Consider the following statements in relation to the motion of a simple pendulum :
(1) The acceleration is always directed towards the mean position
(2) the maximum acceleration occurs at the mean position
(3) The maximum velocity occurs at the mean position
(4) The magnitude of maximum acceleration is equal to the maximum velocity
Which of the following statements are correct ?

Options

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

Answer

To solve the question regarding the motion of a simple pendulum and the correctness of the given statements, we will analyze each statement step by step.

### Step 1: Analyze Statement 1
**Statement 1**: The acceleration is always directed towards the mean position.

- In simple harmonic motion (SHM), the restoring force is always directed towards the equilibrium (mean) position. Since acceleration is directly proportional to the restoring force (F = ma), the acceleration is also directed towards the mean position.

**Conclusion**: This statement is **correct**.

### Step 2: Analyze Statement 2
**Statement 2**: The maximum acceleration occurs at the mean position.

- The maximum acceleration in SHM occurs at the maximum displacement (amplitude), not at the mean position. At the mean position, the displacement is zero, and thus the acceleration is also zero.

**Conclusion**: This statement is **incorrect**.

### Step 3: Analyze Statement 3
**Statement 3**: The maximum velocity occurs at the mean position.

- In SHM, the maximum velocity occurs when the pendulum passes through the mean position. At this point, all potential energy is converted into kinetic energy, resulting in maximum velocity.

**Conclusion**: This statement is **correct**.

### Step 4: Analyze Statement 4
**Statement 4**: The magnitude of maximum acceleration is equal to the maximum velocity.

- The maximum acceleration (A_max) occurs at the extreme positions and is given by A_max = Aω², where A is the amplitude and ω is the angular frequency. The maximum velocity (V_max) occurs at the mean position and is given by V_max = Aω.
- Since A_max = Aω² and V_max = Aω, these two quantities are not equal.

**Conclusion**: This statement is **incorrect**.

### Final Summary of Statements
- Statement 1: Correct
- Statement 2: Incorrect
- Statement 3: Correct
- Statement 4: Incorrect

### Final Answer
The correct statements are **1 and 3**.

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