Oscillations and Simple Harmonic Motion

Overview

Oscillations describe motion that repeats about an equilibrium position. Many physical systems oscillate when displaced from equilibrium and released, because a restoring force tends to bring the system back while inertia carries it past equilibrium.

Common examples include spring-mass systems, pendulums, vibrating strings, tuning forks, vehicle suspensions, and electrical oscillators.

This hub is the canonical topic page for oscillations and SHM. It is intentionally self-sufficient; the linked pages only expand areas that often need separate treatment.

Core Ideas

• Oscillatory motion is periodic motion about an equilibrium position.
• A restoring force acts toward equilibrium.
• In ideal free oscillation, total mechanical energy remains constant.
• In real systems, resistive forces cause damping.
• If an external periodic force acts, forced oscillations may occur.
• Resonance occurs when driving frequency is close to natural frequency, producing maximum response.

Describing Oscillations

The equilibrium position is the position where the resultant force on the object is zero:

$$\sum \vec{F}=\vec{0}$$

In many JC SHM problems, the motion is treated in one dimension after choosing a positive direction. The displacement $x$, velocity $v$, and acceleration $a$ are then signed components of the underlying displacement, velocity, and acceleration vectors along that chosen line.

The signed displacement $x$ is measured from equilibrium:

• $x>0$: one side of equilibrium;
• $x<0$: opposite side.

Amplitude $x_0$ is the maximum magnitude of displacement:

$$|x{|}_{\max }=x_0$$

The period $T$ is the time for one complete oscillation. The frequency $f$ is the number of oscillations per second:

$$f=\frac{1}{T}$$

Angular frequency is the scalar rate of phase change:

$$\omega =2\pi f=\frac{2\pi }{T}$$

Phase describes the stage of motion within a cycle. For two oscillations with the same period, a time lag $\Delta t$ corresponds to phase difference:

$$\Delta \varphi =\frac{2\pi \Delta t}{T}$$

Special cases:

• in phase: $\Delta \varphi =0$;
• antiphase: $\Delta \varphi =\pi$.

See Phase Difference.

Free Oscillations

A free oscillation occurs when a system is displaced and released, then oscillates under its own restoring force without external driving.

Ideal free oscillations have:

• constant amplitude;
• constant period;
• constant total mechanical energy.

Real systems gradually lose energy due to damping.

Simple Harmonic Motion

Simple harmonic motion is oscillatory motion in which acceleration is directly proportional to displacement from equilibrium and always directed toward equilibrium.

Vector form:

$$\vec{a}=-{\omega }^2\vec{s}$$

In the usual one-dimensional signed-component treatment:

$$a=-{\omega }^{2}x$$

The negative sign means acceleration is opposite to displacement:

• if $x>0$, then $a<0$;
• if $x<0$, then $a>0$.

See Simple Harmonic Motion.

Equations of SHM

The displacement can be written as:

$$x=x_0\sin (\omega t+\varphi )$$

or equivalently:

$$x=x_0\cos (\omega t+\varphi )$$

depending on initial condition.

Velocity is:

$$v=\frac{dx}{dt}$$

Here $v$ is the signed velocity component. The speed is $|v|$.

If:

$$x=x_0\sin (\omega t+\varphi )$$

then:

$$v=\omega x_0\cos (\omega t+\varphi )$$

The maximum speed is:

$$v_{\max }=\omega x_0$$

Acceleration is:

$$a=\frac{dv}{dt}=\frac{d^{2}x}{d{t}^2}=-{\omega }^{2}x$$

The maximum acceleration magnitude is:

$$a_{\max }={\omega }^2{x}_0$$

Eliminating time gives:

$$v^2={\omega }^2(x_0^2-x^2)$$

so:

$$v=\pm \omega \sqrt{x_0^2-x^2}$$

Graphical Behaviour in SHM

The displacement-time graph is sinusoidal. It has maximum slope at equilibrium and zero slope at turning points.

The velocity-time graph is also sinusoidal and phase shifted by $\pi /2$ from displacement. Velocity is maximum at equilibrium and zero at extremes.

The acceleration-time graph is sinusoidal and in antiphase with displacement.

The acceleration-displacement graph is a straight line:

$$a=-{\omega }^{2}x$$

with gradient:

$$-{\omega }^2$$

This is a common test for SHM.

The velocity-displacement graph is an ellipse because:

$$v^2={\omega }^2(x_0^2-x^2)$$

Energy in SHM

In ideal SHM, total mechanical energy remains constant.

Kinetic energy is:

$$E_K=\frac{1}{2}m{v}^2$$

Using the SHM velocity-displacement relation:

$$E_K=\frac{1}{2}m{\omega }^2(x_0^2-x^2)$$

The associated potential energy is:

$$E_P=\frac{1}{2}m{\omega }^2{x}^2$$

Total energy is:

$$E_T=E_K+E_P=\frac{1}{2}m{\omega }^2{x}_0^2$$

At equilibrium $x=0$, kinetic energy is maximum and potential energy is minimum. At extremes $x=\pm x_0$, kinetic energy is zero and potential energy is maximum.

