Centripetal force and centripetal acceleration

Centripetal force is the resultant force that keeps an object moving in a circle. It always points towards the centre of the circle, and its size is F = mv²/r, where m is the mass, v the speed and r the radius. It isn't a separate kind of force: gravity, tension, friction or a contact force provides it. Without it, the object would carry on in a straight line.

What is centripetal force?

An object moving in a circle at a steady speed is still accelerating, because its direction keeps changing, and velocity includes direction. By Newton's second law, an acceleration needs a resultant force. For circular motion that force points towards the centre, which is what centripetal means: centre-seeking.

Because the force is always perpendicular to the motion, the speed stays the same and only the direction of travel turns. No work is done by a centripetal force, because the object never moves along the direction of the force.

If the force suddenly stops, for example when a string snaps, the object flies off in a straight line along the tangent, in the direction it was moving at that instant. It does not fly outwards from the centre.

The centripetal force formula

Angular speed is how fast the angle turns. It links to the speed, the frequency f (turns per second) and the time period T (seconds per turn):

Doubling the speed makes the force four times as large, because v is squared. Doubling the radius at the same speed halves the force.

Centripetal acceleration

Dividing the force by the mass gives the centripetal acceleration, which also points towards the centre:

This is just Newton's second law, F = ma, with the acceleration of circular motion. AQA A-level Physics gives all of these equations in 3.6.1.1, which is A-level only, and they are in the data and formulae booklet. The derivation of a = v²/r is not examined.

Practise F = ma itself in this question from the Laws of Motion course.

What provides the centripetal force?

In every case, ask: what is pulling or pushing towards the centre? That force (or the resultant of several forces) is the centripetal force. Never add "centripetal force" as an extra arrow on a free-body diagram alongside them.

Worked examples

Example 1: a car on a roundabout

A car of mass 1100 kg drives round a roundabout of radius 20 m at a steady 9.0 m/s. Find the centripetal force and the friction needed.

Friction between the tyres and the road must provide about 4500 N towards the centre of the roundabout.

Example 2: using angular speed

A 0.15 kg ball on a string 0.80 m long is whirled in a horizontal circle at 2.0 revolutions per second. Find the tension in the string, ignoring gravity.

Example 3: the fastest safe speed

The greatest friction force the road can give a 1500 kg car is 6000 N. What is the fastest the car can take a bend of radius 25 m?

Any faster and friction can't provide enough centripetal force, so the car skids outwards along a wider path. On a wet road the maximum friction is lower, so the safe speed is lower too.

Example 4: a satellite in orbit

At 400 km above the Earth's surface, r = 6.77 × 10⁶ m from the Earth's centre and the gravitational field strength is 8.69 N/kg. Gravity provides the centripetal force, so for a satellite of mass m, mg = mv²/r, and the mass cancels:

One orbit takes T = 2πr ÷ v = 2π × 6.77 × 10⁶ ÷ 7670 ≈ 5550 s, about 92 minutes. The gravitational field strength guide shows where 8.69 N/kg comes from.

Example 5: the top of a vertical circle

A bucket of water is swung in a vertical circle of radius 1.0 m. When the bucket is upside down at the highest point, its weight and the pull of your arm both act downwards, which is towards the centre. The slowest the bucket can go without the water falling out is when the weight alone provides the centripetal force: mg = mv²/r, so v = √(gr) = √(9.81 × 1.0) = 3.1 m/s.

Try your own numbers in the Circular Motion Calculator.

Centripetal vs centrifugal force

When a car turns sharply you feel pushed outwards. This is often called centrifugal force, but no outward force acts on you. Your body is trying to carry on in a straight line (Newton's first law), and the car turns underneath you until the door or seat pushes you inwards. That inward push is the centripetal force. Centrifugal force only appears if you describe the motion from inside the rotating object, as an apparent force; in exam answers about circular motion, stick to the centripetal force.

Check your understanding.

Common mistakes

• Saying the speed is constant so there is no acceleration. The direction changes, so the velocity changes.
• Drawing the force outwards. The centripetal force points to the centre.
• Adding centripetal force as an extra force. It is provided by real forces such as gravity, tension or friction.
• Forgetting to square v in F = mv²/r.
• Using revolutions per second as ω. Multiply by 2π to get rad/s.
• Using height instead of radius for orbits. Add the Earth's radius to the height above the surface.

Practice questions

1. A 0.50 kg mass moves in a circle of radius 0.60 m at 3.0 m/s. Find the centripetal force.
2. Find the angular speed of the mass in question 1.
3. An object moves in a circle of radius 1.5 m with angular speed 2.0 rad/s. Find its centripetal acceleration.
4. What happens to the centripetal force if the speed is tripled and the radius stays the same?

Answers: 1) F = 0.50 × 3.0² ÷ 0.60 = 7.5 N. 2) ω = v ÷ r = 3.0 ÷ 0.60 = 5.0 rad/s. 3) a = ω²r = 2.0² × 1.5 = 6.0 m/s². 4) It becomes 3² = 9 times larger.

Practise with a short timed paper

Circular motion is in the second year of A-level Physics, and orbits come up qualitatively in GCSE Physics. The daily A-level Physics: Year 2 and GCSE Physics Higher papers are short timed papers of 8 questions, with a new one every day.

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Physics · MechanicsCentripetal force calculator: F = mv²/r and mω²rFree centripetal force calculator: find the force, speed, radius or mass for circular motion, from v or ω, with the period, frequency and acceleration.