Centripetal Acceleration

An object moving in a circle at constant speed is still accelerating, because velocity is a vector and its direction is changing continuously.

Centripetal acceleration — the acceleration of a body moving in a circular path, directed towards the centre of the circle.

a = v² / r

and using v = ωr,

a = ω² r

Why it points to the centre — over a short interval the change in velocity Δv points inwards, towards the centre. Since acceleration has the direction of Δv, the acceleration is centripetal: literally "centre-seeking".

The acceleration is perpendicular to the velocity — this is what allows the speed to stay constant while the direction changes. An acceleration component along the velocity would change the speed; one perpendicular to it only turns the object.

Example: the word "centrifugal" describes no force at all in this treatment. A passenger thrown against the side of a turning car feels pushed outwards, but what is actually happening is that the car is turning inwards beneath them while their body continues in a straight line, as Newton's first law requires. The inward force from the door is what makes them turn with the car.

Which form to use — a = v²/r when you know the linear speed, a = ω²r when you know the angular velocity. They are the same statement, related by v = ωr.

Example: a car of speed 12 m s⁻¹ on a bend of radius 30 m has a = 12² ÷ 30 = 4.8 m s⁻². For the same bend at twice the speed the acceleration would be four times as large, since a depends on v².

Increasing with speed, decreasing with radius — a tighter bend or a higher speed both demand a larger acceleration, and therefore a larger force. This is why speed limits fall on sharp bends: the required force rises with the square of the speed and the available friction does not.