For a particle undergoing simple harmonic motion, the velocity is plotted against displacement. The curve will be

Options

A. a straight line
B. a parabola
C. a circle
D. an ellipse

Explanation

To determine the curve that represents the relationship between velocity and displacement for a particle undergoing simple harmonic motion (SHM), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Displacement Equation**:
In SHM, the displacement \( x \) of the particle can be expressed as:
\[ x = A \sin(\omega t) \]
where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( t \) is time.

**Hint**: Remember that displacement in SHM is a function of time and is sinusoidal.

2. **Find the Velocity**:
The velocity \( v \) of the particle is the derivative of displacement with respect to time:
\[ v = \frac{dx}{dt} = A \omega \cos(\omega t) \]

**Hint**: The velocity is the rate of change of displacement, so differentiate the displacement equation.

3. **Relate Velocity and Displacement**:
We can express both \( x \) and \( v \) in terms of \( \sin \) and \( \cos \):
- From the displacement equation, we have \( \sin(\omega t) = \frac{x}{A} \).
- From the velocity equation, we can express \( \cos(\omega t) \) as \( \cos(\omega t) = \sqrt{1 - \sin^2(\omega t)} = \sqrt{1 - \left(\frac{x}{A}\right)^2} \).

**Hint**: Use the Pythagorean identity to relate sine and cosine.

4. **Substitute into the Velocity Equation**:
Substitute \( \cos(\omega t) \) into the velocity equation:
\[ v = A \omega \sqrt{1 - \left(\frac{x}{A}\right)^2} \]

5. **Square Both Sides**:
To eliminate the square root, we square both sides:
\[ v^2 = (A \omega)^2 \left(1 - \left(\frac{x}{A}\right)^2\right) \]
Simplifying gives:
\[ v^2 = (A \omega)^2 - \frac{v^2}{A^2} x^2 \]

6. **Rearrange the Equation**:
Rearranging the equation leads to:
\[ \frac{x^2}{A^2} + \frac{v^2}{(A \omega)^2} = 1 \]

7. **Identify the Curve**:
The equation \( \frac{x^2}{A^2} + \frac{v^2}{(A \omega)^2} = 1 \) represents an ellipse in the \( x-v \) plane.

**Hint**: Recognize that this equation is in the standard form of an ellipse.

### Final Conclusion:
Thus, the curve that represents the relationship between velocity and displacement for a particle undergoing simple harmonic motion is an **ellipse**.