with a dot and drawing an I bar through the dot with its length indicating the range in the uncertainty When you use Excel or the Graphical Analysis data analysis package this step can be done for you with severe limitations Such a package will typically only determine error bars by considering the scatter of the individual data points about the best fit straight line While this is helpful in providing a consistency check for the data it does not tell the complete story of the uncertainties in your data That is unless you use a more advanced feature of such an analysis program it has no way of knowing about the uncertainties that were inherent in your measured values because of the measurement apparatus Only you can decide how accurate
105. s inventors Physics 117 Experiment 3 Fall 2011 16 e Push the button to activate the TRIGGER menu Set TRIGGER Source to CH 1 g Set the TRIGGER Mode to Auto 2 Oscilloscope Signal Deflection We now want to observe how voltages applied to the input of the oscilloscope result in a deflection of the displayed signal a Push the button to activate the DISPLAY menu b Use the appropriate soft key to set DISPLAY Format to XY The yellow trace should turn into a yellow dot at the center of the grid The connectors on the top of your breadboard are called banana connectors Find two banana to BNC adapters and attach one to the CH 1 input of the scope and the other to CH 2 Now connect a wire from the black connector on the breadboard to the black connector of channel one s input connector on the oscilloscope Connect a second wire from the red connector on the breadboard to the red post of channel one s input connector on the oscilloscope When you connect the second wire the dot on the oscilloscope should deflect to the right Try connecting and disconnecting the wire relatively rapidly to send a Morse code signal Use the amount of deflection and the VOLTS DIV setting to determine the EMF voltage from the battery Is your result consistent with what you expect for two C cell batteries Turn the Scale knob associated with CH 1 and observe what happens Explain your observations Now reverse the two leads to the
106. sented by the drawing in Fig 1 3 A circuit which in reality looks something like that shown in Fig 1 4 is represented symbolically like that shown in Fig 1 5 Current by definition flows out the positive end of the battery as shown Physics 117 Experiment 1 Fall 2011 3 Me Cell Battery Figure 1 2 A simple cell left and battery right hill Resistor Figure 1 3 A resistor To measure electric current we use a device called an ammeter which is represented sym bolically in Fig 1 6 Current must flow through an ammeter Fig 1 7 The ammeter will display the current that flows into its red lead and out of its black lead If the measured current is negative then the current is actually flowing into the black lead and out of the red lead Sometimes these meter leads are referred to as positive red and negative black for now let s agree to eschew this convention which often brings with it unnecessary confusion To measure electric potential differences we use a two terminal device called a voltmeter represented symbolically in Fig 1 8 Voltmeters measure the potential difference between points a and b i e if the potential is V at point a and Vp at point b the voltmeter will display Va Vp As with the ammeter there is a red a and a black lead b the price you pay for confusing these two leads is an extra minus sign in the voltage displayed by the met
107. ses per liter of added water Similarly for a capacitor the higher the capacitance the less the potential increases per unit charge The units of capacitance are farads F 1 F is 1 C V The schematic symbol for a capacitor Fig 4 2 should bring to your mind the idea of parallel plates even if most capacitors don t look like this any more Lo Figure 4 2 The schematic symbol for a capacitor Often the space between the conductors is filled with some nonconducting material called a dielectric The effect of the dielectric is to reduce the electric field within its bulk by some factor k which is a property of the dielectric Different materials have different values of k When the electric field between the capacitor plates is reduced by a factor of k the potential difference between the plates is also reduced by this same factor k for the same charge Q on the plates Since C Q AV reducing the potential difference increases the capacitance by and thus the principal effect of placing a dielectric between the conductors of a capacitor is to increase the capacitance by a factor of k For a parallel plate capacitor with dielectric entirely filling the region between the plates Eq 4 3 becomes Keg A d C 4 4 4 2 Charging and Discharging Consider the following circuit consisting of a function generator supplying a voltage V t a resistor R and a capacitor C connected in series I t R V t C Volt Figure 4 3
108. ses shifted by 180 That is we let one wave have its maximum just as the other is at its minimum The total electric field is just the sum of the electric fields associated with each beam in accordance with the principle of superposition So Pohti 9 2 Ey cos 27 G ft Ey cos 27 G ft 9 3 i 9 4 The field associated with the second beam will exactly cancel the first and there is no net field Since there is no field there is no intensity and hence we would see no light This Physics 117 Experiment 9 Fall 2011 51 phenomenon is called total destructive interference Alternatively we could imagine two electromagnetic waves having the same amplitude and phase Then the total field would simply double E By Ey 2p cos 27 G a ft 9 5 Because the intensity of light is proportional to the square of the electric field the intensity would quadruple This phenomenon is called constructive interference 9 1 Young s Double Slit Experiment Today we would like to observe these wavelike properties of light To do this we will do Young s double slit diffraction experiment In this experiment an electromagnetic wave is incident upon two narrow apertures We let the separation between the two apertures be d The electromagnetic wave will exit the two slits with approximately the same amplitude and phase If we place a screen some distance L away from our two slits the electromagnetic waves propagating
