Question

The velocity and acceleration of a particle performing simple harmonic motion have a steady phase relationship. The acceleration shows a phase lead over the velocity in radians of

• A. \( +\pi \)
• B. 0
• C. \( +\pi/2 \)
• D. \( -\pi/2 \)
• E. \( -\pi \)

Hint:

In SHM, velocity leads displacement by π/2 radians, and acceleration leads velocity by π/2 radians. This means acceleration and displacement are out of phase by π radians (180°).

Answer 1

The Correct Option is C

Step 1: Define the equations of motion for a particle in SHM.
Let the displacement of a particle in Simple Harmonic Motion (SHM) be represented by: \[ x(t) = A \sin(\omega t + \phi) \] where \( A \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the initial phase.

Step 2: Derive the expressions for velocity and acceleration.
Velocity \( v(t) \) is the first derivative of displacement with respect to time: \[ v(t) = \frac{dx}{dt} = \frac{d}{dt} [A \sin(\omega t + \phi)] = A\omega \cos(\omega t + \phi) \] Acceleration \( a(t) \) is the first derivative of velocity with respect to time: \[ a(t) = \frac{dv}{dt} = \frac{d}{dt} [A\omega \cos(\omega t + \phi)] = -A\omega^2 \sin(\omega t + \phi) \]

Step 3: Express velocity and acceleration using the same trigonometric function to compare their phases.
We can use the trigonometric identities \( \cos(\theta) = \sin(\theta + \pi/2) \) and \( -\sin(\theta) = \sin(\theta + \pi) \). The velocity equation becomes: \[ v(t) = A\omega \sin(\omega t + \phi + \pi/2) \] The phase of the velocity is \( (\omega t + \phi + \pi/2) \). The acceleration equation becomes: \[ a(t) = A\omega^2 \sin(\omega t + \phi + \pi) \] The phase of the acceleration is \( (\omega t + \phi + \pi) \).

Step 4: Calculate the phase difference.
The phase difference is the phase of acceleration minus the phase of velocity. \[ \text{Phase Difference} = (\text{Phase of } a) - (\text{Phase of } v) \] \[ \text{Phase Difference} = (\omega t + \phi + \pi) - (\omega t + \phi + \pi/2) \] \[ \text{Phase Difference} = \pi - \frac{\pi}{2} = +\frac{\pi}{2} \] A positive phase difference means acceleration leads velocity.

Final Answer: (C) \( +\pi/2 \)