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 solve the problem of determining the shape of the graph plotted between velocity and position 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 between two extreme positions, which we can denote as \( +a \) and \( -a \). The position of the particle at any time can be represented as \( x \).

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

3. **Squaring the Velocity Equation**:
- To analyze the relationship between velocity and position, we can square the velocity equation:
\[ v^2 = \omega^2 (a^2 - x^2) \]

4. **Rearranging the Equation**:
- Rearranging this equation gives:
\[ v^2 + \omega^2 x^2 = \omega^2 a^2 \]
- This can be rewritten as:
\[ \frac{v^2}{\omega^2 a^2} + \frac{x^2}{a^2} = 1 \]

5. **Identifying the Shape of the Graph**:
- The equation we derived, \( \frac{v^2}{\omega^2 a^2} + \frac{x^2}{a^2} = 1 \), is the standard form of the equation of an ellipse.
- Therefore, the graph plotted between velocity \( v \) and position \( x \) will be an ellipse.

6. **Sketching the Graph**:
- On the graph, the maximum position \( x \) will range from \( -a \) to \( a \), and the maximum velocity \( v \) will range from \( -\omega a \) to \( \omega a \).
- The resulting graph will be an ellipse centered at the origin, with the axes corresponding to the maximum values of position and velocity.

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

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