# PHY 123 Lab 6 - Angular Momentum

(updated 10/17/13)

The purpose of this lab is to study torque, moment of inertia, angular acceleration and the conservation of angular momentum.

If you need the .pdf version of these instructions you can get them here.

## Equipment

• rotating table (disk with axle into bearing) with photogate, pulley, and cylinder for winding string around
• disk with handle (with label showing measured mass)
• mass with attached string to be wound around cylinder below the rotating table
• pulley (changes direction of string tension so mass can accelerate vertically under gravity to “spin up” rotating table)
• interface box
• vernier caliper (to measure the diameter of the cylinder)
• meter stick (to measure objects too big for the caliper)
• computer

Figure 1

## Introduction

To prepare for this lab, you should review the material in the online Lecture Notes and Lecture Videos for PHY 121 in Fall 2013. You will also find to be helpful sections 7.4 and 7.5 of the textbook Knight, Jones and Field (2nd ed.), (abbreviated KJF2). If you don't have it, a copy is available in the PHY 121/2/3/4 Help Room, A-131 Physics. It's bolted to the table near the blackboards.

In Part I you measure the angular acceleration $\:\:\alpha$ of an object, a rotating table (also called “platform” below), caused by a known externally applied torque. From this measurement you can obtain the moment of inertia $\:\:I$ of your rotating table. The equation that relates the net torque $\:\:\tau_{net}$ to the moment of inertia and the angular acceleration of a rotating object is (cf., Eq. 7.15 on p. 214 of KJF2):

$\Large \tau_{net}=I\alpha$
(6.1)

You determine the angular acceleration by measuring the angular velocity $\:\:\omega$ as a function of time $\:\:t$. For a constant, externally applied angular acceleration the equation giving the dependence of $\:\:\omega$ on $\:\:t$ is (cf., Tables 7.1 and 7.2 on p. 202 of KJF2):

$\Large \omega=\omega_{0}+\alpha t$
(6.2)

In Part II, after you have determined the moment of inertia $\:\:I$ of the rotating platform, you will investigate conservation of angular momentum. The definition of angular momentum $\:\:L$ is

$\Large L=I\omega$
(6.3)

In the PHY 121 lectures you have already been introduced to the concept of angular momentum. It will help if you review the material on angular momentum in Ch. 9 of KJF2. Since the moment of inertia $\:\:I$ is a scalar, the vectors $\:\:{\vec \omega}$ and $\:\:{\vec L}$ must, necessarily, point in the same direction. Though angular velocity $\:\:{\vec \omega}$ is an important concept and a useful (vector) quantity, it is not something that is “conserved”. In physics we like to emphasize “conserved quantities”, but we must know what the conditions must be in order for them to be conserved. “Conserved” is just a physics word for “remains constant”. If the conserved quantity is a vector, that means that both its direction and its magnitude must remain unchanged.

So Part II of this lab gives you the opportunity to observe the Conservation of Angular Momentum in a rotational system that is not subjected to an externally applied torque (except for friction, which you “correct for” similar to what you did in Part I of this lab).

The symbolic statement of the conservation of angular momentum is

$\Large L = I\omega =I\,^{\prime}\omega\,^{\prime} = L\,^{\prime}$,
(6.4)

where unprimed symbols refer to “initial” values and the primed symbols refer to “final” values. In order to change the moment of inertia of the rotating object from $\:\:I$ to $\:\:I\,^{\prime}$, you drop a disk (the disk with a handle in your array of equipment) that has a moment of inertia $\:\:I_{disk}$, which you must calculate, onto the rotating platform. This changes the moment of inertia of the rotating system from $\:\:I$, for the rotating platform alone, to $\:\:I\,^{\prime} = I + I_{disk}$, for the rotating (platform + disk-with-the-handle) combination. You need to measure the magnitudes of the angular velocity before the drop, $\:\:\omega$, and the angular velocity after the drop, $\:\:\omega\,^{\prime}$. Because the axis of rotation in your “after” system is the same as the axis of rotation in your “before” system, the vectorial directions of $\:\:{\vec \omega}$ and $\:\:{\vec \omega}\,^{\prime}$ are the same. Therefore, the vectorial directions of $\:\:{\vec L}$ and $\:\:{\vec L}\,^{\prime}$ are also the same. This lets you focus on comparing the magnitudes of $\:\:{\vec L}$ and $\:\:{\vec L}\,^{\prime}$.

