3.4.1.2MechanicsCore

Moments and Equilibrium

A force doesn’t just push things along — off-centre it makes them $turn$ . The turning effect is called the $moment$ , and an object is only in full equilibrium when both the forces and the moments on it balance.

40 min Video by Science Shorts 3.4.1 Force, energy and momentum

Moments — AS/A-Level PhysicsWatch the full walkthrough before the notes below.

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What you'll be able to do

Define the moment of a force and give its unit
Apply the principle of moments to a balanced object
Define a couple and calculate its torque
State the two conditions for equilibrium
Locate the centre of mass and use it when taking moments

The moment of a force

The $moment$ of a force about a point is the force multiplied by the $perpendicular distance$ from the point (the pivot) to the line of action of the force. Its unit is the newton metre (N m).

The perpendicular distance is what matters: a force acting straight through the pivot has zero moment, while the same force applied further out, at right angles, has a larger turning effect.

moment = F \times d

Moment (N m) = force (N) × perpendicular distance to the pivot (m).

The principle of moments

For an object in equilibrium, the $principle of moments$ says that about any pivot the total $clockwise$ moment equals the total $anticlockwise$ moment.

This lets you find an unknown force or distance on a balanced beam or seesaw by setting the two turning effects equal.

\sum clockwise moments = \sum anticlockwise moments

The principle of moments for a body in equilibrium.

1Anticlockwise moment from the first child

= 300 \times 1.5 = 450

N m.

2For balance, the second child’s clockwise moment must equal this:

450 \times d = 450

d = 450/450 = 1.0

m from the pivot.

Answer

1.0

m from the pivot

Couples and torque

A $couple$ is a pair of equal, antiparallel forces whose lines of action don’t coincide — think of your two hands turning a steering wheel. A couple produces a pure turning effect with no resultant force.

The moment of a couple is called the $torque$ : one of the forces multiplied by the perpendicular distance between them.

torque = F \times s

Torque of a couple = one force × perpendicular distance between the two forces.

Conditions for equilibrium

An object is in equilibrium only if $two$ conditions are met: the resultant force is zero (so it won’t accelerate) $and$ the resultant moment about any point is zero (so it won’t start to rotate).

When you take moments, the object’s weight acts at its $centre of mass$ . For a uniform beam that’s the midpoint, which is why a beam’s weight is placed at its centre in these calculations.

1The beam’s weight acts at its centre,

2.0

m from the left support.

2Take moments about the left support:

R_{right} \times 4.0 = (200 \times 2.0) + (100 \times 1.0) = 500

N m, so

R_{right} = 125

3Vertical forces balance:

R_{left} + R_{right} = 200 + 100 = 300

N, so

R_{left} = 175

Answer

R_{left} = 175

R_{right} = 125

Tip — Take moments about the point where an unknown force acts — that force then has zero moment and disappears from the equation, leaving one unknown to solve.

Equation recap

moment = F \times d

Moment = force × perpendicular distance to the pivot.

\sum M_{cw} = \sum M_{acw}

Principle of moments in equilibrium.

torque = F \times s

Torque of a couple.

Common mistakes to avoid

Using the distance along the force rather than the perpendicular distance to the pivot.

A moment always uses the perpendicular distance between the pivot and the force’s line of action.

Forgetting the beam’s own weight when taking moments.

Include the weight acting at the centre of mass (the midpoint of a uniform beam).

Checking only that the forces balance and calling it equilibrium.

Equilibrium needs BOTH zero resultant force and zero resultant moment.

Choosing an awkward pivot that keeps several unknowns in the equation.

Take moments about a point where an unknown acts so its moment is zero, simplifying the maths.

Key takeaways

Moment = force × perpendicular distance from the pivot, measured in N m.
In equilibrium, total clockwise moments equal total anticlockwise moments (principle of moments).
A couple is two equal antiparallel forces; its torque = F × distance between them.
Full equilibrium requires zero resultant force AND zero resultant moment, with weight acting at the centre of mass.

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