Uniform circular motion

Velocity has both a size and a direction, and acceleration is defined as the rate of change of either one. An object circling at perfectly constant speed has an unchanging speed but a direction that never stops turning, and that alone is enough to make its acceleration non-zero. Treating "constant speed" as "no acceleration" confuses the size of velocity with velocity itself.

At any instant the velocity points tangent to the circle — along the direction of travel right now, perpendicular to the line from the centre. It never points toward or away from the centre; that would mean moving directly inward or outward, which is exactly what circular motion is not doing. The acceleration, by contrast, points toward the centre at every instant — it has to, since its entire job is to bend the velocity vector back around the curve rather than letting it fly off straight. This is centripetal acceleration, and its perpendicularity to velocity is why it changes direction without ever changing speed, the same logic that governs a magnetic force on a moving charge.

Speed, radius and period are linked by the circle's circumference: one full lap covers a distance 2πr in a time T, so v = 2πr/T. A larger v at fixed radius means each lap is covered faster, so the period is shorter, not longer — speed and period move in opposite directions when radius is held fixed, which is worth checking against intuition before trusting a calculation.