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C.1.2—SHM defining equation

Syllabus
First assessment 2025
Objective
—
Level
HL

Use the SHM Defining Equation

Write the definition

Simple harmonic motion is defined by a=−ω2xa=-\omega^2x, where xx is displacement from equilibrium and ω\omega is angular frequency. The acceleration is proportional to displacement and points in the opposite direction.

Interpret the minus sign

If the particle is displaced to positive xx, acceleration is negative; if it is displaced to negative xx, acceleration is positive. At equilibrium, x=0x=0 and a=0a=0, although the particle may have maximum speed there.

Connect frequency to acceleration

A larger ω\omega gives a larger acceleration for the same displacement. The equation also shows why increasing amplitude does not change the period of ideal SHM: acceleration scales with the displacement.

Common trap

Do not write a=+ω2xa=+\omega^2x or measure xx from an arbitrary origin. The displacement must be relative to equilibrium, and the sign must restore the particle toward it.

C.1.2 Exam Analysis

Assessment in practice

1–3 marks
How it is assessed

Questions use the equation to test phase relationships or speed at a stated displacement. The evidence rewards the correct opposite-direction relationship and consistent use of amplitude and angular frequency.

Command terms

Determine / What is

What earns marks

Write a = −ω²x, define x from equilibrium, and explain the negative sign as a restoring direction. When using a consequence, preserve the same phase relationship: acceleration is opposite to displacement and has magnitude ω²|x|.

Watch for

Ignoring the negative sign and treating acceleration as in phase with displacement.

Representative question

Question 1

[Maximum number: 1]

An object is undergoing simple harmonic motion.

For this object, what is the phase difference between the variation of displacement with time and the variation of acceleration with time?

A

0

B

π4rad\frac{\pi}{4} \mathrm{rad}

C

π2rad\frac{\pi}{2} \mathrm{rad}

D

πrad\pi \mathrm{rad}

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.