Question Detail
Question
The displacement for a particle performing S.H.M. is given by x = A cos(αt + φ). The initial position of the particle is 1 cm and its initial velocity is π cm/s. If the angular frequency of the particle is π rad/s, what will be its initial phase?
Correct Answer
Option D is the correct answer.
Detailed Explanation
At t=0: x(0) = A cos φ = 1. v = dx/dt = -Aω sin(ωt + φ). At t=0: v(0) = -Aω sin φ = π. A = 1/ cos φ. ω = π. v(0) = -(1/cos φ)(π) sin φ = -π tan φ = π => tan φ = -1. φ = 3π/4 or 7π/4. Check x(0) = A cos φ = 1. If φ = 3π/4, cos(3π/4) = -√2/2, so A = -√2, not positive. If φ = 7π/4, cos(7π/4) = √2/2, A = √2, and v(0) = -π tan(7π/4) = -π(-1) = π. So φ = 7π/4.
Hint
Use the initial conditions to solve for the phase constant.