EduNinja

Write the definition

Simple harmonic motion is defined by $a=-\omega^2x$ , where $x$ is displacement from equilibrium and $\omega$ 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 $x$ , acceleration is negative; if it is displaced to negative $x$ , acceleration is positive. At equilibrium, $x=0$ and $a=0$ , although the particle may have maximum speed there.

Connect frequency to acceleration

A larger $\omega$ 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 $a=+\omega^2x$ or measure $x$ from an arbitrary origin. The displacement must be relative to equilibrium, and the sign must restore the particle toward it.

[ 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?

Recognize SHM

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

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.

C.1.2—SHM defining equation

Use the SHM Defining Equation

Retrieve the Core C.1 Simple Harmonic Motion Model

C.1.2 Exam Analysis