Suppose a 64-lb object stretches a spring 8 feet while in equilibrium, and a dashpot provides a damping force of 3 lbs for every foot per second of velocity. The form of the equation of
unforced motion of the object in such a spring-mass system is $my'' + cy' + ky = 0$, where $m$ is the mass of the object, $c$ is the damping constant, and $k$ is the spring constant. Assume that the
spring starts at a height of 1 feet below the horizontal with an upward velocity of -3 feet per second. Use 1 slug = 32 lbs.
Part 1
(i) For what values of $\alpha$ and $\beta$ is the function $y(t) = e^{\alpha t} (C_1 \cos(\beta t) + C_2 \sin(\beta t))$ the general solution of the equation of motion for this spring-mass system? Round your answers to 3
decimal places.
$\alpha = $ ______ and $\beta = $ ______
Part 2
(ii) For what values of the arbitrary constants $C_1$ and $C_2$ does the general solution in (i) satisfy the initial conditions? Round your answers to 3 decimal places.
$C_1 = $ ______ and $C_2 = $ ______
Part 3
(iii) What is the amplitude, R, (in feet) of the solution curve? Round your answer to 3 decimal places.
$R = $ ______ feet
