Please explain and show your work neatly so that I can understand thank you.

33. A uniform rod of length L and mass M is allowed to rotate freely about a frictionless pin mounted as shown in the figure. It is released from rest at a horizontal position. a) What is the initial potential energy of the system? (Take the configuration where the rod is hanging straight down as the zero potential energy reference ). b) What is the kinetic energy of the system when it reaches the vertical configuration C) What is the angular velocity of the rod at the vertical configuration? d) What is the speed of the centre of mass when it passes the lowest point in its trajectory?L O CM . CM34. Two astronauts with mass 100 kg are connected by a 10 m rope. They are isolated in space and are rotating about their centre of mass at a speed of 1 m/S. a) What is their combined angular momentum? b) What is the rotational energy of the system? c) If they pull the rope to shorten the distance between them to 5 m, what is the new angular momentum of the system? d) What is their new angular velocity? e) What is the work done by the astronauts to shorten the rope?B A = 30 C 5 / L fo 1/6 L- W 2 35. Weight of the uniform rod is w1=100 N. w2=200 N. L=3 m. a) Draw a free body diagram of the rod involving forces w1, w2 and forces of support at A and C. b) What are the unknown variables in the problem? c) Generate two equations by balancing forces on the rod in horizontal and vertical directions. d) Generate one equation using the net torque about A. e) Solve the above three equations to obtain the force exerted on the rod at the hinge.36 A sailboat starts from the beach and travels 20 km at an angle of 45 degrees south of east. They then turn and travel an additional 10 km at an angle of 30 degrees north of east. (Consider the initial point as the origin of the coordinate system, the beach along the x-axis and east pointing along the positive x- axis.) (a) Draw a diagram of the route. (b) How far does the boat travel in the x direction during the first stage? (c) How far does the boat travel in the y direction during the second stage of the trip? (d) Express the final displacement of the boat in terms of its x and y components. (e) Determine the final displacement of the boat from the beach in both magnitude and direction