College Physics I Laboratory Angular Momentum

College Physics I Laboratory Angular Momentum Purpose PHSX 206N To investigate conservation of angular momentum by directly measuring the moment of inertia and angular velocities for initial and final positions of a system. Introduction The angular momentum L of a system can be defined as the product of the total moment of inertia I of the system and its angular velocity ω. L = Iω (1) Our system consists of a square mass m on a rotating platform. We will slide the mass from the outermost to innermost position on the platform. Changing the position of the mass relative to the axis of rotation will change the total moment of inertia I of the system. However, as long as the net external torque acting on the system is zero, the total angular momentum of the system remains constant. The angular momentum of the system is said to be conserved. This is a statement of the law of conservation of angular momentum. L i = L f (2) I i ω i = I f ω f (3) Examining the relationship, you will notice that if the value of the moment of inertia goes up then the value of the angular velocity goes down. The reverse is also true, if the value of the moment of inertia goes down then the value of the angular velocity goes up. Introduction Part I During Part I of the experiment, you will be changing the moment of inertia I of the system as it rotates by moving the square mass toward the center of the platform. Even though you are exerting a force on the system, it will be upward along the axis of rotation. Since it is at a zero distance from the axis of rotation, it contributes no torque on the system. This is easy to calculate from the definition of a torque created by a force F acting at a distance r from the axis of rotation (where r = 0 m). τ = rf sin φ (4) The string you pull upward will turns around a pulley and connect horizontally to the square mass. The force of tension pulling the square mass inward along the platform also contributes no torque. The tension force vector acting on the mass is directed along the radius, and thus has no components trying to rotate the mass (sin φ = 0). You will be measuring some initial and final angular velocities in Part I. However, without knowing the initial and final moments of inertia, you will not yet have enough information to show that Equation (2) is valid. Introduction Part II During Part II of the experiment, you will be accelerating the system while it has a constant moment of inertia. You will do this by applying a torque to the platform. The net torque on the system can be expressed in terms of the moment of inertia and the angular acceleration. Στ = Iα (5) In Figure 2, you can see a mass hanging from a string which goes over a pulley and is wrapped around the axis of rotation of the apparatus. The linear acceleration of the string is measured by the photogate, and can be used to calculate the angular acceleration of the platform. a tan = r α (6) Also knowing the value of the mass hanging from the string, the tension can be calculated. Then Equation (4) can be used to find the torque due to the tension in the string. Since the tension is the only force providing a nonzero torque that we can determine, we can then solve the equations to determine the moment of inertia. Angular Momentum 1 of 6 PHSX 206N

Setup 1. Level the apparatus using the square on the track as shown in the leveling instructions in the Assembly Section. 2. Procedure Slide a thumb Part screw I 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 stop screws (2) rotating platform "A" base Figure 4.2: Set-up for conservation of angular momentum 3. With Figure the side 1: of The the apparatus square mass setthat up has for the measuring hole oriented an initial toward angular the center velocity post, ω i slide andthe a final square angular mass onto the velocity track by ωinserting f. Pulling its square the string nut into straight the T-slot, upward but through do not tighten the center the thumb post will screw; pull the the square mass square should mass be free horizontally to slide in toward the T-slot. the central axis of rotation. 4. Slide a second thumb screw and square nut 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. 1. Be sure the apparatus is level. See the last section of this document for leveling instructions. 5. Move the pulley on the center post to its lower position. Remove the spring bracket from the center Make post sure and set theit thumbscrew aside. on top of the square mass is loose so the mass is free to slide in the T-slot on the 2. 6. Attach platform. a string It should to the hole be prevented in the square frommass sliding and offthread the platform it around or the reaching pulley on the the center of post the platform by two and stop pass screws it through which the should indicator be firmly bracket. in place. See Figure 1. For the purposes of this experiment, when the square 7. Mount mass the is against Photogate the outer on the stop rod screw on the itbase willand be in position the initial it so position, straddles andthe when holes thein square the pulley mass is against the on inner the center stop screw rotating it will shaft. be in the final position. 8. 3. Start Check the to DataStudio make sureprogram. the Smart Connect Pulleythe photogate Photogate is to properly a PASCO mounted interface about and the connect holes the on interface shaft. to a Ask computer your instructor (if needed). if you need further instructions on the apparatus set the center rotating up. 4. Open the program on your desktop called Angular Momentum (orange icon with the L). 5. Slowly rotate the pulley wheel to confirm that the red LED blink. There should be a string wrapped around the shaft and secured with tape; be sure that neither the string or the tape interfere with the photogate or obscure any of the holes. 6. Select the Display Part I tab in the program. 7. Do the following steps 3 times. 26! Some of these steps must be done in quick succession. You may need to practice them a few times before you manage to complete them successfully. (a) Be sure the string connected to the square mass goes around the small pulley near the center of the platform and up through the holes of the center post. You should hold the string firmly above the center post for the duration of the trial. (b) Move the square mass to the initial position. Be sure the string does not fall off the small pulley. (c) Give the platform a light spin with your hand. In case you ve accidentally pulled on the string and moved the square mass inward, allow it to slide back to the initial poistion; but don t release the string! (d) Click on the white run arrow icon by the menu bar. The icon will turn into a black run arrow and the displayed values will reset to zero. (e) Allow the platform to complete about two or three revolutions from when you clicked the run arrow icon. Angular Momentum 2 of 6 PHSX 206N

