Instruction Manual Outline
Contents
1. 0 13 8 Oy4 0 0 0 0 0 y h 1 14 Average the values thus obtained for h and h to get the value of h that you will use for the rest of this lab The reason for averaging h and h is that the balance can sometimes act slightly asymmetrically so depending on whether you start on a positive or negative turning point you can get different decay constants The error in h can be calculated as shown below 50 1 WN 1I 1 hy 2h oh B 9 15 As it is not immediately clear how equations 11 13 and 14 came about the derivations are given in Appendix A 1 The effect of the spherical masses when they are rotated to either of their extreme positions is to change the equilibrium angle of the balance to the point where the torque from the wire equals the torque due to gravity Now choose a turning point during the forced oscillation to be your starting point Label that point 6 and just as with the decay label each following turning point up to On where N is an odd number For this part you want N to be as large as possible but you do not want to include points where the growth is overwhelmed by drift or where the growth has ceased Convert the displacements of your turning points into angular displacements with the equation 0 A B x 16 Note that the subscript on the A and B has changed from that in equation 12 Now calculate the change in the equilibrium angle Op w2. The American Book Company 1900 TEL Atomic Cavendish Balance User s Manual available in lab gt J R Taylor An Introduction to Error Analysis 2 Edition University Science Books Sausalito CA 1997 pp 184 188 J R Taylor An Introduction to Error Analysis 2 Edition University Science Books Sausalito CA 1997 pp 75 Appendix A 1 If the differential equation for your oscillator has the form O 2b0 0 0 28 that is if the damping is proportional to the velocity of the oscillator then the oscillator equation may be written as 0 0 4e cos t 29 if we define the phase to be zero at time zero Then h as defined in the Data Analysis section is the factor by which the oscillation decays over one half period So 8 h 02 30 You can add equation 30 with itself over and over for different values of n 0 0 h 0 0 4 9 0 h 0 8 h 9 0 8 h y_ 8 Hence the s will cancel out and you will be left with 0 0 0 6 8 G1 0 0 0 0 0 y1 h This can then be rewritten as equation 13 If one begins with the indices such that the left side has odd numbers first it is possible to end up with equation 14 through the same derivation The uncertainty in h 15 though it looks daunting is actually derived using equation 8 and treating each 6 as an
3. happens move the masses to the neutral position and wait for the oscillations to damp out Continue recording the linear deflection of the turning points until the oscillations have damped out Now take your data from the chart recorder and return all of the equipment to the way you found it Tips The oscillation forcing part of this lab is easier with two people One can stand near the posterboard to say when the spot s motion has reached its max and to record the displacement of the spot at that time while the other moves the masses from one side to the other This technique eliminates the latency between noting the maximum of the oscillation and moving the masses which can lead to blips in the data on the strip chart If you do not have a second person I suggest slightly anticipating the oscillation reaching its peak say to about a millimeter or two going to switch the masses and writing down the value where you think from observations of the spot prior to going back to move the masses the oscillation stopped When I did this experiment the oscillations had a period of about 186 seconds So while your period won t necessarily be exactly the same you will probably need to move the masses every 93 seconds or so except for the first peak which should be reached in about 46 seconds Data Analysis Measure the distance of each peak on your chart from the centerline I suggest using a pair of good calipers to do this Estimate t
4. independent variable but all with the same uncertainty 58 Appendix A 2 When you move the large masses to one side or the other what they really do is change the equilibrium angle of the boom The amount that they change it Op is the most important quantity to allow you to calculate G Consider a turning point When you flip the masses around it seems to the oscillator as though its amplitude has grown by an amount 0 0 depending on whether the amplitude is positive or negative Moreover since the equilibrium point has moved the boom won t start to slow down until it reaches the angle 0 F0 Putting this more quantitatively 8 41 8 I 8 A0 D 8 82 Solving this for Op you obtain _ CD 8 20 0 0 e l h 33 Adding this up for different values of n _ 9 0 h 8 6 K lth 0 0 h 0 90 0 1th 6 98 h 0 9 0 Ith 0 0 h 0 9 eo lth 6 9 h _ 9 0 1 h As you can see the 8 s will cancel out and you will be left with a 0 0 0 0 0 0 0 0 0 34 N 1 1 h 34 can be rewritten as 17 with a little algebra Again getting the uncertainty for Op is just a matter of judiciously applying 8 Appendix A 3 Equation 21 is actually very simple If you recall for an underdamped oscillator b lt the frequency becomes o 0 b 32 And as you s
