(1 point) Suppose a pendulum of length L meters makes an angle of 6 radians with the vertical, as in the figure. It can be shown that as a function of time, @ satisfies the differential equation d*0 g + =sinf =0 dt2 L ' where g = 9.8 m/s? is the acceleration due to gravity. For @ near zero we can use the linear approximation Sin() ~ 6 to get a linear differential equation d?6 48 99, dt? 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 df = 0.2 radians per second. dt o(t) = 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" +cy' +ky=F(t), y(0)=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 n/2 F(t) = { a. Solve the initial value problem, using that the displacement y(t) and velocity y ' () remain continuous when the applied force is discontinuous. For0