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 |
From this equation for a rigid body we can also generate a statement relating external torque and total angular momentum:
τ _{ext} = |
To illustrate this very simple concept, we examine a very simply situation. Consider a bicycle wheel. By pedaling the bike we exert a net external torque on the wheel, causing its angular velocity to increase, and thus its angular momentum to follow suit.
In the next section, we will use the equation relating torque and angular momentum to derive another conservation law: the conservation of angular momentum.