If particle is excuting simple harmonic motion with time period T, then the time period of its total mechanical energy is :-

Options

A. Zero
B. T/2
C. 2T
D. Infinite

Answer

To solve the problem, we need to analyze the total mechanical energy of a particle executing simple harmonic motion (SHM) and determine its time period.

### Step-by-Step Solution:

1. **Understanding Simple Harmonic Motion (SHM)**:
- In SHM, a particle oscillates back and forth around an equilibrium position. The total mechanical energy (E) of a particle in SHM is given by the sum of its kinetic energy (K.E) and potential energy (P.E).

2. **Expression for Total Mechanical Energy**:
- The total mechanical energy in SHM can be expressed as:
\[ E = \frac{1}{2} k A^2 \]
where \( k \) is the spring constant and \( A \) is the amplitude of the motion.

3. **Relating Angular Frequency and Time Period**:
- The angular frequency \( \omega \) is related to the time period \( T \) by the formula:
\[ \omega = \frac{2\pi}{T} \]

4. **Substituting Angular Frequency**:
- The total mechanical energy can also be expressed in terms of angular frequency:
\[ E = \frac{1}{2} m A^2 \omega^2 \]
- Substituting \( \omega = \frac{2\pi}{T} \):
\[ E = \frac{1}{2} m A^2 \left(\frac{2\pi}{T}\right)^2 \]
\[ E = \frac{1}{2} m A^2 \cdot \frac{4\pi^2}{T^2} \]
\[ E = \frac{2 m A^2 \pi^2}{T^2} \]

5. **Observing the Nature of Total Mechanical Energy**:
- The expression for total mechanical energy \( E \) is a constant value because it does not depend on time. It remains constant as the particle oscillates.

6. **Determining the Time Period of Total Mechanical Energy**:
- Since the total mechanical energy does not change with time, the time period of the total mechanical energy is effectively infinite. This means that the energy does not oscillate or vary with time.

### Conclusion:
The time period of the total mechanical energy of a particle executing simple harmonic motion is infinite.

### Final Answer:
**The time period of the total mechanical energy is infinite (Option D).**