Oscillations (A Level)

17. Oscillations

5 min read 11 parts Lesson 10 of 10 Unit quiz

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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 11Describing an oscillation

Oscillation

Oscillation the repetitive variation with time of the displacement of an object about its equilibrium position (x = 0).

Equilibrium position

Equilibrium position the position where no resultant force acts on the object.

Displacement x

Displacement x the distance from equilibrium; a vector that can be positive or negative, in metres.

Amplitude x0

Amplitude x0 the maximum displacement on either side of equilibrium, measured from the equilibrium line to a peak.

Period T

Period T the time for one complete oscillation, in seconds; SHM has a constant period.

Frequency f

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 11Phase difference

1Phase 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.

2In phase

In phase crests (or troughs) line up, with phase difference 0 or 360° (2π rad).

3Antiphase

Antiphase a crest lines up with a trough, with phase difference 180° (π rad).

4Quarter cycle

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 11Conditions for simple harmonic motion

1Simple harmonic motion (SHM)

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.

2Examples

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.

3Not SHM

Not SHM a person on a trampoline, since in the air the restoring force is the constant weight, which is not proportional to displacement.

4Forces

Forces a restoring force returns the object to equilibrium, while a resistive force opposes motion and causes damping.

Section 4 of 11Acceleration and displacement in SHM

Acceleration and displacement in SHM

Acceleration

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

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

Starting at the amplitude the displacement is a cosine, x = x₀ cos ωt.

Section 5 of 11Speed in SHM

Speed in SHM

Velocity

Velocity starting from equilibrium, v = v₀ cos(ωt).

Maximum speed

Maximum speed v₀ = ωx₀, reached at the equilibrium position where x = 0.

Speed against displacement

Speed against displacement v = ±ω√(x₀² − x²); the speed is zero at the amplitude and greatest at the centre.

Amplitude

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 11SHM graphs against time

Graph	Shape (starting at equilibrium)	Link to the previous graph
Displacement	Sine curve	Reference
Velocity	Cosine curve	Gradient of the displacement graph, 90° out of phase
Acceleration	Negative sine curve	Gradient of the velocity graph, 90° out of phase, a mirror image of the displacement graph

Velocity

Velocity it is maximum when the displacement is zero.

Acceleration

Acceleration it is maximum when the displacement is maximum.

Other starting points

Other starting points the curves shift but remain sines or cosines for undamped SHM.

Section 7 of 11Energy in SHM

1Energy 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.

2At the amplitude

At the amplitude potential energy is maximum and kinetic energy is zero.

3At equilibrium

At equilibrium kinetic energy is maximum and potential energy is minimum.

4Total energy

Total energy it is constant, E = E_p + E_k = ½mω²x₀².

5Energy–displacement graphs

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.

6Per period

Per period two complete exchanges occur, because the amplitude is reached twice.

Section 8 of 11Damping

Damping

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

Frequency unchanged the frequency and period of a damped oscillator stay the same while the amplitude falls.

Type	Behaviour	Example
Light damping	Oscillations continue with amplitude decaying exponentially	Pendulum gradually coming to rest
Critical damping	No oscillation; returns to equilibrium in the shortest possible time, with displacement falling quickly to 0	Car suspension, weighing-scale needle
Heavy damping	No oscillation; returns to equilibrium very slowly without passing 0	Door damper stopping slamming

Section 9 of 11Forced oscillations and resonance

Natural frequency f0

Natural frequency f0 the frequency at which a system oscillates freely.

Driving frequency f

Driving frequency f the frequency of the periodic external force.

Resonance

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

Damping and resonance more damping lowers and broadens the resonance peak.

Section 10 of 11Key terms

Term	Meaning
Simple harmonic motion	Oscillation in which acceleration is proportional to displacement and directed towards the equilibrium point.
Amplitude	The maximum displacement from the equilibrium position.
Angular frequency	ω = 2π/T = 2πf, in rad s⁻¹.
Phase difference	The fraction of a cycle by which one oscillation leads or lags another.
Damping	The reduction in amplitude and energy of oscillations by resistive forces.
Critical damping	Damping that returns the system to equilibrium in the shortest time without oscillation.
Natural frequency	The frequency at which a system oscillates freely.
Resonance	Maximum amplitude when the driving frequency equals the natural frequency.

Section 11 of 11Exam technique

Confusing the restoring force with a resistive force — restoring force acts towards equilibrium, resistive force opposes motion.

Leaving the calculator in degree mode for sin ωt — ω is in rad s⁻¹.

Drawing the amplitude arrow from trough to peak — measure it from the equilibrium line to a peak.

Thinking the frequency of a damped oscillator falls with the amplitude — it stays constant.

Saying resonance means the largest force — it means the largest amplitude when f equals the natural frequency.