# Simple Harmonic Motion

## Table of contents

## Activity 1

As described in the appendix below, the period of a pendulum should be independent of the amplitude if it is an ideal (simple harmonic) oscillator.

Put the adjustable mass at the bottom of the pendulum and then test this model. Take data to plot \(T(\theta_0)\) where \(\theta_0\) is the initial angle (amplitude) of oscillation and T is the period (the time to complete one full oscillation). **Note: you should brace the base of the stand for larger angles so that the stand does not rock too much!**

What can you conclude about whether a pendulum can be modelled as a simple harmonic oscillator given your data and its uncertainty? As usual you should justify your conclusions numerically based on a graph of your data.

## Activity 2

As described in the appendix below, if you have a pendulum which consists of a rod (mass m and length L) with a point mass (mass M) placed at a distance r from the pivot point, the period should be \(T=2\pi\sqrt{\frac{\frac{1}{3}mL^2+Mr^2}{\frac{1}{2}mgL+Mgr}}\)if the pendulum can be treated as a simple harmonic oscillator.

You will test the theory. First, check if the period gives the correct value when you remove the adjustable mass (equivalent to setting M=0). Then attach the mass and take data to plot T(r). **In both cases you should have the pendulum swinging only a small amount to get more accurate data**. Note: your graph of T(r) should include your data points with errorbars and the theoretically function above so you can compare them. You should have access to a Python program that will help you with this. If your data is particularly good, you might want to sketch the residuals which the Python program will also graph for you.

What can you conclude about this theory given the uncertainty of your data? Is your apparatus close to being a simple harmonic oscillator?

## Appendix

The most basic mathematical model of a system undergoing periodic oscillatory motion is the "Simple Harmonic Oscillator" model. Ignoring all sorts of possible effects (including friction) gives that the position as a function of time should be given by the equation \(x(t) = A \cos(2\pi \frac{t}{T} + \phi_0)\) where x(t) is any coordinate (which for pendulums is actually an angle instead of a position), T is the period of oscillation, A is the amplitude of oscillation and \(\phi_0\) is the "phase constant". The amplitude and phase constant are set by the "initial conditions" of the system (where and how fast the object is moving at t=0), and are not important for these activities.

What we care about is that the period T should be independent of the initial conditions, and should only depend on the system. For a point mass (M) a distance r from the pivot attached to a rigid rod (m) of lenth L the period should be \(T=2\pi\sqrt{\frac{\frac{1}{3}mL^2+Mr^2}{\frac{1}{2}mgL+Mgr}}\). Note that this is independent of the amplitude and the phase constant.