Answer Happy

Q4) (40 points) Consider the forced pendulum in the figure below. We will neglect the weight of the bar, but not the mass at the end, we treat friction as a damping term, and assume the change in angle is small so that we can approximate.sin(0) = 0. The equation of motion of this pendulum is then given by ml²0+ ye+mgle = M₁(t) Suppose m = 2 kg, l = 0.5 m, y = 0.5 kg-m²/s, g = 1 m/2 (we are on some other planet)

0(1) M,() m (a) (2pts) What are the dependent and independent variables? (b) (4pts) How many initial conditions are needed? What are the initial conditions? (c) (5pts) At some time, say t = 0, the pendulum hangs perfectly vertical, and we use a hammer to impart an instantaneous change in the tangential velocity of the mass to say, 0.5 m/s. Express this mathematically as initial conditions. (d) (10pts) Find the homogeneous solution, x (t). (e) (10pts) Suppose M, (t) = t² + 4t+3. Find the particular solution xp (t). (f) (9pts) Write down the complete solution subject to the following initial conditions: 8 (0) = 0 and 0 (0) = 0