Simple harmonic oscillations: (b) angular frequency ω; ω = 2π/T or ω = 2πf

The idea

Angular frequency ω is the rate of change of phase of an oscillation, measured in radians per second. One complete cycle corresponds to 2π radians, so ω = 2π/T, where T is the period in seconds. Because f = 1/T, it follows that ω = 2πf, where f is the frequency in hertz. For example, an oscillation with period T = 0.50 s has ω = 2π/0.50 = 4π rad s⁻¹ ≈ 12.6 rad s⁻¹; the same oscillation has f = 1/0.50 = 2.0 Hz, and ω = 2π × 2.0 = 4π rad s⁻¹, confirming consistency. Angular frequency links the time description of an oscillation to the sinusoidal description x = A cos(ωt) or x = A sin(ωt).

What earns the marks

• Angular frequency ω is measured in radians per second (rad s⁻¹).
• One complete oscillation corresponds to a phase change of 2π radians.
• ω = 2π/T, where T is the period in seconds.
• ω = 2πf, where f is the frequency in hertz.
• Since f = 1/T, the two forms of the equation are equivalent.
• Angular frequency appears in sinusoidal descriptions such as x = A cos(ωt).

Common mistakes and how to put them right

• Writing ω = 2πT instead of ω = 2π/T.

  Put it right: A longer period must give a smaller angular frequency.

• Forgetting the factor 2π and using ω = f or ω = 1/T.

  Put it right: Angular frequency is in rad s⁻¹, not Hz.

• Mixing units by substituting T in milliseconds without converting to seconds first.

• Assuming ω is measured in hertz.

  Put it right: Hertz measures frequency f, while ω is measured in rad s⁻¹.

Exam tips

• Convert period to seconds before substituting into ω = 2π/T.
• Keep answers in terms of π where possible to avoid premature rounding, then evaluate at the end.
• Check that your value of ω is larger than f by a factor of about 6.28 (2π).