109. st measurements some calculation is necessary to link the measured quantities to the desired result The question then naturally arises How do the uncertainties in the measured quantities affect propagate to the results In other words how do we estimate the uncertainty in the desired result from the uncertainties in the measured quantities F 3 1 High Low Method One way to do this is to carry through the calculation using the extreme values of the measured quantities for example 5 06 cm and 5 18 cm from the previous example to find the range of result values This method is straightforward but quickly becomes tedious if several variables are involved EXAMPLE Suppose that you wish to determine a quantity X which is to be calculated indirectly using the measurements of a b and c together with a theoretical expression X a Suppose further that you have already determined that Physics 117 Uncertainty Analysis Fall 2011 87 a 23 5 0 2 m b 116 3 41 1 N c 8 05 0 03 s The best value of X is 23 5 x 116 3 Xiest gag 339 509 N m s F 9 We ll clean up the significant figures later But X could be about as large as what you get by using the maximum values of a and b and the minimum why value of c 23 7 x 117 4 Rie A E F 10 8 02 And similarly we find 23 3 x 115 2 Koss 332 198 N m s F 11 Notice that Xhigh and Xjow differ from Xhest by about the same amount
110. sured distance to be about 0 05 cm that is we could easily estimate the distance to within of a scale marking F 2 2 Sensitivity Estimate Some measurements are best described as comparison or null measurements in which we balance one or more unknowns against a known quantity For example in the Wheatstone bridge experiment we will determine an unknown resistance in terms of a known precision resistance by setting a certain potential difference in the circuit to zero We can estimate the uncertainty in the resulting resistance by slightly varying the precision resistor to see what range of resistance values leads to a balanced bridge within our ability to check for zero potential difference F 2 3 Repeated Measurement Statistical Technique If a measurement is repeated in independent and unbiased ways the results of the measure ments will be slightly different each time A statistical analysis of these results then it is generally agreed gives the best value of the measured quantity and the best estimate of the uncertainty to be associated with that result The usual method of determining the best value for the result is to compute the mean value of the results If 1 2 IN are the N results of the measurement of the quantity x then the mean value of x usually denoted 7 is defined as N Ti ttate tin _ 1 7 A S ET The uncertainty in the result is usually expressed as the root mea
111. t or exit interview Always answer questions and exercise from a physics point of view Appendix B Graphical Presentation of Data B 1 Introduction Draw a picture is an important general principle in explaining things It s important because most people think visually processing visual information much more quickly than information in other forms Graphing your data shows relationships much more clearly and quickly both to you and your reader than presenting the same information in a table Typically you use two levels of graphing in the lab A graph that appears in your final report is a higher level graph Such a graph is done neatly and almost always with a graphing program following all the presentation guidelines listed below It s made primarily for the benefit of the person reading your report Lower level graphs are rough graphs that you make for your own benefit in the lab room they re the ones the lab assistants will hound you to construct These lower level graphs tell you when you need to take more data or check a data point They re most useful when you make them in time to act on them which means that you should get in the habit of graphing your data in the lab while you still have access to the equipment That s one reason in fact that we recommend that you leave every other sheet in your lab notebook free so you can use that blank sheet to graph your data In graphing your data in the alb you don
112. t source a screen an optical bench and a meter stick Using these set up an experiment to test the validity of Eq 8 1 Plot your data in such a way that if the theory is correct you ought to obtain a straight line graph from which you can obtain the focal length of the lens Again be sure to estimate the uncertainty in your measurement Does your value deduced here agree with that obtained in Part I While making these measurements observe the characteristics of the image produced Is the image larger or smaller than the original object Is the image inverted Part III Repeat Part II with a different lens with a different curvature and hence focal length Which lens has the larger f For which lens are the surfaces more curved i e depart further from being planes Which lens is stronger The strength or power of a lens is defined by the reciprocal of its focal length It is measured in units of diopters 1 meters If you wear glasses or contact lenses your lens prescription is specified in units of diopters What are the strengths of your lenses in diopters Part IV Take one of the negative lenses and sandwich it with one of the converging lenses from either Part II or Part III Use the converging lens that is the stronger of the two i e the higher power in diopters Use tape around the edge to hold the two lenses together and place the composite lens into the stand on the optics bench Now perform
