PASCO Specialty & Mfg. ME-8950A User's Manual
Contents
1. base using the black rod d 2 Connect the Photogate to a PASCO interface and connect Projectile the interface to a computer Cacher 3 Wind a thread around the middle step pulley on the center step pulley Bnet shaft and pass the thread over the pulley with Se Procedure Accounting For Friction Because the theory used to find the rotational inertia experi mentally does not include friction it will be compensated for in this experiment by finding out how much mass over the pulley it takes to overcome kinetic friction and allow the mass to drop at a constant speed Then this friction mass will be subtracted from the mass used to accelerate the apparatus 1 Start the DataStudio program Select Smart Pulley Linear and set up a Digits display to show velocity with three significant figures 2 Hang a small amount of mass such as a few paper clips on the end of the thread that is over the pulley 3 Start monitoring data and then give the Rotating Platform a tap to get it started moving 4 Watch the Digits display to see the velocity 5 Ifthe velocity increases or decreases as the Rotating Platform turns stop monitoring data stop the Rotating Platform and adjust the amount of mass on the thread by adding or removing a paper clip 6 Repeat the process until the velocity stays constant 7 Meas2. e Eea V U Figure 7 1 Assembly for Dropping Ring onto Disk 17 Cilefe 41 scientific Complete Rotational System 012 05293F Procedure 1 Hold the ring just above the center of the disk Give the disk a spin using your hand 2 Start recording data After about 25 data points have been taken drop the ring onto the spinning disk See Figure 7 2 3 Continue to take data after the collision for a few seconds and then stop recording data dropped ring ue ao m photogate head rotational disk 1 A base 7 Figure 7 2 Experiment Setup 4 Examine the Graph display of the rotational speed versus time Use the Autoscale tool to resize the axes if necessary 5 In the Graph display use the Smart Tool to determine the angular velocity immediately before and immediately after the collision Record these values in Table 7 1 6 Weigh the disk and ring and measure the radii Record these values in Table 7 1 Analysis 1 Calculate the expected theoreti Table 7 1 Data and Results cal value for the final angular velocity and record this value in Table 7 1 Initial Angular Speed 2 Calculate the percent difference between the experimental and Final Angular Speed experimental value the theoretical values of the final Xp angular velocity and record in Mass of Disk Table 7 1 Mass of Ring Outer Radius of Ring
3. 10 spoke pulley with photogate head my 300g mass Figure 2 2 Rotational inertia of a point mass Procedure Part I Measurements For the Theoretical Rotational Inertia i Weis the squnke masse find die usd and Table 2 1 Theoretical Rotational Inertia record in Table 2 1 Why is it possible to assume that the square mass acts the same as a point mass Mass M aaa O 2 Measure the distance from the axis of rotation to the Radius R nn center of the square mass and record this radius in Table 2 1 Part Il Measurement For the Experimental Method Accounting For Friction Because the theory used to find the rotational inertia experimentally does not include friction it will be compensated for in this experiment by finding out how much mass over the pulley it takes to overcome kinetic friction and allow the mass to drop ata constant speed Then this frictionmass will be subtracted from the mass used to accelerate the ring 1 Start the DataStudio program Select Smart Pulley Linear and set up a Digits display to show velocity with three significant figures 2 Hanga small amount of mass such as a few paper clips on the end of the thread that is over the pulley 3 Start monitoring data and then give the Rotating Platform a tap to get it started moving 4 Watch the Digits display to se
4. Rotate the apparatus by hand increasing the speed until the orange indicator is centered in the indicator bracket on the center post This indicates that the string supporting the hanging object is once again vertical and thus the hanging object is at the desired radius Table 3 1 Varying the Radius Maintaining this speed use a stopwatch to time ten revolutions Divide the time by ten and Mass of the object record the period in Table 3 1 Mass hanging over the pulley Move the side post to a new radius and repeat Slope from graph the procedure Do this for a total of five radii The weight of the mass hanging over the pulley is equal to the centripetal force applied by the spring Calculate this force by multiplying the mass hung over the pulley by g and record this force at the top of Table 3 2 Calculate the square of the period for each trial and record this in Table 3 1 Plot the radius versus the square of the period This will give a straight line since F di pom r Draw the best fit line through the data points and measure the slope of the line Record the slope in Table 3 1 m PASCO scientific 012 05293F Complete Rotational System 5 Calculate the centripetal force from the slope and record in Table 3 2 6 Calculate the percent difference between the Centripetal Force mg EE two values found for the centripetal force Centripetal Force From Slope LO and record in Table 3 2 Per
5. 7 Conservation of Angular Momentum Using Disk and Ring 41 Technical Support 2 52 itor shot E PERRA nuns ERE REN ERE e Eee Gen Vo xe idus 45 Complete Rotational System 012 05293F Copyright Warranty and Equipment Return Please Feel free to duplicate this manual subject to the copyright restrictions below Copyright Notice The PASCO scientific Model ME 8950A Complete Rotational System manual is copyrighted and all rights reserved However permission is granted to non profit educational institutions for reproduction of any part of this manual providing the reproductions are used only for their laboratories and are not sold for profit Repro duction under any other circumstances without the writ ten consent of PASCO scientific is prohibited Limited Warranty PASCO scientific warrants this product to be free from defects in materials and workmanship for a period of one year from the date of shipment to the customer PASCO willrepair orreplace atits option any part of the product which is deemed to be defective in material or workman ship This warranty does not cover damage to the product caused by abuse or improper use Determination of whether a product failure is the result of a manufacturing defect or improper use by the customer shall be made solely by PASCO scientific Responsibility for the return of equipment for warranty repair belongs to the cus tomer Equipment must be properly packed to prev
6. Assembly 1 Attach the square nut supplied with the Rotational Inertia Accessory to the platform adapter 2 Position the platform adapter at the desired radius as shown in Figure 8 3 Grip the knurled edge of the platform adapter and tighten The rotating disk can be mounted in a variety of positions using any of the four holes on the rotation disk Two D holes exist on the edge of the disk located at 180 from one another e One D hole is located at the center on the top surface the surface with the metal ring channel and the PASCO label of the disk One hole is located at the center on the bottom surface of the disk and is actually the inner race of a bearing This enables the rotational disk to rotate in either direction in addition to other rotating motions applied to your experiment setup PASCC scientific 012 05293F Complete Rotational System Experiment 1 Conservation of Angular Momentum Projectile Version EQUIPMENT NEEDED Rotating Platform ME 8951 Rubber band Projectile Launcher ME 6800 White paper and carbon paper Note If you are Projectile Collision Accessory ME 6815 Thread using a PASPORT Photogate Pulley System ME 6838 Meter Stick SE 6895 interface you will DataStudio Software Mass and Hanger Set ME 8967 also need a Digital Adapter PS 2159 PASCO Interface see Note Calipers SF 8711 Purpose The muzzle velocity
7. Radius of Disk Final Angular Speed theoretical value Difference Between Final Angular Speeds 96 KE Lost 42 PASCO scientific 012 05293F Complete Rotational System Questions 1 Does the experimental result for the angular speed agree with the theory 2 What percentage of the rotational kinetic energy is lost during the collision Calculate this and record the results in Table 7 1 i 3 33 2 i TA JOf KE Lost a 5i FK 2 160 43 scientific Complete Rotational System 012 05293F 44 18 7 X 1610 scientific 012 05293F Complete Rotational System Technical Support Feed Back If you have any comments about this product or this manual please let us know If you have any suggestions on alternate experiments or find a problem in the manual please tell us PASCO appreciates any customer feed back Your input helps us evaluate and improve our product To Reach PASCO For Technical Support call us at 1 800 772 8700 toll free within the U S or 916 786 3800 email techsupp PASCO com Tech support fax 916 786 3292 PASCC scientific 45 Contacting Technical Support Before you call the PASCO Technical Support staff it would be helpful to prepare the following information If your problem is computer software related note Title and Revision Date of software Type of Computer Make Model Speed Type of external Cables Peripherals If your pro
