SHM Chapter 09 v with x

Velocity LO (f)

From x = x_o sin ω t
differentiating we get

$$v=\frac{dx}{dt}=x_0\omega \mathrm{ \; s}i\mathrm{n }\omega t \; =\; v$$₀ cos ω twhere   v₀ = x₀ ω    is the maximum velocity




Variation with time of velocity  

In terms of x:

           
From mathematical identity     cos² ωt + sin² ωt = 1,
rearranging
cos² ωt       = 1 - sin² ωt
$$cos\omega t=\pm \sqrt{(1-si{n}^2\omega }$$  t )
     
since
v       =  x₀ω cos ωt
where x₀ is the maximum displacement
$$v=\pm x_0\omega \sqrt{(1-si{n}^2\omega t)}$$    
 $$v=\pm x_0\omega \sqrt{(1-(\frac{x}{x_0}{)}^2)}$$$$$$
$$v=\pm \omega \sqrt{({x_0}^2-x^2)}$$

    

Variation with displacement of velocity

http://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_SHM09/SHM09_Simulation.xhtml

Model:

http://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_SHM09/SHM09_Simulation.xhtml