Simple Harmonic Motion | Physics Cheat Sheet

Overview

Simple Harmonic Motion (SHM) is periodic motion where the restoring force is directly proportional to displacement. It's the foundation for understanding oscillations and waves.

Definition

Motion is SHM when the restoring force is:

F = -kx

Where $k$ is a positive constant and $x$ is displacement from equilibrium.

General Solution

Position as a function of time:

x(t) = A \cos(\omega t + \phi)

Or equivalently:

x(t) = A \sin(\omega t + \phi')

Where:

$A$ = amplitude (maximum displacement)
$\omega$ = angular frequency
$\phi$ = phase constant
$t$ = time

Key Quantities

Angular Frequency

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

Period

Time for one complete cycle:

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

Frequency

Cycles per second:

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

Velocity and Acceleration

Velocity

v(t) = \frac{dx}{dt} = -A\omega \sin(\omega t + \phi)

Maximum velocity:

v_{\max} = A\omega

Acceleration

a(t) = \frac{dv}{dt} = -A\omega^2 \cos(\omega t + \phi) = -\omega^2 x

Maximum acceleration:

a_{\max} = A\omega^2

Energy in SHM

Kinetic Energy

KE = \frac{1}{2}mv^2 = \frac{1}{2}m\omega^2 A^2 \sin^2(\omega t + \phi)

Potential Energy

PE = \frac{1}{2}kx^2 = \frac{1}{2}kA^2 \cos^2(\omega t + \phi)

Total Energy

E = KE + PE = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2

Total energy is constant and proportional to amplitude squared.

Mass-Spring System

A mass $m$ attached to a spring with constant $k$ :

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

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

f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}

Simple Pendulum

For small angles ( $\theta < 15°$ ):

\omega = \sqrt{\frac{g}{L}}

T = 2\pi\sqrt{\frac{L}{g}}

Properties

Period independent of mass
Period independent of amplitude (for small angles)
Period depends only on length and $g$

Physical Pendulum

Any rigid body oscillating about a pivot:

T = 2\pi\sqrt{\frac{I}{mgd}}

Where:

$I$ = moment of inertia about the pivot
$d$ = distance from pivot to center of mass

Damped Oscillations

With damping force $F_d = -bv$ :

x(t) = Ae^{-\gamma t} \cos(\omega' t + \phi)

Where:

$\gamma = \frac{b}{2m}$ = damping constant
$\omega' = \sqrt{\omega_0^2 - \gamma^2}$ = damped frequency

Types of Damping

Underdamped: $\gamma < \omega_0$ (oscillates while decaying)
Critically damped: $\gamma = \omega_0$ (fastest return without oscillation)
Overdamped: $\gamma > \omega_0$ (slow return, no oscillation)

Driven Oscillations and Resonance

When driven by external force $F_0\cos(\omega_d t)$ :

Resonance

Maximum amplitude when driving frequency equals natural frequency:

\omega_d = \omega_0

At resonance:

A_{\max} = \frac{F_0}{b\omega_0}

Examples

Example 1: Mass-Spring System

A 0.5 kg mass on a spring ( $k = 200$ N/m) is displaced 0.1 m and released.

\omega = \sqrt{\frac{k}{m}} = \sqrt{\frac{200}{0.5}} = 20 \text{ rad/s}

T = \frac{2\pi}{\omega} = 0.314 \text{ s}

f = \frac{1}{T} = 3.18 \text{ Hz}

v_{\max} = A\omega = 0.1 \times 20 = 2 \text{ m/s}

a_{\max} = A\omega^2 = 0.1 \times 400 = 40 \text{ m/s}^2

Example 2: Simple Pendulum

A 1 m pendulum on Earth. Find period.

T = 2\pi\sqrt{\frac{L}{g}} = 2\pi\sqrt{\frac{1}{9.8}} = 2.01 \text{ s}

On Moon ( $g = 1.6$ m/s²):

T = 2\pi\sqrt{\frac{1}{1.6}} = 4.97 \text{ s}

Example 3: Energy in SHM

In Example 1, find total energy and maximum velocity.

E = \frac{1}{2}kA^2 = \frac{1}{2}(200)(0.1)^2 = 1 \text{ J}

At equilibrium (all KE):

\frac{1}{2}mv_{\max}^2 = 1 \text{ J}

v_{\max} = \sqrt{\frac{2}{0.5}} = 2 \text{ m/s}

Example 4: Velocity at Given Position

In Example 1, find velocity when $x = 0.06$ m.

Using energy:

\frac{1}{2}kA^2 = \frac{1}{2}kx^2 + \frac{1}{2}mv^2

\frac{1}{2}(200)(0.01) = \frac{1}{2}(200)(0.0036) + \frac{1}{2}(0.5)v^2

1 - 0.36 = 0.25v^2

v = 1.6 \text{ m/s}

SHM Summary Table

Quantity	Formula	At $x = 0$	At $x = \pm A$
Position	$A \cos(\omega t)$	0	$\pm A$
Velocity	$-A\omega \sin(\omega t)$	$\pm v_{\max}$	0
Acceleration	$-A\omega^2 \cos(\omega t)$	0	$\mp a_{\max}$
KE	$\frac{1}{2}mv^2$	max	0
PE	$\frac{1}{2}kx^2$	0	max