Moment of a force, couple and principle of moments
The turning effect of a force is its moment: force × perpendicular distance from the pivot. In balance, clockwise moments equal anticlockwise moments.
Moment of a force
A force can turn a body about a fixed point called the pivot or axis of rotation. This turning effect is the moment of the force, also called torque (τ). Moment = force × perpendicular distance from the axis to the line of the force: τ = F × d. The SI unit is the newton metre (N m), and torque is a vector. Turning the way a clock's hands move is clockwise; the other way is anticlockwise. Anticlockwise moments are taken as positive and clockwise moments as negative. A door handle is fixed far from the hinges, so a small force gives a large moment.
Moment of a force vs couple
Unlike parallel forces are parallel forces that act in opposite directions.
Principle of moments
Suppose several forces act on a body and it is in equilibrium (balanced, not turning). Then the sum of clockwise moments = sum of anticlockwise moments. For two forces: F₁ × d₁ = F₂ × d₂. A see-saw balances when a lighter child sits farther from the pivot than a heavier child. A beam balance in a shop works the same way.
Applications: gears, torque and steering
- Gears: a gear is a wheel with teeth around its rim. Gears pass torque from one wheel to another. They change the speed of rotation by changing the torque, as in a car, a bicycle and a wall clock.
- See-saw: two children of different weights balance by changing their distances from the pivot.
- Steering wheel: a small force applied on the wide rim gives a large torque to turn the vehicle.
- Spanner: a longer handle gives more torque, so a tight nut opens easily.
- Door: pushing near the hinges needs a large force; pushing at the handle needs only a small force.
Balancing a see-saw
A boy of weight 300 N sits 2 m from the pivot of a see-saw. Where must a girl of weight 400 N sit on the other side to balance it?
- Given: F₁ = 300 N, d₁ = 2 m, F₂ = 400 N, d₂ = ?
- Principle of moments: F₁ × d₁ = F₂ × d₂.
- 300 N × 2 m = 400 N × d₂, so 600 N m = 400 N × d₂.
- d₂ = 600 N m ÷ 400 N = 1.5 m.
The girl must sit 1.5 m from the pivot. The heavier person always sits closer to the pivot.
A long-handled spanner at a cycle shop, the steering wheel of a bus and the gears of a bicycle all use torque. The beam balance used to weigh vegetables or gold works on the principle of moments.
2 marks: “Define moment of a force. Give its SI unit.” The turning effect of a force about an axis, equal to force × perpendicular distance from the axis; SI unit N m.
2 marks: “What is a couple? Give two examples.” Two equal and unlike parallel forces acting at two different points of a body; turning a tap, turning a steering wheel.
4 marks: “State the principle of moments. A 20 N weight hangs 30 cm from the pivot of a lever. What force at 60 cm on the other side balances it?” Sum of clockwise moments = sum of anticlockwise moments; 20 × 30 = F × 60, so F = 600 ÷ 60 = 10 N.
Far is easy: the farther from the pivot you push, the bigger the moment for the same force (door handle, long spanner).