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Simple Harmonic Motion
Eduqas A Level Physics — Revision Notes
Component 1: Newtonian Physics · Topic 6 (Vibrations)
Defining Simple Harmonic Motion
Simple harmonic motion (SHM) is a very specific type of oscillation. It is not just “anything that wobbles” — it has a precise mathematical definition that everything else follows from.
Definition of SHM Simple harmonic motion occurs when the acceleration of an object is:
- proportional to its displacement from the equilibrium position, and
- always directed back towards the equilibrium position (opposite to displacement). Mathematically: a = −ω^2 x where x is displacement from equilibrium and ω is the angular frequency. The minus sign is crucial — it enforces the restoring nature of the acceleration.
A graph of a vs x for SHM is a straight line through the origin with negative gradient −ω^2. If your graph has a positive gradient, it is not SHM.
Displacement, Velocity and Acceleration Equations
If a system starts at maximum displacement A at t = 0, then:
SHM Equations
x = A cos(ωt + ϕ) (general form) x = A cos(ωt) (starts at max displacement) x = A sin(ωt) (starts at equilibrium)
Differentiating displacement gives velocity:
v = −Aω sin(ωt + ϕ)
And differentiating again gives acceleration:
a = −Aω^2 cos(ωt + ϕ) = −ω^2 x ✓
Velocity–Displacement Relationship
Using sin^2 θ + cos^2 θ = 1 with x = A cos(ωt) and v = −Aω sin(ωt):
x^2 A^2
v^2 A^2 ω^2
= 1 =⇒ v = ±ω
A^2 − x^2
This is extremely useful: it gives v directly from x without needing t. Maximum speed occurs at x = 0: vmax = Aω. Speed is zero at x = ±A (the turning points).
Graphical Behaviour
Displacement vs time: cosine (or sine) wave, amplitude A.
Velocity vs time: 90◦^ ahead of displacement (leads by a quarter period). Acceleration vs time: 180◦^ out of phase with displacement.
Acceleration vs displacement: a straight line with negative slope.
Period Formulae
The period T is linked to angular frequency by T = 2π/ω and f = ω/ 2 π.
Periods of Common SHM Systems
Mass on a spring (stiffness k, mass m):
T = 2π
r m k
Simple pendulum (length l, gravitational field strength g):
T = 2π
s l g
Note: the pendulum period does not depend on mass or (for small angles) amplitude.
Where These Come From For a mass-spring system, Hooke’s law gives F = −kx. Using F = ma:
ma = −kx =⇒ a = −
k m
x
Comparing with a = −ω^2 x, we get ω^2 = k/m, so T = 2π/ω = 2π
p m/k. The pendulum derivation is analogous, giving ω^2 = g/l for small angles.
Energy in SHM
As the object oscillates, energy continuously swaps between kinetic energy and potential energy (elastic in a spring, gravitational in a pendulum), but the total mechanical energy stays constant in undamped SHM.
Energy Equations
EK = 12 mv^2 = 12 mω^2 (A^2 − x^2 ) EP = 12 mω^2 x^2 Etotal = EK + EP = 12 mω^2 A^2 = constant Maximum KE (at centre) = Maximum PE (at extremes) = 12 mω^2 A^2
Summary of Key Equations
SHM Formula Sheet
Defining equation: a = −ω^2 x Displacement: x = A cos(ωt + ϕ)
Velocity: v = ±ω
A^2 − x^2 Max velocity: vmax = Aω
Mass-spring period: T = 2π
p m/k
Pendulum period: T = 2π
p l/g
Total energy: E = 12 mω^2 A^2
