Moments, equilibrium, and centre of gravity — for high school physics
A turning force (or moment of a force) is the effect of a force that causes an object to rotate about a point or axis. Everyday examples: opening a door, using a spanner, a seesaw, or a wheel. The rotation depends not only on the size of the force but also on the distance from the point about which the object turns (the pivot).
The pivot (or fulcrum) is the fixed point around which rotation can happen. The greater the distance from the pivot to the line of action of the force, the greater the turning effect.
The moment \(M\) of a force \(F\) about a pivot is:
\(F\) = force (in newtons, N)
\(d\) = perpendicular distance from the pivot to the line of action of the force (in metres, m)
So moment is measured in newton-metres (N⋅m). The perpendicular distance is the shortest distance from the pivot to the line along which the force acts. If the force is at an angle, you must use the component of the distance that is perpendicular to the force, or the component of the force that is perpendicular to the distance.
Door: The hinge is the pivot. Pushing near the handle (large distance from the hinge) gives a big moment, so the door opens easily. Pushing close to the hinge (small \(d\)) gives a small moment — the door is hard to open.
Spanner or wrench: The nut (or bolt head) is the pivot. You apply a force at the end of the handle. The longer the spanner, the larger \(d\), so the same force produces a larger moment and you can tighten or loosen the nut more easily.
Crane: The tower is the pivot. The load at the end of the jib (arm) produces a clockwise moment. A counterweight on the other side produces an anticlockwise moment. To avoid tipping, the crane is designed so that these moments balance (or the counterweight is heavy enough to win).
Bottle opener: The cap rim acts as the fulcrum (pivot). You push down on the handle. The long distance from the pivot to where you push gives a large moment, which overcomes the resistance of the cap — the “force wins” because of the large \(d\).
Wheel, steering wheel, or pedal: For a circular object rotating about its centre, the pivot is the axis. The perpendicular distance from the axis to the line of action of the force is the radius \(r\). So moment \(M = F \times r\). The same force applied further from the centre (larger \(r\)) produces a larger turning effect.
The diagram below shows a force \(F\) applied at a distance \(d\) from the pivot. The moment is \(M = F \times d\). Clockwise and anticlockwise moments are given a sign convention: often we take one direction as positive and the other as negative so that we can add them.
When an object is not rotating (or is rotating at constant angular velocity), the moments about any pivot must balance. This is the principle of moments:
Sum of clockwise moments = sum of anticlockwise moments (about the same pivot). So you can set up an equation to find an unknown force or distance. For the object to be in full equilibrium, the resultant force must also be zero (so it does not accelerate).
Two children on a seesaw: their weights produce moments about the pivot. When the seesaw is balanced, the clockwise moment equals the anticlockwise moment: \(W_1 \times d_1 = W_2 \times d_2\).
Below: a crane (load and counterweight balance about the tower), a bottle opener (force at the handle “wins” because of large \(d\)), and a circular object (wheel or steering wheel — moment = \(F \times r\) where \(r\) is the radius).
Crane
Bottle opener (lever)
Circular object (wheel / steering wheel)
The centre of gravity (CG) of an object is the point through which the entire weight of the object can be considered to act. For a uniform ruler, the centre of gravity is at its midpoint. For equilibrium on a pivot, the pivot is often directly under the centre of gravity so that the weight produces no moment about that pivot. If the line of action of the weight passes through the pivot, the object stays balanced.
An object is more stable if its centre of gravity is low and its base is wide. When an object is tilted, the line of action of its weight may pass outside the base; then the weight produces a moment that tips it over. So: low CG and wide base → line of action of weight stays inside the base for larger tilts → more stable.
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