D7-1
CONSERVATION OF ANGULAR MOMENTUM
Objective
The objective of this experiment is to verify the validity of the principle of conservation of angular
momentum.
Introduction
The angular momentum of a point mass m moving with velocity v, relative to a given axis, is
l = r x p = r x mv .................... (1)
Where the vectors r and p are represented in the figure and, for the
case shown, l is a vector pointing up along the axis of rotation.
If the vectors r and p are perpendicular to each other, then the
magnitude of the angular momentum is simply:
l =rmv .................... (2)
The angular momentum of a rigid body, rotating about a fixed axis, is given by (in magnitude)
L = I ω .................... (3)
where I is the moment of inertia of the body (with respect to the rotation axis) and ω its angular
velocity.
When an external torque τext is applied to a body (or system), the angular momentum of the body (or
system) changes according to
τext = dL/dt .................... (4)
That is, the resultant external torque applied to a body (or system) is equal to the rate of change with
time of the angular momentum of the body (or system).
Conversely, if the resultant external torque applied to a body (or system) is zero then the angular
momentum of the body (or system) remains the same, i.e. it is conserved.
In the experiment described below, we have a system composed of a turntable and a projectile.
Initially, the projectile is mounted onto the table and the sytsem is at rest. Thus, the initial angular
momentum is zero, Li = 0. After the projectile is launched, the angular momentum of the system, Lf,
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is:
Lf = mvR - Itt ω .................... (5)
where mvR is the angular momentum of the projectile (with respect to the axis of the turntable)
right after being launched, i.e. m is the mass of the projectile, v its initial speed and R the radius of
the turntable, and Ittω is the angular momentum of the turntable (set in rotation) right after the
launching. Itt is the turntable's moment of inertia and ω its inital angular speed. Note that the minus
sign in equation (5) indicates the fact that the angular momenta of projectile and turntable have
opposite directions.
In the absence of an external net torque we have:
Li = Lf .................... (6)
and thus:
mvR= Ittω .................... (7)
Procedure
To perform this experiment you will need the following:
-turntable (with a spring gun mounted on it)
-stop clock
-manila paper
-carbon paper
-ball bearing
-ruler
-protractor
-vernier caliper
-scale
-string
-plumb bob
1.
Weigh your ball bearing and record this value (m). Place the ball bearing in the barrel of the
spring gun and measure its height, h, from the bottom of the ball to the table top. Get a sheet of
manila paper and tape it to the table below the spring gun. Use the plumb bob to drop a vertical from
the mouth of the spring gun to the table and mark this on the paper using the carbon paper.
2.
Pull back the plunger in the spring gun and secure it using the pin provided. The first trial
will be used to get an idea of where the ball will land on the table. Place the ball in the spring gun,
set the mouth of the gun over the place marked by the plumb, and quickly pull the pin from the
plunger (try not to interfere with the motion of the turntable when you are doing so). Note where the
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ball lands on the manila paper, and place your carbon paper face down over this general area.
3.
If the metal protractor is not already bolted to the top of the turntable stand do so now.
Make sure that the motion of the turntable is not affected by the protractor and that the protractor is
secure. Reload the spring gun as in step 2 above. Read the position of the turntable’s pointer on the
protractor. Start the stop clock as soon as you pull the pin out of the plunger. Stop timing when the
turntable has come to a stop, and record this value (t). Count the number of full rotations, n, that the
turntable completes during this time. Read the final position of the pointer from the protractor, and
calculate it’s angle of rotation (with respect to its original position), φ (in radians). On the manila
paper, circle the impression left by the ball bearing and number it. Repeat the above for a total of
five trials. When the ball leaves the spring gun we want it’s trajectory to follow a path tangent to the
rotating turntable arm. Check how valid this assumption is with respect to your data.
Replace the manila paper with a clean sheet, and repeat step three above for each member in
your group.
Measure the distance, r, from the centre of rotation of the turntable to the centre of mass of
the ball when it is in the spring gun.
4.
Measurement of Itt: Get a length of string between 1.5 and 2.0 meters long. Attach a 20 gram
mass (M) to one end of the string, and then start wrapping the other end around the aluminum drum
just below the turntable arm. Pass the string over a pulley and continue wrapping the string around
the drum until the mass is just below the pulley. Measure the distance, H, from the bottom of the
mass to the floor. Let go of the mass and measure the time (T) that it takes for the mass to reach the
ground. Repeat this step a total of five times. Measure the radius, R, of the drum using a pair of
vernier calipers.
5.
In order to find the angular momentum of the ball when it leaves the gun, we need to know
its initial velocity, v. This is easy to calculate if we assume that v is in the horizontal direction only,
and we know the vertical height h that the ball fell and the horizontal distance d that the ball
traveled:
⎛ g ⎞
v = d ⎜ ⎟
⎝ 2h ⎠
1/2
................... (8)
Derive the above equation using horizontal and vertical position coordinates for projectile motion.
Analysis
A.
On your manila paper measure the horizontal length l that the ball traveled from the spring
gun to where it hit the table top. Use the equation in part 5 of the procedure to the initial velocity of
the ball. For each trial calculate the angular momentum of the ball.
B.
If we assume that the turntable slows down at a uniform rate (α) we can find the initial
angular speed of the turntable :
ω = 2Φ/t .................... (9)
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where Φ is the total angle through which the turntable rotated (i.e.Φ = 2nπ + φ) and t is the time that
it took for the turntable to come to rest. Find ω for each trial taken in step 2 of the procedure. Also,
calculate α for each trial using the equation:
α = 2Φ/t2 .................... (10)
How valid is the uniform α assumption?
C.
To find Itt, we will use the conservation of energy. The energy of the system in part 4 of the
procedure before the mass is let go is given by the gravitational potential energy of the hanging
mass:
Ei = MgH .................... (11)
When the mass hits the ground, the total energy of the system is given by the equation:
Ef = ½MV2 + ½Itt(V/R)2 + (Ittα)(H/R) .................... (12)
where V is the velocity of the mass when it hits the ground (V = 2H/T), V/R is the angular velocity
of the turntable, and the last term represents the average frictional loss of the system as it rotates
through the angle H/R radians. Equate the terms in the equations above, and solve for Itt using the
values of M, H, R, and T found in part 4 of the procedure (take α to be the average of the five α
values calculated in part B above). Calculate the average value for Itt.
D.
For each trial from part 2 of the procedure, calculate the angular momentum of the turntable,
and compare this value to the angular momentum of the ball bearing. Does your experiment confirm
the conservation of angular momentum? Why or why not? Discuss some of the sources of error in
this experiment and how these could affect your results.
PRE-LAB QUESTIONS
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1. [1] How do you define the angular momentum of a particle and of an extended object?
2. [1] What are the quantities you have to measure in this experiment?
3. [1.5] Using the equations for projectile motion, derive the expression for the speed of the ball
when it leaves the gun, in terms of measurable quantities.
4. [1.5] Using the conservation of energy principle, derive the expression for the moment of inertia
of the turntable in terms of the measurable quantities.
REPORT WRITING INSTRUCTIONS
Read the procedure on previous three pages and prepare a table to help you record all
quantities needed for this experiment.
I. Procedure:
Perform the experiment as instructed in the previous pages of the lab manual and fill in your data
table. Calculate the moment of inertia of the turntable. For each run of the experiment calculate the
angular momentum of turntable and projectile and compare them (7 runs).
II. Conclusions:
1. What influence may the rate of slowing down (α) of the turntable have in the success of your
experiment?
2. Does your experiment confirm the principle of conservation of the angular momentum?
3. Discuss possible errors and how they affect your results?