The shape of graph plotted between velocity and position of a particle executing simple harmonic motion is

Options

A. A straight line
B. An ellipse
C. A parabola
D. A hyperbola

Answer

To determine the shape of the graph plotted between velocity (v) and position (x) of a particle executing simple harmonic motion (SHM), we can follow these steps:

### Step-by-Step Solution:

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

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

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

4. **Identifying the Graph Type**:
- The equation \( \frac{v^2}{\omega^2 A^2} + \frac{x^2}{A^2} = 1 \) resembles the standard form of an ellipse:
\[ \frac{v^2}{b^2} + \frac{x^2}{a^2} = 1 \]
where \( a = A \) and \( b = \omega A \).

5. **Conclusion**:
- Since the equation represents an ellipse, we conclude that the graph plotted between velocity (v) and position (x) of a particle executing SHM is an **ellipse**.

### Final Answer:
The shape of the graph plotted between velocity and position of a particle executing simple harmonic motion is an **ellipse**.