A \(10\)-kg mass is hung on a very large undamped vertical spring and brought to rest at equilibrium, stretching the spring by \(39.2\) centimeters. Then an external upward force of \(F(t)=10t\) N is applied at time \(0\leq t<10\) seconds, followed by no external force (free motion).

(a)
Using \(g=9.8\) m/s\(^2\) for the magnitude of the acceleration due to gravity, let’s determine the spring stiffness \(k\) in N/m.
(b)
Let’s derive an IVP satisfied by \(y(t)\), the displacement of the mass above equilibrium (in meters) at time \(t\geq 0\) seconds.
(c)
Let’s solve this IVP and give the solution as a piecewise defined function.
The displacement of a mass from equilibrium in a Spring-mass system with piecewise continuous forcing. [Picture]
Figure 1: Spring-mass system with piecewise continuous forcing