12.1 Kinematics of uniform circular motion

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.