5. is too high or too low the footpads may be adjusted to fix that Note from KM the footpads should be adjusted to keep the boom level and centered between the small openings on either end the most important thing is to keep the boom and connecting rod from rubbing against the frame Make sure the cross bar labeled beam with hole in Fig 2 is in the neutral position perpendicular to the face of the balance and wait for the oscillations to damp out Once the reflection has stopped moving tape a ruler horizontally to the posterboard so that the lower half of the reflection lies on the ruler proper and the reflection is in the horizontal center of the ruler 15 cm mark or thereabouts See Figure 3 Note this equilibrium position of the reflection and estimate your uncertainty Measure the horizontal distance from the center of the balance to the posterboard and estimate your uncertainty Plug in the power cord for the box labeled TEL Atomic Make sure the chart recorder paper is properly spooled the edge of the paper should be parallel to the paper guide see Figure 3 Use the lever on the right side of the chart recorder to raise the pen holder into the up position Place a pen in the pen holder uncap the pen and use the same lever to move the pen holder back into the down position Now set the speed on the chart recorder 60 cm hr usually works well but be sure to record in your lab book the value you select Set the voltage rang
6. Manual for Determining G with a Cavendish Balance Sean Sullivan May 5 2008 with minor edits by K Minschwaner 2009 Background and History Here will be given a brief account of the history of the Cavendish Experiment and its significance A detailed discussion of the theory due to its extensive length and the fact that the details of how the experiment is conducted have changed since Cavendish s time will be discussed in later parts of this document Nowadays most people think of the Cavendish Experiment as the experiment to determine the universal gravitational constant G However this is not the purpose for which the experiment was originally designed The experiment was originally conceived and built by the geologist John Michell for the purpose of determining the mean density of the Earth The value of G was never an objective of the experiment and in fact the results of the Cavendish Experiment was not used to determine a value for G until nearly 100 years later when C V Boys used it in a paper he presented in 1892 Why was the Cavendish Experiment used for what seems today such a banal purpose It may be illuminating to consider the fact that in the Principia Isaac Newton never wrote his Law of Universal Gravitation in terms of an equation that included G Instead he performed his calculations with ratios But still why was the density of Earth so interesting Scientists wished to know the density of the Earth because Newt
7. e on the chart recorder to around 5V pen position penholder Paper guide voltage range adjustment knob selection lever to move pen up holder up or down chart speed selection Figure 3 This is the strip chart recorder that you will be using during the experiment You can play around with the range and speed settings to see what works best Doing the Experiment Turn on the chart recorder and use the position knob to place the pen on the center line of the paper Place the two large spherical masses on the holes in the crossbar Watch the chart recorder and wait for any resulting oscillations to damp out You are now ready to start the actual experiment Gently swing the masses to one side Get them as close to the glass as you can without striking it If you accidentally tap the glass you re probably still okay as long as you didn t hit it too hard but don t get cavalier After you ve moved the masses watch the reflected laser spot on the cardboard It should be slowly moving to one side Once the movement of the spot stops swing the masses to the other side and record the position read it from the ruler that you taped to the posterboard where the motion of the spot stopped you should do this for every turn around point during this part of the experiment Remember to estimate your uncertainty Continue this process until the magnitude of the oscillation as measured on the ruler stops growing with each cycle Once this
8. en for reference as Figure 2 below Cavendish balance should be set up opposite a piece of white posterboard such that the glass on the outside of the balance is parallel to the posterboard The farther away the balance is from the posterboard the better but a distance of about 2 5 meters should be fine 25 micron tungsten wire mirror small mass boom beam with hole for large mass adjustable footpad Figure 2 This labeled image of the Cavendish balance should be referred to while following the instructions in this manual Set up the neon laser about halfway between the posterboard and the balance The aperture of the laser should be level with the mirror inside the Cavendish balance you will probably need to place books you find in the lab under the laser to accomplish this and there should be a small angle between the laser beam and the normal to the surface of the balance Now turn on the laser and make sure that it is shining on the mirror attached to the boom inside the balance Look for the reflection Ifthe reflection is too far to one side of the posterboard or off the posterboard altogether move it towards the center by turning the balance Do this slowly Chances are that you excited some small oscillations while turning the balance Does the reflection of the laser beam move perfectly along the horizontal If not adjust the screw pads on the base of the balance to correct for any tilt Also if the reflection
