10.9 - Angular momentum of a rigid body
On the right, you see a familiar sight: a rotating compact disc. In the prior section, we
defined the angular momentum of a single particle as the product of its mass, speed
and radial distance from the axis of revolution. A CD is more complex than that. It
consists of many particles rotating at different distances from a common axis of rotation.
The CD is rigid, which means the particles all rotate with the same angular velocity, and
each remains at a constant radial distance from the axis.
We can determine the angular momentum of the CD by summing the angular momenta
of all the particles that make it up. The resulting sum can be expressed concisely using
the concept of moment of inertia. The magnitude of the angular momentum of the CD
equals the product of its moment of inertia, I, and its angular velocity, Ȧ.
We derive this formula for calculating angular momentum below. In Equation 1, you see
one of the rotating particles drawn, with its mass, velocity and radius indicated.
Variables
Strategy
- Express the angular momentum of the CD as the sum of the angular momenta of
all the particles of mass that compose it. - Replace the speed of each particle with the angular velocity of the CD times the
radial distance of the particle from the axis of rotation. - Express the sum in concise form using the moment of inertia of the CD.
Physics principles and equations
The angular momentum of a particle in circular motion
L = mvr
We will use the equation that relates tangential speed and angular velocity.
v = rȦ
The formula for the moment of inertia of a rotating body
I = Ȉmiri^2
Step-by-step derivation
First, we express the angular momentum of the CD as the sum of the angular momenta
of the particles that make it up.
We now express the speed of the ith particle as its radius times the constant angular
velocityȦ, which we then factor out of the sum. The angular velocity is the same for all
particles in a rigid body.
Angular momentum of a rigid
body
Product of moment of inertia, angular
velocityL = IȦ
L = angular momentum
I = moment of inertia
Ȧ = angular velocity
How much angular momentum
does the skater have?
L = IȦ
L = (1.4 kg·m^2 )(21 rad/s)
L = 29 kg·m^2 /s
mass of a particle mi
tangential (linear) speed of a particle vi
radius of a particle ri
angular momentum of particle Li
angular momentum of CD L
angular velocity of CD Ȧ
moment of inertia of CD I
Step Reason
1. Li = miviri definition of angular momentum
2. L = Ȉmiviri angular momentum of object is sum of particles
Step Reason
3. vi = riȦ tangential speed and angular velocity
4. L = Ȉmi(riȦ)ri substitute equation 3 into equation 2
5. L = (Ȉmiri
(^2) )Ȧ