# Lecture 25 - Simple Harmonic Motion

We will now begin to consider oscillatory motion, beginning with the simplest example, simple harmonic motion.

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## A vertical spring

A vertical spring will also execute simple harmonic motion, though it's mean position will be modified by the balance of the gravitational force and the spring force.

## SHM as a function of time

These animations from physclips show the form of displacement, velocity and acceleration of an object in SHM as a function of time.

## Equations of motion for SHM

Starting from Newton's 2nd Law

$ma=\Sigma F$

$m\frac{d^{2}x}{dt^{2}}=-kx$

Based on our previous observations we might guess that the displacement will be able to be expressed as trigonometric function of time.

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

$A$ is the amplitude, $\omega=\frac{2\pi}{T}=2\pi f$ and $\phi$ allows us to change the starting point of the motion.

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

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

$-m\omega^{2}A\cos(\omega t+\phi)=-kA\cos(\omega t+\phi)$

which is true if

$\omega^{2}=\frac{k}{m}$

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

## Energy in SHM

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

$v=\pm\sqrt{\frac{k}{m}(A^{2}-x^{2})}$

## Simple harmonic motion with multiple springs

There is a wikipedia page on multiple springs which has a detailed explanation of the effective spring constants of springs in series and parallel. The key result is that for springs in parallel

$k_{eq}=k_{1}+k_{2}$

and for springs in series

$\frac{1}{k_{eq}}=\frac{1}{k_{1}}+\frac{1}{k_{2}}$

## Simple Pendulum

The restoring force above is $F=-mg\sin\theta$

For small angles $\sin\theta=\theta$ so $F\approx-mg\theta$

and using the relation $s=l\theta$ gives $F\approx-\frac{mg}{L}s$

This is essentially the same as $F=-kx$ with $k=\frac{mg}{L}$

So $\omega=\sqrt{\frac{k}{m}}$ → $\omega=\sqrt{\frac{g}{l}}$

$f=\frac{\omega}{2\pi}=\frac{1}{2\pi}\sqrt{\frac{g}{l}}$

$T=\frac{1}{f}=2\pi\sqrt{\frac{l}{g}}$

## Coming up

Not so simple harmonic motion!

• Physical Pendulum
• Torsional Pendulum
• Damped Harmonic Motion
• Forced Harmonic Motion