9. he uncertainty in your measurements You may notice that there is a slow drift in your chart towards more positive or negative values At some point this drift may even overwhelm the growth of the oscillation in one direction I am not sure why the drift occurs but it may be due to charge traps or some other defect in the balance or peripheral equipment I suggest that for the next section you use only data taken early enough that it is not catastrophically affected by this drift Now from your displacement data calculate your angular displacement for each data point The equation for converting tangential displacements to angles is 6 paeen SF 1 2 L where d is the displacement measured on the ruler do is the equilibrium position of the reflected laser spot on the ruler and L is the distance from the balance to the posterboard The factor of 2 is there because the angular deflection of the laser beam will be twice the amount that the mirror is deflected As you can see for very small displacements which is the regime we should be working in 8 will be linearly related to the displacement d by the small angle approximation Once you have your angular displacements in radians plot the angular displacement versus the displacement that you measured from the strip chart Run a linear regression on this graph the fit should be very good if your data is reliable The equation that the regression returns is the equation to turn the dis
10. hould remember 33 We can then put 32 and 33 together to get K 0 87 7 34 2 which becomes 21 when you remember the fact that i Appendix B Values for the Cavendish Balance Quantity Value Uncertainty M 1 039 kg 0 001 kg ms 0 014545 kg 0 000001 kg I 0 000143 kg m 2 0 000001 kg m 2 R 0 0461 m 0 00016 m m mh 1 fd mb fo 0 01511 kg 0 00004 kg d 0 06665 m 0 00004 m
11. ith the following equation _ A 8 0 0 0 0 49 9 N 1 1 h 17 The error in Op comes from two sources error in h and error in 8 The contributions from these two components can be calculated as shown below ae SOKN 1 1 h 2x 18 3 N 1 1 A _ 8A 2 0 0 9y Oy 0 50 l 7 N D x 19 Add these two errors in quadrature to get 68p The derivation of equation 17 is given in Appendix A 2 The final value that you must measure is the oscillation period T Go to the part of your strip chart where the oscillator is in free decay Choose two positive or negative turning points and measure the horizontal time axis distance between them with your calipers Convert your measurement to a time by using the spooling speed of the strip chart recorder and divide that by the number of periods between your selected turning points in order to get the oscillators natural period Repeat this procedure using several different combinations of turning points and average them to get your best value Remember not to use points on the graph whose amplitude is very low as the period can get distorted in this regime The error in your value for T may be estimated by the standard deviation of your individual measurements where 7 is the mean of your period measurements and T is an individual period measurement From the data you have measured you may calculate the torsion con
12. on had calculated the densities of the Sun Jupiter and Saturn as proportions of the density of Earth Hence if Earth s density was known so were the other three John Michell died in 1793 before he had an opportunity to perform his experiment As a result the balance passed to Francis Wallaston and then to Michell s longtime friend with whom he had kept almost constant correspondence Henry Cavendish Cavendish perhaps as a way of honoring his friend s memory decided to proceed with the experiment much as Michell had first conceived of it However Cavendish was an exceptionally careful experimenter and ended up remaking large parts of the apparatus in order to eliminate everything from air currents to magnetic forces that could possibly affect the result A drawing of the apparatus from Henry Cavendish s paper is shown below in Figure 1 As an aside it is interesting to note that Coulomb used a similar torsion balance in his studies of electrostatic forces However as Cavendish points out in his paper John Michell conceived of the torsion balance design independently of Coulomb since Coulomb had not yet published his results when Michell was designing this experiment Cavendish published his results in 1789 in a paper entitled Experiments to Determine the Density of the Earth and they were astounding Since the force of the Earth on the smaller lead spheres was known their weight and their densities sizes and separa
13. placements that you measured on your strip chart into angular displacements You may be questioning why you would want to do this step After all can t you just get the angular displacements from equation 1 and your displacement data The answer is that you do this step in order to linearize your data and help beat down uncertainties Speaking of uncertainties you now need to calculate the uncertainty in the slope and intercept of your regression line Use the following equations to do this See John Taylor An Introduction to Error Analysis pages 184 188 for more details T Vily 4 Bx Q N 2 i l o 0 5 where A and B are from the equation y A Bx if y is the raw angular displacement and x is the corresponding displacement on the strip chart Call the A and B that you obtain for the growing case A and By I strongly suggest that you use a computer to find A and B However if you insist you may find them with the following equations aA hehehe 6 Pee Nae Sp ey 7 A For most of the error analysis in the rest of this lab you will be using the standard error propagation equation from Taylor which is given as equation 8 below y Fon aa 0 Now we can find the uncertainty in 0 by using equation 8 2 80 6 lxo Ba 42 1 a he 9 5 i Actually only the first three terms in the square root in 9 come from 8 the fourth term is extra and accounts for po
