What is the moment of a force about a point and how is its sign decided?

In one line

The moment of a force about a point is its turning effect, equal to the force times the perpendicular distance from the point to its line of action.

Explained simply

Opening a door is easier when you push at the handle, far from the hinge, than near the hinge. The same push gives more turning when the distance is larger. That turning effect is the moment.

Technically, \( M_O = F \times d \), where d is the perpendicular (shortest) distance from O to the line of action of F. Point O is called the moment centre and d the moment arm. In vector form \( \vec{M}_O = \vec{r} \times \vec{F} \).

How it works

1. Draw the line of action of the force.
2. Drop a perpendicular from O to that line; its length is d.
3. Moment \( = F d \).
4. Decide sense: does the force try to rotate the body clockwise or anticlockwise about O? Pick one as positive and use it throughout.
5. If d is awkward, resolve the force and add moments of the components (Varignon's theorem).
6. A force whose line passes through O has zero moment about O.

Key formula

\( M_O = F \cdot d \): F in N, d in m, M in N·m (or kN·m). Vector form \( \vec{M}_O = \vec{r} \times \vec{F} \), where \( \vec{r} \) is the position vector from O to any point on the line of action.

For a force with components \( F_x, F_y \) applied at (x, y) relative to O: \( M_O = xF_y - yF_x \) (anticlockwise positive).

Quick example

A spanner 0.25 m long with a 200 N force at right angles to it: \( M = 200 \times 0.25 = 50 \) N·m. If the force is at 60° to the spanner, only \( 200\sin 60^\circ = 173.2 \) N is perpendicular, so \( M = 173.2 \times 0.25 = 43.3 \) N·m.

Common mistakes

• Using the distance to the point of application instead of the perpendicular distance to the line of action.
• Changing the sign convention partway through a problem.
• Writing moment units as N/m instead of N·m.

How MU asks it

Theory as a 2 to 3-mark part: 'Define moment of a force; state its units'. Mostly it is used inside every resultant and equilibrium problem. Write the definition with a sketch showing d perpendicular to the line of action, and state your sign convention clearly at the start of each numerical.