113. tay on topic Consistency all elements of the report direct the reader to a single logical conclu sion Avoid illogical erroneous unsubstantiated specious irrelevant statements and contradictions Continuity all elements of the report follow a logical order The discussion is constructed in a sequential manner Avoid incoherent disorganized statements E 4 Questions and Exercises In some experiments specific questions and exercises will be asked The purpose of the questions and exercises is to motivate the discussion Questions and exercises should be answered within the body of the report and always from a physics point of view The report is incomplete without answers to questions and exercises E 5 Some general writing guidelines Do not assume too much about the reader s knowledge of the experiment It s your responsibility to explain the subject matter to the reader Assuming the reader already Physics 117 Formal Reports Fall 2011 81 understands the subject and not providing a complete explanation makes the report seem disjointed Proofread Spelling and grammatical errors are easy to fix otherwise the report appears sloppy Make use of resources dictionary thesaurus Strunk and White etc Check the Units Significant Figures and Uncertainties Values without units are meaningless Make sure all the values have the correct units Checking the unit of a value is also a good cross check of a calculatio
114. te any numerical results in the conclusion for easy review by the reader Other forms of information require specific formatting Equations Equations should be centered on separate lines from the text of the report Each equation should be numbered preferably along the right margin for easy reference An equation is often followed by a sentence that defines the variables in the equation Equations are especially useful when stating the theoretical background of the report Do not include long derivations of equations in the report Instead simply reference which equations are used in the derivation and give final result Keep derivations in your notebooks Figures Diagrams diagrams of experimental setup should be simple yet illustra tive of the experimental setup and apparatus The relevant parts should be labeled and the relevant measured quantities indicated Each diagram should have a figure number Figure 1 and caption below the diagram The caption should be a con cise description of the figure and any important parts Use drafting tools like rules protractors or applications like MS Paint to draw diagrams Do not draw free hand diagrams Figures Graphs The title of a graph should clearly indicate which two quantities are plotted The title convention for a graph is Y vertical axis vs X horizontal axis The axes should be labeled and include units Graphs use the same convention of numbering and captions as diagrams
115. ted thoroughly and to your satisfaction Guidelines to keeping a good notebook 1 Bound Notebook No spiral bound loose leaf or perforated page notebook Lab notebooks are a permanent record of the work done in lab The integrity of the notebook should not be comprised by tearing out pages 2 Keep a record Write down names title time places and dates 3 Be generous with use of pages Start each experiment on a fresh page leave some blank pages between experiments in case you need to add tables or graphs 4 Do not erase redact or scribble out possible mistakes Draw one single line through any value or calculation you suspect may be wrong 5 Define terminology and variables with units Physics 117 Keeping a Lab Notebook Fall 2011 59 10 Sketch the experimental setup Label the relevant parts and indicated measured quantities List equipment used Be complete Every experiment has a introduction pre lab note procedure analysis results and summary Make sure all tables graphs diagrams and calculations are in your notebook for reference Annotate Each section of experiment should begin with a paragraph discussing what the section is about Do not leave it to a reader to guess Be organized neat and legible The reader should not to have struggle to decipher your notes Don t be cryptic Answer all questions and exercises Use your answers to questions build the discussion of your formal repor
116. tercept of a line You will find this semester that you spend a lot of time redrawing curves by employing the method of straight line graphing so that they turn into straight lines for which you can calculate a slope and an intercept This process is so important that although we have a fond hope that you learned how to do this in high school we re going to review it anyway Presumably you have in front of you some graphed data that look pretty linear Start by drawing in by eye the line that you think best represents the trend in your data An analytical procedure exists to draw such a line but in fact your eyeballed line will be pretty close to this analytically determined best line Your job now is to find the slope and intercept of that best line you ve drawn Next we tackle the question of finding the slope and intercept of that line As usual we will assume that the line is described by the equation y mzx b B 1 where m is the slope of the line and b is the y intercept Two points determine a line and a line is also described completely by its slope and inter cept This should make a certain amount of sense You put in two pieces of information you get out two pieces of information Your first task is therefore to choose two points on your line These two points describe the line so they need not and most likely will not be data points They should be far apart on the graph to minimize the effects of the
117. the end of lab You should pay special attention to the clarity and conciseness of your writing In fact if we find that your report would benefit from rewriting we may ask you to submit a revised version of the report before a grade is assigned to the report Guidelines for preparation of formal lab reports are included on a subsequent sheet in this manual Please hand in your lab reports on or before the due date as with the other written exercises in the class late work will not be accepted Physics 117 General Instructions Fall 2011 ix Grading You must complete all of the labs to pass Physics 117 If you have to miss a lab because of illness family difficulties or other legitimate reasons please let your instructor know in advance whenever possible so we can arrange for a make up time You will receive a grade for each of the formal lab reports These grades along with an evaluation of your lab notebook which will be weighted like one formal lab report and an overall evaluation of your performance during the labs will constitute your lab contribution for the course grade Intellectual Responsibility Discussion and cooperation between lab partners is strongly encouraged and indeed of ten essential during the lab sessions However each student must keep a separate record of the data and must do all calculations independently and must write an independent lab report It is strongly advised that students do not communicate
