In damped oscillations, the amplitude of oscillations is reduced to one-third of its initial value a0 at the end of 100 oscillations. When the oscillator completes 200 oscillations, its amplitude must be

Question

In damped oscillations, the amplitude of oscillations is reduced to one-third of its initial value a0 at the end of 100 oscillations. When the oscillator completes 200 oscillations, its amplitude must be (A)  (a0/2)  (B)  (a0/6)  (C)  (a0/12)  (D)  (a0/4)

Tardigrade Admin · Accepted Answer

In damped oscillation, amplitude goes on decaying exponentially

a = a_{0} e^{- b t}

where

b =

damping coefficient.
Initially,

\frac{a _{0}}{3} = a_{0} e^{- b \times 100 T}

where T is time of one oscillation.
Or

\frac{1}{3} = e^{- 100 b T}

.. (i)
Finally,

a = a_{0} e^{- b \times 200 T}

Or

a = a_{0} [e^{- 100 b T}]^{2}

Or

a = a_{0} \times [\frac{1}{3}]^{2}

[from Eq.(i)]
Or

a = \frac{a _{0}}{9}

Q. In damped oscillations, the amplitude of oscillations is reduced to one-third of its initial value $a_{0}$ at the end of $100$ oscillations. When the oscillator completes $200$ oscillations, its amplitude must be

$\frac{a _{0}}{2}$

$\frac{a _{0}}{6}$

$\frac{a _{0}}{12}$

$\frac{a _{0}}{4}$

$\frac{a _{0}}{9}$

Q. In damped oscillations, the amplitude of oscillations is reduced to one-third of its initial value a0​ at the end of 100 oscillations. When the oscillator completes 200 oscillations, its amplitude must be

2a0​​

6a0​​

12a0​​

4a0​​