Maximum Velocity in Simple Harmonic Motion

In simple harmonic motion (SHM), a particle oscillates back and forth around a fixed equilibrium position. The motion is characterized by a restoring force that is proportional to the displacement from the equilibrium position. The equation of motion for SHM is given by:

x = A sin(ωt + φ)

where:
- x is the displacement of the particle from the equilibrium position
- A is the amplitude of the motion
- ω is the angular frequency of the motion
- t is the time
- φ is the phase constant

The velocity of the particle is the derivative of the displacement with respect to time:

v = dx/dt = Aω cos(ωt + φ)

Understanding the Equation
- The amplitude A represents the maximum displacement of the particle from the equilibrium position.
- The angular frequency ω determines how fast the particle oscillates. It is related to the period T of the motion by the equation ω = 2π/T.
- The phase constant φ represents the initial phase of the motion.

Maximum Velocity
To find the maximum velocity, we need to determine the maximum value of the velocity function v. Since the maximum value of cosine function is 1, the maximum velocity occurs when cos(ωt + φ) = 1. This happens when ωt + φ = 0 or ωt + φ = 2π.

Angular Frequency
From the equation of SHM, we can see that the angular frequency ω determines the frequency of the oscillation. Higher values of ω correspond to faster oscillations. Therefore, the option ωr (option B) is the correct answer as it represents a higher angular frequency, which leads to a higher maximum velocity.

Explanation of Other Options
- Option A (ω): This option represents the angular frequency, which determines the frequency of the oscillation but not the maximum velocity.
- Option C (ω^2): This option represents the square of the angular frequency, which is not directly related to the maximum velocity.
- Option D (ω/r): This option represents the ratio of the angular frequency to the amplitude, which is not directly related to the maximum velocity.