Problem Set 5

Physics 5200

Due 2 Mar. 1999

1. (2 points) Consider a rope of length l and mass m (mass per unit length =m/l), lying on a flat surface. Calculate the work required to lift the rope by one end, such that it hangs vertically and the other end is just in contact with the surface. Assume that the rope is lifted extremely slowly, such that at each instant, the force exerted is barely greater than force required to hold the rope steady. Compare this result with the amount of work required to lift a mass m from the surface to the height of the center of mass of the rope (you will have to "find" the center of mass of the rope).
2. (3 points) A block of wood rests on a smooth horizontal table. A gun is fired horizontally at the block and the bullet passes through the block, emerging with half its initial speed just before it entered the block. Show that the fraction of the initial kinetic energy of the bullet that is lost as frictional heat is 3/4 - g /4 where g is the ratio of the mass of the bullet to the mass of the block (g <1).
3. (3 points) Particles of mud are thrown from the rim of a rolling wheel. If the forward speed of the wheel is v₀, and the radius of the wheel is b, show that the greatest height above the ground that the mud can go is
   b + v₀²/2g + gb²/2v₀²
   At what point on the rolling wheel does this mud leave? (Note: It is necessary to assume that v₀²³ bg.)
4. (2 points) A small cube of mass m slides down a circular path of radius R cut into a larger block of mass M, as shown below. M rests on a table and both blocks move without friction. The blocks are initially at rest, and m starts from the top of the quarter circle. Find the velocity v of the cube as it leaves the block.