Question

A simple harmonic motion is described by y = r sin wt . Find the value of displacement at which speed of the body executing shm is half of its maximum speed .

Answer 1

A simple harmonic motion is described as $y=rsin\omega t$

differentiating with respect to time,

$\frac{dy}{dt}=\omega rcos\omega t$

we know, change in position per unit time is called velocity.

so, $v=\omega rcos\omega t$

so, maximum speed, $v=\omega r$

half of maximum speed , $v=\omega r/2=\omega rcos\omega t$

so, $cos\omega t=\frac{1}{2}=cos\frac{\pi}{3}$

so, $\omega t=\frac{\pi}{3}$

now, displacement, y = rsin(π/3) = √3r/2