Question

When an oscillator completes 100 oscillations its amplitude is reduced to $\frac{1}{3}$ of the initial value. What will be its amplitude, when it completes $200$ oscillations?

• A. $\frac{1}{8}$
• B. $\frac{2}{3}$
• C. $\frac{1}{6}$
• D. $\frac{1}{9}$

Answer 1

The correct option is D

$\frac{1}{9}$

Oscillator:

1. An oscillator is a mechanical or electronic device that operates on the principles of oscillation, which is defined as a periodic fluctuation between two objects caused by energy changes.
2. Oscillators are used in a variety of technologies, including computers, clocks, watches, radios, and metal detectors.

Step 1: Given data

$$\begin{gathered} \mathrm{t}=100\mathrm{T} \\ \mathrm{A}=\frac{{\mathrm{a}}_0}{3} \end{gathered}$$

Step 2: Formula used

It's a damped oscillation, in which the oscillation's amplitude at time t is determined by:

$\mathrm{A}={\mathrm{a}}_0{\mathrm{e}}^{-\mathrm{\gamma t}}$.

where $a_0$=initial amplitude of oscillation

$\gamma$ is the damping constant.

Step 3: Calculation

as per requirement

$\frac{{\mathrm{a}}_0}{3}={\mathrm{a}}_0{\mathrm{e}}^{-\frac{100\mathrm{T}}{\mathrm{v}}}...\left(1\right)$

Where,

v is the oscillation frequency.

and

$\mathrm{A}={\mathrm{a}}_0{\mathrm{e}}^{-\frac{200\mathrm{T}}{\mathrm{v}}}...\left(2\right)$

By dividing equation ($2$) by ($1$),

$$\begin{gathered} \frac{\mathrm{A}}{{\mathrm{a}}_0(1/3)}=\frac{{\mathrm{a}}_0{\mathrm{e}}^{-{\displaystyle \frac{200\mathrm{T}}{\mathrm{v}}}}}{{\mathrm{a}}_0{\mathrm{e}}^{-{\displaystyle \frac{100\mathrm{T}}{\mathrm{v}}}}} \\ ={\mathrm{a}}_0{\mathrm{e}}^{-\frac{100\mathrm{T}}{\mathrm{v}}} \\ =\frac{1}{3} \end{gathered}$$

$$\begin{gathered} \mathrm{A}=\mathrm{a}\times \frac{1}{3}\times \frac{1}{3} \\ =\frac{1}{9}{\mathrm{a}}_0 \end{gathered}$$

Thus, the amplitude reduces by $\frac{1}{9}$ (option D) of the initial value.