Wikilab: Energy and the pendulum

By definition, angular frequency and period are related by,

${\displaystyle \omega T=2\pi }$

This lab employs two approximation that are valid when ${\displaystyle \theta <<1}$:

${\displaystyle \sin \theta \approx \theta \qquad \cos \theta \approx 1-{\frac {1}{2}}\theta ^{2}}$

Use the Excel spreadsheet to plot the sine and cosine functions for θ between zero and one, taking approximately 20 increments within that interval.

Copy and paste your page onto the same excel spreadsheet and verify that a better approximation is:

${\displaystyle \sin \theta \approx \theta -{\frac {\theta ^{3}}{3\cdot 2}}\qquad \cos \theta \approx 1-{\frac {1}{2}}\theta ^{2}+{\frac {\theta ^{4}}{4!}}}$

Consider a pendulum of length L situated so that the equilibrium point is at the origin. Make a careful hand drawing showing that:

${\displaystyle x\approx L\theta \qquad \mathrm {and} \qquad y\approx {\frac {1}{2}}L\theta ^{2}}$

Use this to show that an approximate formula for the energy of a pendulum is:

${\displaystyle E={\frac {1}{2}}mv^{2}+{\frac {mg}{2L}}x^{2}}$

Defining ${\displaystyle k=mg/L}$ as the effective spring constant, this energy can be expressed in familiar form:

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