118. to 5ms div With the knob on the aluminum box at its maximum setting that voltage is supposed to be a sinusoid with a frequency of 60 Hz and a peak to peak size of about 3 volts as shown in Fig 3 6 That s an amplitude of 1 5 V by the way if you write the voltage as V t Asin 27 ft then A 1 5 volts Are the size and frequency about right 9 A Mysterious Signal Physics 117 Experiment 3 Fall 2011 19 1 54 1 04 S 4 054 oO gt 4 0 0 O 2 0 54 1 04 1 57 0 20 40 60 80 100 time ms Figure 3 6 The 60 Hz sine wave 10 11 Now disconnect the aluminum box from the oscilloscope Connect one wire to the red post on the oscilloscope input Hold the other end of the wire in your hand Adjust the CH 1 Scale knob until you see a large signal on the screen Where is this signal coming from Hint what is the frequency of the signal Now connect another wire to the black input post and hold one wire in one hand and one wire in the other What happens to the oscilloscope signal Try to explain what is going on Artistic Lissajous Figures Reconnect the 60 Hz box to the CH 1 input and connect the sinusoidal output of the function generator to the CH 2 input Push the button to activate the DISPLAY menu Set DISPLAY Format to XY Now the scope s time base is out of action again The x deflection is controlled by the voltage applied to CH 1 the y deflection by that applied to CH 2 Now observe L
119. uld make it clear how one can go about finding the final uncertainty in a more complicated problem Example 3 Cassandra wishes to know the speed of a cart traveling along a level air track She measures the distance of two photogates from the end of the air track d 18 4 0 2 cm and dz 160 1 0 3 cm and also the times at which the cart triggers each photogate t 0 53 0 01 s and tg 1 88 0 02 s What is the speed of the cart and the uncertainty that Cassandra should quote Answer The expression for the speed is of course da d v gt F 31 to ty First we compute the numerator and its uncertainty da gt di 141 7 0 4 cm F 32 where we applied the rules for addition and subtraction add absolute uncertainties in quadrature We now do a similar calculation for the denominator t2 t 1 35 0 02 s F 33 Finally we calculate v using the rules for multiplication and division on the uncertainties in Eqs F 32 and F 33 add fractional uncertainties in quadrature 7 141 7 0 4 cm L 105 2 F 34 te aroos Vee 5 34 Physics 117 Uncertainty Analysis Fall 2011 F 5 4 Simplified Uncertainty Rules 1 For a sum Add the absolute uncertainties in quadrature i e If A B C then 6A 6B C For a difference Add the absolute uncertainties in quadrature i e If A B C then 6A y B C For a product Add the relative uncertainties i
120. voltage that is alternately zero and Vo switching back and forth between these two values at a rate controlled by the frequency setting on the function generator If for instance the frequency is 500 Hz then for 1 millisecond the function generator acts like the circuit on the left in Fig 4 4 below Then for the next millisecond it acts like the one on the right 3 Vo AS Figure 4 4 At 500 Hz The function generator alternately acts like the circuit on the left for 1 ms and then like the circuit on the right for 1 ms The size of Vo can be adjusted with the amplitude control on the function generator The value of the resistance r for the function generator is about 50 2 Connect the circuit as shown in Fig 4 5 below which is essentially the same as the circuit Physics 117 Experiment 4 Fall 2011 27 shown a few pages back except that the voltage source is now the function generator NOTE The ground terminal of the output of the function generator must be connected to the ground terminal of the oscilloscope 10 kQ Scope Ch 1 Scope Ground Figure 4 5 Connections for your RC circuit Set the function generator to produce a large amplitude square wave at about 3 kHz to begin with Observe the waveform then increase and decrease the frequency and just look at what happens Adjust the scope timebase and vertical sensitivity as desired Set the function generator at a convenient frequency at which you can see an
121. y one frequency or wavelength You will determine the wavelength of this laser light from your double slit interference experiment Part I You have been provided with a photographic plate that has a column of double slits For our lab today we will use at least two of these double slits For ease of measurement we recommend the second and third widest double slit pairs on your film note that in the series of double slits the narrowest is actually only a single slit For at least these two slit pairs measure the distance d between the slits as follows put the whole photographic plate on an overhead projector to magnify the image Measure the slit width of the image as well as the width of the whole photographic plate and its image at the horizontal line through the slits whose width is being measured Because the slits have a finite width you may want to measure edge to edge and average your results for left edges and right edges to get a value for d Repeat this measurement enough times so that you have a good idea of the uncertainty in d Part II Shine your laser beam onto one of the double slits that you have measured Place a screen at large distance L at least a couple of meters away from the double slit You should observe several points of constructive and destructive interference To get maximum sensitivity here it is important that the ambient light be dim Measure the distance between successive minima To get the best precisio

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