17.1.5Oscillations (A Level)

SHM Graphs

5 min read 10 parts Lesson 6 of 10 1 quiz

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Lesson summary

17.1.5 SHM Graphs — AS PHYSICS 9702 notes: Sketch and interpret displacement, velocity and acceleration graphs of SHM against time and displacement.Displacement–time — x = x₀ sin ωt, a sine curve for motion starting at equilibriumVelocity–time — v = ωx₀ cos ωt, a cosine curve sta…

Acceleration–time
Intercepts
v² against x²
Energy graphs
Label amplitudes
Oscillation

Section 1 of 10Graphs against time

Displacement–time

Displacement–time x = x₀ sin ωt, a sine curve for motion starting at equilibrium

Velocity–time

Velocity–time v = ωx₀ cos ωt, a cosine curve starting at its maximum +ωx₀

Acceleration–time

Acceleration–time a = −ω²x₀ sin ωt, an upside-down sine curve of amplitude ω²x₀

Gradients

Gradients velocity is the gradient of the x–t graph and acceleration the gradient of the v–t graph
At a peak of x–t the gradient is zero, so v = 0
Where x–t crosses the axis the gradient is steepest, so |v| is a maximum

Section 2 of 10Phase relationships

1Velocity

Velocity leads displacement by π/2 rad
- (a quarter of a cycle)

2Acceleration

Acceleration in antiphase with displacement
- (π rad out of step)

3Graphs

Graphs all three are 90° out of phase with the next

Position	x	v	a
Equilibrium	0	± maximum	0
Positive extreme	+x₀	0	− maximum
Negative extreme	−x₀	0	+ maximum

Section 3 of 10Acceleration–displacement graph

Shape

Shape a straight line through the origin with negative gradient −ω²

Range

Range from x = −x₀ to +x₀, with end points showing the amplitude and maximum acceleration

Use

Use the cleanest test of SHM and the standard way to find ω from the gradient, then T = 2π/ω

Section 4 of 10Velocity–displacement graph

Equation

Equation v²/ω² + x² = x₀², an ellipse centred on the origin

Intercepts

±x₀ on the x axis and ±ωx₀ on the v axis

Halves

Halves the upper half is motion in the positive direction and the lower half is motion in the negative direction

v² against x²

v² against x² a straight line with gradient −ω² and vertical intercept ω²x₀²

Section 5 of 10Reading information from graphs

x–t graph

x–t graph amplitude from the peaks, period from the gap between peaks, then ω, and velocity from a tangent

v–t graph

v–t graph the peak height gives ωx₀
The area under a half-cycle gives the distance 2x₀ from one extreme to the other

Energy graphs

Energy graphs kinetic and potential energy curves are mirror images and oscillate at twice the motion's frequency

Section 6 of 10Practical: pendulum with a data logger

Setup

Setup a motion sensor records position against time and produces x–t, v–t and a–t graphs

Checks

Checks v–t crosses zero when x–t peaks
a–t is a mirror image of x–t reflected in the time axis and scaled by ω²

Damping

Damping amplitudes fall slightly each cycle while the period stays constant, showing light damping hardly changes the period

Section 7 of 10Sketching rules

Label amplitudes

Label amplitudes x₀, ωx₀ and ω²x₀ on the axes and the period on the time axis

Draw

Draw at least one full cycle with smooth curves

Alignment

Alignment zero crossings and peaks line up, and the a–t curve is below the axis when the x–t curve is above it

Section 8 of 10More from the specification

Phase relations in harmonic waves

Phase relations in harmonic waves phase shifts of ninety degrees between velocity and displacement mirror spatial cycles observed in standing waves.

Simple Harmonic Motion Fundamentals

An oscillating object executes simple harmonic motion (SHM) when its restoring force and acceleration are proportional to displacement from equilibrium and directed towards it (a = -ω²x),
where ω is angular frequency.

Displacement-Time Graph of SHM

On a displacement-time graph of SHM (x = x₀ cos(ωt)), the gradient at any point gives the instantaneous velocity of the oscillating object, with maximum speed v_max = ω·x₀ occurring at zero displacement.

Calculating Velocity and Energy Across Oscillations

The speed of an object in SHM at any displacement x is determined by calculating v = ±ω·√(x₀² - x²);
- mechanical energy continuously exchanges between potential energy and kinetic energy without net loss.

Radians

Angle unit required in W = τθ.

Angular frequency

Ω = 2π/T = √(k/m) for a spring.

Displacement-time graph

A graph of displacement against time whose gradient is velocity.

Restoring force

Force directed toward equilibrium.

Velocity-time graph

A graph of velocity against time whose gradient is acceleration and area is displacement.

Phase difference

How far one oscillation is ahead of or behind another.

Maximum speed

V_max = Aω.

Vector quantity

A quantity with both magnitude and direction.

Oscillation

Repeated back-and-forth motion.

Harmonic

Resonant mode of a stationary wave, with frequency a whole-number multiple of the fundamental where allowed.

Wavelength

Distance between a point on one wave and the same point on the next wave

Pendulum

A weight swinging on a string, exchanging GPE and KE.

Section 9 of 10Key terms

Term	Meaning
Phase lead	velocity leads displacement by π/2
Antiphase	a phase difference of π
Gradient of x–t graph	the velocity
Gradient of v–t graph	the acceleration
Ellipse	the shape of the v–x graph
Light damping	small energy loss with almost unchanged period

Section 10 of 10Exam technique

Making velocity and displacement in phase — velocity leads displacement by a quarter of a cycle.

Drawing acceleration in phase with displacement — it is in antiphase.

Giving a positive gradient for the a–x graph — the gradient is −ω².

Misaligning peaks and zeros between the three graphs — the zero of v lines up with the peaks of x.