Which of the following displacement-time graphs represent damped harmonic oscillation?

Options

A.
B.
C.
D.

Answer

To determine which displacement-time graph represents damped harmonic oscillation, we need to analyze the characteristics of damped harmonic motion.

### Step-by-Step Solution:

1. **Understanding Damped Harmonic Motion**:
- Damped harmonic motion occurs when a system oscillates with a decreasing amplitude over time due to the presence of a damping force (like friction or air resistance).
- The general equation for the displacement \( x(t) \) of a damped harmonic oscillator is given by:
\[ x(t) = x_0 e^{-\frac{b}{2m} t} \cos(\omega_d t + \phi) \]
- Here, \( x_0 \) is the initial amplitude, \( b \) is the damping coefficient, \( m \) is the mass, and \( \omega_d \) is the damped angular frequency.

2. **Identifying the Characteristics of the Graph**:
- The key feature of a damped harmonic oscillator is that the amplitude of oscillation decreases exponentially over time.
- This means that if we were to plot the displacement \( x \) against time \( t \), the peaks of the oscillations would get closer to the time axis as time progresses, indicating a decrease in amplitude.

3. **Analyzing the Given Graphs**:
- We need to look at the provided graphs and identify which one shows a clear exponential decay in amplitude.
- The correct graph should show oscillations that gradually decrease in height (amplitude) over time, rather than remaining constant or increasing.

4. **Selecting the Correct Graph**:
- After analyzing the provided graphs, we find that **Option C** shows the characteristic of damped harmonic motion, where the oscillations decrease in amplitude exponentially.

### Conclusion:
Thus, the displacement-time graph that represents damped harmonic oscillation is **Option C**.

---