When an oscillator completes 100 oscillations its amplitude reduced to (1/3) of initial value. What will be its amplitude, when it completes 200 oscillations?

Question

When an oscillator completes 100 oscillations its amplitude reduced to (1/3) of initial value. What will be its amplitude, when it completes 200 oscillations? (A) (1/8) (B) (2/3) (C) (1/6) (D) (1/9)

Tardigrade Admin · Accepted Answer

Its is a damped oscillation, where amplitude of oscillation at time

t

is given by

A = a_{0} e^{- γ r}

where

a_{0} =

initial amplitude of oscillation

γ =

damping constant
As per question,

\frac{a _{0}}{3} = a_{0} e^{- γ 100/ v}

...(i)
(where

v

is the frequency of oscillation)
and

A = a_{0} e^{- γ 200/ v}

....(ii)
From (i);

\frac{a _{0}}{3} = a_{0} e^{- γ \times 100/ v}

Dividing equation (ii) by (iii), we have

\frac{A}{a _{0} ( 1/3 )} = \frac{e ^{- γ \times 200/ v}}{e ^{- γ \times 100/ v}} = e^{- γ \times 100/ v} = \frac{1}{3}

or

A = a_{0} \times \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} a_{0}

Q. When an oscillator completes $100$ oscillations its amplitude reduced to $\frac{1}{3}$ of initial value. What will be its amplitude, when it completes $200$ oscillations?

$\frac{1}{8}$

$\frac{2}{3}$

$\frac{1}{6}$

$\frac{1}{9}$

Q. When an oscillator completes 100 oscillations its amplitude reduced to 31​ of initial value. What will be its amplitude, when it completes 200 oscillations?

81​

32​

61​

91​

Q. When an oscillator completes $100$ oscillations its amplitude reduced to $\frac{1}{3}$ of initial value. What will be its amplitude, when it completes $200$ oscillations?

$\frac{1}{8}$

$\frac{2}{3}$

$\frac{1}{6}$

$\frac{1}{9}$