For a body executing S.H.M. :
(a) Potential energy is always equal to its K.E.
(b) Average potential and kinetic energy over any given time inteval are always equal.
(c ) Sum of the kinetic and potential energy at any point of time is constant.
(d) Average K.E in one time period is equal to average potential energy in one time period . Choose the most approprate ooption from the options given below :

Options

A. only (c )
B. only (b)
C. (b) and (c )
D. (c ) and (d)

Answer

To solve the question regarding the properties of a body executing Simple Harmonic Motion (S.H.M.), we need to analyze each statement provided:

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- In S.H.M., a body oscillates about an equilibrium position, and its energy can be divided into kinetic energy (K.E.) and potential energy (P.E.).
- The total mechanical energy (E) in S.H.M. is constant and is the sum of kinetic and potential energy at any point in time.

2. **Evaluating Statement (a)**:
- **Statement**: Potential energy is always equal to its K.E.
- **Analysis**: This is incorrect. At different positions in S.H.M., the potential energy and kinetic energy can vary. They are equal at specific instances (like at the mean position), but not always.
- **Conclusion**: **False**.

3. **Evaluating Statement (b)**:
- **Statement**: Average potential and kinetic energy over any given time interval are always equal.
- **Analysis**: This is also incorrect. The average kinetic and potential energies over a random time interval can differ. They are equal only over a complete cycle (one time period).
- **Conclusion**: **False**.

4. **Evaluating Statement (c)**:
- **Statement**: Sum of the kinetic and potential energy at any point of time is constant.
- **Analysis**: This statement is true. The total mechanical energy (E) is constant and is given by \( E = K.E + P.E \). Thus, at any point in time, the sum remains constant.
- **Conclusion**: **True**.

5. **Evaluating Statement (d)**:
- **Statement**: Average K.E. in one time period is equal to average potential energy in one time period.
- **Analysis**: This statement is correct. Over one complete cycle of S.H.M., the average kinetic energy is equal to the average potential energy due to the symmetry of the motion.
- **Conclusion**: **True**.

### Final Answer:
The correct statements are **(c)** and **(d)**.