Homogenous ODE Problem: Range and Solutions for Frictional Spring Motion

[Baartzy89]

Hi all,

I'm struggling with this question - I don't really know where to start. So far I have tried putting arbitrary values for 'a' into a quadratic auxiliary equation but using wolfram to calculate the roots gives me complex conjugates that I can't remember a thing about. Question as follows:

A body of mass 'm' kg attached to a spring moves with friction. The motion is described by the second Newton's law:

m(t).y" + a(t).y' + ky = 0

Where y is the body displacement in m, t is the time in s, a > 0 is the friction coefficient in kg/s and k is the spring constant in kg/s^2. Assuming m=1kg and k=4kg/s^2 find;

A) what is the range of values of a for which the body moves (i) with oscillations, (ii) without oscillation?

B) Find the general solution for any a<4 (Your solution should be a formula depending on the parameter a.) Proof that it follows from the solution obtained that the body slows down to a virtually rest state at large time (ie when t > infinity)?

C) find the particular solution for a=4 subject to the initial conditions y(0)=0, dy/dt= 1m/s at t=0. Plot this solution and determine the largest displacement of the mass usIng calculus.

 

[kurt.physics]

Thanks. </code>A:A) The range of values of a such that the body moves with oscillations is 0 < a < 4. The range of values of a such that the body moves without oscillations is a ≥ 4. B) The general solution for any a<4 is y(t) = c1e^(-(a/2m)t) + c2te^(-(a/2m)t). To prove that the body slows down to a virtually rest state at large time, note that as t → ∞, y(t) → c1, which is a constant. C) The particular solution for a = 4, subject to the initial conditions y(0) = 0 and dy/dt = 1 m/s at t = 0, is y(t) = (1/3)e^(-t/2) + (2/3)te^(-t/2). To determine the largest displacement of the mass, calculate y'(t) and set it equal to 0. Then solve for t to find the value of t that gives the largest displacement. Then plug this value of t into the equation for y(t) to find the largest displacement.