Question

In damped oscillations, the amplitude 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

A. ${\displaystyle \frac{A_0}{2}}$
B. ${\displaystyle \frac{A_0}{4}}$
C. ${\displaystyle \frac{A_0}{6}}$
D. ${\displaystyle \frac{A_0}{9}}$

Answer 1

Hint: We know that in damped oscillation the amplitude, always increases or decreases exponentially. Now using this statement we have to write the mathematical equation to represent this. Now, after representing in mathematical form we have to derive an equation for the amplitude after 100 oscillations and using this equation we need to satisfy the equation that we will be making with the help of amplitude after 200 oscillations.

Formula used:
$A=A_0{e}^{-\gamma t}$

Complete answer:
We know that the initial amplitude is $A_0$ .
We know that in damped oscillation the amplitude always increases or decreases exponentially.
We can represent the above statement using the formula,
$A=A_0{e}^{-\gamma t}$.
Now, if we consider that the time taken for one oscillation is T seconds.
Then the time taken for 100 oscillations must be 100T.
Now, it is given in the question that the amplitude becomes one third at after 100 oscillations, so
$${\displaystyle \frac{A_0}{3}}=A_0{e}^{-100\gamma T}$$
Now,
$${\displaystyle \frac{1}{3}}=e^{-100\gamma T}$$………. Eq.1.
Now for 200 oscillations time taken is 200 T.
And let us consider the amplitude as, $A^{\prime }$
So, according to problem,
$$A^{\prime }=A_0{e}^{-200\gamma T}$$,
$$A^{\prime }=A_0{\left(e^{-100\gamma T}\right)}^2$$, as ($${\displaystyle \frac{1}{3}}=e^{-100\gamma T}$$).
$$A^{\prime }=A_0{\left({\displaystyle \frac{1}{3}}\right)}^2$$,
So,
$$A^{\prime }={\displaystyle \frac{A_0}{9}}$$.

So, the correct answer is “Option D”.

Additional Information:
An oscillation that during oscillating with due time, the amplitude decreases and eventually the oscillation stops and comes to rest. This type of oscillation is known as damped oscillation.

Note:
In the equation $A=A_0{e}^{-\gamma t}$, $\gamma$ is the damping coefficient, and t is the time. We have to find the first equation after doing all the possible calculations or else it will not satisfy the second equation and the result will not come.