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 display of Velocity (rad/s) versus time. Table 4.1: Data The graph shows the angular speed before and after the square (f) mass Quickly is pulled and firmly toward pull the the inner string stop. upward Rescale until the the square mass is in the final Angular position. Speeds The platform will graph if spin necessary. faster. Continue to hold the string so that the square mass remains in the final position. 6. Use (g) the Allow Smart the Cursor platform tool to todetermine completethe about angular two or speed three revolutions Trial Number from when you Initial pulled the string. Final immediately before and immediately after pulling the (h) Click the large STOP button. string. Record these values in Table 4.1. 1 7. Repeat (i) the Record experiment the initial a total angular of three velocity times ω with i of the different rotating platform. 2 initial (j) angular Recordspeeds. the final Record angular these velocity values ω f in of Table the rotating 4.1. platform. 3 (k) Determine the uncertainty δω i of the initial angular velocity based on the precision of the computer display. Setup Part II: Determining the Rotational Inertia (l) Determine the uncertainty δω f of the final angular velocity based on the precision of the computer display. (m) Determine the fractional uncertainty δω i /ω i of the initial angular velocity measurement. (n) Measure Determine the rotational the fractional inertia uncertainty of the apparatus δω f /ωtwice: f of the once final with angular the square velocity mass measurement. in its initial position and once with it in its final position. 8. You should now have 3 pairs of initial and final angular velocities. Do not average any of the angular velocities.? Why would averaging all the initial angular velocities or averaging all the final angular velocities be the 1. Attach a wrong Photogate thingwith to do? Pulley to a mounting rod and attach the mounting rod to the black support rod on the base. 2. Procedure Wind a thread Part around II the pulley on the center shaft and pass the thread over the Pulley. See Figure 4.3. string center post 300g mass rotating platform stop screws (2) 10-spoke pulley with photogate head "A" base hanging mass Figure 4.3: Set-up for determining rotational inertia Figure 2: The apparatus set up for measuring its tangential acceleration. The string wrapped around a spool on the axis of rotation of the platform goes over a pulley and is connected to a hanging mass. The photogate measures the changing tangential velocity of the pulley as the mass falls and calculates the magnitude of the linear acceleration of the string. 27 1. It is necessary to attach a pulley to the Smart Pulley photogate and pass the string over the Smart Pulley from the cylinder on the center shaft. Your laboratory instructor will provide you with details on changing your laboratory set up. Make sure that the orientation of the pulley and photogate is such that the string is not rubbing on the photogate, and that the string is parallel to the photogate post. See Figure 2. 2. Measure the diameter d of the cylinder around which the string is wrapped. 3. Determine the uncertainty δd in the diameter measurement. 4. Calculate the fractional uncertainty δd/d in the diameter measurement. 5. Calculate the radius r of the cylinder around which the string is wrapped. Angular Momentum 3 of 6 PHSX 206N

