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, `alpha` is maximum
B. Value of `alpha` is zero, whatever may be the value of v
C. When v is zero, `alpha` is zero
D. When v is maximum, `alpha` is zero

Answer

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

### Step-by-Step Solution:

1. **Understanding SHM**:
- A particle in SHM can be described by the equation:
\[ x(t) = A \sin(\omega t) \]
where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( x(t) \) is the displacement at time \( t \).

2. **Finding Velocity**:
- The velocity \( v(t) \) of the particle can be found by differentiating the displacement with respect to time:
\[ v(t) = \frac{dx}{dt} = A \omega \cos(\omega t) \]
- The maximum velocity occurs when \( \cos(\omega t) = 1 \) (i.e., when \( \omega t = 0 \)), giving:
\[ v_{\text{max}} = A \omega \]

3. **Finding Acceleration**:
- The acceleration \( a(t) \) can be found by differentiating the velocity:
\[ a(t) = \frac{dv}{dt} = -A \omega^2 \sin(\omega t) \]
- The acceleration is maximum when \( \sin(\omega t) = 1 \) or \( \sin(\omega t) = -1 \) (i.e., at the extreme positions), giving:
\[ a_{\text{max}} = A \omega^2 \]

4. **Analyzing the Relationship**:
- At the mean position (where \( x = 0 \)), \( \sin(\omega t) = 0 \) and thus:
\[ a(t) = 0 \]
Here, the velocity \( v(t) \) is at its maximum (\( v_{\text{max}} = A \omega \)).
- At the extreme positions (where \( x = \pm A \)), the velocity \( v(t) = 0 \) and the acceleration \( a(t) \) is at its maximum (\( a_{\text{max}} = \pm A \omega^2 \)).

5. **Conclusion**:
- From the analysis, we can conclude that:
- When the speed \( v \) is maximum, the acceleration \( a \) is zero.
- Therefore, the correct statement is: **"v is maximum when a is zero."**

### Final Answer:
The correct option is **D: v is maximum when a is zero.**