First slide

Simple harmonic motion

Question

A particle in S.H.M. is described by the displacement function $x (t) = acos (ωt + θ)$ . If the initial $(t = 0)$ position of the particle is 1 cm and its initial velocity is $π cm / s$ . The angular frequency of the particle is $π rad / s$ , then it’s amplitude is

Moderate

A

1 cm
B

$\sqrt{2} cm$
C

2 cm
D

2.5 cm

Solution

$x = acos (ω t + θ)$ ….(i)
and $v = \frac{dx}{dt} = - aωsin (ω t + θ)$ ….(ii)
Given at $t = 0$ , $x = 1 cm$ and $v = π$ and $ω = π$
Putting these values in equation (i) and (ii) we will get $sinθ = \frac{- 1}{a}$ and $cosθ = \frac{1}{a}$
$\Rightarrow \sin^{2} θ + \cos^{2} θ = {(- \frac{1}{a})}^{2} + {(\frac{1}{a})}^{2}$ $\Rightarrow a = \sqrt{2} cm$

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