P20: Moments and Centre of Mass
Moments, levers and centre of mass
Moments And Centre Of Mass
📋 Key Definitions
Moment: The turning effect of a force. The moment depends on the size of the force and the perpendicular distance from the pivot to the line of action of the force.
Principle of moments: For an object in equilibrium (balanced), the total clockwise moment about any pivot equals the total anticlockwise moment.
Centre of mass: The point at which the mass of an object appears to be concentrated. The weight of the object acts through this point.
Lever: A simple machine that uses a pivot (fulcrum) to multiply a force. A small force applied at a large distance from the pivot can balance a large force at a small distance.
📐 Calculating Moments
M = Fd
M = moment (Nm), F = force (N), d = perpendicular distance from pivot to line of action of force (m)
To increase the moment: increase the force or increase the distance from the pivot. This is why longer spanners make it easier to turn tight bolts.
Worked Example 1: Calculating a moment
A force of 20 N acts at a distance of 0.5 m from a pivot. Calculate the moment.
M = Fd = 20 × 0.5 = 10 Nm
Worked Example 2: Finding the force
A moment of 15 Nm is produced by a force acting at 0.3 m from the pivot. Calculate the force.
F = M / d = 15 / 0.3 = 50 N
Worked Example 3: Finding the distance
A force of 40 N produces a moment of 12 Nm. Calculate the distance from the pivot.
d = M / F = 12 / 40 = 0.3 m
⚖ Principle of Moments
When an object is balanced (in equilibrium), the total clockwise moment equals the total anticlockwise moment.
Total clockwise moment = Total anticlockwise moment
(for a balanced object)
Worked Example 4: Balancing moments
A uniform seesaw is 4 m long with the pivot at the centre. A child of weight 300 N sits 1.5 m from the pivot on the left. How far from the pivot on the right must a child of weight 450 N sit to balance the seesaw?
Anticlockwise moment = 300 × 1.5 = 450 Nm
Clockwise moment must = 450 Nm
450 × d = 450
d = 450 / 450 = 1.0 m from the pivot on the right
Worked Example 5: Two forces on one side
A beam is balanced on a pivot. On the left, a force of 100 N acts at 0.3 m and a force of 50 N acts at 0.6 m. On the right, a single force acts at 0.4 m. Calculate the force on the right.
Total anticlockwise moment = (100 × 0.3) + (50 × 0.6) = 30 + 30 = 60 Nm
Clockwise moment = F × 0.4
F × 0.4 = 60
F = 60 / 0.4 = 150 N
🔧 Levers and Gears
Levers
A lever is a force multiplier. By applying a small force at a large distance from the pivot, you can produce a large force at a small distance from the pivot.
| Example | Small force at large distance | Large force at small distance |
|---|---|---|
| Spanner | Hand pushes end of spanner | Bolt turns with large force |
| Scissors | Hand squeezes handles (long) | Blades cut with large force (short) |
| Wheelbarrow | Hands lift handles (long) | Load is lifted with large force (short) |
Gears
Gears are toothed wheels that transmit rotational forces. When gears mesh:
- A larger gear turns slower but with a bigger moment
- A smaller gear turns faster but with a smaller moment
Gears as force multipliers: A small gear driving a large gear multiplies the moment (but the large gear turns more slowly). This is how a car gearbox provides high torque at low speed for starting, or low torque at high speed for cruising.
🎯 Centre of Mass and Stability
Finding the centre of mass
- Symmetrical objects: The centre of mass is at the point of symmetry (e.g. centre of a uniform sphere, midpoint of a uniform rod)
- Irregular objects (suspension method): Suspend the object from a point and let it hang freely. Draw a vertical line from the suspension point downwards. Repeat from a different suspension point. The centre of mass is where the two lines cross.
Stability
| Property | Stable object | Unstable object |
|---|---|---|
| Centre of mass | Low | High |
| Base area | Wide | Narrow |
| When tilted slightly | Returns to original position (line of action of weight falls inside base) | Falls over (line of action of weight falls outside base) |
Why objects topple: An object will topple if the line of action of its weight (drawn straight down from the centre of mass) falls outside the base area. A low centre of mass and a wide base make this less likely, making the object more stable.
❓ Practice Questions
Q1: Foundation A force of 30 N acts at 0.4 m from a pivot. Calculate the moment.
Q2: Foundation State the principle of moments.
Q3: Higher A beam is balanced on a pivot. A 200 N force acts 0.5 m to the left. What force must act 0.8 m to the right to balance it?
Q4: Higher Explain why a racing car has a low, wide shape in terms of centre of mass and stability.
Q5: Foundation Explain how a lever acts as a force multiplier, using a spanner as an example.
✅ Answers
- M = Fd = 30 × 0.4 = 12 Nm
- For an object in equilibrium (balanced), the total clockwise moment about any pivot equals the total anticlockwise moment.
- Anticlockwise moment = 200 × 0.5 = 100 Nm. Clockwise moment must = 100 Nm. F × 0.8 = 100, so F = 100 / 0.8 = 125 N
- A low centre of mass means the line of action of the weight stays well within the wide base area even during sharp turns. A wide base gives a large area for the line of action to fall within. This makes the car very stable and less likely to topple when cornering at high speed.
