Given a rotating body, we state that the body is made up of n single rotating particles, each at a different radius from the axis of rotation. When each particle is considered individually, we can see that each one does in fact have a translational kinetic energy:

K = m1v12 + m2v22 + ... mnvn2

However, we also know from our relation between linear and angular variables that v = . Substituting this expression in, we see that:

K = m1r12σ2 + m2r22σ2 + ... mnrn2σ2

Since all particles are part of the same rigid body, we can factor our σ2 :

K = (mr2)σ2

This sum, however, is simply our expression for a moment of inertia. Thus:

 K = Iσ2

As we might expect, this equation is of the same form as our equation for linear kinetic energy, but with I substituted for m, and σ substituted for v. We now have rotational analogues for nearly all of our translational concepts. The last rotational equation that we need to define is power.

### Power

The equation for rotational power can be easily derived from the linear equation for power. Recall that P = Fv is the equation that gives us instantaneous power. Similarly, in the rotational case:

 P = τσ

With the equation for rotational power we have generated rotational analogues to every dynamic equation we derived in linear motion and completed our study of rotational dynamics. To provide a summary of our results, the two sets of equations, linear and rotational, are given below: Linear Motion:

 F = ma W = Fx K = mv2 P = Fv

Rotational Motion:

 τ = Iα W = τμ K = Iσ2 P = τσ

Equipped with these equations, we can now turn to the complicated case of combined rotational and translational motion.