A pendulum with the 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, then its damping constant (in s$$^{-1}$$) will be

We know that for a damped oscillator the displacement (or amplitude) decreases exponentially with time. Mathematically, if the initial amplitude is $$A_0$$ then at time $$t$$ we have

$$A(t)=A_0 e^{-\beta t}$$

where $$\beta$$ is the damping constant (its SI unit is $$s^{-1}$$). The total mechanical energy of the oscillator is proportional to the square of its amplitude, so

$$E(t)\propto [A(t)]^{2}$$

Substituting the expression for $$A(t)$$, the energy at time $$t$$ becomes

$$E(t)=E_0 e^{-2\beta t}$$

Here $$E_0$$ is the energy at $$t=0$$. Let us now insert the numerical data given in the problem.

At $$t=0$$ the energy is $$E_0=45\text{ J}$$. After the pendulum completes 15 oscillations its energy has dropped to $$E=15\text{ J}$$.

The time taken for one oscillation is the time period $$T=1\text{ s}$$, so the time needed to complete 15 oscillations is

$$t = 15T = 15 \times 1\text{ s} = 15\text{ s}.$$

Now we substitute $$E=15\text{ J}$$, $$E_0=45\text{ J}$$, and $$t=15\text{ s}$$ into the energy-decay equation $$E = E_0 e^{-2\beta t}$$:

$$15 = 45\, e^{-2\beta (15)}.$$

First divide both sides by 45 to isolate the exponential term:

$$\frac{15}{45} = e^{-30\beta}.$$

Simplifying the fraction on the left gives

$$\frac{1}{3} = e^{-30\beta}.$$

To solve for $$\beta$$ take the natural logarithm (base $$e$$) of both sides. Recall the logarithmic identity $$\ln(e^{x}) = x$$.

$$\ln\!\left(\frac{1}{3}\right) = \ln\!\left(e^{-30\beta}\right) = -30\beta.$$

Hence

$$\beta = -\,\frac{1}{30}\,\ln\!\left(\frac{1}{3}\right).$$

We can simplify further because $$\ln\!\left(\frac{1}{3}\right) = -\ln 3$$. Substituting this identity gives

$$\beta = -\frac{1}{30}\,(-\ln 3) = \frac{\ln 3}{30}.$$

Thus the damping constant is

$$\beta = \frac{1}{30}\,\ln 3\;\; \text{s}^{-1}.$$

This matches Option C.

Hence, the correct answer is Option C.