8. a few paper clips on the end of the thread that is over the pulley Start monitoring data and then give the Rotational Disk a tap to get it started moving Watch the Digits display to see the velocity 5 Ifthe velocity increases or decreases as the Rotational Disk turns stop monitoring data stop the y Rotational Disk and adjust the amount of mass on the thread by adding or removing a paper clip 6 Repeat the process until the velocity stays constant 7 Measure the mass on the end of the thread and record it as the Friction Mass in Table 6 2 Finding the Acceleration of Disk and Track Table 6 2 Rotational Inertia Data Fixed Disk and Rotating Disk and Track Combined Track Alone Track Combined BE MEE SSS To find the acceleration put about 50 g record the exact hanging mass in Table 5 2 over the pulley In the DataStudio program set up a Graph display of Velocity versus Time 1 Wind the thread up and hold the Rotating Platform 2 Let the Rotating Platform begin to turn and at the same time start recording data 3 Let the mass descend toward the floor but STOP recording data just before the mass hits the floor 4 Examine your Graph display of Velocity versus Time The slope of the best fit line for your data is the acceleration of the apparatus 5 Record the slope in Table 6 2 PASCO 39 scientific Complete Rotational System 012 05293F Measure the Radius 1 Using calipers measure the
9. in the indicator bracket on the center post This indicates that the string supporting the hanging object is once again vertical and thus the hanging object is at the desired radius Maintaining this speed use a stopwatch to time ten revolutions Divide the time by ten and record the period in Table 3 5 Vary the mass of the object by removing the side masses Keep the radius constant and measure the new period Weigh the object again and record the mass and period in Table 3 5 22 PASCC scientific 012 05293F Complete Rotational System Table 3 5 Varying the Mass of the Object Mass hanging over pulley Centripetal Force mg Radius Mass of Object Period T Coe i ee Difference Analysis 1 The weight of the mass hanging over the pulley is equal to the centripetal force applied by the spring Calculate this force by multiplying the mass hung over the pulley by g and record the result at the top of Table 3 5 2 Calculate the centripetal force for each trial using 2 _ 4n mr T F and record this in Table 3 5 3 Calculate the percent difference between the calculated centripetal force for each trial and mg Record in Table 3 5 Questions 1 When the radius is increased does the period of rotation increase or decrease 2 When the radius and the mass of the rotating object are held constant does increasing the period increase or decrease the centripetal force 3 As the mass of the objec
10. it is possible to vary the mass and radius to see the resulting change in the cen tripetal force The force can also be held constant while other quantities are varied The Centripetal Force Acces sory is powered by hand and the rate of rotation can be counted manually or read by a computer Variable hang ing masses are included Z CA o scientific The ME 8953 Rotational Inertia Accessory includes a disk and a metal ring The disk can be mounted to the rotating base in a variety of positions and at any radius This accessory requires the Rotating Platform ME 8951 to operate The Rotational Inertia Accessory allows you to perform rotational inertia experiments and conserva tion of angular momentum experiments About This Manual The following Equipment section describes each compo nent in detail and the subsequent Assembly section pro vides instructions for component assembly and setup The Experiment section contains several experiments that can illustrate some of the basic theory of centripetal force rotational inertia etc Computer Timing Youcanusea PASCO computerinterface witha PASCO Photogate Head to measure the motion of the apparatus Someofthe experiments describe how to use DataStudio the software program that supports every PASCO com puter interface If you are using a computer interface such as a Science Workshop 150 or ScienceWorkshop 500 you can connect the cable of the Photogate Head directly into t
11. of paper distance along the floor from the release Initial velocity point to the leading edge of the paper Recordin Table 1 1 5 Measure from the leading edge of the paper to each of the ten dots and record these dis ave n TEM 1 2 oo 6 Findthe average of the ten distances and D T Using the vertical distance and the average En UN UEM horizontal distance calculate the time of flight and the initial velocity of the ball E NN EE Record in Table 1 1 and Table 1 4 mE uM ERR Alternate Method Determining the Muzzle po d 4g 1 Attach the photogate bracket to the E0000 1 0 0 Launcher and attach two Photogates to the bracket Plug the Photogates into a com Average 2 Putthe ball into the Projectile Launcher and cock it to the long range position 3 Run the DataStudio program Set it to measure the time between the ball blocking the two Photo gates as it comes out of the Launcher 4 Shoot the ball three times and take the average of these times Record in Table 1 2 PASCO i scientific Complete Rotational System 012 05293F p ae o2 o3 meme Initial Velocity NENNEN Table 1 2 Initial Speed Using Photogates 5 The distance between the Photogates is 10 cm Calculate the initial speed and record it in Table 1 2 and Table 1 4 Part Il Conservation of Angular Momentum Setup rubber band side view of rubber band rotating
12. of the experiment Weigh the object and record its mass in Table 3 1 Hang the object from the side post and connect the string from the spring to the object The string must pass under the pulley on the center post See Figure 3 1 PASCO T scientific Complete Rotational System 012 05293F e Analysis l side post E center post assembly string assembly clamp on rotating ulle pu ey latform AON Pl 3 hanging mass A base U Figure 3 1 Centripetal Force Apparatus Attach the clamp on pulley to the end of the track nearer to the hanging object Attach a string to the hanging object and hang a known mass over the clamp on pulley Record this mass in Table 3 1 This establishes the constant centripetal force Select a radius by aligning the line on the side post with any desired position on the measuring tape While pressing down on the side post to assure that it is vertical tighten the thumb screw on the side post to secure its position Record this radius in Table 3 1 The object on the side bracket must hang vertically On the center post adjust the spring bracket vertically until the string from which the object hangs on the side post is aligned with the vertical line on the side post Align the indicator bracket on the center post with the orange indicator Remove the mass that is hanging over the pulley and remove the pulley
13. of the Projectile Launcher can be determined by shooting the ball into the catcher mounted on the platform and conserving angular momentum during the collision This result can be checked by finding the muzzle velocity of the Launcher by shooting the ball horizontally off the table Theory A ball is launched horizontally and embeds in the catcher mounted on the platform The platform then rotates See Figure 1 1 Angular momentum is conserved during the collision but energy is not conserved The angular Rotating n Platform A base Projectile 31 G Launcher C Y Vo s x O TR eee 1 p Ley im 5 a J E F ae J Projectile Collision Accessory catcher Figure 1 1 Conservation of Angular Momentum momentum before the collision is equal to the angular momentum after the collision L myvgR Io where mp is the mass of the ball v is the muzzle velocity of the ball R is the distance between the ball and the axis of rotation is the rotational inertia of the catcher ball and rotating platform after the collision and o is the angular velocity of the system immediately after the collision Solving for the muzzle velocity of the ball gives PASCO 5 scientific Complete Rotational System 012 05293F Io myR Vo To find the rotational inertia experim
