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

For a deped harmobic oscillator, the amplitude 'a' at any insttant is given by
`a=a_(0)e^(-bt)`
When `a_(0)`= intial amplitude and b is the damping constant. If T is the period of oscillator. Then the time for 100 oscillations is `t_(1)=100T`
In the firt case `a_(1)=(a_(0))/(3)`
and `t_(1)=100T`
`:. (a_(0))/(3)=a_(0) e^(-100T)`
and in the second case, brgt `a_(2)=(a_(0))/(3)=a_(0)e^(-100 bT)`
and the second case,
`a_(2)=(a_(0)e^(-b))^(2)=a_(0)e^(-200b t)`
`=a_(0)(e^(-100 bt))^(2)`
`:. a_(2), =a_(0) ((1)/(3))^(2)=(a_(0))/(9)`
`:.` The amplitude will be reduced to `1//9`th of its inital value.