Consider the following statements regarding particle executing simple harmonic motion :
(1) The total energy of the particle constant
(2) The restoring force is minimum at the extreme positions
(3) the velocity of the particle is minimum at the mean position
Which of the these statements are correct ?

Options

A. 1 and 2
B. 1 and 4
C. 2 and 3
D. 3 and 4

Answer

To determine which statements regarding a particle executing simple harmonic motion (SHM) are correct, we will analyze each statement step by step.

### Step 1: Analyze Statement 1
**Statement 1:** The total energy of the particle is constant.

- In SHM, the total mechanical energy (E) is the sum of kinetic energy (KE) and potential energy (PE). The total energy 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.

- Since \( m \), \( \omega \), and \( A \) are constant for a particle in SHM, the total energy remains constant throughout the motion.

**Conclusion for Statement 1:** True.

### Step 2: Analyze Statement 2
**Statement 2:** The restoring force is minimum at the extreme positions.

- The restoring force (F) in SHM is given by:
\[ F = -kx \]
where \( k \) is the spring constant and \( x \) is the displacement from the mean position.

- At the extreme positions, the displacement \( x \) is maximum (i.e., \( x = A \) or \( x = -A \)). Therefore, the restoring force is maximum at these points, not minimum.

**Conclusion for Statement 2:** False.

### Step 3: Analyze Statement 3
**Statement 3:** The velocity of the particle is minimum at the mean position.

- The velocity \( v \) of a particle in SHM is given by:
\[ v = \omega \sqrt{A^2 - x^2} \]

- At the mean position, \( x = 0 \):
\[ v = \omega \sqrt{A^2 - 0^2} = \omega A \]

- This indicates that the velocity is maximum at the mean position. Conversely, at the extreme positions (\( x = A \) or \( x = -A \)), the velocity becomes zero.

**Conclusion for Statement 3:** False.

### Final Conclusion
- **Statement 1:** True
- **Statement 2:** False
- **Statement 3:** False

Thus, the correct answer is that only Statement 1 is correct.