14. platform end view of rotating platform NS catcher tab of catcher against platform Figure 1 4 Attaching the Catcher to the Track 1 Find the mass of the ball and record it in Table 1 3 2 Attach the ball catcher to the track using a rubber band as shown in Figure 1 4 3 With the Projectile Launcher mounted as it was in Part I aim the launcher directly down the mid dle of the ball catcher using the sights inside the projectile launcher Clamp the launcher to the table 4 Attach the Photogate Head to the base Connect the Photogate Head to a computer and run the DataStudio program 5 Setup the program so that it measures and displays angular speed Procedure 1 Level the rotating platform 2 Load the Launcher with the steel ball on the long range setting 3 Make sure the rotating platform is at rest and fire the ball into the catcher Table 1 3 Angular Speed Record th l d of the platform in Tabl ecord the angular speed of the platform in Table TTE 1 3 Repeat for a total of five shots 4 Measure the distance from the axis of rotation to Distance from axis of rotation R the ball in the catcher and record in Table 1 3 Angular Speed Angular Speed T fe x TIL 8 Ss E PASCO scientific 012 05293F Complete Rotational System Part IIl Determining the Rotational Inertia Setup 1 Attach a Photogate with a pulley and mounting rod to the
15. 012 05293F Complete Rotational System Instruction Manual and 012 05293F Experiment Guideforthe T PASCO scientific Model ME 8950A COMPLETE ROTATIONAL SYSTEM PASCO scientific B A SCC 0 800 772 8700 www pasco com Complete Rotational System 012 05293F 012 05293F Complete Rotational System Table of Contents Section Page Copyright Warranty and Equipment Return eere ii Dhricersn estu PASE NE E ETS 1 Description About this Manual Computer Timing Equipment T P 2 Miscellaneous Supplies ie Ee ote wel aatied mien alone iene 3 Assembly ME 8951 Rotating Platforimn iei rerit eerta tete Pete orae te 4 ME 8952 Centripetal Force Accessory eese 6 ME 8953 Rotational Inertia Accessory eccceeeseceesteceesteceeteeeeseeeenaeeeenes 8 Experiments Using the ME 8951 Rotating Platform Exp 1 Conservation of Angular Momentun Projectile Version 9 Exp 2 Rotational Inertia of a Point Mass eene 15 Experiments Using the ME 8952 Centripetal Force Accesory Exp 3 Centripetal Poree s tea eto tette dereud 19 Exp 4 Conservation of Angular Momentum Using Point Mass 25 Experiments Using the ME 8953 Rotational Inertia Accessory Exp 5 Rotational Inertia of Disk and Ring esses 31 Exp 6 Rotational Inertia of Disk Off Axis Fixed Rotating 37 Exp
16. 2 Why IZEI O 29 scientific Complete Rotational System 012 05293F 30 PASCO scientific 012 05293F Complete Rotational System Experiment 5 Rotational Inertia of Disk and Ring EQUIPMENT REQUIRED Note If you are DataStudio Program Mass aud Hanger Set using a PASPORT PASCO Interface see Note Paper Clips for masses lt 1 g interface you will Rotational Inertia Accessory ME 8953 Balance also need a Digital Photogate Pulley System Calipers Adapter PS 2159 Purpose The purpose of this experiment is to find the rotational inertia of aring and a disk experimentally and to verify that these values correspond to the calculated theoretical values Theory Theoretically the rotational inertia of a ring about its center of mass is given gt by P lapin 5M Rj R3 where M is the mass of the ring R is the inner radius of the ring and R is the outer radius of the ring See Figure 5 1 The rotational inertia of a disk about its center of mass is given by Figure 5 1 Ring 1 sp I MR 2 where M is the mass of the disk and R is the radius of the disk The rotational inertia of a disk about its diameter is given by Disk about center of Mass Figure 5 2 Disk about Diameter To find the rotational inertia experimentally a known torque is applied to the object and the resulting angular acceleration is measured Since 1 IQ T I 2 a PASCC 31 scientific Com
17. SCO 35 scientific Complete Rotational System 012 05293F 36 PASCO scientific 012 05293F Complete Rotational System Experiment 6 Rotational Inertia of Disk Off Axis Fixed Rotating EQUIPMENT REQUIRED DataStudio Program Calipers Mos CIR PASCO Interface See Note Mass and Hanger Set interface you will Rotational Inertia Accessory ME 8953 Paper Clips for masses 1 g also need a Digital Photogate Pulley System Balance Adapter PS 2159 Purpose The purpose of this experiment is to find the rotational inertia of a disk about an axis parallel to the center of mass axis Theory Theoretically the rotational inertia Z of a disk about a perpendicular axis through its center of mass is given by cm 1 2 52 I MR where M is the mass of the disk and R is the radius of the disk The rotational inertia of a disk about an axis parallel to the center of mass axis is given by I 2L 4Md cm where d is the distance between the two axes In one part of this experiment the disk is mounted on its ball bearing side which allows the disk to freely rotate relative to the track So as the track is rotated the disk does not rotate relative to its center of mass Since the disk is not rotating about its center of mass it acts as a point mass rather than an extended object and its rotational inertia reduces from I21 Md tol Md To find the rotational inertia experimentally a kno
18. Second Law for the hanging mass m gives See Figure 4 1 LF mg T ma Solving for the tension in the thread gives T m g a Once the linear acceleration of the mass m is determined the torque and the angular acceleration can be obtained for the calculation of the rotational inertia PASCC 25 scientific Complete Rotational System 012 05293F rotating platform a A base ia mg hanging r d mass Figure 4 1 Rotational Apparatus and Free Body Diagram Part I Conservation of Angular Momentum Setup 1 Level the apparatus using the square on the track as shown in the leveling instructions in the Assembly Section 2 Slide a thumb screw and square nut into the T slot on the top of the track and tighten it down at about the 5 cm mark This will act as a stop for the sliding square mass See Figure 4 2 string center post 300g mass m 2 rotating stop screws 2 platform A base Figure 4 2 Set up for conservation of angular momentum 3 With the side of the square mass that has the hole oriented toward the center post slide the square mass onto the track by inserting its square nut into the T slot but do not tighten the thumb screw the square mass should be free to slide in the T slot 4 Slide a second thumb screw and square n
19. al Force Accessory includes center post that supports an indicator mechanism which consists ofa small pulley a movable spring holder a movable indicator a spring and a plastic indicator disk side post for hanging hooked mass mass 100 g with 3 open hooks 2 additional 50 gram masses clamp on pulley 1 spool of thread 012 05293F Complete Rotational System mass ring 12 7 cm outside diameter rotational disk 25 4 cm diameter ej Pulley and rod rotating platform Ae adapter ME 8953 Rotational Inertia Accessory Equipment The ME 8953 Rotational Inertia Accessory includes Miscellaneous Supplies disk with bearings in the center graph paper ring 12 7 cm diameter carbon paper adapter to connect disk to platform white paper 10 spoke pulley and rod rubber bands Other Equipment Needed paper clips The following is a list of equipment recommended for the ico Compatible PASCO Computer Interfaces experiments described in this manual See the PASCO catalog for more information The Photogate Head connects directly to a Projectile Launcher ScienceWorkshop interface such as the Projectile Collision Accessory i o tanda epn EARE O com USB Link PowerLink Xplorer or Xplorer GLX Mass and Hanger Set Stopwatch String Balance for measuring mass Calipers Meter Stick PASCO scientific ScienceWorkshop 500 and connects through a Digital Adap
20. alue of the rotational inertia of the track alone Subtract the rotational inertia of the track from the rotational inertia of the fixed disk and track This will be the rotational inertia of the fixed disk alone Calculate the experimental value of the rotational inertia of the fixed disk and track com bined Subtract the rotational iner E re Disk tia of the track from the rotational inertia of the free Rotational Inertia for Track Alone disk and track This will be the rotational inertia of the Rotational Inertia for Fixed Disk freedisk alon Off Axis experimental value Calculate the theoretical Rotational Inertia for Free Disk value of the rotational iner and Track Combined Table 6 3 Results tia of the fixed disk off Rotational Inertia for Free Disk axis Alone experimental value Calculate the theoretical Rotational Inertia for Fixed Disk value of a point mass hav Off Axis theoretical value ing the mass of the disk Rotational Inertia for Point Mass Use a percent difference to theoretical value compare the experimental me values to the theoretical values 96 Difference for Free Disk 40 PASCO scientific 012 05293F Complete Rotational System Experiment 7 Conservation of Angular Momentum EQUIPMENT REQUIRED Note If you are DataStudio Program PASCO Interface using a PASPORT Rotational Inertia Accessory ME 8953 Balance interface you will Rotating Platform ME 8951 Pho
21. blem is with the PASCO apparatus note Title and Model number usually listed on the label Approximate age of apparatus A detailed description of the problem sequence of events In case you can t call PASCO right away you won t lose valuable data If possible have the apparatus within reach when call ing This makes descriptions of individual parts much easier f your problem relates to the instruction manual note Part number and Revision listed by month and year on the front cover Have the manual at hand to discuss your questions