## Part I: Measurement of the Moment of Inertia of the Rotating Platform and Attached Cylinder

Figure 2

The apparatus used for this lab is sketched in Fig. 2 above. A small cylinder with string wound around it is attached to a rotating platform. The tension $\:\:T$ due to a weight $\:\:mg$ of mass $\:\:m$ attached to the string provides an external torque, $\:\:\tau_{ext}$. When the platform rotates a photogate registers whenever one of the 4 black radial strips on the plastic rim of the platform blocks the light beam of the photogate. From the 90o angle ($\:\:\pi/2$ radians!) between the strips the angular velocity $\:\:\omega$ is computed. The system is not friction free. A frictional torque, $\:\:\tau_{fr}$, that opposes the external torque from the tension $\:\:T$ causes a smaller net torque, $\:\:\tau_{net}=\tau_{ext}-\tau_{fr}$. The net torque gives the rotating platform an angular acceleration $\:\:\alpha$ that is measured from the rate of increase of the angular velocity $\:\:\omega$. The frictional torque $\large \:\:\tau_{fr}$ causes a frictional angular deceleration (negative angular acceleration) $\large \:\:\alpha_{fr}$ .

The moment of inertia $\:\:I$ of the rotating platform can be determined from

$\huge I\:=\:\frac{mr\,(g\,-\,r\alpha)}{|\alpha_{fr}|\;+\; \alpha}$
(6.5)

where $\:\:g$ is the acceleration of gravity and $\:\:r$ is the radius of the cylinder around which you wind the string (see Fig. 2 above).

Use the vernier caliper to measure the diameter of the cylinder under the rotating platform. (Before asking your TA in case you have questions about how to use the caliper and read its scale, use the informative Wikipedia article entitled Vernier scale. See, espcially, all the animations in the article.) Apply the caliper to the cylinder with a close fit (but don't overdo it and cause damage to the caliper) and remove the caliper without changing the distance between its “jaws”. (This takes a bit of practice.) Reading the caliper scale gives you the diameter of the cylinder from which you can determine its radius $r$ (see Fig. 2 above). Use $\:\:\pm 0.5$ mm as your absolute uncertainty $\:\:\Delta r$ for the radius $\:\:r$ . Enter $\:\:r$ and $\:\:\Delta r$ on your worksheet.

Make sure your string is long enough to reach from the small cylinder (depicted in Fig. 2 and around which you need to wind the string) to not too far above the floor. This will maximize the number of data points you will collect in the computer when the weight of mass $m$ falls toward the floor.

Connect the photogate output to the interface box by plugging its cable into the top socket (labeled “DIG/SONIC 1”) of the black interface box (“LabPro”).

Turn on the computer and check the system by following these instructions: Double click the icon “Exp6_omega_t”. A window with a spreadsheet on the left (having “Time, Velocity, Status 1 columns) comes up. On top is a window “Sensor Confirmation”.

Click “Connect”, again “Connect”. Test the photogate: block the photogate beam with your finger and see the red light on the cross bar of the photogate turn on.

Enter your value for the angular distance (90o) from one black tape to the next in radians ($\:\:\pi /2$) into the program: Click Data→User Parameters: the window “User Parameters” comes up and should show:

Name:Value:Units:Places:Increment:Editable:

Click OK. You are now ready for data taking.

