Before you read this, I suggest you read post 16.50.
Let’s suppose we want to use the (grey) spanner to loosen the (red) nut, in the picture above. We apply a force, F, that makes an angle θ with the axis of the spanner at a distance L from the axis around which the nut turns.
If the nut is difficult to turn, experience tells us that we want to make L as large as possible, F (the magnitude of F) as large as possible and θ as close as possible to 90o. When θ equals 90o, sin θ = 1 which is the highest value that the sine of an angle can have (post 16.50). (Note that F sin θ is the component of F perpendicular to the axis of the spanner, posts 16.50 and 17.2.)
We can summarise the previous paragraph by saying that we must increase FLsin θ to increase the effectiveness of the spanner. T = FLsin θ is called the torque exerted by the force F. It is conventional to represent a torque that induces anti-clockwise motion as a positive number and a clockwise torque as negative.
Torque changes the state of rotational motion of an object in exactly the same way that force changes the state of motion of an object along a line (sometimes called translational motion). Since sin θ is a number with no units, we measure torque in the units of force (N, post 16.13) multiplied by the units of length (m, post 16.12) – torque is measured in N.m.
For example, we are interested in the maximum torque that a car engine can deliver, and not the maximum force that it can generate, because the purpose of the engine is to make the car wheels go round – in other words, to change their state of rotational motion. A modern 2 litre engine will generate a torque of about 350 N.m. Car manufacturers often measure torque in a unit called the foot.pound (ft-lb) where 1 ft-lb = 1.36 N.m.
Sometimes torque is represented as a vector, T. The modulus (post 17.2) of T is given by T = FLsin θ as defined above. The direction of T is the direction of the axis an object will turn around when the torque acts on it. In the picture above, the axis, about which the nut turns, is perpendicular to the plane of the picture. But does T point up or down? To find out, stick up the thumb of your right hand and curl its fingers. Now turn your wrist, so that the fingers point in the direction of rotational motion. In the picture below, my fingers point in the direction in which the nut (in the picture above) turns; my thumb points in the direction of the axis about which an anti-clockwise torque is positive.
So, in the top picture, the force F generates a torque T that points towards you, out of the plane of the paper (just like my thumb).
However, we are usually interested in measuring torque about a fixed axis and so can represent it by a scalar that is positive for a torque that induces anti-clockwise motion and negative for a torque that induces clockwise motion.