12.2 Centripetal acceleration

A force always perpendicular to instantaneous velocity does no work, so speed stays constant while the velocity direction changes.

The force direction must continuously turn with the motion; a single perpendicular impulse is not uniform circular motion by itself.

Tension toward the centre keeps a stone moving around a circle at constant speed while doing no work.

Zero work does not mean zero force or zero acceleration; direction can change with unchanged kinetic energy.

In uniform circular motion, acceleration is centripetal, directed toward the centre, with magnitude a=v²/r=rω².

The velocity is tangential while acceleration is radial; the inward resultant force is F=mv²/r.

Doubling speed at fixed radius quadruples centripetal acceleration, while doubling radius halves it for fixed speed.

“Centripetal force” is not a new force type—it is the name for the inward resultant supplied by tension, gravity, friction or another interaction.

For circular motion, centripetal acceleration is a=rω²=v²/r and points radially inward.

Use angular speed or tangential speed consistently, and remember the acceleration is perpendicular to instantaneous velocity in uniform motion.

At r=0.50 m and ω=4.0 rad s⁻¹, a=rω²=8.0 m s⁻².

The object is accelerating even at constant speed because its velocity direction changes.

The inward resultant force needed for circular motion is F=ma=mrω²=mv²/r.

Identify which real interaction supplies it—tension, gravity, friction or a normal force—and sum forces toward the centre.

A 2.0 kg mass moving at 3.0 m s⁻¹ around radius 1.5 m needs an inward resultant of 12 N.

Centripetal force is not an additional force to draw; it is the name for the inward resultant.