22. celeration of Ring and Disk the disk is rotating as well as the ring it is necessary to determine the acceleration and the rotational inertia of the disk by itself so this rotational inertia can be subtracted from the total leaving only the rotational inertia of the ring 1 To do this take the ring off the rotational apparatus and repeat Finding the Acceleration of Ring and Disk for the disk alone NOTE that it will take less friction mass to overcome the new kinetic friction and it is only necessary to put about 30 g over the pulley in Finding the Acceleration of the Disk Alone Disk Rotating on an Axis Through Its Diameter Remove the disk from the shaft and rotate it up on its side Mount the disk vertically by inserting the shaft in one of the two D shaped holes on the edge of the disk See Figure 5 5 WARNING Never mount the disk vertically using the adapter on the track The adapter is too short for this purpose and the disk might fall over while being rotated Repeat steps Measure the Radius and Finding the Acceleration of the Disk Alone to determine the rotational inertia of the disk about its diameter Record the data in Table 5 2 open rotational disk Eee T 1 1 ees D hole of rotational disk ee A base _ EE erm zas aes U U U U rotating shaft Figure 5 5 Disk mounted vertically Calculations Record t
23. cent Difference cd Part Il Vary Force constant radius and mass Table 3 2 Results varying raduis The radius of rotation and the mass of the hanging object will be held constant for this part of the experiment 1 Weigh the object and record its mass in Table 3 3 Hang the object from the side post and connect the string from the spring to the object The string must pass under the pulley on the center post 2 Attach the clamp on pulley to the end of the track nearer to the hanging object Attach a string to the hanging object and hang a known mass over the clamp on pulley Record this mass in Table 3 3 This determines the centripetal force 3 Select a radius by aligning the line on the side post with any desired position on the measuring tape While pressing down on the side post to assure that it is vertical tighten the thumb screw on the side post to secure its position Record this radius in Table 3 3 4 The object on the side bracket must hang vertically On the center post adjust the spring bracket vertically until the string from which the object hangs on the side post is aligned with the vertical line on the side post 5 Align the indicator bracket on the center post with the orange indicator 6 Remove the mass that is hanging over the pulley and remove the pulley 7 Rotate the apparatus increasing the speed until the orange indicator is centered in the indicator bracket on the center post This indicates that the stri
24. d in Table 3 4 Mass of Object from scale RR Mass of Object from slope LL Percent Difference HEN Part Ill Vary Mass constant radius and force The centripetal force and the radius of rotation will be held constant for this part of the experiment l Weigh the object with the additional side masses in place Record its mass in Table 3 5 Hang the object from the side post and connect the string from the spring to the object The string must pass under the pulley on the center post Attach the clamp on pulley to the end of the track nearer to the hanging object Attach a string to the hanging object and hang a known mass over the clamp on pulley Record this mass in Table 3 5 This establishes the constant centripetal force Select a radius by aligning the line on the side post with any desired position on the measuring tape While pressing down on the side post to assure that it is vertical tighten the thumb screw on the side post to secure its position Record this radius in Table 3 5 The object on the side bracket must hang vertically On the center post adjust the spring bracket vertically until the string from which the object hangs on the side post is aligned with the vertical line on the side post Align the indicator bracket on the center post with the orange indicator Remove the mass that is hanging over the pulley and remove the pulley Rotate the apparatus increasing the speed until the orange indicator is centered
25. diameter of the cylinder about which the thread is wrapped and calcu late the radius Record in Table 6 2 Finding the Acceleration of Track Alone 1 Since in Finding the Acceleration of Disk and Track the track is rotating as well as the disk it is necessary to determine the acceleration and the rotational inertia of the track by itself so this rotational inertia can be subtracted from the total leaving only the rotational inertia of the disk To do this take the disk off the rotational apparatus and repeat Finding the Acceleration of Disk and Track for the track alone NOTE It will take less friction mass to overcome the new kinetic friction and it is only necessary to put about 30 g over the pulley in Finding the Acceleration of Track Alone Disk Using Ball Bearings Free Disk Mount the disk upside down at the same radius as before Now the ball bearings at the center of the disk will allow the disk to rotate relative to the track Repeat Accounting For Friction and Finding the Acceleration of Disk and Track for this case and record the data in Table 6 2 Calculations Record the results of the following calculations in Table 6 3 1 2 3 Subtract the friction mass from the hanging mass used to accelerate the apparatus to determine the mass m to be used in the equations Calculate the experimental value of the rotational inertia of the fixed disk and track combined Calculate the experimental v
26. e the velocity i PASCO scientific 012 05293F Complete Rotational System 5 Ifthe velocity increases or decreases as the platform turns stop monitoring data stop the plat form and adjust the amount of mass on the thread by adding or removing a paper clip 6 Repeatthe process until the velocity stays constant as the mass falls 7 Measure the mass on the end of the thread and record it as the Friction Mass in Table 2 2 Finding the Acceleration of the Point Mass and Apparatus To find the acceleration put about 50 g measure the exact mass and record it in Table 2 2 on the end of the thread over the pulley In DataStudio set up a Graph display of Velocity versus Time 1 Wind the thread up and hold the Rotating Platform 2 Letthe platform begin to turn and at the same time start recording data 3 Letthe mass fall toward the floor but STOP recording data just before the mass hits the floor 4 Examine your Graph display of Velocity versus Time The slope of the best Linear Fit for your data 1s the acceleration of the apparatus 5 Record the slope in Table 2 2 Table 2 2 Rotational Inertia Data Point Mass and Apparatus Apparatus Alone Measure the Radius Hanging Mass 1 Using calipers measure the diameter of the step pulley about which the thread is wrapped and calculate the radius 2 Record the radius in Table 2 2 Finding the Acceleration of the Apparatus Alone Since in Finding the Acce
27. ence to compare the experimental value to the theoretical value Record in Table 2 3 Table 2 3 Results Rotational Inertia for Point Mass and Apparatus Combined Rotational Inertia for Apparatus Alone Rotational Inertia for Point Mass experimental value Rotational Inertia for Point Mass theoretical value i PASCQ scientific 012 05293F Complete Rotational System Experiment 3 Centripetal Force EQUIPMENT NEEDED Centripetal Force Accessory ME 8952 Rotating Platform ME 8951 Stopwatch Balance Graph paper 2 sheets Mass and Hanger Set String Purpose The purpose of this experiment is to study the effects of varying the mass of the object the radius of the circle and the centripetal force on an object rotating in a circular path Theory When an object of mass m attached to a string of length r is rotated in a horizontal circle the centripetal force on the mass is given by 2 F mro r where v is the tangential velocity and o is the angular speed v r o To measure the velocity the time for one rotation the period 7 is measured Then y 2nr T and the centripetal force is given by 2 _ 4n mr Setup Level the A base and rotating platform as described in the ME 8951 assembly section in the introduction to this manual Procedure Part I Vary Radius constant force and mass 1 The centripetal force and the mass of the hanging object will be held constant for this part
