Circular motion
An object moving in a circle at a steady speed is still accelerating, because its direction keeps changing. That acceleration points to the centre, and something must supply the force that causes it.
Part 1 of 3: Learn it
In short
- Angular speed ω = v ÷ r = 2π ÷ T = 2πf, in rad/s.
- Centripetal acceleration a = v² ÷ r = ω²r, directed towards the centre.
- Centripetal force F = mv² ÷ r = mω²r, supplied by tension, friction, gravity or a normal force.
Where this is in your specification
Spec points: AQA 7408 3.6.1.1, OCR A H556 5.2, Edexcel 9PH0 Topic 6
| Board | Topic: Circular motion, oscillations and thermal physics |
|---|---|
| AQA 7408 | 3.6.1 and 3.6.2 |
| Edexcel 9PH0 | Topics 6, 9 and 13 |
| OCR H556 | 5.1, 5.2 and 5.3 |
| Higher C857 76 | Our dynamic Universe: special relativity, the expanding Universe (closest topics) |
Radians and angular speed
One radian is the angle where the arc length equals the radius, so a full turn is 2π rad. Angular speed ω is the angle turned per second. A point on a turning object at radius r moves with linear speed v = ωr.
Centripetal acceleration and force
The speed stays the same but the velocity changes direction, so there is an acceleration. The resultant force causing it points towards the centre and does no work, because it is always perpendicular to the motion.
Where the force comes from
- A car on a flat bend: friction between the tyres and the road.
- A ball whirled on a string: the tension in the string.
- A satellite in orbit: the gravitational pull of the planet.
- A rider at the bottom of a loop: the normal force minus their weight.
How many radians are in one full revolution?
Show the answer
2π.
Part 2 of 3: See it worked
Worked examples
Example 1
A 900 kg car drives round a flat bend of radius 50 m at 15 m/s. Find the frictional force needed.
- F = mv² ÷ r
- F = 900 × 15² ÷ 50 = 900 × 225 ÷ 50
Answer: 4050 N (about 4.1 kN).
Example 2
A ball on a 0.40 m string makes 2.0 revolutions per second. Find ω, its speed and its centripetal acceleration.
- ω = 2πf = 2π × 2.0 = 12.6 rad/s
- v = ωr = 12.57 × 0.40 = 5.03 m/s
- a = ω²r = 12.57² × 0.40 = 63.2 m/s²
Answer: 12.6 rad/s, 5.0 m/s and 63 m/s².
Common mistakes
- Saying the object is not accelerating because its speed is constant.
- Drawing centripetal force as an extra force alongside tension or friction.
- Saying there is an outward centrifugal force pushing the object.
- Leaving angles in degrees, or frequencies in revolutions per minute.
What supplies the centripetal force for the Moon orbiting the Earth?
Show the answer
The Earth's gravitational pull.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
How many radians are in one full revolution?
2π.
What supplies the centripetal force for the Moon orbiting the Earth?
The Earth's gravitational pull.
If the speed doubles at the same radius, what happens to the centripetal force?
It becomes four times as big, because F depends on v².
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Estos apuntes están en inglés porque siguen los programas de examen del Reino Unido.
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