Moments and centre of gravityEdexcel A-Level Physics: Revision notes

The moment of a force about a point measures its turning effect:

moment = F x

where F is the force and x is the perpendicular distance from the pivot to the line of action of the force. The unit is the newton metre (N m). A moment is clockwise or anticlockwise about the pivot. A larger force, or the same force applied further from the pivot, gives a larger moment, which is why a long spanner loosens a stiff nut more easily.

If a force F acts at an angle θ to a rod, only the component perpendicular to the rod turns it:

moment = F sin θ × d

where d is the distance along the rod from the pivot and θ is the angle between the force and the rod. This is the same as F multiplied by the perpendicular distance d sin θ. The component along the rod passes through the pivot and has no moment.

Worked example. A force of 120 N acts at 30° to a spanner of length 0.35 m. Moment = 120 × sin 30° × 0.35 = 21 N m.

The centre of gravity of a body is the point through which the whole weight of the body can be considered to act. For a uniform body it is at the geometric centre, for example the middle of a uniform beam.

For an irregular flat shape, hang it freely from a pin and hang a plumb line from the same pin. At rest, the weight has no moment about the pin only if the centre of gravity is on the vertical line through the pin, so mark the plumb line. Repeat from a second hole. The centre of gravity is where the two lines cross.

For a body to be in equilibrium there must be no resultant force and no resultant moment.

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 the same point.

Unknown forces can be found by taking moments about a point where one or more unknown forces act, so that they have no moment. Then use the fact that the upward forces equal the downward forces.

A uniform seesaw of length 3.0 m is pivoted at its centre. A child of weight 250 N sits 1.2 m from the pivot. Where must a child of weight 400 N sit to balance it?

Clockwise moment = anticlockwise moment: 400 × x = 250 × 1.2, so x = 0.75 m on the other side. The weight of the seesaw acts at the pivot and has no moment.

For a beam on two supports, take moments about one support to find the force at the other, then use upward forces = downward forces. At the point of tipping about a support, the force at the other support is zero.

A body resting on a surface is stable while the vertical line through its centre of gravity falls within its base. The weight then has a moment that keeps it down. If the body is tilted so that this line passes beyond the pivot edge, the weight has an unbalanced moment and the body tips. A low centre of gravity and a wide base increase the angle through which it can be tilted before tipping.