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

Options

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

Answer

D

Explanation

For damped oscillator `r=r_(0)e^(-gamma t)`
where r is damping factor.
Time for 100 oscillations `=(100)/(v)`
`:." "(r_(0))/(3)=r_(0)e^(-gamma(100)/(v))" "…(i)`
and `" "r.=r_(0)e^(-gamma(100)/(v))" "...(ii)`
Dividing (ii) by (i)
`:." "(3r.)/(r_(0))=(e^(-(200gamma)/(v)))/(e^(-(100gamma)/(v)))=e^(-(100gamma)/(v))`
`:." "r.=(1)/(3)r_(0)e^(-(100gamma)/(v))`
`r.=(1)/(3)r_(0)((1)/(3))" "["Using eqn. (i)"]`
`:." "r.=(r_(0))/(9)`
Correct choice is (d).