A point performs simple harmonic oscillation of period T and the equation of motion is given by x = asin(ωt + π6). After the elapse of what fraction of the time period the velocity of the point will be equal to half of its maximum velocity?

Velocity is the time derivative of displacement.Writing the given equation of a point performing SHMx = a sin(ωt+π6)-------(i)Differentiating Eq.(i) w.r.t time, we obtainv = dxdt = a ω cos(ωt+π6)It is given that v = aω2, so thataω2 = a ω cos(ωt + π6)or 12 = cos(ωt+π6)or cosπ3= cos(ωt+π6)or ωt + π6 = π3  ⇒ ωt = π6or t = π6ω= π×π6×2π = T12Thus, at T12 velocity of the point will be equal to half of its maximum velocity.