Question

A body of mass m is suspended from a spring fixed to the ceiling of an elevator car. The stiffness of the spring is x. At the moment t = 0 the car starts going up with an acceleration ω. Neglecting the mass of the spring, find the law of motion y(t) of the body relative to the elevator car if y(0) = 0 and y(0) = 0. Consider the following two cases:

(a) ω = const;

(b) ω = αt, where α is a constant.

Answer 1

(a) As the elevator car is a translating non-inertial frame, therefore the body m will experience an inertial force mw directed downward in addition to the real forces in the elevator' frame. From the Newton' second law in projection form

Fy = mwy for the body in the frame of elevator car:

(Because the initial elongation in the spring is mg/k)

Eqn. (1) show that the motion of the body m is *.H.M and its solution becomes

(b) proceed up to eqn. (1). The solution of this differential eqn be of the form: