Question

How long after the beginning of motion is the displacement of a harmonically oscillating point equal to one half its amplitude,if the period is 24 sec and initial phase is zero?

Answer 1

Given,

Time period: 24 s

If we take the amplitude to be A and the equation of motion to be:

assuming that particle starts from rest, from origin then
$x=Asin\left(\omega t\right)$
Thus at $x=\frac{A}{2}$

$\frac{A}{2}=Asin\left(\right(2\pi /T\left)t\right)$
$1/2=sin\left(\right(2\pi /24\left)t\right)$
We know, $sin\left(\pi /6\right)=1/2$
$\pi /6=\left(2\pi /24\right)t$
$1/6=\left(1/12\right)t$
$t=12/6$
$t=2$ sec.

Hence after 2-sec displacement will be half of the amplitude.