First slide

Simple hormonic motion

Question

At time t = 0, one particle is at maximum positive amplitude and the other particle is at half of the positive amplitude. Their amplitudes and time periods T are same. If they are approaching, find the time by which they cross each other:

Moderate

A

$\frac{T}{6}$
B

$\frac{T}{12}$
C

$\frac{T}{16}$
D

$\frac{T}{24}$

Solution

$x_{1} = A \cos ωt (at t = 0, x_{1} = A)$
$x_{2} = A \sin (ωt + \frac{π}{6})$ $(at t = 0, x_{2} = + \frac{A}{2})$
For $x_{1} = x_{2},$
$\cos ωt = \sin (ωt + \frac{π}{6})$
= $\sin ωt \cos \frac{π}{6} + \cos ωt \sin \frac{π}{6}$
$\cos ωt (1 - \sin \frac{π}{6}) = \sin ωt \cos \frac{π}{6}$
$\tan ωt = \frac{(1 - \sin \frac{π}{6})}{\cos \frac{π}{6}} = \frac{1}{\sqrt{3}}$
$ωt = (\frac{2 π}{T}) t = \frac{π}{6}$
Will give $t = \frac{T}{12}$

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