Torque

A force can cause an object to turn about an axis. The torque is the turning effect of that force. For example, pushing a door near its handle produces a greater turning effect about its hinges than the same push in the same direction near the hinges.

The size of the torque depends on both the force and where its line of action lies relative to the axis. The line of action is an imaginary straight line extending in the direction of the force through the point where the force is applied.

Here, $$T$$ is the magnitude of the torque, $$F$$ is the magnitude of the force, and $$r$$ is the perpendicular distance from the axis to the force’s line of action. Torque is measured in newton metres, $$\mathrm{N,m}$$.

A larger force produces a larger torque if the perpendicular distance stays the same. Likewise, a larger perpendicular distance produces a larger torque if the force stays the same. The distance is not necessarily the distance from the axis to the point where the force is applied: it must be measured at right angles to the line of action.

If the force’s line of action passes through the axis, the perpendicular distance is zero, so the force produces no torque about that axis.

Torque and angular acceleration

A resultant torque changes an object’s rotational motion about an axis. How much angular acceleration it produces depends on the object’s moment of inertia about that axis. Moment of inertia measures the object’s resistance to a change in its rotational motion.

The relationship is:

Here, $$T$$ is the resultant torque about the axis, $$I$$ is the moment of inertia about that axis, and $$\alpha$$ is the angular acceleration.

For a fixed moment of inertia, angular acceleration is proportional to resultant torque: doubling the resultant torque doubles the angular acceleration. For a given resultant torque, a larger moment of inertia gives a smaller angular acceleration. For example, doubling the moment of inertia halves the angular acceleration.

Angular acceleration has the same direction as the resultant torque. If the resultant torque acts opposite to the direction of rotation, the object slows down. If the resultant torque is zero, the angular acceleration is zero; the object may still be rotating at a constant angular velocity.