A skateboarder with his board can be modeled as a particle of mass 78.0 kg, loca
ID: 2144658 • Letter: A
Question
A skateboarder with his board can be modeled as a particle of mass 78.0 kg, located at his center of mass (which we will study in a later chapter). As shown in the figure below, the skateboarder starts from rest in a crouching position at one lip of a half-pipe (point ). The half-pipe is one half of a cylinder of radius 6.90 m with its axis horizontal. On his descent, the skateboarder moves without friction so that his center of mass moves through one quarter of a circle of radius 6.40 m. Find his speed at the bottom of the half-pipe Immediately after passing point , he stands up and raises his arms, lifting his center of mass from 0.500 m to 0.870 m above the concrete (point ). Next, the skateboarder glides upward with his center of mass moving in a quarter circle of radius 6.03 m. His body is horizontal when he passes point , the far lip of the half-pipe. As he passes through point , the speed of the skateboarder is 4.66 m/s. How much chemical potential energy in the body of the skateboarder was converted to mechanical energy in the skateboarder-Earth system when he stood up at point ? How high above point does he rise? Caution: Do not try this stunt yourself without the required skill and protective equipment.Explanation / Answer
Draw a horizontal line across the half pipe. As I understand the problem, that line passes through the skateboarders center of mass ===> The Potential Energy at start and finish is equal.
Conservation of Energy assures that energy conversion is "path independent." ====> It does not matter what the skateboarder does between point A and Point D, all that matters is the start(A) and end (D) state. ====> We need only calculate the Kinetic Energy to find how much energy the skateboarder put into the system.
KE = 0.5*MV^2
KE = 0.5 * 79 kg * (4.66)^2
KE = 39.5 * 21.72 = 857.77 J
EDIT: I can see novice's point. But, I still argue that the book is wrong. The only source of energy being added to the system is the chemical energy the skateboard's muscles. The total energy in excess of the PE at the start and end is the KE and that is 857.77 J. The only source of that energy is the skateboard's muscles and it is all there at the top of the half pipe as KE. The problem states we can neglect friction. Using novice's number, only 286.7 J was added by the skateboarder standing ====> The velocity of 4.66 m/s at the top is a typo. The velocity should be less ~ 2.70 m/s.