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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 = rσ. 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.

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