1) Consider a mass spring system with no damping and external forceF(t)=F0cos0 t. Let's assume m is the mass of the object attached to the spring and k is the spring constant. Let's also assume that x(0) and x'(0) are the location and speed of the object attached to the spring at time t = 0.

a) Find the general solution of the differential equation governing the motion of the spring, in terms of m, k, x(0), x'(0),F0 and0.

b) Assign your own values for m, k, x(0) and x'(0), andF0,and plot the location and speed of the object versus time for four different values of0. Explain the behavior for different values of0.

2) Now, assume that there is damping force proportional to the speed of the object, and the damping coefficient is.

a) Find the general solution of the differential equation governing the motion of the spring, in terms of m, k,,x(0), x'(0),F0 and0. Your solution shall cover all three cases for the solution.

b) Assign your own values for m, k,x(0), x'(0),F0 and0,and plot the location and speed of the object versus time for four different values of. The selected values ofshall cover all three possible cases for the solution.

c) Plot the magnitude of the amplitude of the particular solution versus0 for four different values of and find out0that gives the highest amplitude.