Circular motion

Steady speed, changing velocity

An object circling at constant speed is accelerating, continuously, because velocity is a vector and its direction changes at every instant. The acceleration points to the centre of the circle, always at right angles to the velocity; that perpendicularity is how the direction can change while the speed does not.

Describe the drawing

The dot goes round at constant speed, and the cyan velocity arrow never changes length; but it never stops changing direction, so the velocity is changing and there must be an acceleration. The coral arrow shows it: aimed at the centre at every instant. Restyled, the same motion is a satellite held by gravity or a charge held by a magnetic field.

By Newton's second law, a centre-pointing acceleration demands a centre-pointing resultant force, the centripetal force. Nothing flings you outward. Take a corner in a car and your body continues in a straight line (Newton's first law) while the car turns underneath you; the sensation of being thrown outward is the car door arriving to push you inward. Remove the centripetal force and the object does not fly radially away. It departs along the tangent.

Radian measure and angular speed

Circles are easier in angle language. One radian is the angle whose arc length equals the radius, which gives θ = s/r and makes a full circle 2π radians.

The angular speed ω is the angle swept per second, in rad s⁻¹, and it links to everything else through

Direction of angular velocity is not considered at this level, so ω here means a magnitude. Keep the calculator in radian mode for the whole unit; in degree mode every trigonometric evaluation in this unit gives a wrong value.

The vector triangle behind the acceleration

The centripetal acceleration follows from the velocity vector alone, in an argument short enough to reproduce under exam conditions. Follow the object from one point on the circle to another a small angle Δθ later. The speed never changes, so the two velocity vectors have the same length. Each is tangent to the circle, so swinging the radius through Δθ swings the velocity through the same Δθ.

Describe the drawing

Left, a circle with two positions A and B a small angle apart, each carrying a tangential velocity arrow of the same length, and a coral arrow at the middle of the arc pointing along a dashed line to the centre. Right, the two velocities redrawn from a common tail with the coral change-in-velocity arrow closing the triangle between their tips.

Redraw the two velocities from a common tail and the change in velocity is the third side of the triangle, running from the tip of the first to the tip of the second. Two sides of length v with Δθ between them make an isosceles triangle, so for a small angle the closing side is a short arc of a circle of radius v, and its length is Δv = vΔθ.

The direction follows from the symmetry. The two velocities sit symmetrically about the middle of the arc, so their difference lies exactly along the radius there, pointing inwards. Shrink Δθ and that midpoint closes on the object itself, which is the reason the acceleration at any instant is directed at the centre. The coral arrow on the left of the figure is that same Δv carried back to the circle, and the dashed line shows it running through the centre.

The time to sweep Δθ is Δt = Δθ/ω, straight from the definition of angular speed, so

Δθ cancels, so the result does not depend on the size of the step taken.

Acceleration and force

One substitution each way turns a = vω into the pair the questions use. Put ω = v/r into it for the first form and v = ωr for the second:

Multiply by mass for the force:

Centripetal force is never a new force. The term names the role played by a real force: tension, gravity, friction or a normal contact force, and sometimes several acting together. “What provides the centripetal force?” requires one of those to be named; answering “centripetal force” restates the question.