Theme C — Wave Behaviour

C.1.2 Simple Harmonic Motion (SHM) — IB Physics HL Revision Notes & Topical Questions

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C.1.2 Simple Harmonic Motion (SHM) — IB PHYSICS SL notes: State the conditions for SHM and describe the displacement, velocity and acceleration of an object in SHM using equations and graphs.Equilibrium — the object moves back and forth through a central position, with equal maxi…

Restoring force
Independence of amplitude
What the symbols mean
Minus sign
SHM systems
Two models

Part 1 of 9Conditions for SHM

1Equilibrium

Equilibrium the object moves back and forth through a central position, with equal maximum displacement on each side

2Period

Period each complete vibration takes the same time, so the motion is periodic

3Restoring force

Restoring force the force is always directed towards equilibrium and is directly proportional to the distance from it

4Independence of amplitude

Independence of amplitude the period does not depend on the amplitude
- (small angles for a pendulum)
This is why a pendulum clock keeps time as its swing slowly dies away.

Part 2 of 9The defining equation

Simple harmonic motion

Simple harmonic motion oscillation in which the acceleration is proportional to the displacement and directed towards equilibrium
a = −ω²x
What the symbols mean: a = acceleration (m s⁻²) · ω = angular frequency (rad s⁻¹) · x = displacement from equilibrium (m); the minus sign means a always points back to the centre
a is in m s⁻², x in m and ω in rad s⁻¹.

Minus sign

Minus sign acceleration and displacement always point in opposite directions
Keep the sign of displacement in calculations because it is a vector.

Restoring force

F = −mω²x = −kx with k = mω²

Meaning

Meaning a large displacement gives a large restoring force, while near the centre the force is small
At equilibrium the acceleration is zero but the speed is greatest, so the object overshoots.
The opposite force then slows it, giving regular repeating motion.

Part 3 of 9Examples and non-examples

Examples and non-examples

SHM systems

SHM systems a clock pendulum, a child on a swing, a mass on a spring, a vibrating guitar string, a ruler vibrating off a table edge, a tuning fork prong and molecular vibration

Two models

Two models you must analyse a simple pendulum and a mass–spring system, vertical or horizontal

Simplicity

SHM is easy to analyse and many other oscillations can be broken into a combination of SHMs

Not SHM

Not SHM a trampoline jumper, because the restoring force is the constant weight in the air and not proportional to displacement
A ball bouncing on the floor and a large-angle pendulum swing are also periodic but not SHM.

Damped systems

Damped systems they show decaying SHM-like motion

Part 4 of 9Behaviour through one cycle

1At an extreme

At an extreme the displacement is maximum, the speed is zero and the acceleration is maximum towards the centre

2Moving to the centre

Moving to the centre the bob accelerates towards the midpoint

3At equilibrium

At equilibrium the velocity is maximum and the acceleration is zero

4Past the centre

Past the centre the object slows towards the opposite extreme and v = 0 at x₀, where it reverses

Part 5 of 9Maximum speed and acceleration

Maximum speed

Maximum speed v_max = ωx₀, reached when passing through equilibrium

Maximum acceleration

Maximum acceleration a_max = ω²x₀, reached at the extremes

Larger amplitude

Larger amplitude with the same ω it gives larger speed and acceleration but no change in period

Larger ω

Larger ω a shorter period increases both v_max and a_max

Part 6 of 9Graphs against time

1Shape

Shape undamped SHM graphs are sine or cosine curves, and which one depends on the starting position

2Rate of change

Rate of change velocity is the gradient of the x–t graph and acceleration is the gradient of the v–t graph

3Phase

Phase x, v and a graphs are each 90° out of phase with the next

Part 7 of 9Graphs starting at equilibrium and at the amplitude

Start	Displacement	Velocity	Acceleration
At equilibrium (x = 0)	Sine curve	Cosine curve	Negative sine curve
At the amplitude (x = x₀)	Cosine curve	Negative sine curve	Negative cosine curve

Displacement graph

Displacement graph x is maximum at the extremes and zero at equilibrium

Velocity graph

Velocity graph v is zero at the extremes and maximum in size at equilibrium, a quarter of a period from the x–t curve

Acceleration graph

Acceleration graph it is a reflection of the displacement graph in the time axis, a = −ω²x
When x is positive and at maximum, v = 0 and a is at its most negative.

Part 8 of 9Tests for SHM

1Acceleration–displacement graph

Acceleration–displacement graph a straight line through the origin with negative gradient −ω² is the best test

2Velocity–displacement graph

Velocity–displacement graph it is an ellipse, traced once per cycle

3Algebra test

Algebra test write the net force or acceleration as a = −(constant)x and identify the constant as ω²

4Radians

Radians a calculator must be in radian mode for sine and cosine of ωt

Part 9 of 9More from the specification

Defining equation of simple harmonic motion

Simple harmonic motion (SHM) is characterized by an acceleration directly proportional to displacement and directed towards equilibrium: a = -omega² x.

Energy exchanges in SHM

Total mechanical energy remains constant in an undamped harmonic oscillator, continuously exchanging between kinetic energy and potential energy.
- (maximum at equilibrium)

Isochronous property of harmonic oscillators

The oscillation period of an SHM system is independent of amplitude, remaining constant for small angular displacements.

Energy interchange in SHM

Energy interchange in SHM vibrating mass-spring systems continuously exchange kinetic energy with elastic or gravitational potential energy, keeping total energy constant.

Angular frequency in oscillatory systems

Angular frequency in oscillatory systems angular frequency ω in radians per second relates oscillation frequency f to period via ω = 2πf.

Phase difference on kinematic graphs

Phase difference on kinematic graphs velocity-time graphs lead displacement-time curves by ninety degrees, represented by cosine and sine sinusoidal traces.

Kinetic energy

K = ½mv², energy of motion.

Potential energy

Energy stored by the configuration of a system.

Gravitational potential

Work done per unit mass in bringing a test mass from infinity to the point.

Displacement-time graph

A graph of displacement against time whose gradient is velocity.

Velocity-time graph

A graph of velocity against time whose gradient is acceleration and area is displacement.

Key terms

Term	Meaning
Simple harmonic motion	Motion in which acceleration is proportional to displacement and directed to equilibrium, a = −ω²x.
Restoring force	Force that acts toward the equilibrium position.
Sinusoidal	Having the shape of a sine or cosine curve.
v_max	Maximum speed, ωx₀, at equilibrium.
a_max	Maximum acceleration, ω²x₀, at the extremes.

Avoid these mistakes

Saying acceleration is a maximum at equilibrium — it is zero there and speed is maximum.

Forgetting the minus sign, so the acceleration points away from equilibrium.

Believing the period depends on the amplitude in SHM — it does not.

Calling any repeated motion SHM — the condition a ∝ −x must hold.

Saying velocity is maximum at the extremes — it is zero where the object turns.

Last updated: 15 October 2026

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