Moment of a force

To describe the rotation of rigid bodies we need a new concept: the moment of a force (or torque). It measures not the strength of a force, but its ability to make a body rotate about a point. Anyone who has ever opened a door knows this from experience: pushing far from the hinges is much more effective than pushing close to them, for the same force.

Principle — Moment of a force

The moment of a force $\vec{F}$ about a point O is $M = |\vec{F}| \cdot b$ where $b$ is the arm: the perpendicular distance between the force’s line of action and the point O.

The arm is the real protagonist: what matters is not where the force is applied, but how “wide” its line of action passes relative to point O. Equivalently, using the distance $r$ between O and the point of application and the angle $\theta$ between $\vec{r}$ and $\vec{F}$:

$M = |\vec{F}|\cdot r\cdot\sin\theta$

The $\sin\theta$ extracts exactly the component of $\vec{F}$ perpendicular to $\vec{r}$: only that component causes rotation. A force directed along $\vec{r}$ (towards point O or away from it) has zero moment.

Key formula

$M = F \cdot b \qquad M = F\,r\,\sin\theta$ $M > 0:\ \text{anticlockwise} \qquad M < 0:\ \text{clockwise}$

The sign encodes the direction of rotation: by convention, positive if the force tends to turn anticlockwise, negative if clockwise.

Links

Topics: Statics and equilibrium Concepts: Moment of a force

Related exercises: Worked exercise — The door with two forces · Worked exercise — System with two masses and a pulley · Worked exercise — Square merry-go-round