Centripetal acceleration and forceEdexcel International A Level Physics: Revision notes
Section 1
Describing circular motion
An object moving in a circle at constant speed sweeps out an angle at a steady rate. The angular speed ω is the angle turned per unit time, in rad s⁻¹. For one complete revolution the angle is 2π rad, so ω = 2π/T = 2πf, where T is the period and f the frequency.
The linear speed v along the circle is related to ω by v = rω. Even if v is constant, the direction of the velocity changes continuously, so the velocity (a vector) is not constant.
Convert rev min⁻¹ to rad s⁻¹ by multiplying by 2π and dividing by 60.
Section 2
Deriving the centripetal acceleration
An object moves at constant speed v from P to Q in a short time Δt, turning through a small angle Δθ about the centre of a circle of radius r.
- The velocities at P and Q have equal magnitude v but different directions.
- Draw both vectors from one point. They form an isosceles triangle with apex angle Δθ, so for a small angle Δv = vΔθ.
- The arc length PQ is vΔt = rΔθ, so Δθ = vΔt/r.
- Acceleration a = Δv/Δt = vΔθ/Δt, giving a = v²/r.
As Δt tends to zero, Δv becomes perpendicular to the velocity and points towards the centre. Using v = rω gives a = rω².
Section 3
Centripetal force
By Newton's second law a resultant force must act in the direction of the acceleration. A centripetal force is the resultant force, directed towards the centre of the circle, needed to produce and maintain circular motion:
F = ma = mv²/r = mrω²
It is not a new kind of force. It is whatever real force or combination of forces supplies the inward resultant: tension in a string, friction between tyres and road, gravity for a satellite, or the normal contact force from a wall. The force does no work because it is perpendicular to the velocity, so the speed stays constant.
Never draw a 'centripetal force' as an extra arrow on a free-body diagram, and avoid 'centrifugal force'. Label the real force providing the inward resultant.
Section 4
Using the equations
The force needed depends on the square of the speed (or angular speed). Doubling v multiplies the required force by four. If the available force (for example maximum friction) is less than mv²/r, the object cannot follow the circle and moves off along a straighter path.
Worked example: a 1200 kg car takes a bend of radius 45 m. The maximum friction force is 7.8 kN. Then v² = Fr/m = 7800 × 45 ÷ 1200 = 292.5, so the maximum speed is v = 17 m s⁻¹.
Choose the form of the equation that matches the data: mv²/r when you have speed, mrω² when you have angular speed or frequency.
Must know
- a = v²/r = rω², directed towards the centre
- F = mv²/r = mrω², a resultant force towards the centre
- v = rω and ω = 2πf = 2π/T
- The vector-diagram derivation: Δv = vΔθ and Δθ = vΔt/r
- Doubling the speed quadruples the centripetal force needed
That's the notes covered.
Carry on to the next subtopic.
Exam questions on Centripetal acceleration and force
- A ball of mass 0.25 kg is attached to a light string and whirled in a horizontal circle of radius 0.80 m on a frictionless table. The other end of the string is fixed at the centre of the circle and the ball moves at a constant speed of 4.0 m s⁻¹.The ball moves at constant speed. Explain why a resultant force acts on it and state the direction of that force.2 marks
- A laboratory centrifuge spins at a steady 3000 revolutions per minute. A sample tube is held so that its contents are at a distance of 0.12 m from the axis of rotation.The tube wall must provide the force needed to keep 5.0 g of sediment moving in the circle at this radius. Calculate this force.2 marks
- A car of mass 1200 kg drives round a flat, circular bend of radius 45 m. On a dry road the maximum sideways friction force the tyres can provide is 7.8 kN.Calculate the maximum speed at which the car can take the bend without skidding.3 marks
Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).