6. Measure the mass m of the mass hanger. 7. Determine the uncertainty δm in the mass measurement of the hanging mass. 8. Determine the fractional uncertainty δm/m in the mass measurement of the hanging mass. 9. Select the Display Part II tab in the program. 10. Remove the tape from the cylinder, pull the string out, run it over the pulley, and attach the hanging mass to the end. 11. Do the following steps 2 times. Once for the initial position and once for the final position. (a) Put the square mass into position on the platform and tighten it down.! Please don t break the plastic knob on the thumb screw of the square mass. The strength of Hercules is not required to secure the square mass firmly in place. (b) Do the following steps 5 times. i. Be sure the string is wound around the center rotating cylinder and the hanging mass is suspended a few centimeters below the photogate and pulley. ii. Release the platform and allow the hanging mass fall about 30 cm. iii. Click on the white run arrow icon by the menu bar. The icon will turn into a black run arrow. The program will automatically stop when data has been recorded.! If the hanging mass hits the floor before program stops, then you will need to redo the current trial. iv. Record the linear acceleration a of the string.? How does the linear acceleration relate to the displayed graph? (c) Calculate the average linear acceleration a avg for the current position. (d) Determine the uncertainty δa in the linear acceleration by taking the difference between the largest and smallest values you have for the linear acceleration. (e) Calculate the fractional uncertainty δa/a avg of the linear acceleration. (f) Calculate the angular acceleration α of the platform. δa = a max a min (7) α = a avg r (8) (g) Calculate the tension F t in the string. If we choose a coordinate system where upward is the positive y-direction, then a y = a avg (assuming you ve calculated a avg as a positive quantity). (h) Calculate the torque τ t that the tension exerts on the platform.? What angle does φ t represent? What is its value? ΣF y = F gy + F ty (9) ma y = mg + F t (10) ma avg = mg + F t (11) m(g a avg ) = F t (12) τ t = rf t sin φ t (13) (i) Calculate the moment of inertia I for the current position. Because the only torque on the platform is due to the tension, it is the net torque. Στ = τ t (14) Iα = τ t (15) I = τ t α Angular Momentum 4 of 6 PHSX 206N (16)

Determining your Final Values and Uncertainties in Your Final Values 1. Do the following steps 3 times. Once for each pair of initial and final angular velocities you have from Part I. (a) Calculate the initial angular momentum L i of the platform. See Equation (1). (b) Calculate the final angular momentum L f of the platform. See Equation (1). In the results section of your notebook, state the results of your experiment in the form L i ± δl i and L f ± δl f. Note, δl i should be equal to the largest fractional uncertainty from your values of the initial angular velocity ω i, initial linear acceleration a i,avg, mass m, or diameter d multiplied by your value of L i. Note, δl f should be equal to the largest fractional uncertainty from your values of the initial angular velocity ω f, initial linear acceleration a f,avg, mass m, or diameter r multiplied by your value of L f. You should also address the following questions: ( δωi δl i = L i max, ω i ( δωf δl f = L f max, ω f δa i, δm a i,avg m, δd ) d δa f a f,avg, δm m, δd d 1. Do your results for L i equal your results for L f within their uncertainties? Be sure to clearly state the quantitative values you are comparing. If there are any large discrepancies, quantitatively comment on their possible origin. 2. If there are not any large discrepancies, but one or two sets of results match while the other does not, comment on this inconsistency. 3. Are the assumed uncertainties in your angular velocity values in Part I too small? Are the uncertainties in your linear acceleration values too big in Part II? Should the uncertainties for these values in both parts be from the computer s precision? ) (17) (18) Leveling the Apparatus Some experiments require the apparatus to be extremely level. If the track is not level, the uneven performance will affect the results. Perform the following steps to be sure the rotating platform of the apparatus is level. Figure 3: The two positions you will need to place the rotating platform during the leveling process. 1. Purposely make the apparatus unbalanced by attaching the 300 g square mass onto either end of the aluminum track. Tighten the screw so the mass will not slide. Angular Momentum 5 of 6 PHSX 206N

2. Adjust the leveling screw on one of the legs of the base until the end of the track with the square mass is aligned over the leveling screw on the other leg of the base. See Figure 3. 3. Rotate the track 90 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. Angular Momentum 6 of 6 PHSX 206N