The equation of motion of a particle started at t = 0 is given by x = 5 sin (20 t + π/3), where x is in centimeter and t in second. When does the particlea) first come to restb) first have zero accelerationc) first have maximum speed?

Sol. . x = 5 sin (20t + π/3) a) Max. displacement from the mean position = Amplitude of the particle. At the extreme position, the velocity becomes ‘0’. ∴ x = 5 = Amplitude. ∴ 5 = 5 sin (20t + π/3) sin (20t + π/3) = 1 = sin (π/2) ⇒ 20t + π/3 = π/2 ⇒ t = π/120 sec., So at π/120 sec it first comes to rest. b) a = ω2x = ω2 [5 sin (20t + π/3)] For a = 0, 5 sin (20t + π/3) = 0 ⇒ sin (20t + π/3) = sin (π) ⇒ 20 t = π – π/3 = 2π/3 ⇒ t = π/30 sec. c) v = A ω cos (ωt + π /3) = 20 × 5 cos (20t + π/3) when, v is maximum i.e. cos (20t + π/3) = –1 = cos π ⇒ 20t = π – π/3 = 2 π/3 ⇒ t = π/30 sec.