The big idea: Swing a ball on a string round in a circle. You feel the string constantly pulling your hand inward — and if it snaps, the ball flies off in a straight line.
So circular motion always needs a force pulling toward the centre. Even at a steady speed the direction keeps changing, so there's an acceleration (and force) pointing to the centre — centripetal.
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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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