In linear simple harmonic motion of a particle at mean position :

Question

Accepted Answer

In linear simple harmonic motion (SHM), the behavior of a particle at the mean position is characterized by specific properties of velocity and acceleration. Let's explore why Option 1 is the correct answer and why the other options are incorrect.

1. **Explanation for Option 1: Velocity is maximum and acceleration is minimum**

In simple harmonic motion, the mean position is the equilibrium position where the particle experiences no net force. At this point, the particle's velocity is at its maximum because it has been accelerated from one extreme position to the mean position. The acceleration, which is proportional to the displacement from the mean position (as given by Hooke's Law, $a = -\omega^2 x$), is zero at the mean position because the displacement $x$ is zero. Therefore, at the mean position, the velocity is maximum, and the acceleration is minimum (zero).

2. **Explanation for why Option 2 is incorrect: Velocity is minimum and acceleration is maximum**

This option is incorrect because, at the mean position, the velocity is not minimum; it is maximum. The acceleration is not maximum at the mean position; it is zero. The maximum acceleration occurs at the extreme positions, where the displacement is greatest.

3. **Explanation for why Option 3 is incorrect: Both velocity and acceleration are minimum**

This option is incorrect because, while the acceleration is indeed minimum (zero) at the mean position, the velocity is not minimum. The velocity is at its maximum at the mean position. Therefore, it is incorrect to say both are minimum.

4. **Explanation for why Option 4 is incorrect: Both velocity and acceleration are maximum**

This option is incorrect because, at the mean position, the velocity is maximum, but the acceleration is not. The acceleration is zero at the mean position. The maximum acceleration occurs at the extreme positions, not at the mean position.

In conclusion, Option 1 is the correct answer because it accurately describes the conditions at the mean position in linear simple harmonic motion: the velocity is maximum, and the acceleration is minimum (zero).