Please help answer the following questions

Instructions Complete these problems on a separate sheet of paper (or on a tablet/iPad). Your work should be neat and the problems should be completed in the correct order. Submit the assignment as a PDF on Blackboard. . You may use a calculator or online app to find the zeros of the characteristic equation. Make sure that your answers includes the characteristic polynomial p(r) and its roots. Do not approximate r. . You may use your calculator or an online app to find the coefficients. Make sure that your answer includes the linear system you used to find your coefficients. Problems 1. Consider the spring-mass system: 0.1x" + 10x = 3 cos(wt), x(0) = 0, x'(0) = 0 (a) Solve for x(t) when w = 11. (b) Rewrite your answer using the trig identity: cos(B) - cos(A) = 2 sin (A, B) sin( (AtB) (c) Sketch the graph of your solution. 2. Consider the spring-mass system (from Problem 1): 0.1x" + 10x = 3 cos(wt), x(0) = 0, x'(0) =0 (a) For which value of w does resonance occur? (b) Solve for a(t) using your value of w from (a). (c) Sketch the graph of your solution. 3. A 6-kg mass is attached to a spring with spring constant k = 6 N/m. Suppose the system is immersed in a medium with damping coefficient c = 8 N.s/m and an external force F(t) = 8 cos(wt) N acts on the system. (a) Find the steady-state solution Up when w = 3 and identify the amplitude C(3). (b) Identify the practical resonance frequency Wmax. (c) Identify the practical resonance amplitude C(Wmax). Scanned with CamScanner