To study the dissipations of the energy of a simple pendulum, student plots a graph between square root of time and amplitude . The graph would be a

Options

A. straight line
B. hyperbola
C. parabola
D. exponential

Answer

To solve the problem of determining the type of graph that would be plotted between the square root of time and amplitude for a simple pendulum experiencing energy dissipation, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Damped Oscillation**:
- When a simple pendulum oscillates, it gradually loses amplitude due to energy dissipation (like air resistance). The amplitude decreases over time.

2. **Amplitude Formula**:
- The amplitude \( x \) at time \( t \) can be expressed as:
\[ x(t) = A e^{-\frac{rt}{2m}} \]
where \( A \) is the initial amplitude, \( r \) is the damping coefficient, and \( m \) is the mass of the pendulum bob.

3. **Rearranging the Amplitude Equation**:
- To analyze the relationship between amplitude and time, we can rearrange the equation:
\[ \frac{x}{A} = e^{-\frac{rt}{2m}} \]
- Taking the natural logarithm of both sides gives:
\[ \ln\left(\frac{x}{A}\right) = -\frac{rt}{2m} \]

4. **Expressing Time in Terms of Amplitude**:
- Rearranging the equation to solve for \( t \):
\[ t = -\frac{2m}{r} \ln\left(\frac{x}{A}\right) \]

5. **Introducing Square Root of Time**:
- We need to plot the graph of square root of time \( \sqrt{t} \) against amplitude \( x \). To do this, we take the square root of both sides:
\[ \sqrt{t} = \sqrt{-\frac{2m}{r} \ln\left(\frac{x}{A}\right)} \]

6. **Identifying the Type of Graph**:
- The equation shows that \( \sqrt{t} \) is related to the logarithm of the amplitude. This indicates that as the amplitude decreases, the square root of time increases.
- The relationship is not linear, and since the logarithm function is involved, the graph will not be a straight line, parabola, or exponential function.

7. **Conclusion**:
- The graph between \( \sqrt{t} \) and \( x \) will resemble a hyperbolic shape. Therefore, the correct answer is that the graph will be a hyperbola.

### Final Answer:
The graph between the square root of time and amplitude will be a **hyperbola**.

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