# PHY 141 Lab 6 - Angular Momentum

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

## Equipment

• rotating table with photo gate and pulley , with cylinder for string
• disk with handle
• mass with string attached which winds around cylinder attached to the rotating platform
• interface box
• vernier caliper: to measure the diameter of the cylinder
• computer

## Introduction

In Part I we measure the angular acceleration $\alpha$ of an object, a rotating table under a known external torque and obtain its moment of inertia $I$. The equation which relates the net torque $\tau_{net}$ to the moment of inertia and the angular acceleration of a rotating object is

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

We measure the angular acceleration by measuring the angular velocity $\omega$ as a function of time $t$. For a constant angular acceleration the equation which gives the dependence of $\omega$ on $t$ is

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

In Part II, after the moment of inertia $I$ of the rotating platform has been determined, Conservation of Angular Momentum is investigated. The definition of Angular Momentum $L$ is

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

We test Conservation of Angular Momentum using the relation

$\Large I\omega=I'\omega '$
(6.4)

where the dashed quantities refer to the angular momentum and angular velocity in the final state, and the undashed quantities refer to same things in the initial state. In order to change the moment of inertia of the rotating object we drop a disk (the disk with a handle in our equipment) onto the rotating platform, thus changing the moment of inertia, and measure the angular velocity $\omega$ before and after the drop.

## Measurement of the Moment of Inertia of the Rotating Platform and Attached Cylinder

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 of mass $m$ attached to the string provides an external torque, $\tau_{ext}$. When the platform rotates a photo gate registers whenever one of the 4 black radial strips on the plastic rim of the platform blocks the light beam of the photo gate. From the 90o angle between the strips the angular velocity $\omega$ is computed. The system is not friction free. A frictional torque, $\tau_{fr}$ , opposes the external torque thus causing a smaller net torque, $\tau_{net}=\tau_{ext}-\tau_{fr}$. The net torque gives the rotating platform an angular acceleration $\alpha$ which is measured from the rate of increase of the angular velocity. The frictional torque $\tau_{fr}$ causes a frictional deceleration $\alpha_{fr}$ .

The moment of inertia $I$ of the rotating platform can be calculated 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).

You should derive this equation by considering that the net torque is $\tau_{net}=\tau_{ext}-\tau_{fr}=I\alpha$, the frictional torque is $\tau=I\alpha_{fr}$ and the applied torque is $\tau_{ext}=Tr=m(g-a)r$.

## Procedure

Use the vernier caliper to measure the diameter (ask your TA in case you have questions about the caliper scale) of the cylinder under the rotating platform. Apply the caliper to the cylinder, snuggly, and remove it. From the diameter, get the radius $r$ of the cylinder (see Fig. 2 above). Use +/-0.5 mm as your absolute error in the radius $r$ . Record $r$ and its absolute error $\Delta r$ in your notebook.

You will be using the computer in “MOTION TIMER” mode. While you setup your experiment the interface box should be in “STOP” mode. When you want to collect data press “SET” to send data from the interface box of the photo gate to the computer.

Measurement of $\alpha_{fr}$

For this part make sure the string is not attached (it can get tangled). Start the table rotating slowly, press “SET” and count for 15 to 20 seconds. Then hit “STOP” on the interface box. Hit “ENTER” on the keyboard to see a data table of time values. Make sure that you have ~15 rows of data. Hit “ENTER”, select “Graph Data”, and select “Velocity vs. Time”, and then select “Other (specify the length)”.Your computer screen will show “Enter length (in meters)”, this is actually the distance between timings, ignore the “in meters” on the computer screen and enter the angular distance of $\pi/2$ in radians (1.571), the distance between the two pieces of tape on the rotating disk. This allows the computer to measure the angular velocity, $\omega$. Once you have put in the angular distance, hit “ENTER” to bypass the “SELECT GRAPH STYLE” screen. Then select “Automatic Scaling, Axis Starts at 0” for both horizontal and vertical scaling. Your graph should show a smooth straight line with a negative slope. To get out of the graph screen, hit “ENTER” and select “Display Table of Values”. On the screen, you should see a table of values of time t and angular velocity ω (the units for angular velocity is in radians/s, just ignore the units for linear velocity on your computer screen). Copy the first fifteen rows of data in to your lab notebook.

