If a graph is plotted between velocity (v) and displacement (y) of a particle executing SHM from mean position, then the nature of the graph is

Options

A. Straight line
B. Parabola
C. Ellipse
D. Hyperbola

Answer

To determine the nature of the graph plotted between velocity (v) and displacement (y) of a particle executing Simple Harmonic Motion (SHM) from the mean position, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between velocity and displacement in SHM**:
The velocity \( v \) of a particle in SHM can be expressed in terms of its displacement \( x \) from the mean position. The formula is:
\[ v = \omega \sqrt{A^2 - x^2} \]
where \( \omega \) is the angular frequency and \( A \) is the amplitude of the motion.

2. **Square both sides of the equation**:
To eliminate the square root, we square both sides:
\[ v^2 = \omega^2 (A^2 - x^2) \]

3. **Rearrange the equation**:
Rearranging gives:
\[ v^2 = \omega^2 A^2 - \omega^2 x^2 \]
This can be rewritten as:
\[ v^2 + \omega^2 x^2 = \omega^2 A^2 \]

4. **Identify the form of the equation**:
The equation \( v^2 + \omega^2 x^2 = \omega^2 A^2 \) resembles the standard form of an ellipse equation:
\[ \frac{v^2}{\omega^2 A^2} + \frac{x^2}{A^2} = 1 \]
This indicates that the relationship between \( v \) and \( x \) is elliptical.

5. **Conclusion**:
Therefore, when we plot the graph of velocity \( v \) against displacement \( x \), the nature of the graph is an ellipse.

### Final Answer:
The nature of the graph plotted between velocity (v) and displacement (y) of a particle executing SHM from the mean position is an **ellipse**.

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