Uniform Circular Motion and Centripetal Acceleration

Constant speed can still require acceleration

A particle travelling steadily around a circle covers equal arc lengths in equal times. Its speed is constant. Its velocity is not, because the tangent direction changes from point to point. Acceleration measures change in velocity, so it can be nonzero even when a speedometer reading stays fixed. Uniform circular motion means a circular path with constant speed, not a constant velocity vector.

Let the circle have radius R and centre O. At every point, velocity is tangent to the circle and perpendicular to the radius. Over a short interval, compare the initial and final velocity arrows after placing their tails together. Their difference points toward the inside of the circle. As the interval approaches zero, the velocity-change direction approaches the radial inward direction at the selected point.

Derive the magnitude using similar triangles

Take nearby positions P and P′ separated by angle Δθ at the centre. Their position vectors from O both have magnitude R. The two velocity vectors both have magnitude v and are rotated by the same angle Δθ, because rotating each radius through 90° gives the corresponding tangent direction. The triangle formed by the two radii and the position chord is therefore similar to the triangle formed by the two velocity vectors and their difference.

As Δt tends to zero, the chord and travelled arc approach the same length to first order. The limiting chord length per time is the speed v. Thus the acceleration magnitude is (v/R) × v = v²/R. The argument also supplies the direction: inward toward O. This is called centripetal, meaning centre-seeking, acceleration.

An equivalent small-angle calculation

For a finite angular separation, the velocity-difference magnitude is 2v sin(Δθ/2), while the position-chord magnitude is 2R sin(Δθ/2). For small angles in radians, sin(Δθ/2) approaches Δθ/2. Dividing the velocity change by elapsed time therefore gives a_c = v × (dθ/dt). Because v = R(dθ/dt), the result is again v²/R. The limiting step matters: a chord is not exactly an arc for a finite interval.

Magnitude can stay fixed while the vector changes

If v and R are constant, v²/R is constant. The inward direction nevertheless rotates as the object moves. Acceleration is therefore not a constant vector. Do not put this changing vector into r = r₀ + ut + ½at² as if its direction remained fixed. Each point has an inward acceleration, but a single fixed arrow cannot point inward from every point on the circle.

Averaging over a complete revolution

After one complete uniform revolution, final velocity equals initial velocity as a vector. Average acceleration is (v_final − v_initial)/T = 0. This does not make instantaneous acceleration zero: it remains nonzero and inward throughout. Similarly, average velocity over a full revolution is zero because displacement is zero, while average speed is the positive constant speed. Averaging vector directions can produce zero even when the magnitude never vanishes.

If speed changes, acceleration has another part

For nonuniform circular motion, the radial inward component still has magnitude v²/R at that instant, but a tangential component changes the speed. The total acceleration is then not purely radial. A particle speeding up around a circle has acceleration partly forward along the tangent and partly inward. A particle slowing down has a backward tangential component. “Circular” alone is insufficient to conclude that total acceleration points exactly toward the centre.