Kinematics of circular motion: (c) angular velocity ω; ω = 2π/T or ω = 2πf

The idea

Angular velocity ω is the rate of change of angle for an object moving in a circle, measured in radians per second (rad s⁻¹). One complete revolution corresponds to an angle of 2π radians, so ω = 2π/T, where T is the period in seconds. Because f = 1/T, this is equivalent to ω = 2πf, where f is the frequency in hertz. For example, a wheel with T = 0.50 s has ω = 2π/0.50 ≈ 12.6 rad s⁻¹; the same wheel has f = 2.0 Hz and ω = 2π × 2.0 ≈ 12.6 rad s⁻¹. Angular velocity describes how fast the angle changes and is the same for all points on a rigid rotating body, even though points further from the axis move faster in a straight-line sense.

What earns the marks

• Angular velocity ω is the angle swept per unit time, measured in radians per second (rad s⁻¹).
• One complete revolution is 2π radians, giving ω = 2π/T.
• Since f = 1/T, the equivalent form is ω = 2πf.
• Angular velocity is the same for all points on a rigid body rotating about a fixed axis.

Common mistakes and how to put them right

• Using degrees instead of radians.

  Put it right: One revolution is 2π rad, so ω = 2π/T.

• Mixing up T and f in the formula.

  Put it right: ω = 2π/T, so a longer period gives a smaller angular velocity.

• Forgetting the unit: the error is quoting ω in Hz or s; the correction is that angular velocity is measured in rad s⁻¹.

• Treating ω as a linear speed.

  Put it right: ω is an angular quantity in rad s⁻¹, while linear speed v = ωr is in m s⁻¹.

Exam tips

• Convert revolutions per minute to hertz first, then use ω = 2πf.
• Keep π in your working until the final step to reduce rounding error, then round to the required significant figures.
• Check that your answer is in rad s⁻¹; if it is in s⁻¹ you have probably calculated frequency instead.