14. ssible systematic error due to uncertainties in your balance posterboard distance measurement don t worry you don t need to include any extra terms in your analysis for the rest of the lab Use Microsoft Excel or some other spreadsheet program to calculate and average your uncertainties X18 gy 60 N Call the average fractional uncertainty calculated this way for the growing oscillation case 50 Now repeat the above analysis but for the case where the oscillator is damping out Only use turning points with relatively large amplitudes You should have N 11 points Give the values a subscript d so Aa Ba etc Note you will need to repeat the error analysis as well ie calculate a new Oa A etc Before we can go further in our analysis we are going to want to find the amount that the oscillations of the boom decay over one half oscillation period during free damping call this value h h is related to other constants by h eT 11 where b is the decay constant and T is the period of the oscillation Convert the displacements from your stripchart for the damping oscillation into angles using the equation 0 A Bax 12 Now pick a peak on your stripchart after you began letting the oscillation damp out and label it 0 Number each peak after it in the same manner up to On where N should be an odd number gt 11 h can then be calculated with the following equations 8 0y h 1 0 0 0 0
15. stant K of the tungsten wire An K gt e 21 where I is the moment of inertia of the boom plus the small spheres and b is the decay constant which can be calculated from h see equation 11 Equation 21 is derived in Appendix A 3 2 b In h 22 A 22 The uncertainty in this measurement is then from 8 2 i aN ae eae iar 23 5b 2 ms ar 23 I strongly suggest that you use the value listed in Appendix B for I since to do otherwise would entail taking apart the balance which risks damage to the apparatus However 10 since it is the instructor s discretion whether or not this is done I may be calculated as the sum of the moments of inertia of the two small spheres I and the boom I Loa Oe ee 24 Ss Ss 5 S and l ae w 12 where m is the mass of a small sphere d is the distance from the center of the boom to the center of a small sphere r is the radius of a small sphere m is the mass of the boom and w and l are the boom s width and length respectively Equations 23 and 24 can be derived from standard mechanics and the error may be gotten by using equation 8 The error in I would then be the errors in I and I added in quadrature Again using 8 we can find the uncertainty in K le ar sano Fo 0 y 26 An equation for G can be found by equating the torque on the balance due to the large masses when they are located in one of the extreme positions to the
16. tions were also known it was possible to use a ratio of the forces on the smaller spheres due to the Earth and due to the larger lead spheres to find the mean density of the Earth The number thus obtained was 5 48 0 038 times the density of water which is within one percent of the currently accepted value of 5 52 Deriving G from these results gives a value of G 6 74x107 kg m s which is again very close to the currently accepted value of G 6 73x10 kg m s Think you can do better In this experiment you will perform a version of the Cavendish Experiment except with a modern and much smaller balance with the objective of obtaining a value for G If you follow the instructions set forth in this manual you should be able to get a value for G within 3 of the accepted value in a reasonable amount of time Figure 1 This diagram is taken from Cavendish s 1798 paper and shows in detail the experimental setup he used to conduct his measurements A description of each component would be too space consuming but you probably get the general idea However if you are interested Cavendish describes every piece of the apparatus at great length in his paper Procedure These instructions follow the method that I used when I ran this experiment For alternate approaches or more information please refer to the TEL Atomic users manual Setting up A picture of the balance with parts labeled is giv
17. torque from the tungsten wire when the boom is displaced by the change in equilibrium angle 8p However two corrections must be made to this basic equation One is a correction for the attraction of the aluminum boom to the large mass The second is a correction for the fact that each large mass acts not only on the small mass that it is placed next to but also on the small mass on the other side of the balance The resulting equation is given below 25 b b K oe K R 2M m m f1 m f ld 27 where m is the mass of the hole in the boom where the small mass sits M is the mass of the large sphere and fa and f terms are corrections for the attraction of the large masses to the distant small sphere and boom respectively With the instructor s permission you may take the value for the correction from Appendix B For further discussion of the correction factors including the derivation please refer to the TEL Atomic user s manual In order to calculate the uncertainty in G I suggest turning all of your uncertainties into fractional uncertainties so that you can just add them instead of needing to use 8 assuming that you have the uncertainty for the bracketed expression in the denominator as a whole 11 S P Lally The Phys Teacher 37 34 1999 B E Clotfelter Am J Phys 55 3 210 1987 gt H Cavendish in Scientific Memoirs The Laws of Gravitation edited by A S MacKenzie
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