If x, v and a denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period T, then, which of the following does not change with time?

Options

A. `(aT)/(v)`
B. `aT+2piv`
C. `(aT)/(x)`
D. `a^(2)T+4pi^(2)v^(2)`

Explanation

To solve the problem, we need to analyze the expressions given in the options and determine which one does not change with time for a particle executing simple harmonic motion (SHM).

### Step-by-step Solution:

1. **Understanding the Variables**:
- Displacement (x): The position of the particle in SHM.
- Velocity (v): The rate of change of displacement.
- Acceleration (a): The rate of change of velocity.

2. **Formulas in SHM**:
- The acceleration in SHM is given by:
\[ a = -\omega^2 x \]
- The velocity is given by:
\[ v = \omega \sqrt{A^2 - x^2} \]
- The time period (T) is related to angular frequency (ω) by:
\[ T = \frac{2\pi}{\omega} \]

3. **Analyzing Each Option**:
- **Option 1: \( \frac{aT}{v} \)**
Substituting \( a = -\omega^2 x \) and \( v = \omega \sqrt{A^2 - x^2} \):
\[ \frac{aT}{v} = \frac{-\omega^2 x \cdot T}{\omega \sqrt{A^2 - x^2}} = -\frac{\omega T x}{\sqrt{A^2 - x^2}} \]
This expression depends on x, which changes with time. So, this option is not constant.

- **Option 2: \( aT + 2\pi v \)**
Substituting the values:
\[ aT + 2\pi v = -\omega^2 x \cdot T + 2\pi (\omega \sqrt{A^2 - x^2}) \]
This expression also depends on x, which changes with time. Thus, this option is not constant.

- **Option 3: \( \frac{aT}{x} \)**
Substituting the values:
\[ \frac{aT}{x} = \frac{-\omega^2 x \cdot T}{x} = -\omega^2 T \]
Here, -ω²T is a constant because both ω and T are constants. Therefore, this option does not change with time.

- **Option 4: \( a^2T + 4\pi^2 v^2 \)**
Substituting the values:
\[ a^2T + 4\pi^2 v^2 = (\omega^4 x^2)T + 4\pi^2 (\omega^2 (A^2 - x^2)) \]
This expression depends on x, which changes with time. So, this option is not constant.

4. **Conclusion**:
The only expression that does not change with time is:
\[ \frac{aT}{x} = -\omega^2 T \]
Thus, the correct answer is **Option 3**.