Moments & the Principle of Moments

Explained

Forces that turn things

A force does not only push an object along a line. Applied away from a pivot, it turns it, and that turning effect is called a moment.

The moment equals the force multiplied by the perpendicular distance from the pivot to the line of action of the force. Its unit is the newton metre, written N m, which is a force unit times a distance unit and not a unit of energy despite the joule also being a newton metre.

Why distance matters as much as force

Because the two are multiplied, doubling either one doubles the turning effect.

That is why a long spanner loosens a bolt a short one will not, why a door handle is fitted at the edge furthest from the hinges, and why pushing a door near its hinge barely moves it. In each case the force is unchanged and only the distance has altered.

It is also why a small child can balance a much heavier adult on a see-saw by sitting further out.

Perpendicular distance

The word perpendicular is in the definition and is worth including when you state it.

The distance is measured at right angles from the pivot to the line along which the force acts, not along the object itself. When the force is applied at right angles to a bar, which is the usual case, those two are the same thing. When it is applied at an angle, the perpendicular distance is shorter, so the moment is smaller, which is why pushing a spanner sideways works less well than pushing it square.

The principle of moments

When an object is balanced and not turning, the sum of the clockwise moments about a pivot equals the sum of the anticlockwise moments.

Balanced here means in rotational equilibrium: it is not turning, which is different from not moving. Both conditions are needed for something to be fully in equilibrium, and this one concerns turning only.

To apply it, work out every moment separately as force times distance, put all the clockwise ones on one side of an equation and all the anticlockwise ones on the other, and solve for whatever is missing.

Where the weight of the beam acts

A beam with mass has its own weight, and that weight acts at its centre of gravity, which for a uniform beam is its midpoint.

If the pivot is at the midpoint, that weight acts through the pivot, so its distance is zero and it produces no moment at all. It can be ignored entirely.

If the pivot is anywhere else, the beam's weight must be included as another force, at its own distance from the pivot. Leaving it out is the usual reason a calculation comes out wrong when every individual moment looks right.

Everyday applications

A lever multiplies force by applying it far from the pivot and delivering it close to one. Scissors, spanners, wheelbarrows, crowbars and bottle openers all work this way.

Nothing is gained for free. The effort moves a long way and the load moves a short way, so the work done is the same at both ends. A lever trades distance for force rather than creating it.