CA - 1
CONSERVATION OF ANGULAR MOMENTUM USING AIR MOUNTED DISKS
Objective: The purpose of this experiment is to demonstrate that, in the absence of external torques,
the angular momentum of a system is conserved. The basic apparatus used in this experiment is the
same as that used in the experiment to measure angular acceleration using air mounted disks. Refer
to the write-up of that experiment regarding use of the basic equipment.
Background:
Angular Velocity:
ω =2πn/200
(1)
The linear velocity (v) of a point on the disk’s surface is related to the angular velocity of the cylinder
(typically denoted by the Greek letter omega: ω) by:
v =Rω
(2)
Similarly, the linear tangential acceleration of a point on the disk’s surface (a) is related to the angular
acceleration of the disk by:
Angular Momentum:
Rotational Kinetic Energy:
where I is the Moment of Inertia.
a =Rα
(3)
L = Iω
E = ½Iω2
(4)
(5)
The Moment of Inertia of a disk is:
I = 0.5mR2
Where m is the mass and R is the radius of the disk.
(6)
Apparatus: Rotational dynamics equipment
PART I
Procedure: Insert the valve pins so that the two disks of the apparatus are free to rotate
independently. The general idea is to set the two disks rotating at different speeds, and then to
remove the pin from the upper disk so that the two disks rotate together, having undergone an
inelastic collision. Measure the diameter of the disks and using the masses printed on the apparatus,
calculate the moments of inertia of the rotating disks. Measure the angular velocity of each disk
before the collision, and that of the combination after the collision. Compare the total angular
momentum before and after the collision. Repeat this procedure four times, as follows.
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1. Start with the lower disk with ω = 0, and the upper disk in motion.
2. Start with the upper disk with ω = 0, and the lower disk in motion.
3. Have both disks in motion rotating in the same direction.
4. Have the two disks rotating in opposite directions.
For each case calculate the total angular momentum and kinetic energy before the collision
and these quantities after the collision. Compare the pre and post collision values. Which quantities
are conserved?
PART II
Procedure: In this part of the experiment we will again be considering conservation of angular
momentum. This time, the angular momenta before the collision are that of a ball launched
horizontally and that of the disk with a ball catcher attached (Figure 1). A steel ball will be
permitted to roll down a small ramp with a horizontal launching track. The ball is released from the
top of the ramp and it will leave the launch at some speed determined by the height of the ramp and
its own radius. The launching speed of the ball can be measured using the following technique. The
ramp is placed near the edge of the laboratory table so that the ball, when launched, goes over the
edge of the table and lands on the floor. Knowledge of the height of the launch point above the floor
and the horizontal distance traversed by the ball during its fall allow determination of its initial
velocity (Figure 2). Calling the initial velocity v0, the height h and the horizontal distance d we find
v0 = (gd2/2h)½
(7)
To record the horizontal position at which the ball hits the floor, we tape a piece of paper to
the floor at the approximate point of impact, and place on it a sheet of carbon paper. When the ball
hits the carbon paper, it produces a mark on the paper. Repetition of this process a number of times
allows us to get a reasonable value for the launch speed of the ball.
After determining the launch speed of the ball, attach the small spool, with the cord holder
underneath, and the ball catcher to the upper disk, using a grey topped thumb screw. Measure α, the
angular acceleration of the disk, and from this determine the moment of inertia of the disk and ball
catcher combination as was done in the previous experiment. Use a suspended weight of 10 gram
for this measurement.
Measure and record the mass of the ball.
The next step is to permit the ball to roll off the launcher and be caught by the catcher. The
total angular momentum of the system, ball and disk, should be conserved. Hold the disk stationary
and position the launcher so that the ball will be caught at a radius of 4 cm from the center of
rotation of the disk. This radius is called the impact parameter b. Release the ball from the top of
the ramp as you did when measuring the launch velocity and permit the catcher to catch the ball.
Move the launcher out of the way. The disk will begin rotating. The angular velocity of the disk
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can then be determined from the digital readout.
Make five measurements at each of the impact parameters 2, 4, 6, and 8 cm.
The equation describing the relation between ω and b is
ω=
bp
I + mb 2
(8)
where I is the moment of inertia of the disk and catcher, p is the linear momentum of the ball, and m
is the mass of the ball.
Plot ω vs. b, where ω is the angular velocity after the collision and b is the impact parameter.
Do an order of magnitude calculation to show that mb2 << I, so that the graph of ω vs b is
essentially a straight line. Compare the linear momentum of the ball before the collision based on
the slope of the line with that calculated by launching the ball over the edge of the table.
Questions to be answered in the report:
1. Why is angular momentum conserved in these collisions (Parts I and II)? Note that linear
momentum is not conserved in Part II. How then is angular momentum conserved?
2. Derive Equation 8.
3. Is kinetic energy conserved in Part II? If not, calculate the fractional loss of kinetic
energy for at least one of the cases you measured.
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Figure 1 - Equipment Setup
Figure 2
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