# a)    Variation with time of energy in simple harmonic motion

 http://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_SHM17/SHM17_Simulation.xhtml

If the variation with time of displacement is as shown, then the energies should be drawn as shown.

recalling Energy formula
$KE=\frac{1}{2}m{v}^{2}$
$PE=\frac{1}{2}k{x}^{2}$ in terms of time t,
x = x0 sin(ωt)
differentiating with t gives
v = v0 cos (ωt)

$KE=\frac{1}{2}m{v}^{2}=\frac{1}{2}m\left({x}_{0}\omega cos\left(\omega t\right){\right)}^{2}=\frac{1}{2}m{\mathrm{\left(x}}_{0}^{2}{\omega }^{2}\mathrm{\right)c}o{s}^{2}\left(\omega t\right)\right)$similarly
$PE=\frac{1}{2}k{x}^{2}=\frac{1}{2}k\left({x}_{0}sin\left(\omega t\right){\right)}^{2}=\frac{1}{2}\left(m{\omega }^{2}\right){x}_{0}^{2}si{n}^{2}\left(\omega t\right)\right)$
therefore total energy is a constant value in the absence of energy loss due to drag (resistance)
$TE=KE+PE=\frac{1}{2}m{\omega }^{2}{x}_{0}^{2}\left(si{n}^{2}\omega t+co{s}^{2}\omega t\right)=\frac{1}{2}m{\omega }^{2}{x}_{0}^{2}$

this is how the x vs t looks together of the energy vs t graphs

the table shows some of the common values

 general energy formula SHM energy formula when $t=0$ when  $t=\frac{T}{4}$ when $t=\frac{T}{2}$ when $t=\frac{3T}{4}$ when $t=T$ $KE=\frac{1}{2}m{v}^{2}$ $K$ $\frac{1}{2}m{\omega }^{2}{x}_{0}^{2}$ 0 $\frac{1}{2}m{\omega }^{2}{x}_{0}^{2}$ 0 $\frac{1}{2}m{\omega }^{2}{x}_{0}^{2}$ $PE=\frac{1}{2}k{x}^{2}$ $PE=\frac{1}{2}\left(m{\omega }^{2}\right){x}_{0}^{2}si{n}^{2}\left(\omega t\right)\right)$ 0 $\frac{1}{2}m{\omega }^{2}{x}_{0}^{2}$ 0 $\frac{1}{2}m{\omega }^{2}{x}_{0}^{2}$ 0 $TE=\frac{1}{2}m{\omega }^{2}{x}_{0}^{2}$ $\frac{1}{2}m{\omega }^{2}{x}_{0}^{2}$ $\frac{1}{2}m{\omega }^{2}{x}_{0}^{2}$ $\frac{1}{2}m{\omega }^{2}{x}_{0}^{2}$ $\frac{1}{2}m{\omega }^{2}{x}_{0}^{2}$ $\frac{1}{2}m{\omega }^{2}{x}_{0}^{2}$

## Model:

http://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_SHM17/SHM17_Simulation.xhtml