Moments, couples and centre of massAQA A-Level Physics: Revision notes

The moment of a force about a point is the force multiplied by the perpendicular distance from the point to the line of action of the force:

$\text{moment} = F \times d_\perp$

Its unit is the newton metre (N m). A moment has a sense, clockwise or anticlockwise, and is a measure of the turning effect of the force.

If the force is not at right angles to the line from the pivot, use the perpendicular distance: for a force at angle $\theta$ to a spanner of length $l$, the moment is $F\,l\sin\theta$.

Worked example. 120 N at right angles to a spanner at 0.35 m gives $120 \times 0.35 = 42$ N m.

A couple is a pair of equal and opposite coplanar forces whose lines of action do not coincide. Because the forces are equal and opposite, the resultant force is zero, but they have a resultant turning effect.

The moment of a couple is one force multiplied by the perpendicular distance between the lines of action of the two forces:

$\text{moment of couple} = F \times d$

This moment is the same about every point. Worked example. Two 8.0 N forces on opposite sides of a 0.36 m wheel give $8.0 \times 0.36 = 2.9$ N m.

The principle of moments states that, for a body in equilibrium, the sum of the clockwise moments about any point equals the sum of the anticlockwise moments about that point.

For complete equilibrium of a body acted on by coplanar forces, two conditions are needed:

• the resultant force is zero
• the resultant moment about any point is zero

Choose the pivot or a support as the point: any force through that point has zero moment, which removes an unknown from the equation.

The centre of mass of an object is the point through which its whole weight may be considered to act. For a uniform regular solid (for example a uniform beam, sphere or block) it is at its geometric centre.

When taking moments, the weight of a uniform beam is drawn as a single force at the middle of the beam. If a body is suspended or supported at its centre of mass, the weight has no moment about the support and it can balance.

A uniform plank of length 5.0 m and weight 400 N rests on supports P (at its left end) and Q (4.0 m from P). A 700 N person stands 1.0 m from P.

1. Total upward force $R_P + R_Q = 1100$ N.
2. Moments about P: $R_Q \times 4.0 = 400 \times 2.5 + 700 \times 1.0 = 1700$, so $R_Q = 425$ N.
3. $R_P = 1100 - 425 = 675$ N.

The plank is about to tip when one support force falls to zero. Taking moments about the other support then gives the critical position.