Answer to Question #84261 in Classical Mechanics for Robin

Question

To a massless spring of length LO a mass m is hung, which causes a periodic change in the length L(t) of the spring.
(a) In case the suspended mass is released from rest with unstretched spring [L(t=0)=L0}, determine the change in the length L(t) as a function of time and set L(t) in a diagram.

Answer 1

Since the mass is hung, the spring will be stretched as soon as the mass is set free, and according to Hooke's law:

$$ΔL=\frac {mg}{k}.$$

This is a magnitude of oscillations also. Thus, the function length from the ceiling to the mass vs. time is

$$L(t)=L_0+ΔL\cdot sin(ωt),$$ $$ω=\sqrt{\frac{k}{m}},$$ $$L(t)=L_0+\frac {mg}{k}\cdot sin(\sqrt{k/m}\cdot t).$$