First, "LesleyIlic" post has nothing to do with this topic- it is spam. Moderators, please delete it and ban LesleyIlic.
A 3 lb weight has mass 3/32 (from f= ma again- weight is force and a= 32 ft/sec^2) so the differential equation involved is (the "-" is because if we stretch the spring, making x larger, the force is back against the stretch). The problem, as you wrote t, does not give an initial x value but putting t into the u gives x= -1/12. So apparently u(0)= -1/12 (one inch and x is in feet) and u'(0)= 2 ft/sec. Do you know how to solve u''= -12u with initial values u(0)= -1/12 and u'(0)= 2?
Can you find the general solution to the differential equation? Recall that for a second order linear equation with constant coefficients, if the solutions to the characteristic equation are then the general solution is of the for and will be "critically damped" if the characteristic equation has a "double" root- that is b= 0 and will be "overdamped" if the characterstic equation has two real roots.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)
Okay, if the equation is critically damped, the characteristic equation has a double root. What is the general solution to the differential equation? There is no "physics" involved here.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
The characteristic equation is which has roots so the general solution isProblem: 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)
Now u(0)= 2 and u'(0)= v let you solve for A and B (in terms of v). What are the period and amplitude of u, in terms of k and v? Set the equal to [tex]\pi[/itex] and 3, respectively, and solve for k and v. There is no "physics" involved here.
Let u be the depth to which the block sinks in the water. Then the water displaced (assuming u is not greater than l) is and the much water has weight and, by Archmede's principle, that is the upward force on the block: "mass times acceleration equals force" becomes . Can you solve that differential equation?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 :-/ )