Choose the correct statement(s) regarding SHM

Options

A. All the three quantities displacement , velocity and acceleration show harmonic variation with time, having same periods
B. The velocity amplitude is `omega` times the displacement amplitude.
C. The acceleration amplitude is `omega` times the displacement amplitude
D. All of these

Answer

To solve the question regarding the correct statements about Simple Harmonic Motion (SHM), let's analyze each option step by step.

### Step-by-Step Solution:

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

2. **Velocity in SHM**:
The velocity \( V \) can be derived from the displacement equation by differentiating with respect to time:
\[ V(t) = \frac{dx}{dt} = A \omega \cos(\omega t + \phi) \]
Here, the amplitude of velocity is \( A \omega \).

3. **Acceleration in SHM**:
The acceleration \( a \) is found by differentiating the velocity:
\[ a(t) = \frac{dV}{dt} = -A \omega^2 \sin(\omega t + \phi) \]
Thus, the amplitude of acceleration is \( A \omega^2 \).

4. **Time Period of SHM**:
The time period \( T \) of oscillation is related to angular frequency by:
\[ T = \frac{2\pi}{\omega} \]
Since the angular frequency \( \omega \) is the same for displacement, velocity, and acceleration, it follows that all three quantities have the same time period.

5. **Analyzing the Options**:
- **Option 1**: "All the three quantities displacement, velocity, acceleration show harmonic vibration with time having the same period."
- **Correct**: As derived, all have the same time period.

- **Option 2**: "The velocity amplitude is omega times the displacement amplitude."
- **Correct**: \( A_v = A \omega \).

- **Option 3**: "The acceleration amplitude is omega times the displacement amplitude."
- **Incorrect**: The acceleration amplitude is \( A \omega^2 \), not \( \omega A \).

- **Option 4**: "All of these."
- **Incorrect**: Since Option 3 is incorrect, this option cannot be true.

### Conclusion:
The correct statements regarding SHM are:
- Option 1 and Option 2 are correct.
- Option 3 is incorrect.
- Option 4 is incorrect.

### Final Answer:
The correct statements are:
- 1 and 2.