28. ent damage and shipped postage or freight prepaid Damage caused by improper packing of the equipment for return shipment will not be covered by the warranty Shipping costs for returning the equipment after repair will be paid by PASCO scientific Credits This manual authored by Ann and Jon Hanks This manual edited by Ann and Jon Hanks Equipment Return Should this product have to be returned to PASCO sci entific for whatever reason notify PASCO scientific by letter or phone BEFORE returning the product Upon notification the return authorization and shipping instructions will be promptly issued NOTE NO EQUIPMENT WILL BE ACCEPTED FOR RETURN WITHOUT AN AUTHORIZATION When returning equipment for repair the units must be packed properly Carriers will not accept responsibility for damage caused by improper packing To be certain the unit will not be damaged in shipment observe the following rules 1 The carton must be strong enough for the item shipped 2 Make certain there is at least two inches of packing material between any point on the apparatus and the inside walls of the carton 3 Make certain that the packing material can not shift in the box or become compressed thus letting the instrument come in contact with the edge of the box Address PASCO scientific 10101 Foothills Blvd P O Box 619011 Roseville CA 95678 9011 Phone 916 786 3800 FAX 916 786 8905 email techsupp
29. entally a known torque is applied to the object and the resulting angular acceleration is measured Since t Ja rat Qa where a is the angular acceleration which is equal to a r and x is the torque caused by the weight hanging from the thread which is wrapped around the base of the apparatus and t rT where r is the radius ofthe step pulley about which the thread is wound and Tis the tension in the thread when the apparatus is rotating Applying Newton s Second Law for the hanging mass m gives See Figure 1 2 XF mg T ma EN Ns Tm a Platform i mg A base c E hanging mass ez a d Figure 1 2 Rotational Apparatus and Free Body Diagram Solving for the tension in the thread gives T m g a Once the linear acceleration of the mass m is determined the torque and the angular acceleration can be obtained for the calculation of the rotational inertia For comparison the initial speed muzzle velocity of the ball is determined by shooting the ball horizontally off the table onto the floor and measuring the vertical and horizontal distances through which the ball travels For a ball shot horizontally off a table with an initial speed v the horizontal distance traveled by the ball is given by x vot where t is the time the ball is in the air No air friction is assumed The vertical distance the ball drops in time t is
30. f the spring Insert the spring bracket into the slot on the center post and tighten the thumb screw Tieoneendofastring about 30cm long to the bottom of the indicator disk and tie a loop in the other end of the string Insert the indicator bracket into the slot on the center post placing it below the spring bracket Tighten the thumb screw Attach the pulley in the higher of the two holes on the center bracket Insert the thumb screw at the bottom of the center post and attach the square nut spring bracket center post indicator spring disk indicator bracket pulley X reference mark C thumbscrew center of post Q square nut Figure 5 Center Post Assembly PASCC scientific 012 05293F Complete Rotational System Side Post Assembly Assemble the side post as shown in Figure 6 1 Insert the thumb screw at the bottom of the side post single length and attach the square nut 2 Usinga string about 30 cm long tie the string around the screw head on the top ofthe side post Then thread the other end of the string down through one of the holes in the top of the side post and then back up through the other hole Do not pull the string taut 3 Loosen the screw on the top of the side post and wrap the loose end of the string around the threads of the screw and tighten the screw Threading the Centripetal Force Accessory 1 Mount the center post in the T slot on the side
31. g 4 Watch the Digits display to see the velocity 5 Ifthe velocity increases or decreases as the Rotational Disk turns stop monitoring data stop the Rotational Disk and adjust the amount of mass on the thread by adding or removing a paper clip 6 Repeat the process until the velocity stays constant 7 Measure the mass on the end of the thread and record it as the Friction Mass in Table 5 2 Table 5 2 Rotational Inertia Data Ring and Disk Disk Alone Disk Vertical Friction Mass Slope Radius Finding the Acceleration of Ring and Disk To find the acceleration put about 50 g record the exact hanging mass in Table 5 2 over the pulley In the DataStudio program set up a Graph display of Velocity versus Time 1 Wind the thread up and hold the Rotating Platform 2 Letthe Rotating Platform begin to turn and at the same time start recording data 3 Letthe mass descend toward the floor but STOP recording data just before the mass hits the floor 4 Examine your Graph display of Velocity versus Time The slope of the best fit line for your data is the acceleration of the apparatus 5 Record the slope in Table 5 2 PASCO 33 scientific Complete Rotational System 012 05293F Measure the Radius 1 Using calipers measure the diameter of the cylinder about which the thread is wrapped and calcu late the radius Record in Table 5 2 Finding the Acceleration of the Disk Alone Since in Finding the Ac
32. given by y gh The initial velocity of the ball can be determined by measuring x and y The time of flight of the ball can be found using g and then the muzzle velocity can be found using v x t m PASCO scientific 012 05293F Complete Rotational System Part I Determining the initial velocity of the ball Projectile E Setup Launcher 1 Clamp the Projectile Launcher to a sturdy table near one end of the table 2 Adjustthe angle ofthe Projectile Launcher to zero degrees so the ball will be shot off horizontally See Figure 1 3 Procedure 1 Put the ball into the Projectile Launcher and cock it to the long range position Fire one shot to locate where the ball hits the floor At this position tape a piece of white clamp paper to the floor Place a piece of carbon paper carbon side down on top of this E NN paper and tape it down When the ball hits the floor it will leave a mark on the white paper Figure 1 3 Projectile 2 Fire about ten shots Launcher Setup 3 Measure the vertical distance from the bottom of the ball as it leaves the barrel this position is marked on the side of the barrel to the floor Record this distance in Table 1 1 Table 1 1 Determining the Initial Velocity 4 Use a plumb bob to find the point on the floor that is directly beneath the release Vertical distance point on the barrel Measure the horizontal Horizontal distance to edge
33. he interface If you are using a PASPORT interface such as a USB Link PowerLink Xplorer or Xplorer GLX you will need to use a PASPORT Digital Adapter PS 2159 to connect the Photogate Head to the interface See the PASCO web site at www pasco com for more information on PASCO interfaces sensors and software Complete Rotational System 012 05293F photogate g mounting od En accessory mounting rod cast iron A base E rings 2 1 extra ME 8951 Rotating Platform Equipment The ME 8951 Rotating Platform includes the follow ing PASCO cast iron A base with rotating shaft and pulley with 10 holes aluminum track two square masses about 300 g with thumb screw and square nut 300g square mass 2 with thumbscrews and aluminum rotating square nuts platform D LD low profile thumb screw assemblies rotating vertical shaft with 10 hole pulley two additional low profile screws and square nuts to act as stops for the square mass in the Conser vation of Angular Momentum experiment accessory mounting rod for mounting a 10 spoke pulley with or without the optional PASCO Pho togate Head photogate mounting rod for mounting a PASCO Photogate Head ME 9498A spool of g clamp on pulley plastic indicator disk side post 50g masses 2 100g mass with 3 open hooks center post ME 8952 Centripetal Force Accessory Equipment The ME 8952 Centripet
34. he results of the following calculations in Table 5 3 1 Subtract the friction mass from the hanging mass used to accelerate the apparatus to determine the mass m to be used in the equations 2 Calculate the experimental value of the rotational inertia of the ring and disk together 3 Calculate the experimental value of the rotational inertia of the disk alone 4 Subtract the rotational inertia of the disk from the total rotational inertia of the ring and disk This will be the rotational inertia of the ring alone 5 Calculate the experimental value of the rotational inertia of the disk about its diameter 6 Calculate the theoretical value of the rotational inertia of the ring 34 PASCO scientific 012 05293F Complete Rotational System 7 Calculate the theoretical value of the rotational inertia of the disk about its center of mass and about its diameter 8 Use a percent difference to compare the experimental values to the theoretical values Table 5 3 Results Rotational Inertia for Ring and Disk Combined Rotational Inertia for Disk Alone experimental value Rotational Inertia for Ring experimental value Rotational Inertia for Vertical Disk experimental value Rotational Inertia for Disk theoretical value Rotational Inertia for Ring theoretical value Rotational Inertia for Vertical Disk theoretical value Difference for Disk Difference for Ring Difference for Vertical Disk PA
