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A pendulum with a time period of 1 sec is losing energy due to damping. At a certain time its energy is 45J. If after completing 15 oscillations, its energy has become 15 J its damping constant (in ${s^{ - 1}}$) is:
A) $\dfrac{1}{2}$ 
B) $\dfrac{1}{{15}}\ln 3$ 
C) $\dfrac{1}{{30}}\ln 3$ 
D) 2

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Last updated date: 19th Jun 2024
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Answer
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Hint: In case of pendulum oscillating with decreasing amplitude/damping oscillation due to a damping force being applied on it loses energy gradually due loss of amplitude at an exponential rate.

Formula Used:
${A_t} = {A_0}{e^{ - bt/2m}}$
${A_t}$ : Amplitude of oscillation at time ${t}$
${A_0}$ : Amplitude of oscillation at time ${t=0}$
$\dfrac{b}{m}$ : damping coefficient

Complete step by step answer:
Step-1:
 A simple pendulum oscillating with initial amplitude of motion as ${A_o}$ is decreased in the presence of dissipative forces after time ‘t’ it is given as,
${A_t} = {A_0}{e^{ - bt/2m}}$ ………… (1)
Where $b/m$ is the damping constant.
Step-2:
Now the Initial energy of oscillation can be given as
${E_0} = \dfrac{1}{2}k{A_0}^2 = 45J$  (assuming complete energy in form of potential energy)
Here,   [K = constant]
And after time t =15 secs that is after 15 oscillations (as period of oscillation is 1 second) its energy will be
${E_t} = \dfrac{1}{2}K{({A_0}{e^{ - bt/2m}})^2} = 15J$
Using equation (1) in above relation ${A_t} = {A_0}{e^{ - bt/2m}}$
${E_t} = \dfrac{1}{2}K{A_0}^2{e^{ - 2bt/2m}}$
Submitting values now we have
$15 = 45{e^{ - bt/m}}$  
$\Rightarrow \dfrac{1}{3} = {e^{ - 15b/m}}$
Taking logarithm both sides $\ln \dfrac{1}{3} =  - 15\dfrac{b}{m}$
$ - \ln (3) =  - 15\dfrac{b}{m}$
Therefore, $\dfrac{b}{m} = \dfrac{{\ln (3)}}{{15}}$ is the answer.
Hence, option (B) is correct.

Additional Information:
A system may be so damped that it cannot vibrate. There are many types of mechanical damping. Friction, also called in this context dry, or Coulomb, damping, arises chiefly from the electrostatic forces of attraction between the sliding surfaces and converts mechanical energy of motion, or kinetic energy, into heat.

Note: At any instant the total energy of a system is generally the sum of its kinetic and potential energies. Also the total energy is equal to the maximum potential energy and maximum kinetic energy. In this problem we used this trick of writing the complete energy as maximum potential energy, which is obtained when amplitude is maximum.