The displacement of a particle is represented by the equation y = sin3 ωt. The motion is a) non-periodic b) periodic but not simple harmonic c) simple harmonic with period 2π/ω d) simple harmonic with period π/ω
Correct option is B) Given the equation of displacement of the particle, y=sin3ωt We know sin3θ=3sinθ−4sin3θ Hence, y=4(3sinωt−4sin3ωt)⇒4dtdy=3ωcosωt−4×[3ωcos3ωt]⇒4×dt2d2y=−3ω2sinωt+12ωsin3ωt⇒dt2d2y=4−3ω2sinωt+12ωsin3ωt⇒dt2d2y is not proportional to y. Hence, the motion is not SHM. As the expression is involving sine function, hence it will be periodic. Also sin3ωt=(sinωt)3=[sin(ωt+2π)]3 =[sin(ωt+2π/ω)]3 Hence, y=sin3ωt represents a periodic motion with period 2π/ω.
