A particle of mass m moves along a straight frictionless track under the influence of a restoring force that produces an acceleration a(x)=−kx, where k is a positive constant and x is the displacement from equilibrium.
At time t=0, the particle is released from rest at position x=L>0.
(a) Derive an expression for the particle’s velocity v as an explicit function of its position x as it moves toward the equilibrium point. Express your result in terms of k, L, and x.
(b) Obtain a formula for the time t required for the particle to travel from x=L to x=0. Show all steps in your derivation.
(c) Discuss whether the motion of the particle is periodic. If it is periodic, justify your conclusion and determine the period; if not, explain why the motion fails to be periodic.