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Homework Statement
Consider the illustration of 3 springs:
In A, we hang a very light spring and pan from a hinge. The pan and spring are so light, we can neglect any stretching of the original length ##l_{0}##. In B we add a weight ##mg## which force is balanced by ##kl## (Hooke's Law; the spring tension is proportional to the displacement ##l##); i.e. ##mg=kl##.
In C, we displace the pan by a further ##y(0) = y_{0}## and give the pan an initial velocity ##v_{0} = \frac{dy}{dt}|_{t=0}##.
Let ##y(t),v(t)=\frac{dy}{dt}## be the displacement and velocity of the pan. ##y(t)## satisfies the equation (Newton's 2nd Law): ##m\frac{d^2y}{dt^2}=mg-k(y+l)-B\frac{dy}{dt}##
or, since ##mg = kl##, and dividing across by ##m##,
##(1) ## ##\frac{d^2}{dt^2}+\frac{B}{m}\frac{dy}{dt}+\frac{k}{m}y=0##
Given ##\frac{B}{m} = 4(sec)^{-1}## and ##\frac{k}{m} = (a.) 3, (b.) 4, (c.) 7##,
with ##y(0) = 1, \frac{dy}{dt}(0) = 0##, solve ##(1)## for the three cases, ##a, b, c##.
Write the solution in each case for ##y(t), v(t)##.
Homework Equations
##y(0) = 1##
##\frac{dy}{dt}(0) = 0##
The Attempt at a Solution
I think I've solved two of the cases they wanted me to solve, but I'm stuck on this part:Case ##(a.)##
As ##t→∞##, the vector ##\left(\frac{y(t)}{v(t)}\right)## approaches ##\left(\frac{0}{0}\right)## along what direction?
Could someone explain what this means? What does 'approaches along what direction' mean? The three possible answers are ##\left(\frac{1}{1}\right), \left(\frac{1}{-1}\right), \left(\frac{-1}{3}\right)##. (They aren't fractions, I just don't know how to put a vertical vector in a post using LaTex)