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Oscillations (A Level)

17. Oscillations — Cambridge AS & A Level Physics 9702 Revision Notes & Topical Questions

5 min read 9 parts Lesson 10 of 10 Unit quiz

📚 Cambridge AS & A Level Physics 9702 Hub / Topical Revision
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17. Oscillations — Cambridge AS & A Level Physics 9702 — CAMBRIDGE AS & A LEVEL PHYSICS 9702 — COMPLETE NOTES notes: Describe oscillations and simple harmonic motion, use the SHM equations and graphs, explain energy changes, and distinguish damping types and resonance.Oscillation…

  • Equilibrium position
  • Displacement x
  • Amplitude x0
  • Frequency f
  • Angular frequency ω
  • Phase difference

Section 1 of 9Describing an oscillation

  • Oscillation = the repetitive variation with time of the displacement of an object about its equilibrium position (x = 0).
  • Equilibrium position
    • the position where no resultant force acts on the object.
  • Displacement x
    • the distance from equilibrium; a vector that can be positive or negative, in metres.
  • Amplitude x0
    • the maximum displacement on either side of equilibrium, measured from the equilibrium line to a peak.
  • Period T
    • the time for one complete oscillation, in seconds; SHM has a constant period.
  • Frequency f
    • the number of oscillations per second, in hertz, f = 1/T.
      What the symbols mean: T = period, the time for one complete wave (s) · f = frequency (Hz)
  • Angular frequency ω
    • ω = 2π/T = 2πf, in rad s−1.
      What the symbols mean: ω = angular frequency (rad s−1) · f = frequency (Hz) · T = period (s)

Section 2 of 9Phase difference

  • Phase difference = how far one oscillation is ahead of or behind another of the same frequency; a full cycle is 360° or 2π rad.
  • In phase
    • crests (or troughs) line up, with phase difference 0 or 360° (2π rad).
  • Antiphase
    • a crest lines up with a trough, with phase difference 180° (π rad).
  • Quarter cycle
    • a lead of ¼ of a cycle is 90° (π/2 rad), and the leading oscillation reaches each peak first.

Section 3 of 9Conditions for simple harmonic motion

  • Simple harmonic motion (SHM) = oscillation in which the acceleration is proportional to the displacement from a fixed point and always directed towards it, a ∝ −x.
    • There is a central equilibrium point.
    • A restoring force acts towards it, with a magnitude proportional to the displacement.
    • The restoring force gives the acceleration, so a and x always point in opposite directions.
  • Examples
    • a clock pendulum, a swing, a mass on a spring, guitar strings, a ruler vibrating off a table edge, a marble in a bowl and electrons in alternating current.
  • Not SHM
    • a person on a trampoline, since in the air the restoring force is the constant weight, which is not proportional to displacement.
  • Forces
    • a restoring force returns the object to equilibrium, while a resistive force opposes motion and causes damping.

Section 4 of 9Acceleration and displacement in SHM

  • Acceleration
    • a = −ω2x.
      What the symbols mean: a = acceleration (m s−2) · ω = angular frequency (rad s−1) · x = displacement from equilibrium (m); the minus sign means a always points back to the centre
    • The magnitude of a is greatest at maximum displacement (x = ±x0).
    • The minus sign means a is opposite to x.
    • The a–x graph is a straight line through the origin with gradient −ω2.
  • Displacement
    • if the motion starts at x = 0 when t = 0, then x = x0 sin(ωt).
    • The displacement equals x0 when sin ωt = ±1.
    • The calculator must be in radian mode, and the sign of a negative displacement is kept.
  • Starting at the amplitude
    • the displacement is a cosine, x = x0 cos ωt.

Section 5 of 9Speed in SHM

  • Velocity
    • starting from equilibrium, v = v0 cos(ωt).
  • Maximum speed
    • v0 = ωx0, reached at the equilibrium position where x = 0.
  • Speed against displacement
    • v = ±ω√(x02 − x2); the speed is zero at the amplitude and greatest at the centre.
  • Amplitude
    • a larger amplitude means a larger distance in the same time, so a larger maximum speed.
  • Conversions
    • T, f and ω must be converted using ω = 2π/T = 2πf.

Section 6 of 9SHM graphs against time

GraphShape (starting at equilibrium)Link to the previous graph
DisplacementSine curveReference
VelocityCosine curveGradient of the displacement graph, 90° out of phase
AccelerationNegative sine curveGradient of the velocity graph, 90° out of phase, a mirror image of the displacement graph
  • Velocity
    • it is maximum when the displacement is zero.
  • Acceleration
    • it is maximum when the displacement is maximum.
  • Other starting points
    • the curves shift but remain sines or cosines for undamped SHM.

Section 7 of 9Energy in SHM

  • Energy interchange
    • SHM swaps energy between kinetic and potential forms.
    • A pendulum swaps gravitational potential and kinetic energy.
    • A horizontal mass on a spring swaps elastic potential and kinetic energy.
    • A vertical mass on a spring involves elastic potential, gravitational potential and kinetic energy.
  • At the amplitude
    • potential energy is maximum and kinetic energy is zero.
  • At equilibrium
    • kinetic energy is maximum and potential energy is minimum.
  • Total energy
    • it is constant, E = Ep + Ek = ½mω2x02.
  • Energy–displacement graphs
    • potential energy is a "U"-shaped curve, zero at x = 0 and maximum at ±x0.
    • Kinetic energy is an inverted "U" ("n" shape), maximum at x = 0 and zero at ±x0.
    • The total energy is a horizontal line above both curves.
  • Per period
    • two complete exchanges occur, because the amplitude is reached twice.

Section 8 of 9Damping

  • Damping = the reduction in energy and amplitude of oscillations due to resistive forces such as friction or air resistance.
    • It continues until the oscillator comes to rest at the equilibrium position.
  • Frequency unchanged
    • the frequency and period of a damped oscillator stay the same while the amplitude falls.
TypeBehaviourExample
Light dampingOscillations continue with amplitude decaying exponentiallyPendulum gradually coming to rest
Critical dampingNo oscillation; returns to equilibrium in the shortest possible time, with displacement falling quickly to 0Car suspension, weighing-scale needle
Heavy dampingNo oscillation; returns to equilibrium very slowly without passing 0Door damper stopping slamming

Section 9 of 9Forced oscillations and resonance

  • Natural frequency f0
    • the frequency at which a system oscillates freely.
  • Driving frequency f
    • the frequency of the periodic external force.
  • Resonance = the maximum-amplitude response that occurs when the driving frequency equals the natural frequency.
    • The driver supplies energy most efficiently, so the system gains maximum energy.
    • If f is close to f0, the amplitude rises but less than at resonance.
    • A child on a swing goes higher with small pushes timed to the swing's own frequency.
  • Damping and resonance
    • more damping lowers and broadens the resonance peak.

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