A Classical Approach of Newtonian Mechanics

8 ROTATIONAL MOTION 8.7 Torque

f

torque



P

r

O

pivot

Figure 81: Torque about a fixed point.

between the directions of r and f. What is the vector torque τ acting on the

object about an axis passing through the pivot point? The magnitude of this

torque is simply

τ = r f sin θ. (8.59)

In Fig. 81 , the conventional direction of the torque is out of the page. Another

way of saying this is that the direction of the torque is mutually perpendicular

to both r and f, in the sense given by the right-hand grip rule when vector r is

rotated onto vector f (through an angle less than 180 ◦ degrees). It follows that

we can write

τ = r × f. (8.60)

In other words, the torque exerted by a force acting on a rigid body which pivots

about some fixed point is the vector product of the displacement of the point of

application of the force from the pivot point with the force itself. Equation (8.60)

specifies both the magnitude of the torque, and the axis of rotation about which

the torque twists the body upon which it acts. This axis runs parallel to the

direction of τ, and passes through the pivot point.