What Makes Motion Simple Harmonic? | CAIE 9702 Paper 4 Notes

Paper 4

Physics9702

What Makes Motion Simple Harmonic?

Every simple harmonic motion is an oscillation, but not every oscillation is simple harmonic.

The test uses acceleration and displacement. It does not depend on whether the object is a spring, pendulum or floating block.

The two-mark definition

Definition

2 marks

What is meant by simple harmonic motion?

Model answer: Motion in which acceleration is directly proportional to displacement from a fixed point and is always in the opposite direction to the displacement.

The two ideas receive separate marks: proportionality, then opposite direction (9702/42/M/J/25 Q5(a)) .

a=-\omega^2x

Symbols

$a$ = acceleration (m s⁻²)
$x$ = displacement from equilibrium (m)
$\omega$ = angular frequency (rad s⁻¹)

The minus sign is essential. If displacement is positive, acceleration is negative. The acceleration therefore points back towards equilibrium.
At equilibrium, displacement and acceleration are both zero. At an extreme, the magnitude of acceleration is greatest.

Swipe left or right to see the whole diagram →

For s.h.m., an a-x graph is a straight line through the origin with negative gradient −ω². If it is curved, does not pass through the origin, or has positive gradient, the motion is not simple harmonic.

A straight line through the origin shows direct proportionality. A negative gradient shows opposite directions.
These are the exact two graph marks used in recent papers (9702/42/M/J/24 Q4(a)) (9702/42/O/N/25 Q5(a)(i)).

Use the gradient of an acceleration graph

\text{gradient}=-\omega^2

Symbols

$\omega$ = angular frequency (rad s⁻¹)

Worked example

Find the period from the graph

At a displacement of 1.2 cm, the acceleration is −13 cm s⁻². Find the period.

The magnitude of the gradient is the acceleration divided by displacement.
$\omega^2=\dfrac{13}{1.2}=10.8\ \mathrm{s^{-2}}$
Take the square root, then convert angular frequency to period.
$T=\dfrac{2\pi}{\sqrt{10.8}}=1.9\ \mathrm{s}$

Answer

T=1.9\ \mathrm{s}

This is the latest 2025 Paper 4 method (9702/42/O/N/25 Q5(a)(ii)).

What Makes Motion Simple Harmonic?

Every simple harmonic motion is an oscillation, but not every oscillation is simple harmonic.

The test uses acceleration and displacement. It does not depend on whether the object is a spring, pendulum or floating block.

The two-mark definition

Definition

2 marks

What is meant by simple harmonic motion?

Model answer: Motion in which acceleration is directly proportional to displacement from a fixed point and is always in the opposite direction to the displacement.

The two ideas receive separate marks: proportionality, then opposite direction (9702/42/M/J/25 Q5(a)) .

a=-\omega^2x

Symbols

$a$ = acceleration (m s⁻²)
$x$ = displacement from equilibrium (m)
$\omega$ = angular frequency (rad s⁻¹)

The minus sign is essential. If displacement is positive, acceleration is negative. The acceleration therefore points back towards equilibrium.
At equilibrium, displacement and acceleration are both zero. At an extreme, the magnitude of acceleration is greatest.

Swipe left or right to see the whole diagram →

A straight line through the origin shows direct proportionality. A negative gradient shows opposite directions.
These are the exact two graph marks used in recent papers (9702/42/M/J/24 Q4(a)) (9702/42/O/N/25 Q5(a)(i)).

Use the gradient of an acceleration graph

\text{gradient}=-\omega^2

Symbols

$\omega$ = angular frequency (rad s⁻¹)

Worked example

Find the period from the graph

At a displacement of 1.2 cm, the acceleration is −13 cm s⁻². Find the period.

The magnitude of the gradient is the acceleration divided by displacement.
$\omega^2=\dfrac{13}{1.2}=10.8\ \mathrm{s^{-2}}$
Take the square root, then convert angular frequency to period.
$T=\dfrac{2\pi}{\sqrt{10.8}}=1.9\ \mathrm{s}$

Answer

T=1.9\ \mathrm{s}

This is the latest 2025 Paper 4 method (9702/42/O/N/25 Q5(a)(ii)).