Theme C — Wave Behaviour

C.1.7 Calculating Energy Changes in SHM — IB Physics HL Revision Notes & Topical Questions

5 min read 10 parts Lesson 9 of 37 1 quiz

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Lesson summary

C.1.7 Calculating Energy Changes in SHM — IB PHYSICS HL notes: Derive and use EP = ½mω²x², EK = ½mω²(x0² − x²) and ET = ½mω²x0² for any simple harmonic oscillator.Restoring force — in SHM F = −mω2xEffective constant — k = mω2So the same energy formulae apply to springs, pendulums…

Maximum kinetic energy
Total energy
Same result
Dependence
Derivation
Substitution

Part 1 of 8Effective spring constant

Restoring force

Restoring force in SHM F = −mω²x

Effective constant

Effective constant k = mω²
So the same energy formulae apply to springs, pendulums and any other SHM

Part 2 of 8Total energy

Maximum kinetic energy

Maximum kinetic energy from v = ωx₀ cos ωt, E_K = ½mω²x₀² cos²ωt
Its maximum is ½mω²x₀²

Total energy

E_T = ½mω²x₀²
At that point the potential energy is zero

Same result

Same result using v = −ωx₀ sin ωt gives the same expression

Dependence

E_T is proportional to the square of the amplitude and the square of the frequency

Part 3 of 8Potential energy

Equation

E_P = ½mω²x²

Derivation

E_P = E_T − E_K = ½mω²x₀²(1 − cos²ωt) = ½mω²x₀² sin²ωt

Substitution

Substitution x = x₀ sin ωt gives E_P = ½mω²x²

Maximum

E_P,max = ½mω²x₀² = E_T = E_K,max

Part 4 of 8Kinetic energy–displacement relation

Substitution

Substitution put v = ±ω√(x₀² − x²) into E_K = ½mv²

Equation

E_K = ½mω²(x₀² − x²)

Values

Values maximum ½mω²x₀² at x = 0 and zero at x = ±x₀

Sum check

E_K + E_P = E_T for every x

Part 5 of 8Energy fractions

Kinetic fraction

E_K/E_T = 1 − (x/x₀)²

Potential fraction

E_P/E_T = (x/x₀)²

At half amplitude

25% potential and 75% kinetic

Equal energies

Equal energies occur at x = x₀/√2 ≈ 0.71x₀

Ratios

Ratios mass and ω cancel in fractions

Part 6 of 8Energy against time

Potential energy

E_P = ½mω²x₀² sin²ωt

Kinetic energy

E_K = ½mω²x₀² cos²ωt

Frequency

Frequency each varies at twice the frequency of the motion

Average

Average each has an average of E_T/2 over a cycle

Part 7 of 8Graphs and damping

EK against x2

EK against x2 a straight line with gradient −½mω² and intercept E_T
It can be used to find ω experimentally

Energy graph intercepts

Energy graph intercepts the energy-axis intercept is E_T and the x-intercepts are ±x₀

Damping

Damping the energy changes with the square of the amplitude ratio

Pendulum

Pendulum k = mg/L and ω² = g/L for small angles

Part 8 of 8More from the specification

Simple Harmonic Motion Defining Relation

For an oscillating object, acceleration is directly proportional to its displacement from equilibrium and directed towards that equilibrium position: a = -ω^2 x,
where ω is the angular frequency.

Velocity and Kinetic Energy in SHM

When an object passes through its central equilibrium position (x = 0), velocity and kinetic energy reach maximum values,
while displacement and potential energy become zero.

Time Period and Angular Frequency

The oscillation period T of an SHM system is independent of amplitude (isochronous) and relates to angular frequency by T = 2π / ω = 1 / f.

Phase Difference in Oscillating Systems

In simple harmonic motion, velocity leads displacement by a phase angle of π/2 radians (90°),
while acceleration is in complete antiphase relative to displacement.
- (π radians or 180° phase difference)

Simple Harmonic Motion Defining Relation

For an oscillating object, acceleration is directly proportional to its displacement from equilibrium and directed towards that equilibrium position: a = -ω^2 x,
where ω is the angular frequency.

Velocity and Kinetic Energy in SHM

When an object passes through its central equilibrium position (x = 0), velocity and kinetic energy reach maximum values,
while displacement and potential energy become zero.

Time Period and Angular Frequency

The oscillation period T of an SHM system is independent of amplitude (isochronous) and relates to angular frequency by T = 2π / ω = 1 / f.

Phase Difference in Oscillating Systems

In simple harmonic motion, velocity leads displacement by a phase angle of π/2 radians (90°),
while acceleration is in complete antiphase relative to displacement.
- (π radians or 180° phase difference)

Angular frequency

Ω = 2π/T = √(k/m) for a spring.

Gravitational potential

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

Angular displacement

Angle turned, in radians.

Phase angle

The angle φ in x = x₀ sin(ωt + φ) that fixes the state of the oscillation at t = 0.

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.

Displacement

Change in position.

Position

Location relative to a reference point.

Period

Time for one revolution.

Equilibrium

State with zero net force.

Velocity

Rate of change of displacement.

Acceleration

Rate of change of velocity.

Oscillation

Repeated back-and-forth motion.

Key terms

Term	Meaning
Kinetic energy in SHM	E_K = ½mω²(x₀² − x²).
Potential energy in SHM	E_P = ½mω²x².
Total energy in SHM	E_T = ½mω²x₀².
Effective spring constant	k = mω² for any SHM.
Energy fraction	E_P/E_T = (x/x₀)².

Avoid these mistakes

Using E_P = mgh or ½kx² with the wrong constants — use ½mω²x² for any SHM.

Forgetting to square the amplitude.

Stating E_K = ½mω²x² — that is the potential energy.

What the symbols mean:

Mixing up ω and f in energy formulae — ω = 2πf.

What the symbols mean: ω = angular frequency (rad s⁻¹) · f = frequency (Hz) · T = period (s)

Thinking equal energies occur at x = x₀/2 — they occur at x₀/√2.

Last updated: 18 October 2026