KCSE 2025 Physics P1 Q17 — Circular Motion: Centripetal Force & Acceleration
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The Question
“This is a Section B question on circular motion. (a)(i) An object moves from A to B along a circle of radius r; explain what is meant by the angular displacement θ and the linear (arc) displacement S. (a)(ii) Starting from the definitions of angular velocity and linear velocity, derive the relationship v = rω. (b) A pendulum bob of mass 20 g moves in a circle of radius 1.0 m at a linear speed of 0.5 m/s; calculate the centripetal acceleration and the centripetal force acting on the bob. (c) Explain why a car must reduce its speed when going round a sharp bend.”
(a)(i) Describe the angular and linear displacement
As the object moves from A to B along the circle, the angular displacement θ is the angle A–centre–B swept out at the middle of the circle. The linear displacement S is the actual distance travelled along the curved path — the length of the arc from A to B.
(a)(ii) Write angular and linear velocity
Angular velocity ω is the angular displacement divided by the time taken. Linear velocity v is the arc length travelled divided by the same time. Because the arc length equals the radius multiplied by the angle, r × θ, the linear velocity can be written as rθ over t.
(a)(ii) Combine to get v = rω
In the expression rθ over t, the group θ over t is exactly the angular velocity ω. Replacing it gives the compact relationship between linear and angular speed: v equals r times ω.
(b) Find the centripetal acceleration
The bob has mass 20 g, which is 0.02 kg, and moves at 0.5 m/s on a radius of 1.0 m. The centripetal acceleration always points to the centre and has magnitude v squared over r. Substituting the values gives the acceleration.
(b) Find the centripetal force
By Newton's second law, the force needed to keep the bob on its circular path is its mass multiplied by the centripetal acceleration. Multiplying 0.02 kg by 0.25 m/s² gives the centripetal force.
(c) Explain why a car slows on a sharp bend
The centripetal force needed to round a bend is m v squared over r. On a sharp corner the radius r is small, and at high speed the required force becomes larger than the friction the tyres can get from the road. Without enough friction the car would skid outward, so the driver must reduce speed to lower the force the bend demands.
Final Result
The centripetal acceleration of the bob is 0.25 m/s² and the centripetal force is 0.005 N (5 × 10⁻³ N), directed toward the centre of the circle. The derived link between linear and angular speed is v = rω.
Why this method works
An object moving in a circle is constantly changing direction, so even at steady speed its velocity is changing — that change is an acceleration directed toward the centre, of size v²/r. Newton's second law then says a real force, the centripetal force F = mv²/r, must supply that acceleration; for the bob it comes from the string, and for the car it comes from friction between tyres and road. The bend equation shows why speed matters so much: because v is squared, doubling the speed quadruples the force needed, while a tighter bend (smaller r) raises it further. Friction has a ceiling, so once the demand exceeds it the car slides — which is why slowing down, not steering harder, is the safe response.
The force is tiny because the bob is light: F = mv²/r = 0.02 × 0.5² / 1.0 = 0.02 × 0.25 = 0.005 N, matching F = ma exactly.