1. Problem 1: Consider a weakly damped oscillator with mass m, natural frequency (on, and damping constant l3=muf3, Suppose the driving force per unit mass, t], is given by the function f(ti:sin[3wot), so obviously fo=1, a. Find on, the transient frequency, and the large time amplitude A and phase: 5 of the system b. What is x(t] at large times after the transient term has gone to zero? Problem 2: this is a variant of Problem 1, but do not use Fourier techniques because we now want to describe the motion starting from t:0. After t:[}, the mass is subject to a force, f(t] is given by fit): cos3itoot]. The system is underdamped with same damping constant as in problem 1. a. The usual way to start a problem where the forcing function is a power of sin or cosine is to convert to a function of a single power of cosine but with terms of different frequencies. This way, you can use what you have learned in class directly. One way to convert is to write cositpoti as (1,1'2)[eitut + e'\"\"\"']. Then cube it, where (A+B}3 = A3+3AZB+3ABE+B3, and recall for example {eiei3=e'39, (one of the mathematical reasons I like exponentials), and re-arrange to find a function with two terms: coslcoot), cos(3motl. Determine the new format of the forcing term. b. Find the complete solution of xlt) for t>0. Hint: find A and 8 for each forcing term. Then use principle ofsuperposition. What does this mean? The transient solution for either forcing term is the same, and it contains the two unknown constants. You then add the long term solution for each of the forcing terms to the transient solution. You do not need to solve for initial conditions