12. Motion in a circle

One radian is the angle at the centre subtended by an arc whose length equals the circle’s radius; a full turn is 2π radians.

Use θ=s/r for angular displacement, with arc length s and radius r in consistent units.

An arc of 0.50 m on a circle of radius 2.0 m subtends θ=0.25 rad.

Radians are not centimetres, and 360° must be converted to 2π rad before using rotational equations.

Angular speed ω is angular displacement per unit time: ω=∆θ/∆t, measured in rad s⁻¹.

For uniform rotation use a constant value; for changing rotation distinguish average from instantaneous angular speed.

A wheel turning through 12 rad in 3.0 s has average angular speed 4.0 rad s⁻¹.

Angular speed is not ordinary linear speed, and radians are treated as dimensionless in units but still encode rotation.

One revolution takes period T, so ω=2π/T. A point at radius r has tangential speed v=rω.

Use the radius of the point being tracked and keep period, frequency and angular speed distinct: ω=2πf.

A point 0.20 m from the axis completing one revolution in 0.50 s has ω=12.6 rad s⁻¹ and v=2.51 m s⁻¹.

All points on a rigid rotating disc share ω but not v; farther points move faster linearly.

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.