For a particle performing `SHM`

Options

A. The kinetic energy is never equal to the potential energy
B. the kinetic energy is always equal to the potential energy
C. The average kinetic in one time period is equal to the average potential in this period
D. The avarage kinetic energy in any time interval is equal to average potential energy in that interval.

Answer

To solve the problem regarding a particle performing Simple Harmonic Motion (SHM) and to identify the correct statements from the given options, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Basics of SHM**:
- In SHM, a particle oscillates back and forth around an equilibrium position. The motion can be described in terms of kinetic energy (KE) and potential energy (PE).

2. **Analyzing the Statements**:
- We need to evaluate each statement provided in the question.

3. **Statement A**: "The kinetic energy is never equal to the potential energy."
- This statement is **incorrect**. In SHM, there are instances where the kinetic energy can equal the potential energy, particularly when the particle is at the midpoint of its motion.

4. **Statement B**: "The kinetic energy is always equal to the potential energy."
- This statement is also **incorrect**. While kinetic energy can equal potential energy at certain points, it does not hold true for all positions in SHM.

5. **Statement C**: "The average kinetic energy in one period is equal to the average potential energy in that period."
- This statement is **correct**. Over one complete cycle of SHM, the average kinetic energy is equal to the average potential energy due to the conservation of mechanical energy.

6. **Statement D**: "The average kinetic energy in any interval of time is equal to the average potential energy in that interval."
- This statement is **incorrect**. The averages may not be equal in arbitrary intervals of time, as they depend on the specific positions and velocities of the particle during that interval.

7. **Conclusion**:
- After evaluating all statements, the only correct statement is **Statement C**.

### Final Answer:
The correct option is **C**.

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