35. le 1 5 Results Average Angular Speed O Rotational Inertia Calculated Initial Velocity vo ssid Measured Initial Velocity Using the average angular speed the rotational inertia and the distance r calculate the muzzle velocity of the ball and record it in Table 1 5 Calculate the percent difference between the muzzle velocities found in Parts I and I Record in Table 1 5 Questions 1 What percentage of the kinetic energy is lost in the collision Use the masses and velocities to calculate this percentage Lost KE pefore KE afer gy KE pefore n PASCQ scientific 012 05293F Complete Rotational System Experiment 2 Rotational Inertia of a Point Mass EQUIPMENT NEEDED DataStudio Software PASCO Interface see Note Paper clips for masses lt 1 g Balance Purpose The purpose of this experiment is to find the rotational inertia of a point mass experimentally and to Mass and Hanger Set Photogate Pulley System Calipers verify that this value corresponds to the calculated theoretical value Theory Note If you are using a PASPORT interface you will also need a Digital Adapter PS 2159 Theoretically the rotational inertia of a point mass is given by MR2 where M is the mass R is the distance the mass is from the axis of rotation To find the rotational inertia experimentally a known torque is applied to the object and the resulting ang
36. leration of the Point Mass and Apparatus the apparatus is rotating as well as the point mass it is necessary to determine the acceleration and the rotational inertia of the apparatus by itself so this rotational inertia can be subtracted from the total leaving only the rotational inertia of the point mass 1 Take the point mass off the rotational apparatus and repeat Finding the Acceleration of the Point Mass and Apparatus for the apparatus alone NOTE that it will take less friction mass to overcome the new kinetic friction and it is only necessary to put about 20 g on the end of the thread over the pulley 2 Record the data in Table 2 2 Calculations 1 Subtract the friction mass from the hanging mass used to accelerate the apparatus to determine the mass m to be used in the equations 2 Calculate the experimental value of the rotational inertia of the point mass and apparatus together and record in Table 2 3 3 Calculate the experimental value of the rotational inertia of the apparatus alone Record in Table 2 3 PASCC scientific 17 Complete Rotational System 012 05293F 4 Subtract the rotational inertia of the apparatus from the combined rotational inertia of the point mass and apparatus This will be the rotational inertia of the point mass alone Record in Table 2 3 5 Calculate the theoretical value of the rotational inertia of the point mass Record in Table 2 3 6 Usea percent differ
37. lley System to the base and connect it to an interface 4 Run the DataStudio program EINE z u rotating rotational disk m platform platform __ 10 spoke pulley with adapter L photogate head A base hanging P Figure 6 2 Set up for Disk Off Axis Measurements For the Theoretical Rotational Inertia Record these measurements in Table 6 1 1 Weigh the disk to find the mass M 2 Measure the diameter and calculate the radius R 3 Measure the distance d from the axis of rotation to the center of the disk 38 PASCO scientific 012 05293F Complete Rotational System Table 6 1 Theoretical Rotational Inertia Mass of Disk Distance Between Parallel Axis Radius of Disk Measurements For the Experimental Method Accounting For Friction Because the theory used to find the rotational inertia experimentally does not include friction it will be compensated for in this experiment by finding out how much mass over the pulley it takes to overcome kinetic friction and allow the mass to drop ata constant speed Then this friction mass will be subtracted from the mass used to accelerate the apparatus 1 In the DataStudio program select Smart Pulley Linear and set up a Digits display to show velocity with three significant figures 2 Hang a small amount of mass such as
38. ng supporting the hanging object is once again vertical and thus the hanging object is at the desired radius 8 Maintaining this speed use a stopwatch to time ten revolutions Divide the time by ten and record the period in Table 3 3 9 To vary the centripetal force clamp the pulley to the track again and hang a different mass over the pulley Keep the radius constant and repeat the procedure from Step 4 Do this for a total of five different forces Analysis 1 The weight of the mass hanging over the pulley is equal to the centripetal force applied by the spring Calculate this force for each trial by multiplying the mass hung over the pulley by g and record the results in Table 3 3 2 Calculate the inverse of the square of the period for each trial and record this in Table 3 3 3 Plot the centripetal force versus the inverse square of the period This will give a straight line since 2 _ 4n mr T F 4 Draw the best fit line through the data points and measure the slope of the line Record the slope in Table 3 3 PASCC scientific 21 Complete Rotational System 012 05293F 5 6 Table 3 3 Varying the Centripetal Force Mass of the object Radius Slope from graph Calculate the mass of the object from the slope and record in Table 3 4 Calculate the percent difference between the two values found for Table 3 4 Results varying the centripetal force the mass of the object and recor
39. ofthe track that has the rule Align the line on the center post with the zero mark on the rule and tighten the thumb screw to secure it in place Then mount the side post on the same side of the track See Figure 7 2 Hang the 100 g mass from the string on the side post and adjust the height ofthe mass so the string coming from the center post will be level when the mass is hanging straight down PASCC scientific holes for thread nylon thumbscrew of thread 30cm long side post thumbscrew 100g mass square nut Figure 6 Side Post Assembly ae side post center post 100g mass indicator disk E indicator bracket S pulley rotating platform reference mark center of post zero mark of rule Figure 7 Threading the Centripetal Force Accessory Complete Rotational System 012 05293F rotational disk square D hole on top sur face of rotational platform adapter bearing of nut platform adapter rotating platform Figure 8 Rotational Inertia Accessory Including Platform Adapter Assembly ME 8953 Rotational Inertia Accessory Rotational Inertia Accessory Assembly Little assembly is required to use the Rotational Inertia Accessory Therotational disk can be placed directly onto the axle of the rotating base or can be used with the rotating platform via the included platform adapter Platform Adapter
40. onal Figure 3 Using the Photogate Mount Rod With the Photogate Head Make sure that the Photogate Head does not rub against the 10 spoke pulley or any other part of the apparatus 5 Connectthecabletothe Photogate Headanda PASCO interface Complete Rotational System 012 05293F To use the Super Pulley and Photogate Head with the Pulley Mounting Rod l Insertthe pulley mounting rod into the hole When the Photogate Head is powered by a computer interface you can tell when the photogate is blocked by watching the LED indicator on the photogate Attach the Super Pulley and the Photogate Head if needed to the Pulley Mounting Rod in the Accessory Mounting Rod and tighten the thumb screw See Figure 4 Rotate the accessory mounting rod so that a string from the pulley on the center shaft will be aligned with the groove on the Super Pulley Adjust the position of the base so the string accessory mounting rod nylon thumb screw D pulley mounting rod Photogate Head A base s optional passing over the Super Pulley will clear the Figure 4 Using the Accessory Mounting Rod With the Pulley edge of the table Mounting Rod Super Pulley and Photogate Head ME 8952 Centripetal Force Accessory Center Post Assembly Assemble the center post as shown in Figure 5 l Attach one end ofthe spring to the spring bracket and connect the indicator disk to the other end o
41. pasco com web www pasco com 012 05293F Complete Rotational System Introduction PASCO s Complete Rotational System provides a full range of experiments in centripetal force and rotational dynamics The system consists of three separate compo nents Description The ME 8951 Rotating Platform consists of a sturdy 4 kg base with low friction bearings and a rotating arm which serves as a versatile base for rotation experiments This platform is a general purpose base upon which you may mount anything having a mass under 3 kg you wish to rotate The T slots in the track supply a convenient way to mount objects to the track using thumbscrews and square nuts To usethe Centripetal Force Accessory ME 8952 or the Rotational Inertia Accessory ME 8953 each must be mounted on this base A photogate pulley mount and two 300 g masses are also included The ME 8952 Centripetal Force Accessory is comprised of two vertical posts which can be mounted to the Rotat ing Platform with thumbscrews These posts are adjust able and can be positioned virtually anywhere along the length ofthe platform The radius indicator is at the center of the apparatus so it can be clearly seen while the apparatus isrotating This accessory requires the Rotating Platform ME 8951 to operate The PASCO Centripetal Force Accessory can be used to experiment with centrip etal force and conservation of angular momentum For the centripetal force experiments
