(Solved): Consider a mass M hanging vertically by two identical springs. The springs have spring constants k ...

Consider a mass $$M$$ hanging vertically by two identical springs. The springs have spring constants $$k$$ and $$2k$$, and both have nominal lengths of $$l$$ when there are no forces on them. The top and bottom walls are $$2l$$ apart. Ignore damping, and assume the mass only moves along the vertical direction. (1) What is the equilibrium position of the mass? Clear state or label your coordinate system. $$(5\mathrm{pts})$$ (2) The mass is initially at rest at the equilibrium position. At $$t=0$$, the connection between the mass and the bottom spring breaks, and the mass starts to oscillate. Calculate the angular frequency and the period of oscillation. (5 pts) (3) What is the vertical displacement of the mass $$y(t)$$ at times $$t>0$$ ? (5 pts) (4) What is the kinetic and potential energy of the mass as a function of time? (5 pts)