Chapter 4.3, Problem 32E

Vibrating Spring without Damping. A vibrating spring without dampening can be modelled by the initial value problem (11) in Example 3 by taking $b=0$.

a. If $m=10\text{ kg}$, $k=250{\text{ kg/sec}}^{\text{2}}$, $y\left(0\right)=0.3\text{ m}$, $y^{\prime }\left(0\right)=-0.1\text{ m/sec}$, find the equation of motion for this undamped vibrating string.

b. After how many seconds will the mass in part (a) first cross the equilibrium point?

c. When the equation of motion is of the form displayed in $\left(9\right)$, the motion is said be oscillatory with frequency $\beta /2\pi$. Find the frequency of oscillation for the spring system of part (a).

Example 3

In Section 4.1 we discussed the mechanics of the mass-spring oscillator (Figure 4.1, page 152), and we saw how Newton’s second law implies that the position $y\left(t\right)$ of the mass $m$ is governed by the second order differential equation.

$m{y}^{″}\left(t\right)+b{y}^{\prime }\left(t\right)+ky\left(t\right)=0$, … $\left(11\right)$