The final concept we develop for rotational motion is that of angular
momentum. We will give the same treatment to angular momentum that we did to
linear momentum: first we develop the concept for a single particle, then
generalize for a system of particles.
Angular Momentum for a Single Particle
Consider a single particle of mass m travelling with a velocity v a radius r
from an axis, as shown below.
The angular momentum of the single particle, then, is defined as:
l = rmv sinθ
Notice that this equation is equivalent to l = rp sinθ, where p is the
linear momentum of the particle: a particle does not need to move in a circular
path to possess angular momentum. However, when calculating angular momentum,
only the component of the velocity moving tangentially to the axis of rotation
is considered (explaining the presence of sinθ in the equation).
Another important aspect of this equation is that the angular momentum
is measured relative to the origin chosen. This choice is arbitrary, and our
origin can be chosen to correspond to the most convenient calculation.
Because angular momentum is the cross
product of position and linear momentum,
the angular momentum formula is expressed in vector notation as:
l = r×p
This equation provides the direction of the angular momentum vector: it always
points perpendicular to the plane of motion of the particle.
Angular Momentum and Net Torque.
It is possible to derive a statement relating angular momentum and net torque.
Unfortunately, the derivation requires quite a bit of calculus, so we will
simply revert to the linear analogue. Recall that: F = . In a
A net torque changes a particle's angular momentum in the same way that a net
force changes a particle's linear momentum.
In circumstances of rotational motion, however, we usually deal with rigid
bodies. In such cases the definition of the angular momentum of a single
particle is of little use. Thus we extend our definitions to systems of
Angular Momentum of Systems of Particles
Consider a rigid body rotating about an axis. Each particle in the body moves
in a circular path, implying that the angle between the velocity of the particle
and the radius of the particle is 90o. If there are n particles, we
find the total angular momentum of the body by summing the individual angular
L = l1 + l2 + ... + ln
Now we express each l in terms of the particle's mass, radius and
L = r1m1v1 + r2m2v2 + ... + rnmnvn
We now substitute σ for v using the equation v = σr:
L = m1r12σ1 + m2r22σ2 + ... + mnrn2σn
However, in a rigid body, each particle moves with the same angular
Here we have a concise equation for the angular momentum of a rigid body. Note
the similarity to our equation of p = mv for linear momentum.