- A spanner has a long handle. You apply a small force at the end of the handle (large distance from the pivot/bolt). This creates a large moment (M = Fd, and d is large). The bolt experiences a large turning force, even though your input force is relatively small. The lever multiplies your force.
🎯 Exam Tips
- When calculating moments, always use the perpendicular distance from the pivot to the line of action of the force
- In balancing problems, set the total clockwise moment = total anticlockwise moment and solve
- For levers, the key idea is: small force × large distance = large force × small distance
- Gears: bigger gear = bigger moment but slower rotation; smaller gear = smaller moment but faster rotation
- Centre of mass of a uniform object is at its geometric centre — this is commonly assumed in exam questions
- Stability questions: always mention line of action of weight and whether it falls inside or outside the base
🔢 Maths Skills
Mathematical Skills
Use M = Fd to calculate moments (force × perpendicular distance from pivot). For a balanced object: total clockwise moment = total anticlockwise moment. You may need to add moments on the same side and then set equal to the other side. Always use the perpendicular distance — if a force acts at an angle, use F × d × cos θ for the effective moment.
Maths Example
A uniform beam is 3 m long with the pivot 0.5 m from the left end. A 200 N weight hangs at the left end and a 50 N weight hangs at the right end. Is the beam balanced? Anticlockwise moment = 200 × 0.5 = 100 Nm. Clockwise moment = 50 × 2.5 = 125 Nm. No — clockwise moment is greater, so the beam tips to the right. To balance it: add a force F at 0.5 m left of pivot: 100 + F × 0.5 = 125, so F = 50 N (downward on the left side, or upward on the right).
⚠️ Common Misconceptions
Watch Out!
1. Wrong: The moment is calculated using the distance along the object, even if the force is at an angle. Correct: You must use the PERPENDICULAR distance from the pivot to the LINE OF ACTION of the force. If the force is at an angle, only the perpendicular component creates a moment.
2. Wrong: A larger force always produces a larger moment. Correct: Moment = force × distance. A small force at a large distance can produce a larger moment than a large force at a small distance (e.g. a long spanner).
3. Wrong: An object is stable if it is heavy. Correct: Stability depends on having a LOW centre of mass and a WIDE base. A heavy object with a high centre of mass and narrow base (like a tall thin vase) is unstable.
✍️ 6-Mark Question
Extended Answer
6 marks: A uniform seesaw is 4 m long with the pivot at the centre. A child of weight 400 N sits 1.2 m from the pivot on the left. An adult of weight 800 N sits on the right side. Calculate where the adult must sit to balance the seesaw. Explain the principle you have used.
The principle of moments states that for an object in equilibrium, the total clockwise moment equals the total anticlockwise moment. The child's anticlockwise moment = 400 × 1.2 = 480 Nm. For balance, the adult's clockwise moment must also be 480 Nm. Clockwise moment = 800 × d = 480 Nm. Therefore d = 480 / 800 = 0.6 m from the pivot on the right side. This makes sense because the adult weighs twice as much as the child, so they must sit half as far from the pivot. The seesaw is balanced because the total clockwise moment (480 Nm) equals the total anticlockwise moment (480 Nm).
Mark scheme: 1 mark for stating the principle of moments; 1 mark for calculating anticlockwise moment (400 × 1.2 = 480 Nm); 1 mark for setting clockwise moment = anticlockwise moment; 1 mark for correct calculation (d = 0.6 m); 1 mark for interpreting the answer (adult closer to pivot because heavier); 1 mark for showing both moments are equal. (6 marks total)
📊 AO3: Analyse & Evaluate
Analysis and Evaluation
A delivery worker uses a wheelbarrow to carry a load. The wheelbarrow has a load of weight 600 N at a distance of 0.4 m from the wheel (pivot). The worker lifts the handles at a distance of 1.2 m from the wheel.
(a) Calculate the minimum force the worker must apply to lift the wheelbarrow.
(b) The load is moved 0.1 m further from the wheel. Calculate the new lifting force and explain why the force increases.
(c) Evaluate the wheelbarrow as a lever. Explain why it is designed with the load close to the wheel and long handles.
Answers: (a) Clockwise moment (load) = 600 × 0.4 = 240 Nm. Anticlockwise moment (lift) = F × 1.2 = 240, so F = 200 N. The worker lifts with only 200 N to support a 600 N load. (b) New clockwise moment = 600 × 0.5 = 300 Nm. F × 1.2 = 300, so F = 250 N. The force increases because moving the load further from the pivot increases its clockwise moment, so a greater anticlockwise moment (and force) is needed to balance it. (c) The wheelbarrow is a force multiplier lever. The load (large force) is placed close to the pivot (small distance), creating a moderate moment. The handles are long (large distance), so the worker only needs a small force to create the balancing moment. This design principle (small force × large distance = large force × small distance) makes it possible for one person to move heavy loads. Moving the load even closer to the wheel would further reduce the effort needed.
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