Question

A particle is performing damped oscillation with frequency \[5\,{\text{Hz}}\]. After every \[10\] oscillations its amplitude becomes half. Find time from the beginning after which the amplitude becomes \[\dfrac{1}{{1000}}\] of its initial amplitude.
A. \[10\,{\text{s}}\]
B. \[20\,{\text{s}}\]
C. \[25\,{\text{s}}\]
D. \[50\,{\text{s}}\]

Answer 1

Hint: First of all, we will determine the time period and total time taken for the number of oscillations. We will use the initial and final amplitude in the formula and manipulate accordingly, to find the time required.

Complete step by step answer:
In the given problem,
Frequency is \[5\,{\text{Hz}}\].
So, the time period can be calculated as:
$ T = \dfrac{1}{f} \\
T = \dfrac{1}{5}\,{\text{s}} \\ $
Since, the number of oscillations mentioned in the question \[10\].
So, the time required to complete \[10\] oscillations:
$ t = T \times 10 \\
t = \dfrac{1}{5} \times 10 \\
t = 2\,{\text{s}} \\ $

It is the case of damped oscillation, which means the amplitude gradually decreases with the function of time.

For damped oscillation, we have a formula:
$ A = {A_0}{\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{2}}}$ …… (1)
Where,
$ A$ indicates amplitude after time $t $.
$ {A_0}$ indicates initial amplitude.
According to question:
Amplitude becomes \[\dfrac{1}{{1000}}\] of its initial amplitude. So, we can write:
$ A = \dfrac{{{A_0}}}{{1000}} $
Using the above value in equation (1), we get:
$ \dfrac{{{A_0}}}{{1000}} = {A_0}{\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{2}}} \\
  \dfrac{1}{{1000}} = {\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{2}}} \\
  {10^{ - 3}} = {\left( {\dfrac{1}{2}} \right)^{\dfrac{t}{2}}} \\
  {10^{ - 3}} = {\left( {{2^{ - 1}}} \right)^{\dfrac{t}{2}}} \\ $
Now, we take logarithm on both the sides, we get:
 $ {10^{ - 3}} = {2^{\dfrac{{ - t}}{2}}} \\
  \log {10^{ - 3}} = \log {\left( 2 \right)^{\dfrac{{ - t}}{2}}} \\
   - 3\log 10 = \dfrac{{ - t}}{2}\log 2 \\
  3 = \dfrac{t}{2} \times 0.301 \\ $
Again, simplifying further, we get:
$ t = \dfrac{6}{{0.301}} \\
  t = 19.93\,{\text{s}} \\
  {\text{t}} \sim {\text{20}}\,{\text{s}} \\ $
Hence, the required time is ${\text{20}}\,{\text{s}} $.

So, the correct answer is “Option B”.

Additional Information:
An oscillator is something which has a periodic rhythmic effect. A damp oscillation means an oscillation that, over time, eventually ends. The oscillatory motion in which with the passage of time the amplitude continuously decreases is known as damped oscillation.
Let’s take an example, if you take a pendulum in motion by only giving it energy one time, then you will realize that after some time, the pendulum ceases its motion when some of its energy is lost in overcoming air resistance and therefore loses energy continuously over time.

Note:
While solving this problem, it is important to find the time period of the oscillation. The time used in the formula is the total time taken for the given number of oscillations, but not the time period. Again, remember that the logarithm used here is the common logarithm not the natural logarithm.