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

Answer

D

Explanation

In case of damped vibration, amplitde at any instant is
`a = a_(0)e^(-b)`
where `a_(0) =` initial amplitude
`b =` damping constant
`Ist` caes : `t = 100 T` and `a = (a_(0))/(3)`
`:. (a_(0))/(3) = a_(0)e^(-b(100T)) rArr e^(-100bT) = (1)/(3)`
`IInd` cases : `t = 200 T`
`a = a_(0)e^(-bt)=a_(0)e^(-b(200 T)) = a_(0) (e^(-100 bT))^(2) = a_(0) xx ((1)/(3))^(2) = (a_(0))/(9)`