In linear simple harmonic motion of a particle, at the extreme position of particle :

Question

Accepted Answer

In linear simple harmonic motion (SHM), the behavior of a particle at its extreme positions is characterized by specific properties of velocity and acceleration. Let's explore why Option 4 is the correct answer and why the other options are incorrect.

### Explanation for Option 4: Velocity is minimum and acceleration is maximum

1. **Velocity at Extreme Positions:**
   - In SHM, the velocity of a particle is zero at the extreme positions. This is because the particle momentarily comes to rest before reversing its direction of motion. Therefore, the velocity is at its minimum (zero) at these points.

2. **Acceleration at Extreme Positions:**
   - The acceleration in SHM is directly proportional to the displacement from the equilibrium position and is directed towards the equilibrium position. At the extreme positions, the displacement is maximum, and hence the acceleration is also maximum. The acceleration is at its peak because it is trying to bring the particle back towards the equilibrium position.

### Explanation for Why Other Options are Incorrect:

1. **Option 1: Both velocity and acceleration are maximum**
   - This option is incorrect because, at the extreme positions, the velocity is not maximum; it is actually zero. The velocity is maximum at the equilibrium position, not at the extremes.

2. **Option 2: Both velocity and acceleration are minimum**
   - This option is incorrect because, while the velocity is indeed minimum (zero) at the extreme positions, the acceleration is not minimum. The acceleration is maximum at these points due to the maximum displacement from the equilibrium position.

3. **Option 3: Velocity is maximum and acceleration is minimum**
   - This option is incorrect because, at the extreme positions, the velocity is not maximum; it is zero. Additionally, the acceleration is not minimum; it is maximum at these points.

In summary, Option 4 correctly describes the conditions at the extreme positions in linear simple harmonic motion: the velocity is minimum (zero), and the acceleration is maximum. This aligns with the fundamental principles of SHM, where the restoring force (and thus acceleration) is greatest at the maximum displacement, and the particle momentarily stops before reversing direction.