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Lecture 25 - Simple Harmonic Motion

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

If you need a pdf version of these notes you can get it here

Video of lecture

Restoring force of a spring

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

$ma=\Sigma F$


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



Energy in SHM



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


and for springs in series


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



Coming up

Not so simple harmonic motion!

  • Physical Pendulum
  • Torsional Pendulum
  • Damped Harmonic Motion
  • Forced Harmonic Motion
phy141/lectures/25.txt · Last modified: 2013/11/02 14:35 by mdawber
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