Physics 422 -- Fall 2006

Homework #1, due Wednesday Sep. 6 at beginning of class

1.         [3pts] In the Brachistochrone problem, the bead travels a distance A in the horizontal direction and B in the vertical direction. Find the range of A/B for which the fastest path goes below the level of the end point.

2.         [3pts] In the Brachistochrone problem, the bead travels a distance A in the horizontal direction and B in the vertical direction. If A/B = 5, find the minimum travel time. (Express your answer in terms of B and g, using numerical methods to solve the necessary equation.)

3.         [3pts] Marion & Thornton: problem 6.6.

4.         [3pts] Marion & Thornton: problem 6.10. Do this problem in two different ways: using a Lagrange multiplier, and without using a Lagrange multiplier. (The answer to this problem must be well-known in the School of Packaging, since it determines the ideal proportions for a tin can.)

5.         [3pts] Marion & Thornton: problem 6.11. The "equation of constraint" asked for in this problem should be answered in the form of an expression for the angle of rotation of the disk as a function of x. The second (easier) part of this problem is to find the condition on R/a that lets the disk always fit inside the parabola.

6.         [3pts] Marion & Thornton: problem 6.12. Hint: Use x, y, and z coordinates and use a Lagrange multiplier to enforce the constraint that (x,y,z) lies on a sphere of radius R. Derive a differential equation that relates x(z) and y(z), with z taken to be the independent variable. You DO NOT have to solve the equation -- the algebra that would be necessary is unpleasant!

7.         [3pts] Marion & Thornton: problem 6.14. Hint: Use polar coordinates x = r cos(theta), y = r sin(theta). The equation of the conical surface then becomes z = 1 - r. You can use this equation to eliminate the variable z, so there is no need to use a Lagrange Multiplier in this problem.