Chapter 4.3, Problem 33E

Vibrating Spring with Damping. Using the model for a vibrating spring with dampening discussed in Example 3.

a. Find the equation of motion for the vibrating spring with dampening if $m=10\text{ kg}$, $b=60\text{ kg/sec}$, $k={\text{250 kg/sec}}^{\text{2}}$, $y\left(0\right)=0.3\text{ m}$, and $y\left(0\right)=-0.1\text{ m/sec}$.

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

c. Find the frequency of oscillation for the spring system of part (a). [Hint: See the definition of frequency given in Problems 32(c).]

d. Compare the results of Problems 32 and 33 and determine what effect the dampening has on the frequency of oscillation. What other effects does it have on the solution?

32. 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 (9), the motion is said to be oscillatory with frequency $\beta /2\pi$. Find the frequency of oscillation for the spring system of part (a).