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C.1.4—Period, frequency and angular frequency

Syllabus
First assessment 2025
Objective
—
Level
HL

Convert Period, Frequency and Angular Frequency

Three equivalent measures

T=1f=2πω,ω=2πfT=\frac1f=\frac{2\pi}{\omega},\qquad \omega=2\pi f

Use seconds for TT, hertz for ff, and radians per second for ω\omega.

Worked example from the mapped local textbook

A guitar-string point oscillates at f=196 Hzf=196\,\mathrm{Hz}.

T=1196=5.10×10−3 sT=\frac1{196}=5.10\times10^{-3}\,\mathrm s

ω=2π(196)=1.23×103 rad s−1\omega=2\pi(196)=1.23\times10^3\,\mathrm{rad\,s^{-1}}

Common trap

Do not mix this general conversion objective with the separate spring and pendulum period models. Also, ω\omega is 2π2\pi times ff, not f/(2π)f/(2\pi).

C.1.4 Exam Analysis

Assessment in practice

1–3 marks
How it is assessed

Questions ask you to infer a period from a graph or determine how a pendulum frequency changes when length changes. The evidence rewards the correct square-root dependence for the model and the correct T–f–ω conversion.

Command terms

Determine / What is

What earns marks

Write T = 1/f = 2π/ω before substituting. Keep T in seconds, f in hertz and ω in rad s−1. For a pendulum or spring, first calculate the model’s period, then convert to the requested frequency.

Watch for

Using a direct inverse-length relationship for a pendulum instead of f ∝ 1/√l, or forgetting the factor 2π when converting f to ω.

Representative question

Question 1

[Maximum number: 4]

Determine the time period of the system when a is small.

Retrieve the Core C.1 Simple Harmonic Motion Model

Recognize SHM

SHM requires a restoring acceleration a=−ω2xa=-\omega^2x about equilibrium. Track displacement, amplitude, period, frequency and angular frequency with T=1/f=2π/ωT=1/f=2\pi/\omega.

Track one cycle

At an extreme, potential energy is maximum and kinetic energy is zero; at equilibrium, kinetic energy is maximum and potential energy is minimum. Total energy remains constant in ideal SHM.

Read the motion

The gradient of a displacement–time graph is velocity. Acceleration is opposite to displacement. Use the sign of displacement and the gradient to identify direction at any instant.

Final check

Measure displacement from equilibrium, keep amplitude distinct from peak-to-peak distance, and name the relevant potential-energy form for the oscillator.