A Classical Approach of Newtonian Mechanics

8 ROTATIONAL MOTION 8.8 Power and work

8.8 Power and work

Consider a mass m attached to the end of a light rod of length l whose other end

is attached to a fixed pivot. Suppose that the pivot is such that the rod is free

to rotate in any direction. Suppose, further, that a force f is applied to the mass,

whose instantaneous angular velocity about an axis of rotation passing through

the pivot is ω.

Let v be the instantaneous velocity of the mass. We know that the rate at which

the force f performs work on the mass—otherwise known as the power—is given
by (see Sect. 5.8)

P = f·v. (8.61)

However, we also know that (see Sect. 8.4)

v = ω × r, (8.62)

where r is the vector displacement of the mass from the pivot. Hence, we can
write

(note that a·b = b·a).

P = ω × r · f (8.63)

Now, for any three vectors, a, b, and c, we can write

a × b · c = a · b × c. (8.64)

This theorem is easily proved by expanding the vector and scalar products in

component form using the definitions (8.11) and (8.13). It follows that Eq. (8.63)

can be rewritten

However,

P = ω · r × f. (8.65)

τ = r × f, (8.66)

where τ is the torque associated with force f about an axis of rotation passing

through the pivot. Hence, we obtain

P = τ·ω. (8.67)