Chapter 4.4, Problem 31P

In the absence of damping, the motion of a spring-mass system satisfies the initial value problem

$\begin{array}{ccc}my\text{'}\text{'}+ky=0,& y\left(0\right)=a,& y\text{'}\left(0\right)=0\end{array}$.

Show that the kinetic energy initially imparted to the mass is $m{b}^2/2$ and that the potential energy initially stored in the spring is $k{a}^2/2$, so that initially the total energy in the system is $\left(k{a}^2+m{b}^2\right)/2$.

Solve the given initial value problem.

Using the solution in part(b), determine the total energy in the system at any time $t$. Your result should confirm the principle of conservation of energy for this system.