A particle executes simple harmonic motion described by the equation x = A ( t + ) , where the initial phase is in the interval [0, 2 ] . The particle first comes to rest at t = 1 s, and its acceleration first becomes zero at t = 4 s. The time period T and the initial phase of the motion respectively are

Step-by-step solution

The equation of motion is x = A ( t + ) . The particle comes to rest when its velocity is zero, which occurs at the extreme positions where the phase is 2 , 3 2 , etc. Since [0, 2 ] , the first time it comes to rest is at phase 2 . (1) + = 2 ... (i) The acceleration becomes zero when the particle passes through the mean position, which occurs at phase , 2 , etc. The first time this happens after t=1 s is at phase . (4) + = ... (ii) Subtracting equation (i) from (ii): 3 = - 2 = 2 = 6 rad/s The time period is T = 2 =