Choose the correct statement regarding SHM

Options

A. The velocity is ahead of displacement by a phase angle of `pi`
B. The velocity of ahead of displacement by a phase angle of `(pi)/(2)`
C. The velocity is behind displacement by a phase angle of `(pi)/(2)`
D. The velocity is behind displacement by a phase angle of `pi`.

Answer

To solve the question regarding the phase difference between velocity and displacement in Simple Harmonic Motion (SHM), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Basic Equations of SHM**:
- The standard equation for displacement in SHM is given by:
\[ x(t) = A \sin(\omega t + \phi) \]
where:
- \( x(t) \) is the displacement,
- \( A \) is the amplitude,
- \( \omega \) is the angular frequency,
- \( t \) is the time,
- \( \phi \) is the phase constant.

2. **Find the Velocity Equation**:
- The velocity \( v(t) \) is the time derivative of displacement:
\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}(A \sin(\omega t + \phi)) \]
- Using the chain rule, we get:
\[ v(t) = A \omega \cos(\omega t + \phi) \]

3. **Convert the Velocity Equation to Sine Form**:
- To compare the phase of displacement and velocity, we can convert the cosine function into sine:
\[ \cos(\theta) = \sin\left(\theta + \frac{\pi}{2}\right) \]
- Thus, we can rewrite the velocity equation as:
\[ v(t) = A \omega \sin\left(\omega t + \phi + \frac{\pi}{2}\right) \]

4. **Identify the Phase Difference**:
- Now we can compare the phases of displacement and velocity:
- The phase of displacement \( \phi_x = \omega t + \phi \)
- The phase of velocity \( \phi_v = \omega t + \phi + \frac{\pi}{2} \)
- The phase difference between velocity and displacement is:
\[ \Delta \phi = \phi_v - \phi_x = \left(\omega t + \phi + \frac{\pi}{2}\right) - \left(\omega t + \phi\right) = \frac{\pi}{2} \]

5. **Conclusion**:
- The correct statement regarding the phase difference between velocity and displacement in SHM is that the velocity is ahead of displacement by a phase angle of \( \frac{\pi}{2} \).

### Final Answer:
The correct option is: **Velocity is ahead of displacement by phase angle \( \frac{\pi}{2} \)**.

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