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Topic 1 · 1.5

Forces, moments and centre of gravity

Core and Extended · Papers 1–6

🎯What you need to be able to do

  • Know that forces change size and shape; plot and interpret load–extension graphs; use \( k = \dfrac{F}{x} \) and the limit of proportionality EXTENDED.
  • Find the resultant of forces along a line; describe how a resultant force changes velocity; use \( F = ma \) EXTENDED.
  • Describe friction and drag; describe circular motion qualitatively EXTENDED.
  • Define the moment of a force and apply the principle of moments, including several forces on each side EXTENDED.
  • State what centre of gravity is, find it for a plane lamina, and relate it to stability.

📚The physics

Stretching a spring

Forces can change the size and shape of an object. Hang loads on a spring, measure its length each time against a vertical ruler, and plot extension (new length − original length) against load.

Left: a spring hangs from a clamp stand with slotted masses on its end and a pointer against a vertical ruler. Right: a graph of extension in centimetres against load in newtons. Up to 4 N it is a straight line through the origin, 4 cm per newton; beyond the limit of proportionality at 4 N and 16 cm it curves upwards.
A straight line through the origin means extension is proportional to load.

EXTENDED The spring constant is force per unit extension, \( k = \dfrac{F}{x} \) (N/m or N/cm): the gradient of a force–extension graph. The limit of proportionality is the point where the graph stops being a straight line; beyond it, extension is no longer proportional to load.

Resultant force

Forces along the same line add (same direction) or subtract (opposite directions) to give the resultant force. An object remains at rest, or continues in a straight line at constant speed, unless a resultant force acts. A resultant force changes the velocity: its speed, its direction, or both.

A car with a driving force of 900 N forwards, air resistance of 300 N and friction of 150 N backwards. The resultant is 900 minus 300 minus 150, which is 450 N forwards, so the car accelerates.
Opposite directions subtract.

Friction between two solid surfaces may impede motion and produces heating. Drag is friction on an object moving through a liquid or a gas (air resistance).

\[ F = ma \]

EXTENDED The force and the acceleration are in the same direction. A force always at right angles to the motion makes an object move in a circle:

A ball moving on a circular path, with its velocity along the tangent and the force pointing towards the centre. A box lists: with the same mass and radius, a larger force means a higher speed; with the same mass and speed, a larger force means a smaller radius; a larger mass needs a larger force for the same speed and radius.
EXTENDED Qualitative only: the equation \( F = \dfrac{mv^2}{r} \) is not required.

Turning effect of forces

The moment of a force measures its turning effect (a spanner, a door handle, a see-saw):

\[ \text{moment} = \text{force} \times \text{perpendicular distance from the pivot} \]

The principle of moments: for an object in equilibrium, the sum of the clockwise moments about any point equals the sum of the anticlockwise moments about that point. When there is no resultant force and no resultant moment, an object is in equilibrium. EXTENDED You can check this with a metre rule balanced on a pivot, loads hung at measured distances on both sides: the moments on each side agree.

Centre of gravity and stability

The centre of gravity is the point through which all of an object’s weight appears to act.

An irregular card hangs freely from a pin through a hole, with a plumb line hanging from the same pin; a line is drawn along the thread. The card is hung from a second hole and a second line is drawn. The centre of gravity is where the two lines cross.
The card hangs with its centre of gravity directly below the pin.
Two tilted blocks. A wide, low block tilted by 25 degrees: the vertical line through its centre of gravity still falls inside its base, so it falls back. A tall, narrow block tilted by the same angle: the line of its weight falls outside the base, so it topples over.
An object topples when the line of its weight passes outside its base: a low centre of gravity and a wide base make it stable.

✏️Worked example

A light plank is balanced on a pivot at its centre. Loads of 300 N and 200 N act at 1.6 m and 0.8 m to the left of the pivot. (a) EXTENDED Calculate the force F needed 1.6 m to the right of the pivot to balance the plank. [3] (b) State the upward force of the pivot on the plank. Explain. [2]
A plank on a central pivot. A 300 N force acts 1.6 m left of the pivot and a 200 N force 0.8 m left of the pivot; an unknown force F acts 1.6 m right of the pivot.

