# Physics equations/07-Work and Energy/A:pendulum

## Wikilab: Energy and the pendulum

By definition, angular frequency and period are related by,

$\omega T=2\pi$ This lab employs two approximation that are valid when $\theta <<1$ :

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

$\sin \theta \approx \theta -{\frac {\theta ^{3}}{3\cdot 2}}\qquad \cos \theta \approx 1-{\frac {1}{2}}\theta ^{2}+{\frac {\theta ^{4}}{4!}}$ Consider a pendulum of length L situated so that the equilibrium point is at the origin. Make a careful hand drawing showing that:

$x\approx L\theta \qquad \mathrm {and} \qquad y\approx {\frac {1}{2}}L\theta ^{2}$ Use this to show that an approximate formula for the energy of a pendulum is:

$E={\frac {1}{2}}mv^{2}+{\frac {mg}{2L}}x^{2}$ Defining $k=mg/L$ as the effective spring constant, this energy can be expressed in familiar form:

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