What does the minus sign in a = −ω²x mean?

That the acceleration always points opposite to the displacement — back towards the equilibrium position. It is the restoring nature of the force, and stating it explicitly is required for the definition marks.

Where are speed and acceleration greatest in SHM?

Speed is greatest at the equilibrium position, where displacement is zero. Acceleration is greatest at the extremes of the motion, where displacement is maximum. Energy questions hinge on knowing both.

What is resonance and when is it a problem?

When a periodic driving force matches a system's natural frequency, energy transfer is most efficient and the amplitude grows to a maximum. Useful in microwave ovens and MRI; destructive in bridges and buildings — which is why engineers add damping.

← Edexcel IAL Physics revision

Edexcel IAL Physics revision · A2 — Mechanics & fields

Oscillations

Simple harmonic motion: the definition is an equation, and the examiner wants it with the minus sign explained. Everything else — energy interchange, damping, resonance — follows from that one line.

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What the syllabus demands

—Define simple harmonic motion: a = −ω²x
—Use x = x₀ sin ωt, v = ±ω√(x₀² − x²), and v_max = ωx₀
—Describe the interchange between kinetic and potential energy in SHM
—Describe light, critical and heavy damping
—Describe forced oscillations and resonance; sketch amplitude-frequency curves

Definitions that earn marks

Clear definitions to practise — check your course mark scheme

Simple harmonic motion: Motion in which the acceleration is proportional to the displacement from a fixed point and always directed towards that point: a = −ω²x.
Damping: The removal of energy from an oscillating system by resistive forces, causing the amplitude to decrease over time.
Resonance: The condition in which a system oscillates with maximum amplitude, occurring when the driving frequency equals the natural frequency of the system.

The equations

Defining equationa = −ω² x

Displacementx = x₀ sin(ωt)

Velocityv = ±ω √(x₀² − x²)

Maximum speedv_max = ω x₀

More equations to practise: the Edexcel IAL formula sheet.

Where the marks die

Common mistakes to check

01
Defining SHM without both conditions: acceleration proportional to displacement AND directed towards the equilibrium position (the meaning of the minus sign). One without the other loses the mark.
02
Maximum speed placed at the extremes. Speed is maximum at the centre (x = 0); acceleration is maximum at the extremes (x = ±x₀). They are out of phase by 90°.
03
Claiming damping changes the frequency dramatically. Light damping reduces amplitude steadily but leaves the period almost unchanged — and shifts the resonance peak only slightly (to a lower frequency, with a lower, broader peak).
04
Resonance answers without 'natural frequency'. The mark scheme wants: driving frequency = natural frequency, maximum energy transfer, maximum amplitude.

One worked example, done properly

Question

A mass oscillates in SHM with amplitude 5.0 cm and period 2.0 s. Find its maximum speed and its speed at x = 3.0 cm.

Method

1.ω = 2π/T = π rad/s.
2.v_max = ωx₀ = π × 0.050 = 0.157 m/s.
3.At x = 0.030 m: v = ω√(x₀² − x²) = π√(0.0025 − 0.0009) = π × 0.040.

v_max ≈ 0.16 m/s; v ≈ 0.13 m/s at x = 3.0 cm

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