(a) Anticlockwise moments: \( 300 \times 1.6 + 200 \times 0.8 = 480 + 160 = 640 \) N m. Clockwise: \( F \times 1.6 \). So \( F = \dfrac{640}{1.6} = \) 400 N.

(b) 300 + 200 + 400 = 900 N upwards: the plank is in equilibrium, so there is no resultant force.

Check it. F is at the same distance as the 300 N load but must also balance the 200 N load, so it must be larger than 300 N — 400 N is sensible.
Measuring distances from the end of the plank. Moments are always taken about the pivot: every distance is measured from it.

📝Practise

In the style of the multiple-choice, theory and practical papers. EXTENDED marks Supplement content.

1. (Multiple choice.) Which statement completely describes any object in equilibrium? A: No forces act on it. B: There is no resultant force. C: There is no resultant moment. D: There is no resultant force and no resultant moment.
D.
2. (Theory.) A nut needs a moment of 36 N m to loosen it. A mechanic uses a wrench 0.45 m long. Calculate the smallest force needed. [2] (Modelled on 0625/12 June 2026 Q6.)
F = moment / distance = 36 / 0.45 = 80 N, applied at the end of the wrench, at right angles to it.
3. (Multiple choice.) EXTENDED A uniform rod 30 cm long, weight 0.80 N, is pivoted 10 cm from its left end. A 2.0 N model hangs from the right end. What weight at the left end balances it? A: 2.2 N. B: 4.0 N. C: 4.4 N. D: 4.8 N. (Modelled on 0625/22 June 2026 Q5.)
C. The rod’s weight acts at its centre, 5 cm right of the pivot. Clockwise: 0.80 × 5 + 2.0 × 20 = 44 N cm. W × 10 = 44, so W = 4.4 N.
4. (Theory.) EXTENDED A rocket and its satellite have a mass of 2.0 × 104 kg. The rocket engines give an upward force of 3.2 × 105 N. Calculate the initial acceleration. [3] (Modelled on 0625/42 June 2026 Q2(a).)
Weight = mg = 2.0 × 104 × 9.8 = 1.96 × 105 N. Resultant force = 3.2 × 105 − 1.96 × 105 = 1.24 × 105 N. a = F/m = 1.24 × 105 / 2.0 × 104 = 6.2 m/s2. (Forgetting the weight gives 16 m/s2.)
5. (Theory.) EXTENDED A satellite moves in a circular orbit at constant speed. State the direction of the force on it, and explain why it is accelerating. [2]
Towards the centre of the orbit (the centre of the Earth), at right angles to its motion. Its direction of motion keeps changing, so its velocity changes: it is accelerating even though its speed is constant.
6. (Theory.) EXTENDED A spring stretches by 6.0 cm when a 1.5 N load hangs from it, within the limit of proportionality. Calculate its spring constant, and the extension with a 2.5 N load. [3]
k = F/x = 1.5 / 6.0 = 0.25 N/cm (25 N/m). Extension = 2.5 / 0.25 = 10 cm.
7. (Theory.) EXTENDED A model car of mass 0.40 kg starts from rest. A resultant force of 1.2 N acts on it for 4.0 s. Calculate its final speed. [3]
a = F/m = 1.2 / 0.40 = 3.0 m/s2; v = at = 3.0 × 4.0 = 12 m/s.
8. (Practical.) Describe how to find the centre of gravity of a flat, irregular piece of card. [4]
Make a hole near the edge; hang the card from a pin through it so it swings freely. Hang a plumb line from the same pin and mark its line on the card. Repeat from a second hole far from the first. The centre of gravity is where the lines cross (check with a third hole).

🔗Go deeper — other people’s work

These are external resources, not mine. If one stops working, tell me and everything above it on this page still stands.

  • PhET “Hooke’s Law” and “Balancing Act” — springs and moments
  • PhET “Forces and Motion: Basics” — resultant force and acceleration