Question

• Jacob is standing in the middle of a circular boat of radius l. He throws the shark that landed on Aaron’s head in the last review sheet back into the water. The shark’s mass is m. The mass of the boat and everyone on it is M. If the boat shifts a distance d as a result of the throw, how far did he throw the shark? What is the farthest the boat can shift such that the shark won’t fall off the boat?

• What is the center of mass of a thickening stick of length L, if λ(x) = kx is its density function?

• What is the center of mass of a cone with a circular base relative to its height L? How about a square-based pyramid of height L? How do these compare to the answer for a thickening stick in the previous question?

• Where is the center of mass of a semicircle, assuming constant density? How about a quarter-circle? What about 1/N of a circle? What happens when N → ∞?

• If a block of metal has density function ρ(x, y, z) = x + 3y + 2z 2 , where is its center of mass? (hint: consider the center of mass along each axis separately)

Answer 1