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

Options

A. `2`
B. `1/15"ln"3`
C. `1/2`
D. `1/3"ln"3`

Explanation

To find the damping constant of the pendulum, we can follow these steps:

### Step 1: Understand the energy decay in a damped oscillator
The energy of a damped oscillator decreases exponentially over time. The relationship can be expressed as:
\[ E(t) = E_0 e^{-bt} \]
where:
- \( E(t) \) is the energy at time \( t \),
- \( E_0 \) is the initial energy,
- \( b \) is the damping constant,
- \( t \) is the time.

### Step 2: Identify the given values
From the problem, we have:
- Initial energy \( E_0 = 45 \, J \)
- Energy after 15 oscillations \( E(15) = 15 \, J \)
- Time period \( T = 1 \, s \)
- Therefore, the time for 15 oscillations \( t = 15 \, s \).

### Step 3: Set up the equation
Using the energy decay formula, we can set up the equation:
\[ 15 = 45 e^{-b \cdot 15} \]

### Step 4: Simplify the equation
Dividing both sides by 45:
\[ \frac{15}{45} = e^{-b \cdot 15} \]
\[ \frac{1}{3} = e^{-b \cdot 15} \]

### Step 5: Take the natural logarithm of both sides
Taking the natural logarithm gives:
\[ \ln\left(\frac{1}{3}\right) = -b \cdot 15 \]

### Step 6: Solve for the damping constant \( b \)
Rearranging the equation to solve for \( b \):
\[ b = -\frac{\ln\left(\frac{1}{3}\right)}{15} \]

### Step 7: Calculate \( b \)
Using the property of logarithms:
\[ \ln\left(\frac{1}{3}\right) = -\ln(3) \]
Thus:
\[ b = \frac{\ln(3)}{15} \]

### Step 8: Final calculation
Using the approximate value \( \ln(3) \approx 1.0986 \):
\[ b \approx \frac{1.0986}{15} \approx 0.07324 \, s^{-1} \]

### Conclusion
The damping constant \( b \) is approximately \( 0.0732 \, s^{-1} \).

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