PRACTICE PROBLEM 1

Consider a spring with a weight attached to it set in motion by an external force given by f(t) = cos(t). In this problem we will neglect damping and are given that the differential equation governing the position of the weight attached to the end of the spring is given by

where k > 0 is related to the spring constant described in Hooke’s Law.

State the analytical solution to the differential equation for k equal to one and for k not equal to one (k not equal to one means give the solution in terms of k) assuming that the weight starts out at rest and in the equilibrium position, i.e.,

Describe in words what is happening as k gets very close to one.

In the animation given below the graph in red corresponds to k = 1 and the animated graph in blue corresponds to k varying from 0 to 2.

Here is the animation and a Quicktime version of the same animation.