Position the rotating wheel such that the light beam of the photogate is just to the right or the left of a piece of black tape, making sure that the tape is not obstructing the light beam. You will spin the rotating wheel either clockwise or counterclockwise so that initially (very close to) the full distance between two pieces of tape sweeps past the photogate.

Measurement of $\large \:\:\alpha_{fr}$

Spin the rotating table slowly, click the green “Collect” icon, and collect approximately 15 velocity-time pairs of data; then hit STOP. On the screen, you should see a table of values of time $\:\:t$ and magnitude of angular velocity $\:\:\omega$ and a falling, approximately linear, graph. Warning: The S.I. unit for angular velocity is radians/s; ignore the units for linear velocity on your computer screen. If there is a kink in the graph of your data, ignore any points before it and copy the first 10 velocity – time pairs after the kink into Table 1 on your worksheet.

Enter the values in Table 1 without any uncertainties into the plotting tool below. The fit should give you a negative slope for your line. The value of this slope is $\:\:\alpha_{fr}$.

x axis label (include units):
y axis label (include units):
Check this box if the fit should go through (0,0).
(Don't include (0,0) in your list of points below, it will mess up the fit.)
What kind of errors are you entering below?
x1: +/-    y1: +/-
x2: +/-    y2: +/-
x3: +/-    y3: +/-
x4: +/-    y4: +/-
x5: +/-    y5: +/-
x6: +/-    y6: +/-
x7: +/-    y7: +/-
x8: +/-    y8: +/-
x9: +/-    y9: +/-
x10: +/-    y10: +/-

Measurement of $\:\:\alpha$

Attach the 200 gram mass, $\:\:m$, to the free end of the string and wind the string around the cylinder. Loop the string over the pulley and position the photogate just as you did above for the measurement of $\:\:\alpha_{fr}$. Click the green “Collect” icon and release the weight. Push “STOP” when the mass is done falling or before the mass touches the floor.

Make sure you have collected approximately 15 velocity-time pairs of data. Copy the first 10 velocity – time pairs into Table 2 on your worksheet. Did you make sure that your photogate was oriented so that the light beam was really intercepted by the black tape? Use the plotting tools below to plot your values and obtain the slope as you did in the previous measurement. (The slope of this line should be positive this time. Do you understand why?) Enter your slope value, which is $\:\:\alpha$, on your worksheet.

x axis label (include units):
y axis label (include units):
Check this box if the fit should go through (0,0).
(Don't include (0,0) in your list of points below, it will mess up the fit.)
What kind of errors are you entering below?
x1: +/-    y1: +/-
x2: +/-    y2: +/-
x3: +/-    y3: +/-
x4: +/-    y4: +/-
x5: +/-    y5: +/-
x6: +/-    y6: +/-
x7: +/-    y7: +/-
x8: +/-    y8: +/-
x9: +/-    y9: +/-
x10: +/-    y10: +/-

Calculation of I

You have now measured everything needed to calculate the moment of inertia of the rotating platform system using Eq. 6.5. The only uncertainty you will take into account is the uncertainty in $\:\:r$. Assume that the relative uncertainty in $\:\:r$ is the same as the relative uncertainty in $\:\:I$ (because $\:\:g$ is much greater than $\:\:r\alpha$, and the relative uncertainty in the measured value of the mass is very small). Therefore, if you multiply your value for $\:\:I$ by the relative uncertainty in $\:\:r$ you can obtain an estimate for $\:\:\Delta I$, the absolute uncertainty in $\:\:I$. Write down both your value for $\:\:I$ and your estimate for $\:\:\Delta I$ on your worksheet.

## Part II: Testing conservation of angular momentum

You have already determined the moment of inertia $\:\:I$ of the rotating platform system. You will now perform another experiment that increases its mass by dropping an additional disk (see the “disk with handle” in Fig. 1) onto the rotating platform. If done carefully, this drop does not cause an external torque that would modify the initial angular momentum $\:\:{\vec L}$.

