Damping — Quick Summary

Mr Toogood's Physics · Periodic motion

AQA 3.6.1.3
Eₖ = ½mv²
Kinetic energy
Eₚ = ½kx²
Potential energy
E_T = ½kA²
Total energy
A = A₀e^(−λt)
Amplitude decay

Energy in oscillators

Graph of kinetic energy, potential energy and total energy against displacement for an oscillator

Eₖ and Eₚ trade off as parabolas; their sum (E_T) stays constant.

For an (approximately) undamped oscillator, energy is continuously exchanged between kinetic and potential forms, but the total is conserved:

E_T = Eₖ + Eₚ = ½kA²
  • At equilibrium (x=0): Eₖ is maximum, Eₚ = 0.
  • At maximum displacement (x=±A): Eₖ = 0, Eₚ is maximum.

Combining these gives kinetic energy at any displacement:

Eₖ = E_T − Eₚ = ½k(A²−x²)

Types of damping

Graph comparing light, critical and heavy damping, amplitude against time

Light (blue), critical (red), heavy (green) damping compared.

  • Light damping: each oscillation's amplitude is fractionally smaller than the last. Most real oscillators (springs, pendulums) are lightly damped.
  • Critical damping: returns to equilibrium in the shortest possible time without overshooting — used where oscillation is undesirable (car suspension, compass needles).
  • Heavy damping: returns to equilibrium slowly, without oscillating.

Finding the decay constant

Graph of natural log of amplitude against time, a straight line used to find the decay constant

Plotting lnA against t linearises the exponential decay.

Taking natural logs of A = A₀e^(−λt) gives a straight-line form:

ln A = ln A₀ − λt
Reading the graph: y-intercept = ln A₀; gradient = −λ. A steeper (more negative) gradient means a larger decay constant — faster damping.

Exam essentials

Key equations

  • Eₖ=½mv²   Eₚ=½kx²
  • E_T=½kA²
  • A=A₀e^(−λt)

Damping quick ID

  • Light: gradually shrinking sinusoid.
  • Critical: fastest return, no overshoot.
  • Heavy: slow return, no overshoot.

Common slips

  • Total energy is only constant for the undamped/lightly-damped case over a short time — damping removes energy overall.
  • Plot ln A against t (not ln t) to linearise the decay.
  • Gradient of the ln A–t graph is −λ — watch the sign when reading it off.