Anything moving in a circle at constant speed is constantly changing direction — which means it’s constantly accelerating. Newton’s second law says there must be a resultant force causing that acceleration. That force is the centripetal force, and it always points towards the centre of the circle.
📘 What you need to know
The centripetal force is the resultant force needed to keep an object moving in a circle — it acts towards the centre, perpendicular to the velocity.
F = mv² / r = mrω² (both forms required).
The centripetal force is not a new type of force — it is always provided by an existing force (tension, friction, gravity, normal force …).
Because it is perpendicular to the velocity, the centripetal force does no work and does not change the object’s speed.
Remove the centripetal force and the object flies off in a straight line tangent to the circle (Newton’s first law).
Why there must be a centripetal force
Velocity is a vector — it has both size and direction. Even at constant speed, the direction of motion changes at every instant as the object goes round the circle. A changing velocity means an acceleration, and Newton’s second law demands a resultant force to cause it. That resultant is the centripetal force.
Crucially, the centripetal force always points towards the centre — perpendicular to the object’s velocity. Because it never has a component along the direction of motion, it does no work and the speed stays constant.
At every point the velocity is tangent to the circle; the centripetal force points inward, always perpendicular to v.
The centripetal force equations
From Newton’s second law (F = ma) and the centripetal acceleration a = v² / r:
Centripetal forceF = mv² / r = mrω²
The two forms are equivalent because v = rω — use whichever the question gives you. The force grows with mass, with the square of the speed, and is larger for a smaller radius (tighter circle).
What provides the centripetal force?
Centripetal force is not a new type of force. It is a label for whichever existing force (or resultant of forces) is directed towards the centre:
Ball on a string → tension in the string.
Car rounding a bend → friction between tyres and road.
Planet orbiting the Sun → gravitational force.
Electron orbiting a nucleus → electrostatic force.
In each case, equate the real force to mv² / r and solve for whatever the question asks.
The centripetal force is always an existing force pointing inward — tension, friction, or gravity depending on the situation.
Quick reference:F = mv²/r = mrω² • points towards the centre • perpendicular to v → does no work → speed stays constant • it is always an existing force (tension, friction, gravity …).
Worked examples
WE 1
Maximum speed of a ball on a string
A 300 g ball is swung in a horizontal circle of radius 0.80 m on the end of a string. The string will break if the tension exceeds 60 N. Find the maximum speed of the ball.
A 1200 kg car travels around a roundabout of radius 40 m at 12 m s⁻¹. Calculate the centripetal force needed and identify what provides it.
Use F = mv² / rF = (1200 × 12²) ÷ 40 = (1200 × 144) ÷ 40F = 4320 Nprovided by friction between the tyres and the road
WE 3
Using the angular speed form
A 0.50 kg mass moves in a horizontal circle of radius 0.30 m at an angular speed of 4.0 rad s⁻¹. Find the centripetal force.
Use F = mrω²F = 0.50 × 0.30 × 4.0²= 0.50 × 0.30 × 16F = 2.4 N
💡 Top tips
Identify what provides the centripetal force first — tension, friction, gravity — then set it equal to mv² / r.
Use mv² / r when given speed; use mrω² when given angular speed.
The centripetal force does no work — it never changes kinetic energy, only direction.
Faster or tighter both demand more centripetal force — v enters squared, so doubling speed quadruples the required force.
If the string is cut, the object goes straight — tangent to the circle at that instant (Newton’s first law).
⚠ Common mistakes
Treating centripetal force as a separate force to add to a free-body diagram — it is always one of the forces already there, acting inward.
Thinking the object “feels” an outward force. In an inertial frame there is no outward centrifugal force — only the inward centripetal force.
Using the diameter instead of the radius in mv² / r.
Forgetting to square the speed — F = mv² / r, not mv / r.
There is no “centrifugal force” in an inertial frame — only the inward centripetal force exists. The sensation of being pushed outward in a car is because your body is trying to go straight while the car turns beneath you. Up next: Centripetal Acceleration, the inward acceleration that goes hand-in-hand with this force.
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