Which one of the following statement is true for the speed `v` and the acceleration a of a particle executing simple harmonic motion?

Options

A. when v is maximum, a is zero
B. when x is zero, a is zero
C. when v is maximum, a is maximum
D. value of a is zero, whatever may be the value of v.

Explanation

To determine which statement is true regarding the speed \( v \) and the acceleration \( a \) of a particle executing simple harmonic motion (SHM), we can analyze the relationship between these quantities mathematically and conceptually.

### Step-by-Step Solution:

1. **Understanding Simple Harmonic Motion (SHM)**:
- In SHM, the position \( x \) of a particle can be described as:
\[ x(t) = A \cos(\omega t + \phi) \]
where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase constant.

2. **Velocity in SHM**:
- The velocity \( v \) of the particle is the derivative of the position with respect to time:
\[ v(t) = \frac{dx}{dt} = -A \omega \sin(\omega t + \phi) \]
- The maximum speed \( v_{\text{max}} \) occurs when \( \sin(\omega t + \phi) = \pm 1 \), giving:
\[ v_{\text{max}} = A \omega \]

3. **Acceleration in SHM**:
- The acceleration \( a \) of the particle is the derivative of the velocity:
\[ a(t) = \frac{dv}{dt} = -A \omega^2 \cos(\omega t + \phi) \]
- The maximum acceleration \( a_{\text{max}} \) occurs when \( \cos(\omega t + \phi) = \pm 1 \), giving:
\[ a_{\text{max}} = A \omega^2 \]

4. **Analyzing the Statements**:
- **Statement 1**: "When \( v \) is maximum, acceleration is zero."
- This is true because maximum velocity occurs at the equilibrium position (where \( x = 0 \)), and at this point, the acceleration \( a \) is zero.
- **Statement 2**: "When displacement is zero, acceleration is zero."
- This is also true because at the equilibrium position (displacement \( x = 0 \)), the velocity is maximum, and the acceleration is zero.
- **Statement 3**: "When velocity is maximum, acceleration is maximum."
- This is false because when the velocity is maximum, the displacement is zero, leading to zero acceleration.
- **Statement 4**: "Value of acceleration is zero, whatever may be the value of velocity."
- This is false because acceleration can be zero only at a specific point (equilibrium position) and not for all values of velocity.

5. **Conclusion**:
- The correct statements are:
- Statement 1: "When \( v \) is maximum, acceleration is zero."
- Statement 2: "When displacement is zero, acceleration is zero."

### Final Answer:
The true statements are:
- When \( v \) is maximum, acceleration is zero.
- When displacement is zero, acceleration is zero.