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 Total Mechanical Energy in SHM**:
- The total mechanical energy (E) of a particle in simple harmonic motion is given by the formula:
\[ E = \frac{1}{2} m \omega^2 A^2 \]
where:
- \( m \) is the mass of the particle,
- \( \omega \) is the angular frequency,
- \( A \) is the amplitude of the motion.

2. **Identifying Constants**:
- In the formula for total mechanical energy, \( m \), \( \omega \), and \( A \) are all constants. This means that the total mechanical energy does not change over time as the particle oscillates.

3. **Understanding Variation of Total Energy**:
- Since the total mechanical energy (E) remains constant, it does not vary with time. Therefore, it does not have a frequency of change.

4. **Relating Frequency to Time Period**:
- The frequency (f) of a quantity is defined as the number of cycles per unit time. If the total mechanical energy is constant (i.e., it does not change), its frequency of change is zero:
\[ f = 0 \]

5. **Calculating Time Period**:
- The time period (T) is the reciprocal of frequency:
\[ T = \frac{1}{f} \]
- Since \( f = 0 \):
\[ T = \frac{1}{0} = \infty \]

6. **Conclusion**:
- Therefore, 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**.

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