Velocity of a particle undergoing simple harmonic motion

Simple harmonic motion is a type of periodic motion in which the restoring force is directly proportional to the displacement of the particle from its equilibrium position. This type of motion can be observed in various systems such as a mass-spring system, a pendulum, or the motion of a particle in a circular path.

In simple harmonic motion, the velocity of the particle changes as it moves back and forth between its extreme positions. At different points in the motion, the velocity of the particle can have different values. Let's analyze the options given:

a) Constant velocity: This option is incorrect because in simple harmonic motion, the velocity of the particle is not constant. The particle speeds up as it moves towards the equilibrium position and slows down as it reaches the extreme positions.

b) Maximum at x = 0 and minimum at the extreme positions of the oscillations: This option is correct. At the equilibrium position (x = 0), the particle momentarily stops and changes direction. This means the velocity is zero at this point. As the particle moves away from the equilibrium position, it gains velocity and reaches its maximum value at the extreme positions of the oscillations.

c) Maximum at the extreme positions of the motion and minimum at x = 0: This option is incorrect. The velocity of the particle is maximum at x = 0 (equilibrium position) and minimum at the extreme positions of the motion.

d) Varies with time: This option is also correct. The velocity of the particle undergoing simple harmonic motion varies with time. It follows a sinusoidal pattern, changing from positive to negative as the particle moves back and forth.

In conclusion, the correct options are b) Maximum at x = 0 and minimum at the extreme positions of the oscillations, and d) Varies with time. The velocity of a particle undergoing simple harmonic motion is not constant but instead changes with time. It is maximum at the equilibrium position and minimum at the extreme positions of the motion.