42. pe m of the best fit line for your data is the acceleration use Fit gt Linear Fit Record the slope in Table 4 2 Repeat the procedure for the mass at the inner stop Record results in Table 4 2 Measure Radius 1 Using calipers measure the diameter of the step pulley about which the thread is wrapped and calculate the radius 2 Record the radius in Table 4 2 Table 4 2 Rotational Inertia Data Mass at Outer Stop Mass at Inner Stop Frona sm Rotational Inertia 28 PASCO scientific 012 05293F Complete Rotational System Analysis Calculate the rotational inertias e Subtract the friction mass from the hanging mass used to accelerate the apparatus to determine the mass m to be used in the equations Calculate the experimental values of the rotational inertia and record it in Table 4 3 2 Calculate the expected theoretical values for the final angular velocity and record these values in Table 4 3 Table 4 3 Results Trial 1 Trial 2 Trial 3 3 For each trial calculate the percent difference between the experimental and the theoretical val ues of the final angular velocity and record these in Table 4 3 Questions Calculate the rotational kinetic energy before the string was pulled 1 KE 519i Then calculate the rotational kinetic energy after the string was pulled KE 1y yop 1 Which kinetic energy is greater
43. play of Velocity rad s versus time Table 4 1 Data The graph shows the angular speed before and after the square mass is pulled toward the inner stop Rescale the Angular Speeds graph if necessary 6 Use the Smart Cursor tool to determine the angular speed Trial Number immediately before and immediately after pulling the string Record these values in Table 4 1 MEE MEN RE ee 7 Repeatthe experiment a total of three times with different E o oa J o o Jo initial angular speeds Record these values in Table 4 1 LOS Part Il Determining the Rotational Inertia Measure the rotational inertia of the apparatus twice once with the square mass in its initial position and once with it in its final position Setup 1 Attach a Photogate with Pulley to a mounting rod and attach the mounting rod to the black sup port rod on the base 2 Wind a thread around the pulley on the center shaft and pass the thread over the Pulley See Fig ure 4 3 string n center post 300g mass i rotating o platform 10 spoke pulley with stop screws 2 photogate head A base a hanging mass Figure 4 3 Set up for determining rotational inertia PASCO 2 scientific Complete Rotational System 012 05293F Procedure Accounting For Friction Because the theory used to find the rotational inertia experimentally does not include friction it will be compensated for in this experiment by finding out how much mass over the pulley it
44. plete Rotational System 012 05293F where a is the angular acceleration which is equal to a r and t is the torque caused by the weight hanging from the thread which is wrapped around the base of the apparatus t rT where r is the radius of the cylinder about which the thread is wound and T is the tension in the thread when the apparatus is rotating Applying Newton s Second Law for the hanging mass m gives See Figure 5 3 XF mg T ma CENE MEM rotational diee T A base i hanging e gm iis S Figure 5 3 Rotational Apparatus and Free Body Diagram Solving for the tension in the thread gives T m g a Once the linear acceleration of the mass m is determined the torque and the angular acceleration can be obtained for the calculation of the rotational inertia Setup 1 Remove the track from the Rotating Platform and place the mass ring l disk directly on the center shaft as shown in Figure 5 4 The rotational pede E witi side of the disk that has the indentation for the ring should disk l pnotogate head be up 2 Place the ring on the disk seating it in this indentation 3 Mount the Photogate Pulley System to the base and connect it to a PASCO interface A base 4 Attach a thread to the top step of the three step pulley on the mass and Ro
45. rm rotated 90 as shown 300g square 7 mass base until the end ofthe track with the square mass is aligned over the leveling screw on the other leg ofthe base See Figure 2 then adjust this foot 3 Rotate the track 90 degrees so it is parallel to one side of the A and adjust the other leveling screw until the track will stay in this position 4 The track is now level and it should remain at rest regardless of its orientation Using an Optional Photogate Head The optional Photogate Head ME 9498A can be mounted to the Rotating Platform in two ways Photogate Mount Rod can be used to mount the Photogate Head directly to the base as shown in Figure 3 Accessory Mounting Rod can to be used to mount a Pulley Mounting rod and a Super Pulley along with the Photogate Head to the base in order to run a string over the Super Pulley See Figure 4 To Mount the Photogate Head Only 1 Mountthe Photogate Head on the threaded end ofthe photogate mount rod 2 Remove the swivel clamp from the top of the Photo gate Head 3 Slide the non threaded end of the photogate mount rod into a hole in the A base and clamp it in place with the thumbscrew 4 Adjust the Photogate Head so that its infrared beam can be interrupted by the 10 spoke pulley on the vertical shaft as the shaft turns PASCC scientific 5 10 spoke pulley on vertical shaft accessory p mounting rod Photogate Head opti
46. t is increased does the centripetal force increase or decrease PASCC scientific 23 Complete Rotational System 012 05293F m PASCO scientific 012 05293F Complete Rotational System Experiment 4 Conservation of Angular Momentum Using a Point Mass EQUIPMENT REQUIRED Note If you are DataStudio Program Rotating Platform ME 8951 using a PASPORT PASCO Interface see Note Photogate Pulley System interface you will Rotational Inertia Accessory ME 8953 Balance also need a Digital Adapter PS 2159 Purpose A mass rotating in a circle is pulled in to a smaller radius and the new angular speed is predicted using conservation of angular momentum Theory Angular momentum is conserved when the radius of the circle is changed L 10 Or where is the initial rotational inertia and is the initial angular speed So the final rotational speed is given by I i Wr O I To find the rotational inertia experimentally a known torque is applied to the object and the resulting angular acceleration is measured Since 1 IQ I 1 Qa where is the angular acceleration which is equal to a r and T is the torque caused by the weight hanging from the thread which is wrapped around the base of the apparatus T amp T where r is the radius of the cylinder about which the thread is wound and T is the tension in the thread when the apparatus is rotating Applying Newton s
47. takes to over come kinetic friction and allow the mass to drop at a constant speed Then this friction mass will be subtracted from the mass used to accelerate the apparatus 1 Start the DataStudio program Select Smart Pulley Linear and set up a Digits display to how velocity with three significant figures 2 Hanga small amount of mass such as a few paper clips on the end of the thread that is over the pulley Make sure that the thread is wound around the step pulley Start monitoring data and then give the Rotating Platform a tap to get it started moving Watch the Digits display to see the velocity 5 Ifthe velocity increases or decreases as the platform turns stop monitoring data stop the plat form and adjust the amount of mass on the end of the thread 6 Repeatthe process until the velocity stays constant 7 Measure the mass on the end of the thread and record it as Friction Mass in Table 4 2 EE ed Finding the Acceleration of the Apparatus To find the acceleration put about 30 g record the exact hanging mass in Table 4 2 over the pulley In the DataStudio program set up a Graph display of Velocity versus Time 1 Wind the thread up and hold the Rotating Platform 2 Letthe Rotating Platform begin to turn and at the same time START recording data 3 Letthe mass descend toward the floor but STOP recording data just before the mass hits the floor 4 Examine your graph of velocity versus time The slo
