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  4. A pendulum with time period of 1 s is losing energy due to damping. At certain time its energy is 45 J. If after completing 15 oscillations, its energy has become 15 J, its damping constant (in s-1) is :

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Q. A pendulum with time period of $1\,s$ is losing energy due to damping. At certain time its energy is $45\, J$. If after completing $15$ oscillations, its energy has become $15\, J$, its damping constant (in $s^{-1}$) is :

JEE MainJEE Main 2015Oscillations

Solution:

Amplitude of a damped oscillator
$A=A o e^{-k t}$
where $b$ is the damping constant.
Energy $E=\frac{1}{2} K A^{2} E=\left(\frac{1}{2} K A_{0}^{2}\right) e^{-2 bt }$
$\Rightarrow 15=45 e^{-2b( 1\times 15)} 3=e^{30 b }$
$\Rightarrow b=\frac{1}{30} \ln (3)$