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.
Consider a single particle of mass m travelling with a velocity v a radius r from an axis, as shown below.
l = rmv sinθ |
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 |
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 similar way,
τ = |
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 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 90^{ o } . If there are n particles, we find the total angular momentum of the body by summing the individual angular moments:
Now we express each l in terms of the particle's mass, radius and velocity:
We now substitute σ for v using the equation v = σr :
However, in a rigid body, each particle moves with the same angular velocity. Thus:
L | = | ( mr ^{2})σ | |
= | Iσ |