Hi all, this is my first time posting here. I'm in diff eq at uni, and so far things have been going smoothly, except for this section! I don't have a strong physics background, so maybe that's what's tripping me up? In any case, I could really use some help with these problems:
Problem: A mass weighing 3 lb stretches a spring 3 in. If the mass is pushed upward, contracting the spring, and then set in motion with a downward velocity of 2ft/sec, and if there is no damping, find the position u of the mass at any time t. Determine the frequency, period, ,amplitude, and phase of the motion.
Answer: u = (1/4 (root 2))sin(8(root 2t)) - (1/12)cos(8(root 2t) ft, t in seconds;
omega = 8(root 2) radians/seconds
T = (pi/(4 root 2)) seconds
R = (root (11/288)), or approx .1954 ft
delta = pi - arctan (3/(root 2)), or approx. 2.0113
Problem: Assume that the system described by the equation mu" + &u' + ku = 0 is either critically damped or overdamped. Show that the mass can pass through the equilibrium position at most once, regardless of the initial conditions.
(The hint they give: determine all possible values of t for which u = 0)
Problem: Assume that the system described by the equation mu" + (gamma)u' + ku = 0 is critically damped and that the initial conditions are u(0) = u(sub)0, u'(0) = v(sub)0. If v(sub)0 = 0, show that u approaches 0 as t approaches infinity, but that u is never zero. If u(sub)0 is positive, determine a condition on v(sub)0 that will lensure that the mass passes through its equilibrium position after it is released.
answer: v(sub)0 < -(gamma)u(sub)0/2m
Problem: The position of a certain spring-mass system satisfies the intiial value problem:
(3/2)u" + ku = 0, u(0)=2, u'(0)=v
If the period and amplitude of the resulting motion are observed to be pi and 3, respectively, determine the values of k and v.
answer: k = 6, v = + or - 2(root 5)
Problem: A cubic block of side l and mass density p per unit volume is floating in a fluid of mass density p(sub)0 per unit volume, where p(sub)0 > p. If the block is slightly depressed and then released, it oscillate in the vertical direction. Assuming that the viscous damping of the fluid and air can be neglected, derive the differential equation of motion and determine the period of the motion.
(The hint they give: Use Archimedes' principle: An object that is completely or partially submerged in a fluid is acted on by an upward (buoyant) force equal to the weight of the displaced fluid).
answer: (p)(l)(u") + (p(sub)0)(g)(u) = 0
Any help given would be appreciated. I don't really understand how this relates, and I've yet to take a course in calc based physics (or any physics, for that matter). And I don't just need the answers (they're all in the back of the book), but the process to get to there.
(oh, and I'm not sure how to make any Greek symbols or subscript or anything, sorry about that, if it's confusing :-/ )