4.5.4The moment of a force
Physics only — not assessed in Combined Science: Trilogy.
A force or a system of forces may cause an object to rotate. You should be able to describe examples in which forces cause rotation — a spanner turning a nut, a door swinging on its hinges, a see-saw tipping.
The turning effect of a force is called the moment of the force. The size of the moment is defined by the equation:
moment of a force = force × distance M = F d
moment of a force, M, in newton-metres, Nm; force, F, in newtons, N; distance, d, is the perpendicular distance from the pivot to the line of action of the force, in metres, m.
You should be able to recall and apply this equation.
That word perpendicular is the whole difficulty of this sub-topic. The distance is not from the pivot to where you happen to be holding the spanner — it is the shortest distance from the pivot to the line along which the force acts. If the force is applied at an angle rather than at right angles to the spanner, the perpendicular distance is smaller than the length of the spanner, and the moment is smaller too.
This is also why a longer spanner turns a stiff nut more easily: the same force at a greater perpendicular distance gives a larger moment.
Worked example
A force of 40 N is applied at right angles to a spanner, 0.25 m from the centre of the nut. Calculate the moment.
- Write the equation: M = F d.
- The force is at right angles to the spanner, so the perpendicular distance is simply 0.25 m.
- Substitute: M = 40 × 0.25.
- = 10.
- Give the unit: newton-metres.
Answer: 10 Nm
Common mistake
The newton-metre here is not a joule, even though 1 J = 1 Nm for work done. Work done and the moment of a force are different quantities, and a moment is always quoted in Nm.
- Moment of a force
- the turning effect of a force, calculated as force × distance normal to the direction of the force from the pivot
4.5.4Balanced objects
Physics only — not assessed in Combined Science: Trilogy.
If an object is balanced, the total clockwise moment about a pivot equals the total anticlockwise moment about that pivot.
You should be able to calculate the size of a force, or its distance from a pivot, acting on an object that is balanced.
The method is always the same: work out the moment you can, set the two totals equal, and solve for the unknown.
Worked example
A child of weight 300 N sits 1.8 m from the pivot of a see-saw. Where must a 450 N child sit on the other side to balance it?
- Calculate the moment you know: 300 × 1.8 = 540 Nm anticlockwise.
- For balance, the total clockwise moment must equal the total anticlockwise moment, so the clockwise moment is also 540 Nm.
- Write the clockwise moment: 450 × d = 540.
- Rearrange: d = 540 ÷ 450.
- Check it is sensible: the heavier child should sit closer to the pivot, and 1.2 m is less than 1.8 m.
Answer: 1.2 m from the pivot
Worked example
A uniform beam is balanced on a pivot. A 20 N force acts 0.60 m to the left, and two forces act to the right: 15 N at 0.40 m and an unknown force at 0.50 m. Calculate the unknown force.
- Anticlockwise moment: 20 × 0.60 = 12 Nm.
- Known clockwise moment: 15 × 0.40 = 6 Nm.
- Total clockwise must equal total anticlockwise: 6 + (F × 0.50) = 12.
- So F × 0.50 = 6.
- F = 6 ÷ 0.50.
Answer: 12 N
Common mistake
Add up all the moments on each side before setting them equal. A question with two forces on one side is testing exactly that — comparing only one force from each side gives the wrong answer.
- Principle of moments
- if an object is balanced, the total clockwise moment about a pivot equals the total anticlockwise moment about that pivot
4.5.4Levers and gears
Physics only — not assessed in Combined Science: Trilogy.
A simple lever and a simple gear system can both be used to transmit the rotational effects of forces.
You should be able to explain how levers and gears transmit the rotational effects of forces.
A lever: the effort is applied at a greater distance from the pivot than the load. Because the moment is force × distance, a small effort at a large distance produces the same moment as a large force at a small distance — so the lever allows a smaller force to move a larger load. A crowbar, a wheelbarrow and a pair of scissors all work this way. What is not gained is distance: the effort end moves much further than the load end.
A gear: two toothed wheels of different sizes mesh so that the teeth of one turn the teeth of the other. Because the force at the teeth is the same for both wheels, the moment produced depends on each wheel's radius. A small gear driving a large gear produces a larger moment on the large wheel, but turns it more slowly. A large gear driving a small one does the reverse — a smaller moment, but a faster rotation.
Worked example
Explain why a low gear on a bicycle makes it easier to climb a hill.
- In a low gear, a small gear wheel at the pedals drives a large gear wheel at the back.
- The force at the meshing teeth acts at a larger perpendicular distance from the centre of the larger wheel.
- Since the moment is force × distance, a larger moment is produced at the rear wheel for the same force at the pedals.
- That larger turning effect is what is needed to climb the hill.
- The trade-off is that the rear wheel turns more slowly, so the bicycle moves more slowly for each turn of the pedals.
Answer: The gear system produces a larger moment at the rear wheel for the same effort at the pedals, so the hill can be climbed — at the cost of moving more slowly.
Before you practise
- This sub-topic is physics only — it is not assessed in Combined Science: Trilogy.
- The moment of a force is its turning effect: M = F d, to be recalled.
- d is the perpendicular distance from the pivot to the line of action of the force, in metres.
- Moments are measured in newton-metres — which here are not joules.
- For a balanced object, the total clockwise moment about a pivot equals the total anticlockwise moment.
- Add up all the moments on each side before setting them equal.
- A lever lets a smaller effort at a greater distance balance a larger load close to the pivot.
- Gears transmit rotation: a small gear driving a large one gives a bigger moment but slower turning.