The big idea: Swing a ball on a string. You can feel the ball tugging your hand outwards — so you have to keep pulling inwards to hold it in its circle.
Let go and it does not fly outwards. It shoots off straight ahead, exactly the way it was already moving.
So circular motion always needs a force pointing at the centre — the centripetal force. Even at a steady speed the ball is accelerating, because its direction never stops changing.
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Spot it: Centripetal = 'toward the centre'. The force and the acceleration both point inward, along the radius — never along the direction of motion.
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The data booklet gives the centripetal acceleration and the speed. Combine them with F = ma to get the force that must point to the centre.
- centripetal acceleration — points to the centre (m s⁻²)
- speed around the circle (m s⁻¹)
- radius of the circle (m)
- angular speed — radians turned per second (rad s⁻¹)
- period — time for one full lap (s)
- speed around the circle (m s⁻¹)
- radius of the circle (m)
- period — time for one full lap (s)
- angular speed (rad s⁻¹)
Put them together: Newton's second law (F = ma, also given) with a = v²/r gives the centripetal force:
Fc = mv²/r — the net force pointing to the centre.
- centripetal force — the NET force toward the centre (N)
- mass of the object (kg)
- speed around the circle (m s⁻¹)
- radius of the circle (m)
The acceleration a = v² ÷ r (v = speed, r = radius). Cover the one you want — two side by side → multiply; one above the other → divide.
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A 1200 kg car drives round a flat bend of radius 50 m at 15 m s⁻¹. Find the centripetal force needed.
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Free-body diagram of a car on a flat bend: weight down, normal up (these cancel), and friction toward the centre — the friction IS the centripetal force Fc = mv²/r.
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How this is tested — circular motion is examined two main ways:
Paper 1A
- Pick the correct free-body diagram — e.g. a car on a banked road.
- Which way Fc points.
Paper 2
- A structured vertical-circle calculation.
- Find the string tension at the lowest point.
The classic trap: Fc is the net force toward the centre, not an extra force you add to the diagram. At the bottom of a vertical circle: tension − weight = mv²/r, so the tension is bigger than the weight.
Vertical circle, lowest point: At the bottom, the tension pulls up (toward the centre) and the weight pulls down (away from it).
The net upward force is the centripetal force:
T − mg = mv²/r, so T = mg + mv²/r.
| Force at the lowest point | Direction | Toward centre? |
|---|---|---|
| Tension T (string) | Up — toward the centre | + (helps) |
| Weight mg | Down — away from the centre | − (opposes) |
| Net = T − mg | Up | = mv²/r |
A 0.40 kg ball on a string is swung in a vertical circle of radius 0.80 m. At the lowest point its speed is 6.0 m s⁻¹. Find the tension in the string there. (g = 9.8 m s⁻².)
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