  SHM09

### 1.2.3 Velocity LO (f)

From x = xo sin ω t

differentiating we get

$v=\genfrac{}{}{0.1ex}{}{dx}{dt}={x}_{0}\omega \left(sin\omega t\right)={v}_{0}cos\left(\omega t\right)$

where   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

$cos\omega t=±\sqrt{\left(1-si{n}^{2}\omega t\right)}$

since

v       =  x0ω cos ωt

where x0 is the maximum displacement

$v=±{x}_{0}\omega \sqrt{\left(1-si{n}^{2}\omega t\right)}$
$v=±{x}_{0}\omega \sqrt{\left(1-\left(\frac{x}{{x}_{0}}{\right)}^{2}\right)}$

$v=±\omega \sqrt{\left({{x}_{0}}^{2}-{x}^{2}\right)}$ Variation with displacement of velocity ### Translations

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