5.3ThermalStretch

Simple Harmonic Motion

A pendulum, a mass bobbing on a spring, a vibrating guitar string — completely different systems obeying exactly the same mathematics. Whenever a restoring force is proportional to displacement, you get simple harmonic motion, and one equation describes the whole thing.

45 min Video by Physeeks Module 5.3 Oscillations

Equations of Simple Harmonic Motion (All) — Oscillations, A Level PhysicsWatch the full walkthrough before the notes below.

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What you'll be able to do

State the defining equation of SHM, $a = - ω^{2} x$
Use $x = A cos (ω t)$ and $v = \pm ω A^{2} - x^{2}$
Use the time period equations for a mass-spring system and a simple pendulum
Describe energy exchange between kinetic and potential energy during SHM

The defining equation

A system undergoes $simple harmonic motion$ if its acceleration is always directed toward a fixed equilibrium point and proportional to its displacement from it.

a = - ω^{2} x

Defining equation of SHM.

Displacement, velocity and time period

Released from maximum displacement, an oscillator’s displacement follows $x = A cos (ω t)$ ; velocity is greatest at equilibrium and zero at the extremes.

x = A cos (ω t) v = \pm ω A^{2} - x^{2} T = 2 π \frac{m}{k} (spring) T = 2 π \frac{l}{g} (pendulum)

SHM displacement, velocity, and time period formulas.

1Use

v_{m a x} = ω A

v_{m a x} = 5.0 \times 0.060

Answer

v_{m a x} = 0.30

m s⁻¹

Tip — Time period never depends on amplitude for genuine SHM — a pendulum swinging through a larger (small) angle keeps the same period.

Energy in SHM

Energy continuously exchanges between kinetic and potential forms; the total stays constant (ignoring damping).

E_{total} = \frac{1}{2} m ω^{2} A^{2}

Total energy of an SHM oscillator.

1Use

E = \frac{1}{2} m ω^{2} A^{2}

E = 0.5 \times 0.25 \times 6. 0^{2} \times 0.08 0^{2} = 0.5 \times 0.25 \times 36 \times 0.0064

Answer

E \approx 0.0288

Equation recap

a = - ω^{2} x

Defining equation of SHM.

v_{m a x} = ω A

Maximum speed.

T = 2 π \frac{m}{k}, T = 2 π \frac{l}{g}

Time period formulas.

Common mistakes to avoid

Forgetting the minus sign in a = −ω²x.

The minus sign shows the acceleration always opposes displacement (a restoring effect) — essential to the definition.

Assuming maximum speed and maximum acceleration occur at the same point.

Maximum speed is at equilibrium (x=0); maximum acceleration is at the extremes (x=±A) — opposite ends of the motion.

Believing amplitude affects the time period of SHM.

Time period depends only on the system’s physical properties (m, k or l, g), never on amplitude.

Key takeaways

SHM: a = −ω²x, acceleration proportional to (and opposing) displacement.
x = Acos(ωt); maximum speed ωA occurs at equilibrium.
T = 2π√(m/k) for a spring; T = 2π√(l/g) for a pendulum — independent of amplitude.
Total energy (½mω²A²) is constant, exchanging between kinetic and potential forms.

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Simple Harmonic Motion

45 min Video by Physeeks Module 5.3 Oscillations