A Forced Spring System Later in the semester, we will see that the motion of a damper], forced spring like that shown in Figure 1 can be modelled by the initial value problem d2}; dy . k =Fi' 1 md+r~dt+y o (} emgj euwu. J Pure ill-E) where yt) is the position of the mass at the end of the spring (relative to its resting place}, Ft} is the force that \"\"351 alt-raga , spring is is applied to the mass. m is the mass. c is the damping ' intrhoJ moss m coefcient and h is the spring constant. during-n3 r C) The initial conditions for the problem are at\") = in: and ") = in In this project, weill assume the forcing term is oscillatory so that F} 2 F0 cos(wt) where n' is the forcing frequency.r and F]; is a constant. This project has 4 parts. Part 1: An undamped spring with external forcing For an undamped spring. the spring constant is c = [1 so that the model equation reduces to a!\" Indg + a = so} (2} where Fm is as given above. in this section._ we'll use the initial conditions y} = and {HO} 2 (3} Question 1 Carefuliy expioin what each of the initial conditions in equation [3} mean. What is the behaviour of the moss when t = 0? Later in the semester: we'll learn how to solve these types of equations. The solution is y(t} = % (cosm cos ( $ t)) (4} For the following questions. you should not substitute any values for the parameters. Question 2 Find the derivative, y'(t), of the solution given in (4). What does the derivative tell you about the motion of the mass? Question 3 By substituting solution (4) into the model equation (2), show that this is indeed a solution, as long as w # 5. You should also show that both initial conditions are satisfied