17.1.1Oscillations (A Level)

Describing Oscillations

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17.1.1 Describing Oscillations — AS PHYSICS 9702 notes: Define displacement, amplitude, period, frequency, angular frequency and phase difference for an oscillation.Oscillation — the repetitive variation with time of the displacement of an object about its equilibrium positionEqu…

Why it repeats
Displacement x
Frequency f
Relationship
What the symbols mean

Section 1 of 9What is an oscillation?

Oscillation

Oscillation the repetitive variation with time of the displacement of an object about its equilibrium position

Equilibrium position

Equilibrium position where the resultant force on the object is zero

Why it repeats

Why it repeats a restoring influence pulls the object back and inertia carries it past equilibrium

Examples

Examples a child on a swing, a mass on a spring, tuning fork prongs and charges in an aerial

Section 2 of 9Displacement, amplitude, period and frequency

Displacement x

Displacement x the distance from equilibrium in a stated direction, a vector that can be positive or negative, in m

Amplitude x₀

Amplitude x₀ the maximum magnitude of the displacement, always positive, in m
A pendulum swinging through a total arc of 16 cm has amplitude 8 cm

Period T

Period T the time for one complete oscillation, in s
Simple harmonic oscillations have a constant period

Frequency f

Frequency f the number of complete oscillations per unit time, in hertz (s⁻¹)

Relationship

Relationship f = 1/T,
so 5 oscillations per second means f = 5 Hz and T = 0.2 s
What the symbols mean: T = period, the time for one complete wave (s) · f = frequency (Hz)

Section 3 of 9Angular frequency

Definition

Definition the rate of change of angular displacement with time, in rad s⁻¹

Equation

ω = 2π/T = 2πf
What the symbols mean: ω = angular frequency (rad s⁻¹) · f = frequency (Hz) · T = period (s)

Meaning

ω is 2π times the frequency, not the number of oscillations per second

Use

Use it appears inside the sine and cosine functions describing the motion
A 50 Hz oscillation has ω ≈ 314 rad s⁻¹

Section 4 of 9Phase and phase difference

1Phase difference φ

Phase difference φ how far one oscillation is ahead of or behind another, of the same frequency

2Equations

φ = (Δt/T) × 2π in radians, or (Δt/T) × 360° in degrees
Δt is the time between corresponding points

3In phase

φ = 0 (or 2π), both reach maxima together

4Antiphase

φ = π rad (180°), one is at a maximum when the other is at a minimum

5Quarter cycle

φ = π/2 rad (90°), one is at an extreme when the other is at equilibrium

6Terms

Terms the leading oscillation is ahead and the lagging one is behind

Section 5 of 9Reading an oscillation from a graph

1Amplitude

Amplitude the height of the highest point on a displacement–time graph

2Period

Period the horizontal distance between two successive peaks, read on the time axis

3Frequency

Frequency f = 1/T, then ω = 2πf

4Crossings of the axis

Crossings of the axis the object passes through equilibrium and moves fastest where the curve is steepest
At the peaks it is momentarily at rest

5Marking diagrams

Marking diagrams arrows for T and amplitude must run from top of one peak to top of the next, and from the axis to a peak

6Measuring phase

Measuring phase measure the horizontal shift between neighbouring peaks, divide by T and multiply by 2π

Section 6 of 9Scales and links

Examples

Examples a guitar string at 330 Hz has T of about 3 ms, an earthquake can have f below 1 Hz and atoms in a solid vibrate at about 10¹³ Hz

Waves

Waves points along a wave oscillate with the same frequency but different phase
Waves in phase reinforce and waves in antiphase cancel

Difference from waves

Difference from waves an oscillator has no wavelength and its motion stays within the oscillation

Section 7 of 9More from the specification

Gradient relationships on kinematic graphs

Gradient relationships on kinematic graphs calculating the gradient of a displacement-time graph determines instantaneous velocity,
while velocity-time slope gives instantaneous acceleration.

Velocity and displacement phase relationship

Velocity and displacement phase relationship maximum oscillatory speed occurs as the oscillator passes equilibrium,
- where displacement equals zero on displacement-time graphs.

Oscillatory velocity-time sinusoidal curves

Oscillatory velocity-time sinusoidal curves velocity-time graphs for undamped simple harmonic motion form cosine waves leading displacement sine curves by a quarter cycle.

Parameters describing simple harmonic motion

Periodic oscillations are quantified by amplitude, period (T, duration of one cycle),
- (x0, maximum displacement)
and frequency.
- (f = 1 / T, in Hz)

Angular frequency and phase relationships

Angular frequency is omega = 2 pi f = 2 pi / T; in SHM, velocity leads displacement by pi/2 radians (90°),
and acceleration leads displacement by pi radians (180°).

Displacement-time graphical analysis

A sinusoidal displacement-time curve yields a cosine velocity-time curve via its instantaneous gradient,
and an inverted sine acceleration-time curve.

Vector quantity

A quantity with both magnitude and direction.

Acceleration

Rate of change of velocity.

Velocity

Rate of change of displacement.

Gradient

Slope of a graph line.

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.

Maximum speed

V_max = Aω.

− 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

Section 8 of 9Key terms

Term	Meaning
Oscillation	repeated variation of displacement about an equilibrium position
Amplitude	the maximum magnitude of displacement
Period	the time for one complete oscillation
Frequency	the number of oscillations per second, in hertz
Angular frequency	ω = 2π/T = 2πf, in rad s⁻¹
Phase difference	the fraction of a cycle (in radians or degrees) by which one oscillation leads or lags another

Section 9 of 9Exam technique

Using the peak-to-peak distance as the amplitude — the amplitude is measured from the equilibrium position.

Treating ω as the number of oscillations per second — ω = 2πf.

Reading the period from the vertical axis — read it from the time axis.

Stating a phase difference for oscillations of different frequency — it is only meaningful at the same frequency.