Figure 3

You need to calculate the moment of inertia $\:\:I_{disk}$ of the disk with handle from its mass $\:\:M$ (given to you on a label attached to it) and its radius $\:\:R$ (which you must measure), using the formula.

$\Large I_{disk}\:=\:\frac{1}{2}MR^2$
(6.6)

You may neglect the small contribution of the handle on the dropped disk to its moment of inertia $\:\:I_{disk}$. Measure the radius $\:\:$R of the disk with handle (to be dropped) with the ruler and use 1 mm for the absolute uncertainty. Read the value of $\:\:M$ from the label and use 1 gram for its absolute uncertainty. Enter all values on your worksheet. Using Eqs. (E.1), (E.2), (E.3), and (E.8) from the Lab 1 manual, Uncertainty, Error and Graphs, with Eq. (6.6), above, you may show that the relative uncertainty $\:\:\frac{\Delta I_{disk}}{I_{disk}}$ in $\:\:I_{disk}$ is twice the relative uncertainty $\:\:\frac{\Delta R}{R}$ in $\:\:R$. Then Use Eq. (E.4) from the Lab 1 manual, Uncertainty, Error and Graphs, to calculate the value of the absolute uncertainty $\:\:\Delta I_{disk}$ in $\:\:I_{disk}$, which you should record on your worksheet.

The final moment of inertia $\:\:I’$ is equal the sum of $\:\:I$ (the moment of inertia of the platform measured in Part I) and $\:\:I_{disk}$ (the moment of inertia that you calculated in this part for the dropped disk ). In symbols, the final moment of inertia is $\:\:I’ = I + I_{disk}$. Write your value for $\:\:I'$ on your worksheet.

Warning: This is only valid if the axis of rotation goes through both the center of the rotating platform and the center of the dropped disk because when you calculated $\:\:I_{disk}$ you used an equation that was valid only if the axis of rotation went through its center. To find the uncertainty in $\:\:I'$ use Eq. (E.6) from the Lab 1 manual, Uncertainty, Error and Graphs.

Remove the string from the rotating table. Get the computer ready for data taking as you have done previously and then spin the rotating table. Click the green “Collect” icon. Collect some data points and, while the rotating platform is spinning, carefully drop the disk-with-handle onto it from a small height (less than 1 cm) above the platform. It should be clear that the orientation of the handle is “up” and that you want the rim of the disk-with-handle to match the outer rim of the rotating platform as closely as possible.

You should get a graph that looks roughly like this:

The higher line corresponds to the initial system configuration rotating with its initial angular velocity $\:\:\omega$, and the lower line corresponds to the final system configuration rotating with its final angular velocity $\:\:\omega\,^{\prime}$. From the spreadsheet on the PC Monitor enter each group of 3 points next to the discontinuity $\:\:\omega \rightarrow \omega\,^{\prime}$ into Table 3 on your worksheet.

Calculate the average value of $\:\:\omega$ and $\:\:\omega\,^{\prime}$ and use the difference between the maximum and minimum values of $\:\:\omega$ and $\:\:\omega\,^{\prime}$ as your estimate for the uncertainty in these two quantities. Write these values in your worksheet.

You are now ready to work out $\:\:L$ and $\:\:L\,^{\prime}$ from Eq. (6.3). You should also calculate the uncertainty in $\:\:L$ and $\:\:L\,^{\prime}$ using Eq. (E.7) in the Lab 1 manual, Uncertainty, Error and Graphs. Once you have done this, see if $\:\:L$ and $\:\:L\,^{\prime}$ are consistent, i.e., agree within experimental uncertainty. From this conclude whether or not angular momentum was conserved in your experiment. If it was not conserved, can you think of possible reasons why not? As only one example, if you don't drop the disk so that its center coincided with the rotation axis (center) of the rotating platform, will this make a significant difference? Discuss these things with your lab partner and, together, with your TA. 