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(D) The energy of a damped oscillator at time $t$ is given by $E(t) = E_0 e^{-\gamma t}$,where $\gamma = \frac{b}{m}$ is the damping constant.Given,in

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$\frac{1}{15} \ln 3$

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(D) The energy of a damped oscillator at time $t$ is given by $E(t) = E_0 e^{-\gamma t}$,where $\gamma = \frac{b}{m}$ is the damping constant.Given,initial energy $E_0 = 45\, J$ and final energy $E = 15\, J$.The time taken for $15$ oscillations with a time period $T = 1\, s$ is $t = 15 	imes T = 15 	imes 1 = 15\, s$.Substituting the values into the equation: $15 = 45 e^{-\gamma (15)}$.Dividing by $45$: $\frac{1}{3} = e^{-15\gamma}$.Taking the natural logarithm on both sides: $\ln(1/3) = -15\gamma$.$-\ln 3 = -15\gamma$.Therefore,$\gamma = \frac{1}{15} \ln 3\, s^{-1}$.

$A$ pendulum with a time period of $1\, s$ is losing energy due to damping. At a 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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