The velocity of a particle executing simple harmonic motion is

Options

A. `omega^(2)sqrt(A^(2)+x^(2))`
B. `omegasqrt(A^(2)-x^(2))`
C. `omegasqrt(A^(2)+x^(2))`
D. `omega^(2)sqrt(A^(2)-x^(2))`

Explanation

To find the velocity of a particle executing simple harmonic motion (SHM), we start with the basic principles of SHM and derive the formula step by step.

### Step-by-Step Solution:

1. **Understanding Simple Harmonic Motion (SHM)**:
- In SHM, a particle oscillates about an equilibrium position. The maximum displacement from this equilibrium position is called the amplitude (A).
- The particle moves back and forth through this equilibrium position, reaching maximum displacement (extreme positions) on either side.

2. **Displacement in SHM**:
- The displacement (x) of a particle in SHM can be described by the equation:
\[ x(t) = A \sin(\omega t) \]
- Here, \(A\) is the amplitude, \(\omega\) is the angular frequency, and \(t\) is the time.

3. **Finding Velocity**:
- The velocity \(v\) of the particle is the time derivative of displacement:
\[ v(t) = \frac{dx}{dt} \]
- Differentiating the displacement equation:
\[ v(t) = \frac{d}{dt}(A \sin(\omega t)) = A \omega \cos(\omega t) \]

4. **Expressing Velocity in Terms of Displacement**:
- To express velocity in terms of displacement \(x\), we can use the identity \( \cos^2(\theta) + \sin^2(\theta) = 1 \).
- From the displacement equation, we have:
\[ \sin(\omega t) = \frac{x}{A} \]
- Therefore, we can express \(\cos(\omega t)\) as:
\[ \cos(\omega t) = \sqrt{1 - \sin^2(\omega t)} = \sqrt{1 - \left(\frac{x}{A}\right)^2} \]

5. **Substituting Back into the Velocity Equation**:
- Now substituting \(\cos(\omega t)\) back into the velocity equation:
\[ v = A \omega \sqrt{1 - \left(\frac{x}{A}\right)^2} \]
- Simplifying this, we get:
\[ v = \omega \sqrt{A^2 - x^2} \]

6. **Final Result**:
- The final formula for the velocity of a particle executing simple harmonic motion in terms of its displacement \(x\) is:
\[ v = \omega \sqrt{A^2 - x^2} \]