PHYSICS 123 EXPERIMENT NO. 6
CONSERVATION OF ANGULAR MOMENTUM
Introduction:
In this experiment we study torque, moment of inertia, and angular momentum conservation with a rotating platform.
Equipment:
One rotating platform with photogate, one pulley with clamp, one iron disk with handle,
one interface box, one computer with timing program, small masses, vernier caliper.
Computer Terminal
Rotating table
Photo gate
Interface Box
Pulley
Mass with string
Vernier Caliper
Disk with handle
Cylinder for wrapping
string around
Method:
Observing the angular momentum of an object, the rotating platform, under an external
torque will enable us to measure the moment of inertia of the object using the equation
external = I
Where external is the net external torque,  is the angular acceleration, and I is the
moment of inertia. After the moment of inertia is determined, conservation of angular
momentum will be investigated. We drop a mass onto the rotating platform and measure
the angular velocity ,  , before and after the "drop". We test angular momentum
conservation using the relation
Iii= Iff
where i and f standard for initial and final.
Part I Measurement of the Moment of Inertia of the Rotating Wheel and Attached
cylinder:
large wheel
axis of rotation
pulley
pulley
cylinder with string
wound around it
mass m
The apparatus used for this lab is sketched above. A small cylinder with string wound
around it is attached to a large wheel. A mass m attached to the string provides an
external torque, τexternal, which gives the system, cylinder-wheel, an angular acceleration
photo gate mounted on the wheel registers whenever one of the 4 black radial strips on
the plastic rim of the wheel blocks the light beam of the photo gate. From the 900 angle
between the strips the angular velocity is computed. The system is not friction free, the
spinning wheel will stop after a period of time. A frictional torque, τfric, opposes the
external torque thus making a smaller net torque, τnet.
First you measure fric by measuring the rate of decrease of the angular velocity of
the initially spinning wheel. You use fric in a subsequent measurement of the moment of
inertia I of the system, cylinder-wheel. I can be calculated from the following formula:
mr ( g  r 



I
|  fric | 
where m is mass attached to the string, r is the radius of the cylinder, g is the
gravitational accelaration and  is the angular acceleration of the wheel due to the net
torque acting on the wheel, external torque minus frictional torque.
Connect the photogate to the interface box and set up the computer in MOTION
TIMER mode (see procedure 3. to 5. in lab “Acceleration" ). Measure the angular
velocity, , of the freely turning table for 20 or 30 seconds. Hit “ENTER” on the
keyboard to see a data table of time values. 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 (in radians)
between the two pieces of tape on the rotating disk. 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).
Graph  vs. time in your lab book, fit a best line through your data points and
calculate the angular acceleration (deceleration actually), fric , due to the frictional
torque.
Question 1: Is the frictional torque dependent on the velocity ?
II. MEASUREMENT OF MOMENT OF INERTIA, I: Wrap a piece of string around the
cylinder and measure the length of the string to determine the cylinder’s
circumference. Use the circumference to find the radius of the cylinder. Attach a
mass to the free end of the string, and wind the string neatly around the cylinder.
Loop the string over the pulley and release the mass. Record angular velocities as
above and calculate the angular acceleration,  . Calculate the moment of inertia, I
,from the equation
mr ( g  r 
I
|  fric | 
Derive this equation.
III. CONSERVATION OF ANGULAR MOMENTUM:
Remove the string from the rotating table. Get the computer and interface box ready for
data taking as you have done previously and then spin the rotating table. While the table
is spinning, drop the iron disk onto it from a small height (less than 1cm) above the table.
You want the rim of the disk to match the rim of the wheel as closely as possible.
Obtain the graph of angular velocity ω vs. time from the computer by hitting “ENTER”,
select “Graph Data”, and select “Velocity vs. Time”, and then select “Other (specify the
length)”. Make sure that your angular distance, which you put in earlier as 2п/4, is still
correct in the computer. Hit “ENTER” to bypass the “SELECT GRAPH STYLE” screen.
Then select “Automatic Scaling, Axis Starts at 0” for both horizontal and vertical scaling.
You should get a graph that looks roughly like this:
 i: take the average of 4 points here, get the
error from (max-min)/2; do the same for  f
ignore these points
f
t
the negative slope is caused by
 fr
Given the mass ,M, of the iron disk and measuring its radius R, calculate its moment
of inertia using
I = 1/2MR2
Compare the initial angular momentum of the table/disk system, Iii , with the final
angular momentum after the drop, Iff . Determine whether your experiment (within its
error limits) confirms angular momentum conservation.
Questions 2:
What assumptions are your calculations of the moment of inertia based on ? If you
drop the disk a couple of cm off center, what effect will this have on your result ?
Explain Qualitatively.
What you need:
1. Radius of rotating table
2. Angular distance between tape pieces
3. Graph of  vs t
4. Calculation of slope of graph to find fric
5. Calculation of  and I
6. Derivation of Inertia equation
7. Radius and mass of disk, calculation of its moment of inertia
8. Measurement of i and f
9. Check of conservation of angular momentum
10. Answers to the questions