This note explains the turning effect of a force (its moment), the principle of moments for objects in balance, and how to find and use the centre of gravity. These ideas explain how levers, seesaws, cranes and stable vehicles work.
Key ideas
Moment of a force
A force can make an object turn about a fixed point, called the pivot. The turning effect is the moment of the force. It depends on two things: the size of the force and the perpendicular distance from the pivot to the line along which the force acts.
That is why a long spanner turns a stiff nut more easily than a short one, and why the door handle is placed far from the hinges. The same force applied further from the pivot gives a bigger moment. The moment is calculated as force × perpendicular distance, with units newton metres (N m).
A moment can be clockwise or anticlockwise. When you add moments, give each direction its own total.
Principle of moments
When an object is in equilibrium (balanced, not accelerating or turning), the total clockwise moment about any pivot equals the total anticlockwise moment about the same pivot.
Equilibrium needs two conditions:
- The resultant force on the object is zero (forces up equal forces down, and left equal right).
- The resultant moment about any point is zero (clockwise moments equal anticlockwise moments).
To solve a balance problem, choose the pivot, list the forces on each side, calculate each moment, and set the clockwise total equal to the anticlockwise total. If you choose the pivot where an unknown force acts, that force has zero moment and drops out of the equation, which often saves work. The support force at a pivot is found afterwards from the condition that all the forces balance.
Centre of gravity
The centre of gravity is the point where the whole weight of an object seems to act. For a uniform, symmetrical object such as a metre rule, it is at the middle. For an irregular shape, it can be found by experiment.
To find the centre of gravity of an irregular flat sheet (a lamina):
- Make small holes near the edge of the sheet.
- Hang the sheet freely from a pin through one hole, and hang a plumb line (a string with a weight) from the same pin.
- When the sheet is still, draw a line on it along the plumb line.
- Repeat with at least two other holes.
- The lines cross at the centre of gravity.
It works because a freely hanging sheet comes to rest with its centre of gravity directly below the pivot.
Stability
An object is stable if it returns to its original position after a small tilt. It is unstable if it falls over. When an object is tilted, it topples once the vertical line down from its centre of gravity falls outside its base. Stability is improved by:
- a wide base, so it can tilt further before the line leaves the base;
- a low centre of gravity, so it must be tilted more before the line leaves the base.
Racing cars have wide, low bodies. A double-decker bus keeps heavy parts, such as the engine, low down. A tall, narrow box with its load packed on top is much easier to tip over.
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| Pivot | The fixed point about which an object can turn. |
| Moment = F × d | Turning effect in N m. d is the perpendicular distance from the pivot to the line of the force. |
| Principle of moments | In equilibrium, total clockwise moment = total anticlockwise moment. |
| Equilibrium | Resultant force = 0 and resultant moment = 0. |
| Centre of gravity | The point where the weight of an object appears to act. |
| Stable / unstable | Returns to rest after a small tilt / topples after a small tilt. |
Worked examples
Example 1 A mechanic pushes at right angles to a spanner with a force of 85 N. The force acts 0.25 m from the centre of the nut. Find the moment about the nut.
Step 1: moment = F × d = 85 × 0.25.
Step 2: 85 × 0.25 = 21.25 N m.
Answer: 21.3 N m.
Example 2 A seesaw is balanced on its central pivot. A child of weight 300 N sits 1.8 m from the pivot on the left. A second child sits 2.4 m from the pivot on the right. Ignore the weight of the plank. Find the weight of the second child and the force from the pivot.
Step 1: anticlockwise moment (left child) = 300 × 1.8 = 540 N m.
Step 2: clockwise moment (right child) = W × 2.4.
Step 3: balance: W × 2.4 = 540, so W = 540 ÷ 2.4 = 225 N.
Step 4: the forces must balance, so the upward force from the pivot = 300 + 225 = 525 N.
Answer: second child 225 N; pivot force 525 N.
Example 3 A uniform metre rule of weight 1.2 N is balanced on a pivot at the 40 cm mark. A load W hangs from the 15 cm mark. Find W and the force from the pivot.
Step 1: the weight of a uniform rule acts at its centre, the 50 cm mark. This is 0.10 m from the pivot, on the right of the pivot (clockwise).
Step 2: clockwise moment = 1.2 × 0.10 = 0.12 N m.
Step 3: the load is 0.25 m from the pivot (40 − 15 = 25 cm), on the left (anticlockwise). Moment = W × 0.25.
Step 4: balance: W × 0.25 = 0.12, so W = 0.12 ÷ 0.25 = 0.48 N.
Step 5: pivot force = 1.2 + 0.48 = 1.68 N upwards.
Answer: W = 0.48 N; pivot force 1.68 N.
Common errors
- Measuring the distance along a tilted lever or to the wrong point instead of using the perpendicular distance from the pivot to the line of the force.
- Writing the unit of a moment as N or N/m. It is N m.
- Forgetting the weight of the rule or plank, which acts at its centre if it is uniform, or leaving out the pivot force when balancing the forces.
- Using centimetres on one side and metres on the other. Use the same unit on both sides.
- Saying that a low centre of gravity alone makes an object stable. A wide base also matters, and the key idea is whether the line from the centre of gravity falls inside the base.
Exam tip: Label clockwise and anticlockwise on your diagram, then write "clockwise moments = anticlockwise moments" before substituting numbers. It keeps the working organised.
Check yourself
- A force of 40 N acts at right angles 0.30 m from a pivot. What is the moment?
Answer
Moment = 40 × 0.30 = 12 N m. - A 20 N weight is 0.50 m to the left of a pivot. At what distance on the right must a 25 N weight be placed for balance?
Answer
20 × 0.50 = 10 N m anticlockwise. So 25 × d = 10, d = 0.40 m. - Why does a plumb line hang through the centre of gravity of a freely hanging sheet?
Answer
A freely hanging sheet rests with its centre of gravity directly below the pivot, and a plumb line hangs vertically below the same pivot, so the line passes through the centre of gravity. - Give two ways to make a free-standing bookcase more stable.
Answer
Make the base wider, or lower the centre of gravity (for example by putting the heaviest books on the bottom shelves).