5. (5 points) The simplified pitch dynamics of a rocket are given by J = k k2 sin(0) + I + T sin() - where 0 is the pitch angle, T is thrust, and is the engine gimbal angle. The parameters of the model are J, k, k2, and I. For the questions below assume thrust T is constant, the input to the system is the engine gibmal angle , and the output is the pitch angle 0. (a) Linearize the above equations of motion to obtain a linear differential equation. (b) Find the transform P(s) = (s)/B(s) for 0(0) = 0 (0) = 0 and (0) = (0) = 0 where e(s) = L[0] and B(s) L[B]. (c) What is 0 (t) when (t) = sin(t)? Use the following (unitless) model parameters in your analysis J = I = T = 3 and initial conditions from (b). Explain what is happening physically and why your answer is peculiar. 1, k = 4, k2 = Hint: With a sinusoidal input (t) what would you expect to see in the output (t)?