First slide

Simple hormonic motion

Question

A particle performs harmonic oscillations along a straight line with a period T and amplitude a. The mean velocity of the particle averaged over the time interval during which it travels a distance $\frac{a}{2}$ starting from the extreme positions is

Moderate

A

$\large \frac{a}{T}$
B

$\large \frac{{2a}}{T}$
C

$\large \frac{{3a}}{T}$
D

$\large \frac{a}{{2T}}$

Solution

Let the particle start its motion at extreme position.
$\large \therefore \,x\, = a\,\cos \omega t$
$\large V\, = \, \frac{{dx}}{{dt}} = - a\omega \sin \omega t$
when $\large x\, = \,\frac{a}{2}$ , $\large \frac{a}{2}\, = \,a\cos \omega t$
$\large \therefore \omega \tau \, = \,\frac{\pi }{3}$
$\large \Rightarrow \,\tau \, = \,\frac{T}{6}$
$\large < v > \, = \,\frac{{\int\limits_0^{\frac{T}{6}} {\left( { - a\omega \sin \omega t} \right)} }}{{\int\limits_0^{\frac{T}{6}} {dt} }}\, = \,\frac{{3a}}{T}$
$\large \therefore \,Magnitude\, = \,\frac{{3a}}{T}$

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