Consider a vertical mass-spring system as shown in the figure below....

Consider a vertical mass-spring system as shown in the figure below.

Before the mass is placed on the end of the spring, the spring has a natural length. After the mass is placed on the end of the spring, the system has a new equilibrium position, which corresponds to the position where the force on the mass due to gravity is equal to the force on the mass due to the spring.

(a) Assuming that the only forces acting on the mass are the force due to gravity and the force of the spring, formulate two different (but related) second-order differential equations that describe the motion of the mass. For one equation, let the position y₁(t) be measured from the point at the end of the spring when it hangs without the mass attached. For the other equation, let y₂(t) be measured from the equilibrium position once the mass is attached to the spring.

(b) Rewrite these two second-order equations as first-order systems and calculate their equilibrium points. Interpret your results in terms of the mass-spring system.

(c) Given a solution y₁(t) to one system, how can you produce a solution y₂(t) to the second system?

(d) Which choice of coordinate system, y₁ or y₂, do you prefer? Why?