When a dampled harmonic oscillator completes 100 oscillations, its amplitude is reduced to `(1)/(3)` of its initial value. When will be its amplitude when it completes 200 oscillations?

Options

A. `(1)/(5)`
B. `(2)/(3)`
C. `(1)/(6)`
D. `(1)/(9)`

Answer

D

Explanation

In the case of a damped harmonic oscillator, the amplitude 'a' at any ins"tan"t is given by
`a=a_(0)c^(-bt)`
where `a_(0)`= initial amplitude and b is the damping constant If T is the period of oscillator, then the time for 100 oscillations is `t_(1)=100 T`
In the first case, `a_(1)=(a_(0))/(3) and t_(1)=100 T`
` therefore (a_(0))/(3) =a_(0)e^(-100 bT) therefore e^(-100bT)=(1)/(3)`
and in the second case,
`a_(2)=a_(0)e^(-bt_(2))=a_(0)e^(-200b t)=a_(0)(e^(-100b t))^(2)`
`therefore a_(2)=a_(0)((1)/(3))^(2)=(a_(0))/(9)`
`therefore` The amplitude will be reduced to `1//9`th of its initial value.