Chapter 3.7, Problem 30P

(a)

To determine

To prove: if the motion of a spring–mass system satisfies the initial value problem $m{u}^{″}+ku=0$, $u\left(0\right)=a,u^{\prime }\left(0\right)=b$ in the absence of damping then the kinetic energy initially imparted to the mass is $\frac{m{b}^2}{2}$, the potential energy initially stored in the spring is $\frac{k{a}^2}{2}$ and the total energy in the system is $\frac{k{a}^2+m{b}^2}{2}$.

(b)

To determine

The solution of the initial value problem $m{u}^{″}+ku=0$, $u\left(0\right)=a,u^{\prime }\left(0\right)=b$.

(c)

To determine

The total energy in the system at any time t.