Angular Momentum and Torque
Physics 100L Exercise 4
Introduction
Rotational motion is very analogous to linear motion. In linear motion, we normally define the
following terms: distance, velocity, acceleration, force, momentum. In rotational motion we
define analogous terms. Refer to the table below:
Linear Term
Symbol
Rotational Term
Symbol
Position
Velocity
Acceleration
Force
Mass
Momentum
x
v
a
F
m
p
Angle
Angular velocity
Angular acceleration
Torque
Moment of Inertia
Angular momentum
I
L




Note that, with the exception of L and I we tend to use Greek symbols for rotational quantities.
The letter N is also used to symbolize torque. The beauty of these analogies is that they allow us
to use the same equations that we developed for linear motion with only a replacement of
symbols. In particular, we know that in linear motion
p = mv and F = ma.
In rotational motion, we simply look up the analogous quantities in our table and write
L = I and  = I.
Also, just as momentum is conserved in linear motion if there are no external forces, in rotational
motion angular momentum is conserved if there are no external torques. We will look at these
relationships in today’s experiment.
Direction of Angular Momentum and Torque
Since angular momentum and torque are, like the counterparts in linear motion, vectors, there is
often some confusion over which way these vectors point. In linear motion, the momentum
vector simply points along the direction of the motion, and the force vector simply points along
the direction of the force. For rotational quantities, the directions of the vectors are given by the
following right-hand-rule. If you take your right hand and curl your fingers in the direction of
the rotation, then the direction of your thumb is the direction of the angular momentum vector.
For the torque vector, your fingers curl in the direction in which the torque pushes the rotation.
This is not the first right-hand-rule we will see – there are more in electricity and magnetism.
Fortunately, however, there are no left-hand-rules, so it is easy to keep things straight!
Angular Momentum Part I
In this part of the experiment we will see one of the consequences of conservation of angular
momentum. One person in your group will stand on the rotating platform holding a spinning
bicycle wheel. Make sure you hold the platform steady as you impart a spin to the wheel.
Measure how long it takes for the wheel to make one revolution – you may need to measure
several revolutions and divide by the number you took because it will be difficult to time a single
revolution. Then have the person turn the wheel over. What happens? The reason for this is
that by turning the wheel over, you have made its angular momentum reverse direction. Since
the system as a whole (i.e. you and the wheel together) must have a constant angular momentum,
the change in the wheel’s angular momentum forces a change in your angular momentum – in
the opposite direction. Remember this, because it will be analogous to another situation we will
see in a few weeks.
Record your results for the times in Table 1. Also calculate the angular velocity of both the
wheel and the person from the equation   1t and record this in your table as well.
Angular Momentum Part II
In this part of the experiment you will use the same apparatus to see how the distance of the mass
from the axis of rotation affects the moment of inertia. Have one person in your group stand on
the platform with their arms extended. Push them so that they start spinning slowly. Measure
how long it takes for the person to make one revolution as you did in the last experiment. Then
have the person bring their arms to their side. What happens? This is because the distribution of
an object’s mass affects its moment of inertia. With the mass concentrated around the axis of
rotation, the moment of inertia gets smaller, and because of conservation of angular momentum,
the angular velocity gets bigger. This is how ice skaters accelerate when they bring their arms in
during a spin. Measure the time for one revolution now, and use the conservation equation,
I i i  I f  f
where i represents initial and f represents final, to find the ratio between the initial and final
moments of inertia.
Torque
For the final part of the experiment we will look at torque – in particular, we will see how a
balance scale works using the method of moments. Balance your ruler on the stand with nothing
attached to it. We will then hang masses on either side and get the ruler to balance again. For
one mass on either side, the equilibrium condition that the sum of torques equals zero implies
that
m1d1  m2 d 2
You will put a given mass at a given position on one side and find where the other mass goes to
balance the apparatus. You can then check to see if this is what you would expect from the
equation above.
You will then do the same thing for three masses. You will put two of them at given positions
and move the third around until it balances. You can then calculate the expected position from
the method of moments and compare it to what you got. For three masses, the above equation
needs only a slight modification. You will add in a term m3d3. The side on which it is added
reflects where the mass was put – if it was on the same side as m2, then add the term on that side,
and vice versa for m1. For example, if it was on m1’s side, then your equation becomes
m1d1  m2 d 2  m3 d 3 .
Finally, you will “weigh” your keys using this apparatus. Place a known mass at a known
position on one side, then put your keys on the other side. Move them around until the ruler
balances. You can now calculate the mass of your keys using the method of moments. Compare
this to what you get on the scale.