Spring-Mass System

For a horizontal spring:

$$\vec{F}=-k\vec{x}$$

Using Newton’s second law:

$$m\vec{a}=-k\vec{x}$$

Thus:

$$\vec{a}=-\frac{k}{m}\vec{x}$$

Comparing with $\vec{a}=-{\omega }^2\vec{x}$:

$$\omega =\sqrt{\frac{k}{m}}$$

Hence:

$$T=2\pi \sqrt{\frac{m}{k}}$$

Simple Pendulum

For small angular displacement:

$$\sin \theta \approx \theta$$

Then pendulum motion approximates SHM. The angular frequency is:

$$\omega =\sqrt{\frac{g}{l}}$$

and the period is:

$$T=2\pi \sqrt{\frac{l}{g}}$$

where $l$ is the pendulum length and $g$ is gravitational field strength. The ideal small-angle period is independent of mass.

See Pendulum Motion.

Damped and Forced Oscillations

Real systems lose energy due to friction, air resistance, internal resistance, or other dissipative effects. Damping causes:

• amplitude to decrease with time;
• total energy to decrease;
• oscillations to eventually stop unless energy is supplied.

Light damping allows oscillations to continue with decreasing amplitude. Critical damping returns the system to equilibrium in the shortest time without oscillation. Heavy damping gives slow non-oscillatory return.

If a periodic external force acts on a system:

$$\vec{F}(t)={\vec{F}}_0\cos {\omega }_{d}t$$

where ${\omega }_d$ is the driving angular frequency. The system undergoes forced oscillation. At steady state, it oscillates at the driving frequency.

Resonance occurs when the driving frequency is close to the natural frequency:

$$f_{\text{driving}}\approx f_0$$

Greater damping causes a lower, broader resonance peak. See Damping and Resonance.

Circular Motion Connection

The projection of uniform circular motion onto one diameter gives SHM. If a particle moves in a circle of radius $r$ with angular velocity vector $\vec{\omega }$, its angular speed is $\omega =|\vec{\omega }|$, and its projection executes:

$$x=r\sin \omega t$$

Thus SHM can be viewed as a one-dimensional projection of circular motion.

Problem-Solving Checklist

If given period:

$$f=\frac{1}{T},\omega =\frac{2\pi }{T}$$

If given amplitude and position:

$$v=\pm \omega \sqrt{x_0^2-x^2}$$

If asked whether motion is SHM, check whether:

$$\vec{a}\propto -\vec{x}$$

or in one-dimensional signed form:

$$a\propto -x$$

If energy is involved:

$$E_T=\frac{1}{2}m{\omega }^2{x}_0^2$$

Exam Relevance

Common exam tasks include:

• stating the SHM definition and explaining the negative sign;
• using $a=-{\omega }^{2}x$ in one-dimensional signed problems;
• interpreting $x$-$t$, $v$-$t$, $a$-$t$, $a$-$x$, and energy graphs;
• deriving $T=2\pi \sqrt{m/k}$ or $T=2\pi \sqrt{l/g}$;
• explaining why a pendulum formula requires small angles;
• comparing damping types;
• explaining resonance curves and practical resonance examples.

Common mistakes:

• forgetting the negative sign in $a=-{\omega }^{2}x$;
• confusing amplitude with instantaneous displacement;
• using the pendulum formula for large angles;
• forgetting velocity is zero at turning points;
• assuming resonance is always beneficial.

Summary

Quantity	Expression
Frequency	$f=1/T$
Angular frequency	$\omega =2\pi f$
SHM condition	$a=-{\omega }^{2}x$
Displacement	$x=x_0\sin (\omega t+\varphi )$
Velocity	$v=\omega x_0\cos (\omega t+\varphi )$
Max speed	$v_{\max }=\omega x_0$
Max acceleration	$a_{\max }={\omega }^2{x}_0$
Total energy	$\frac{1}{2}m{\omega }^2{x}_0^2$
Spring period	$2\pi \sqrt{m/k}$
Pendulum period	$2\pi \sqrt{l/g}$

Links

• Prerequisite: Kinematics
• Prerequisite: Forces
• Prerequisite: Work, Energy and Power
• Related: Circular Motion
• Related: Waves
• Concept: Simple Harmonic Motion
• Concept: Damping and Resonance
• Concept: Pendulum Motion
• Concept: Phase Difference

Provenance

• source file: 1_PDFsam_08_Oscillations.pdf
• generated by: bridging_tools/ingest_JC_phy_wiki.py
• manifest entry: inbox/lecture_notes/1_PDFsam_08_Oscillations.pdf
• source hash: b21f19dd5702c1868bffa66ed50a27d4edbf80149c2c5b2daeafc268d642185c