For a particle executing S.H.M. which of the following statements holds good `:`

Options

A. the total energy of the particle always remains the same
B. the restoring force is always directed towards a fixed point
C. the restoring force is maximum at the extreme positions
D. the velocity of the particle is minimum at the centre of motion of the particle

Explanation

To determine which statements hold true for a particle executing Simple Harmonic Motion (S.H.M.), we will analyze each statement step by step.

### Step 1: Analyze the First Statement
**Statement:** The total energy of a particle always remains the same.

**Explanation:**
In S.H.M., the total mechanical energy (E) is given by the formula:
\[ E = \frac{1}{2} k A^2 \]
where \( k \) is the spring constant and \( A \) is the amplitude of the motion. This total energy is conserved and does not change over time, as it is independent of the position and velocity of the particle. Therefore, this statement is **true**.

### Step 2: Analyze the Second Statement
**Statement:** The restoring force is always directed towards a fixed point.

**Explanation:**
The restoring force \( F \) in S.H.M. is given by:
\[ F = -kx \]
where \( x \) is the displacement from the mean position. The negative sign indicates that the force acts in the opposite direction of the displacement. Thus, if the particle is displaced to the right (positive \( x \)), the force acts to the left (negative \( x \)), and vice versa. This means the restoring force is always directed towards the mean position (fixed point). Therefore, this statement is **true**.

### Step 3: Analyze the Third Statement
**Statement:** The restoring force is maximum at the extreme position.

**Explanation:**
The restoring force is given by:
\[ F = -kx \]
The maximum displacement occurs at the extreme positions, where \( x = \pm A \). Substituting this into the formula gives:
\[ F_{\text{max}} = kA \]
Thus, the restoring force is indeed maximum at the extreme positions. Therefore, this statement is **true**.

### Step 4: Analyze the Fourth Statement
**Statement:** The velocity of the particle is minimum at the center of motion of the particle.

**Explanation:**
The velocity \( v \) of a particle in S.H.M. is given by:
\[ v = \omega \sqrt{A^2 - x^2} \]
where \( \omega \) is the angular frequency and \( x \) is the displacement. At the center of motion (mean position), \( x = 0 \):
\[ v = \omega \sqrt{A^2 - 0^2} = \omega A \]
This is the maximum velocity, not the minimum. The velocity is minimum (zero) at the extreme positions where \( x = \pm A \). Therefore, this statement is **false**.

### Conclusion
The true statements regarding a particle executing S.H.M. are:
1. The total energy of a particle always remains the same. (True)
2. The restoring force is always directed towards a fixed point. (True)
3. The restoring force is maximum at the extreme position. (True)
4. The velocity of the particle is minimum at the center of motion of the particle. (False)

### Final Answer
The correct statements are 1, 2, and 3. The fourth statement is incorrect.

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