48. tational Apparatus shaft and suspend the string over the hanger pulley of the Photogate Pulley System Attach a hanger and j j mass to the end of the thread 5 Start the DataStudio program Figure 5 4 Set up for Disk and Ring Procedure Measurements for the Theoretical Rotational Inertia 1 Weigh the ring and disk to find their masses and record these masses in Table 5 1 2 Measure the inside and outside diameters of the ring and calculate the radii R and R2 Record in Table 5 1 3 Measure the diameter of the disk and calculate the radius R and record it in Table 5 1 32 IZEI O scientific 012 05293F Complete Rotational System Table 5 1 Theoretical Rotational Inertia Measurements for the Experimental Method Accounting For Friction Because the theory used to find the rotational inertia experimentally does not include friction it will be compensated for in this experiment by finding out how much mass over the pulley it takes to overcome kinetic friction and allow the mass to drop at a constant speed Then this friction mass will be subtracted from the mass used to accelerate the apparatus 1 In the DataStudio program select Smart Pulley Linear and set up a Digits display to show velocity with three significant figures 2 Hang a small amount of mass such as a few paper clips on the end of the thread that is over the pulley 3 Start monitoring data and then give the Rotational Disk a tap to get it started movin
49. ter PS 2159 to a PASPORT interface such as a Complete Rotational System 012 05293F Assembly ME 8951 Rotating Platform Assembling the Rotating Platform 1 Insertthe cylindrical end ofthe shaft into the bearings 2 Mount the track to the shaft and tighten the thumb on the top side of the A shaped iron base Secure the screw against the flat side ofthe D on the shaft See shaft in place by inserting the E ring in the slot at Figure 1 the bottom of the shaft See Figure 1 rotating platform flat of vertical shaft vertical shaft Figure 1 Attaching the Vertical Shaft to the Base and Rotating Platform Assembly i IPASCC scientific 012 05293F Complete Rotational System rotating plat form 300g square mass leveling feet first Leveling the Base Some experiments such as the Centripetal Force exper iments require the apparatus to be extremely level If the track is not level the uneven performance will affect the results To level the base perform the following steps 1 Purposely make the apparatus unbalanced by attach ing the 300 g square mass onto either end of the aluminum track Tighten the screw so the mass will not slide If the hooked mass is hanging from the side post in the centripetal force accessory place the square mass on the same side 2 Adjust the leveling screw on one of the legs of the adjust this foot Figure 2 Leveling the Base rotating platfo
50. togate Pulley System also need a Digital Adapter PS 2159 Purpose A non rotating ring is dropped onto a rotating disk and the final angular speed of the system is compared with the value predicted using conservation of angular momentum Theory When the ring is dropped onto the rotating disk there is no net torque on the system since the torque on the ring is equal and opposite to the torque on the disk Therefore there is no change in angular momentum Angular momentum is conserved L 10 Op where is the initial rotational inertia and 0 is the initial angular speed The initial rotational inertia is that of a disk 7 IMP and the final rotational inertia of the combined disk and ring is 1 2 1 2 2 Ij 5M R 5M r So the final rotational speed is given by M R oO 0 ij 2 3 2 4 MR My ri r3 Setup 1 Level the apparatus using the square mass on the track 2 Assemble the Rotational Inertia Accessory as shown in Figure 7 1 The side of the disk with the indentation for the ring should E Y TE be up 3 Mount the Photogate on the metal rod on the base and position it Rotational Disk so it straddles the holes in the pulley on the center rotating shaft indentation up 4 Start the DataStudio program Select Smart Pulley Rota A base tional as the sensor 5 Setup a Graph display of Velocity rad s versus Time s p q
51. ular acceleration is measured Since t Ja f gla where a is the angular acceleration which is equal to a r and t is the torque caused by the weight hanging from the thread which is wrapped around the step pulley below the rotating platform and t T where r is the radius ofthe step pulley about which the thread is wound and T is the tension in the thread when the apparatus is rotating Applying Newton s Second Law for the hanging mass m gives see Figure 2 1 XF mg T ma a rotating platform A base Li eR Figure 2 1 Rotational Apparatus and Free Body Diagram Solving for the tension in the thread gives PASCC scientific T m g a 15 hanging S mass mg Complete Rotational System 012 05293F Once the linear acceleration of the mass m is determined the torque and the angular acceleration can be obtained for the calculation of the rotational inertia Setup 1 Level the rotating platform 2 Attach the square mass point mass to the track on the rotating platform at any radius you wish 3 Mount the Photogate Pulley system to the base and connect the photogate through an interface to a computer See Figure 2 2 4 Attach a thread to the middle step of the step pulley and hang the thread over the 10 spoke pulley Allow the string to reach to the floor rotating Pa platform A base
52. ure the mass on the end of the thread and record it as the Friction Mass in Table 1 4 Finding the Acceleration of the Apparatus To find the acceleration put about 30 g record the exact hanging mass in Table 1 4 over the pulley In the DataStudio program set up a Graph display of Velocity versus Time 1 Wind the thread up and hold the Rotating Platform 2 Letthe Rotating Platform begin to turn and at the same time start recording data 3 Letthe mass descend toward the floor but STOP recording data just before the mass hits the floor 4 Examine your Graph display of Velocity versus Time The slope of the best fit line for your data is the acceleration of the apparatus 5 Record the slope in Table 1 4 Table 1 4 Rotational Inertia Data Fron mes Slope Radus PASCC scientific 13 Complete Rotational System 012 05293F Measure the Radius l 2 Using calipers measure the diameter of the step pulley about which the thread is wrapped and calculate the radius Record the radius in Table 1 4 Analysis l 2 e Calculate the average of the angular speeds in Table 1 3 and record the result in Table 1 5 Calculate the rotational inertia Subtract the friction mass from the hanging mass used to accelerate the apparatus to determine the mass m to be used in the equations Calculate the experimental value of the rotational inertia and record it in Table 1 5 Tab
53. ut into the T slot and tighten it down at about the 20 cm mark Now the square mass is free to slide between the two limiting stops 5 Move the pulley on the center post to its lower position Remove the spring bracket from the cen ter post and set it aside 6 Attach a string to the hole in the square mass and thread it around the pulley on the center post and pass it through the indicator bracket 7 Mount the Photogate on the rod on the base and position it so it straddles the holes in the pulley on the center rotating shaft 8 Start the DataStudio program Connect the Photogate to a PASCO interface and connect the inter face to a computer if needed 26 PASCO scientific 012 05293F Complete Rotational System Procedure l Select Smart Pulley Rotational as the type of sensor Set up a Graph display of Velocity rad s versus time 2 Hold the string just above the center post With the square mass against the outer stop give the track a spin using your hand 3 Click Start to begin recording data After about 20 data points have been taken pull up on the string to cause the square mass to slide from the outer stop to the inner stop 4 Continue to hold the string up and take about 20 data points after pulling up on the string Click Stop to end recording data 5 Examine the Graph dis
54. wn torque is applied to the object and the resulting angular acceleration is measured Since 1 JO gla where o is the angular acceleration which is equal to a r and T is the torque caused by the weight hanging from the thread which is wrapped around the three step pulley on the shaft of the apparatus t rT where r is the radius of the step pulley about which the thread is wound and 7 is the tension in the thread when the apparatus is rotating PASCO a scientific Complete Rotational System 012 05293F Applying Newton s Second Law for the hanging mass m gives See Figure 6 1 XLF mg T ma caus HOM otating plat _ form rotational disk platform adapter l b SAT base e 0 andi rj WESS pr B d mg iv Figure 6 1 Rotational Apparatus and Free Body Diagram Solving for the tension in the thread gives T m g a Once the linear acceleration of the mass m is determined the torque and the angular acceleration can be obtained for the calculation of the rotational inertia Setup 1 Setup the Rotational Accessory as shown in Figure 6 2 Mount the disk with its bearing side up Use the platform adapter to fasten the disk to the track at a large radius 2 Mount two square masses on the opposite end of the platform to act as a counterweight Be sure to tighten the screws 3 Mount the Photogate Pu
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