(1 point) Suppose a pendulum of length L. meters makes an angle of 8 radians with the vertical, as in the figure. It can be shown that as a function of time, @ satisfies the differential equation d?6 29 - doingu; dt? L where g = 9.8 m/ s? is the acceleration due to gravity. For @ near zero we can use the linear approximation sin(f) ~ 6 to get a linear differential equation d*0 \" d9-0. dt2 L Use the linear differential equation to answer the following questions. (a) Determine the equation of motion for a pendulum of length 0.5 meters having initial angle 0.4 radians and initial angular velocity 9 dt = 0.2 radians per second. 0(t) = ' 0.452cos(sqrt(9.8/0.5 1)) + 0.452sin(sqrt(9.8/0.5t)) radians (b) What is the period of the pendulum? That is, what is the time for one swing back and forth? Period = 1.418 | seconds (1 point) Consider the initial value problem my\" +ecy' +ky=F(t), y0) =0, y'(0)=0 modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F'(t), where the unit of force is the Newton (N). Assume that m = 2 kilograms, = 8 kilograms per second, k = 80 Newtons per meter, and the applied force in Newtons is 40 if0 /2. a. Solve the initial value problem, using that the displacement y() and velocity y ' () remain continuous when the applied force is discontinuous. For0 /2, y(t) = | | help (formulas) b. Determine the long-term behavior of the system. Is lim y() = 07 If it is, enter t00 zero. If not, enter a function that approximates y(t) for very large positive values of . For very large positive values of , y(t) o~ 0 help (formulas)