Physics CS 33	Set # 1
Winter 2006
Due date	Wed. April 12th
Read HR&K:	Chapter 15
Read K&K:	None
HR&K Problems:	Chapter 15  Excercise 17,  Problems 1 (ignore part c),  3,  4,  9, 12
K&K Problems:	None
1.	A thin-walled cone (height H, base radius R) is filled to the top with water.  a) Find the total weight of the water in the cone. b) Find the total force exerted on the base of the cone by the water.  Hint: Is the water pressure near the base of the cone uniform, or is it stronger in the center? c) Based on your results for part b), what distance does the "h" in  P = ρ g h  refer to?
2.	Using subscript notation, prove: a) The curl of the gradient of any scalar field is zero. b) The divergence of the curl of any vector field is zero.
3.	A solid cube (side length 2.0 meters) of an unknown material floats in still water with half of its volume submerged. a) What is the mass of the object?   b) You push down on top of the object. How much force is required to have 60% of the object's volume submerged? c) With the object still 60% submerged and at rest, you stop pushing down.  Show that the cube's motion is simple harmonic, and find the amplitude and frequency.  Ignore any damping effects the water may have.
4.	Consider a spherical star of mass M and radius R with uniform density.  a) Find the potential φ (i.e. potential energy per unit mass) everywhere inside and outside the star as a function of r, the distance from the center of the star.  Assume the potential goes to zero infinitely far away from the star.   Hint: you will have to use the shell theorem. b) Use the hydrostatic equation to find the pressure as a function of r inside the star.  What is the pressure at the center of the star if it has the same mass and radius as the sun?
5.	Starting from the definition of the gradient in Cartesian ( x, y, z ) coordinates, use the chain rule to derive the gradient in cylindrical ( ρ , θ , z )  coordinates.  See the links from the class website to check your answer.