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 question regarding the graph between the square root of time and amplitude for a simple pendulum, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to analyze the relationship between the amplitude of a simple pendulum and the square root of time. The amplitude of the pendulum decreases over time due to energy dissipation (like air resistance and friction).

2. **Identify the Variables**:
- Let \( A \) be the amplitude of the pendulum.
- Let \( t \) be the time.
- We are interested in the relationship between \( \sqrt{t} \) and \( A \).

3. **Energy Considerations**:
The potential energy \( PE \) of the pendulum at its maximum displacement (amplitude) is given by:
\[ PE = \frac{1}{2} k A^2 \]
where \( k \) is a constant related to the spring constant of the pendulum.

4. **Dissipation of Energy**:
As time progresses, the amplitude \( A \) decreases due to energy dissipation. This can be expressed as:
\[ A \propto e^{-kt} \]
where \( k \) is a constant that represents the rate of energy dissipation.

5. **Relating Amplitude and Time**:
Since \( A \) decreases exponentially, we can express \( A \) in terms of \( \sqrt{t} \):
\[ A \propto \sqrt{t} \]
However, since \( A \) decreases, we can say:
\[ A = A_0 e^{-kt} \]
where \( A_0 \) is the initial amplitude.

6. **Graphing the Relationship**:
When we plot \( A \) against \( \sqrt{t} \), we need to consider that as \( t \) increases, \( A \) decreases. The graph will not be a straight line but rather a curve that decreases, resembling a hyperbola.

7. **Conclusion**:
Therefore, the graph between the square root of time and amplitude will show a decreasing trend, indicating that as time progresses, the amplitude decreases. The correct option for the graph is a hyperbola.

### Final Answer:
The graph would be a hyperbola.