Measurement of $\alpha$

Make sure your string is long enough to reach from the small cylinder in Fig. 2 (which you wind the string around) to close above the floor, in order to collect enough data points when the weight of the mass $m$ falls toward the floor.

Attach a 200g mass m to the free end of the string and wind the string neatly around the cylinder. Loop the string over the pulley and position the photo gate as you have done above in the measurement of $\alpha_{fr}$. Push the “SET” button on the gray interface box and release the mass. Push “STOP” on the interface box when the mass is done falling or before the mass touches the floor. Obtain from the computer the table of values of time t and angular velocity ω as you have done above in the measurement of $\alpha_{fr}$.

Make plots of both sets of data using the plotting tool. The slope of each graph of angular velocity vs time is the angular acceleration, in the first case due to friction and in the second due the torque exerted by the falling weight. Record your values of $\alpha_{fr}$ and $\alpha$ in your notebook.

Calculation of I

We have now measured everything we need to calculate the moment of inertia of the system using equation 6.5. The only error we will take into account is the error in $r$, and we will consider that the relative error in $r$ is the same as the relative error in $I$.(because $g$ is much greater than $r\alpha$ and the relative error in the mass measurement is very small). Therefore if you multiply your value for $I$ by the relative error in $r$ you can obtain an estimate for the absolute error in $I$. Write down both your value for $I$ and your estimate for the error in this value in your notebook.

## Testing conservation of angular momentum

Now that we know what the moment of inertia of the system is we will perform an experiment in which we change it by increasing its mass. This is done by dropping an additional disk (see the “disk with handle” in Fig. 1) on top of the rotating platform. This drop (if done carefully) does not cause an external torque which would modify the initial angular momentum.

We need to calculate the moment of inertia $I_{disk}$ of the disk with handle from its mass $M$ and its radius $R$, using the formula.

$\Large I=\frac{1}{2}MR^2$
(6.6)

You can neglect the small contribution of the handle on the dropped disk to the moment of inertia. Measure the radius R of the disk with handle (to be dropped), with the ruler and use 1mm for its absolute error. Obtain the value of M and use 1 gram for its error. Enter all values on your worksheet. As before you can assume that the relative error in $I$ is the same as the relative error in $R$, which you can use to calculate the value of the absolute error in $I$, $\Delta I$ which you should record on you worksheet.

The final moment of inertia $I’$ is equal the sum of $I$ and $I_{disk}$, the moment of inertia of the dropped disk. For the total moment of inertia $I’$ of the combined object (platform + dropped disk) take simply the sum of the two moments of inertia, $I’ = I + I_{disk}$. Write your value for $I'$ on your worksheet. This is only valid if the axis of rotation goes through both the center of the rotating table and the center of the dropped disk (When you calculated $I_{disk}$ you used an equation which was valid only if the axis of rotation went through its center). To find the error in $I'$ use equation (1.6) of Lab 1.

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. Press the “SET” button“ Collect some points and, while the rotating table is spinning, drop the disk with handle onto it from a small height (less than 1cm) above the platform. You want the rim of the disk to match the rim of the black cylinder on the platform as closely as possible. When you have the computer plot the angular velocity against time, you should get a graph that looks roughly like this:

The higher line group corresponds to the initial angular velocity $\omega$ and the lower line group corresponds to the final angular velocity $\omega’$. From the table of values record the angular velocity for 3 on either side of the jump $\omega \rightarrow \omega '$ in your notebook.

Calculate the average value of $\omega$ and $\omega '$ and use the difference between the maximum and minimum values of $\omega$ and $\omega '$ as an estimate of the error in these two quantities. Fill these values in on your worksheet.

You are now ready to work out $L$ and $L'$ from equation (6.3). You should also calculate the error in $L$ and $L'$ using equation (1.7). Once we have done this we can see if $L$ and $L'$ are the same within experimental uncertainty. Was angular momentum conserved in your experiment? If it wasn't can you think of possible reasons? For example, if you don't drop the disc right on the center will it make a big difference?