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

Options

A. `(1)/(8)`
B. `(2)/(3)`
C. `(1)/(6)`
D. `(1)/(9)`

Answer

C

Explanation

This is a case of damped oscillations as the amplitude of ocillations is decreasing with time.
Amplitude of oscillations at any instant t is `A=A_(0)e^(-bt//2m)` where `A_(0)` is the initial amplitude of oscillations and b is the damping constant.
Now, when `t=100T,A=(A_(0))/(3)` (T is time period)
Let the amplitude be A, at `t=200T` i.e. after completing 200 oscillations.
`therefore A=(A_(0))/(3)=A_(0)e^(-100Tb//2m)" "...(i)`
and `A.=A_(0)e^(-200Tb//2m)" "...(ii)`
From (i). `(1)/(3)=e^(-100Tb//2m)`
From (ii), `(A.)/(A_(0))=e^(-200Tb//2m)=(1)/(9)orA.=(A_(0))/(9)`
`therefore` The amplitude will be reduced to 1/9 of initial value.