A body executes simple harmonic motion. The potential energy (P.E), the kinetic energy (K.E) and energy (T.E) are measured as a function of displacement `x`. Which of the following staements is true?

Options

A. PE is maximum when x=0
B. KE is maximum when x=0
C. TE is zero when x=0
D. KE is maximum when x is maximum

Explanation

To solve the problem, we need to analyze the potential energy (P.E), kinetic energy (K.E), and total energy (T.E) of a body executing simple harmonic motion (SHM) as a function of displacement \( x \).

### Step-by-Step Solution:

1. **Understanding the Energy Expressions**:
- The kinetic energy (K.E) of a body in SHM is given by the formula:
\[ K.E = \frac{1}{2} m \omega^2 (A^2 - x^2) \]
where \( m \) is the mass, \( \omega \) is the angular frequency, \( A \) is the amplitude, and \( x \) is the displacement from the mean position.
- The potential energy (P.E) is given by:
\[ P.E = \frac{1}{2} m \omega^2 x^2 \]
- The total energy (T.E) in SHM is the sum of kinetic and potential energy:
\[ T.E = K.E + P.E = \frac{1}{2} m \omega^2 A^2 \]
This shows that the total energy is constant and does not depend on \( x \).

2. **Analyzing the Statements**:
- **Statement 1**: "Potential energy is maximum when \( x = 0 \)".
- When \( x = 0 \), \( P.E = \frac{1}{2} m \omega^2 (0)^2 = 0 \). Thus, this statement is **incorrect**.

- **Statement 2**: "Kinetic energy is maximum when \( x = 0 \)".
- When \( x = 0 \), \( K.E = \frac{1}{2} m \omega^2 (A^2 - 0^2) = \frac{1}{2} m \omega^2 A^2 \), which is the maximum value of K.E. Thus, this statement is **correct**.

- **Statement 3**: "Total energy is zero when \( x = 0 \)".
- The total energy is constant and equal to \( \frac{1}{2} m \omega^2 A^2 \), which is not zero. Thus, this statement is **incorrect**.

- **Statement 4**: "Kinetic energy is maximum when \( x \) is maximum".
- When \( x = A \) (the maximum displacement), \( K.E = \frac{1}{2} m \omega^2 (A^2 - A^2) = 0 \). Thus, this statement is **incorrect**.

3. **Conclusion**:
- The only correct statement is **Statement 2**: "Kinetic energy is maximum when \( x = 0 \)".

### Final Answer:
The correct statement is: **Kinetic energy is maximum when \( x = 0 \)**.

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