(A) : Projection of a uniform circular motion on the diameter of the circle is simple harmonic.
(B) : The energy of a body executing SHM is directly proportional to square of its amplitude.

Options

A. Both A and B are correct
B. Both A and B are wrong
C. A is correct but B is wrong
D. A is wrong but B is correct

Answer

To solve the question, we need to analyze both statements (A) and (B) regarding simple harmonic motion (SHM) and the energy of a body executing SHM.

### Step-by-Step Solution:

**Step 1: Analyze Statement (A)**
Statement (A) claims that the projection of a uniform circular motion on the diameter of the circle is simple harmonic motion.

- **Explanation**:
- Consider a point moving in a circle with uniform speed. If we project this point onto a diameter of the circle, the projection will oscillate back and forth along the diameter.
- As the point moves around the circle, its projection on the diameter will move from one extreme (maximum displacement) to the other extreme and back, creating a to-and-fro motion.
- This motion is periodic and can be described by a sine or cosine function, which is characteristic of SHM.

**Conclusion for (A)**: Statement (A) is correct.

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**Step 2: Analyze Statement (B)**
Statement (B) states that the energy of a body executing SHM is directly proportional to the square of its amplitude.

- **Explanation**:
- In SHM, the total mechanical energy (E) is the sum of kinetic energy (KE) and potential energy (PE).
- The potential energy in SHM can be expressed as \( PE = \frac{1}{2} k x^2 \), where \( k \) is the spring constant and \( x \) is the displacement from the mean position.
- The maximum displacement (amplitude) is denoted by \( A \). At maximum displacement, \( x = A \), thus the potential energy becomes \( PE = \frac{1}{2} k A^2 \).
- The kinetic energy varies, but the total energy remains constant and can be expressed as \( E = \frac{1}{2} k A^2 \).
- Therefore, the total energy is directly proportional to the square of the amplitude \( A^2 \).

**Conclusion for (B)**: Statement (B) is correct.

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### Final Conclusion:
Both statements (A) and (B) are correct.

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