A body of mass m kg attached to a spring moves with friction. The motion is described by the second Newton's law: m(d^2y/dt^2) + a(dy/dt) + ky = 0 where y is the body displacement in m, t is the time in s, a > 0 is the friction coeffcient in kg/s and k is the spring constant in kg/s2. Assuming m = 1 kg and k = 4 kg/s2,
find: a) What is the range of values of a for which the body moves
(i) with oscillations,
(ii) without oscillations?
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 (i.e. when t --> infinity)?
c) Find the particular solution for a = 4 subject to the initial conditions y(0) = 0, dy/dt = 1 m/s at t = 0. Plot this solution and determine the largest displacement of the mass using calculus.