Velocity v/s position graph of a body performing SHM is

Options

A. elipse
B. circle
C. parabola
D. straight line

Answer

To solve the problem of determining the velocity vs. position graph of a body performing Simple Harmonic Motion (SHM), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Simple Harmonic Motion (SHM)**:
- In SHM, a body oscillates about an equilibrium position. The maximum displacement from this equilibrium position is called the amplitude (A).

2. **Velocity in SHM**:
- The velocity (v) of a body in SHM can be expressed as:
\[ v = \omega \sqrt{A^2 - x^2} \]
where:
- \( \omega \) is the angular frequency,
- \( A \) is the amplitude,
- \( x \) is the displacement from the equilibrium position.

3. **Rearranging the Velocity Equation**:
- Squaring both sides gives:
\[ v^2 = \omega^2 (A^2 - x^2) \]
- Rearranging this, we find:
\[ v^2 + \omega^2 x^2 = \omega^2 A^2 \]

4. **Identifying the Graph**:
- This equation resembles the standard equation of an ellipse:
\[ \frac{x^2}{A^2} + \frac{v^2}{\frac{\omega^2 A^2}{\omega^2}} = 1 \]
- Here, \( A \) is the semi-major axis and \( \frac{A \omega}{\omega} \) is the semi-minor axis.

5. **Conclusion**:
- The graph of velocity (v) versus position (x) for a body performing SHM is an ellipse centered at the origin.

### Final Answer:
The velocity vs. position graph of a body performing Simple Harmonic Motion is an **ellipse**.

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