Chapter 4.1, Problem 28P

(a) Numerical simulations as well as intuition based on an understanding of the origin and significance of each term in the equation suggest that solutions of theundamped spring-mass equation

$my\text{'}\text{'}+ky=0$ (i)

are oscillatory. Use simulations to determine a value of $r$ such that $y_1=\cos rt$ and $y_2=\sin rt$ are both solutions of Eq.(i). Then show that $y=c_1{y}_1+c_2{y}_2$ is also solution of $my\text{'}\text{'}+ky=0$ for any constants $c_1$ and $c_2$.

(b) Use the results of part (a) to find analytical solutions of the following initial value problems for $my\text{'}\text{'}+ky=0$:

i. $m=1\text{kg},k=1{\text{kg/s}}^{\text{2}},y\left(0\right)=1\text{m},y\text{'}\left(0\right)=-1\text{m/s}$ (weak spring)

ii. $m=1\text{kg},k=25{\text{kg/s}}^{\text{2}},y\left(0\right)=1\text{m},y\text{'}\left(0\right)=-1\text{m/s}$ (stiff spring)

(c) Plot graphs of the two solutions obtained above on the same set of coordinate axes and discuss the amplitude and frequency of oscillations.