8 ROTATIONAL MOTION 8.8 Power and work
In other words, the rate at which a torque performs work on the object upon
which it acts is the scalar product of the torque and the angular velocity of the
object. Note the great similarity between Eq. (8.61) and Eq. (8.67).
Now the relationship between work, W, and power, P, is simply
dW
P =. (8.68)
dtLikewise, the relationship between angular velocity, ω, and angle of rotation, φ,
is
ω =dφ. (8.69)
dt
It follows that Eq. (8.67) can be rewritten
dW = τ·dφ. (8.70)Integration yields
W =∫
τ·dφ. (8.71)Note that this is a good definition, since it only involves an infinitesimal rotation
vector, dφ. Recall, from Sect. 8.3, that it is impossible to define a finite rotation
vector. For the case of translational motion, the analogous expression to the
above is
W =∫
f·dr. (8.72)Here, f is the force, and dr is an element of displacement of the body upon which
the force acts.
Although Eqs. (8.67) and (8.71) were derived for the special case of the ro-tation of a mass attached to the end of a light rod, they are, nevertheless, com-
pletely general.
Consider, finally, the special case in which the torque is aligned with the an-gular velocity, and both are constant in time. In this case, the rate at which the
torque performs work is simply
P = τ ω. (8.73)