Simple Harmonic Motion

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Concept 1

Defining SHM

A system performs SHM if its restoring force — and therefore its acceleration — has one specific property.

Hooke's law (springs)

F = −ky

restoring force is proportional to displacement, opposite in direction

SHM condition

a = −ω²y

acceleration proportional to displacement, opposite in direction

Learn this word for word

The definition of simple harmonic motion (asked 2016, 2020, 2022): "The unbalanced force, and therefore the acceleration, are proportional to the displacement from the rest position and act in the opposite direction to it."

Watch out: the 2020 marking instructions do NOT accept "force is proportional to displacement" on its own — direction must be stated too.

Concept 2

The equations of motion

Starting from rest at maximum displacement gives the cosine solution; starting at the centre gives the sine solution. Both are on the relationships sheet.

Starts at the centre

y = A sin ωt

displacement, sine case

Starts at maximum displacement

y = A cos ωt

displacement, cosine case

Angular frequency

ω = 2πf

f = 1/T

Practice 1

A pendulum bob starts at the centre of its swing and completes 4.0 oscillations in 5.0 s. Find ω. 2 marks

Answer

T = 5.0 / 4.0 = 1.25 s ω = 2π/T = 2π/1.25 = 5.0 rad s⁻¹

Concept 3

Speed and energy in SHM

These forms work for both the sine and cosine cases — they only need the amplitude and the displacement at the instant in question.

Speed

v = ±ω√A² − y²

maximum at the centre, zero at the extremes

Kinetic energy

E_k = ½mω²(A² − y²)

maximum at the centre

Substitute & solve

A ball-bearing on a track makes 1.5 oscillations in 2.5 s. Its amplitude is 0.20 m. Find ω and the maximum speed. 4 marks

T=2.5 / 1.5 = 1.67 s

ω=2π/1.67 = 3.8 rad s⁻¹

v=±ωA = ±3.8 × 0.20

v=±0.76 m s⁻¹

Practice 2

A mass on a spring has amplitude 0.15 m and ω = 4.0 rad s⁻¹. Find its speed when y = 0.090 m. 3 marks

Answer

v = ±ω√(A² − y²) v = ±4.0 × √(0.15² − 0.090²) v = ±4.0 × √(0.0225 − 0.0081) = ±4.0 × 0.120 v = ±0.48 m s⁻¹

Concept 4

Damping

In a real system, friction transfers energy out — and since E_tot = ½mω²A², the amplitude falls as the energy falls. How quickly depends on the damping.

Underdamped — the system oscillates, and the amplitude dies away.
Critically damped — it does not oscillate at all, and reaches rest in the shortest possible time.
Overdamped — it does not oscillate either, but takes longer to come to rest than a critically damped system.

Check yourself

Sources & credits: notes written by Mr Stewart. Past-paper questions and marking instructions reproduced for educational use, © Qualifications Scotland (SQA).