Physics 422 -- Fall 2012

Homework #1, due Friday Sep. 7 at beginning of class

1.         [4pts] A hoop of mass m and radius R rolls without slipping down an inclined plane of mass M, which makes an angle theta with the horizontal direction. The inclined plane slides without friction on a horizontal surface.
(a) Using the horizontal position of the bottom of the inclined plane, and the contact point of the hoop (measured along the inclined plane from the bottom of the inclined plane) as the two generalized coordinates, find the Lagrangian.
(b) Use the Lagrangian to obtain the equations of motion.
(c) Solve the equations of motion.
(d) On physical grounds, there are two obvious constants of the motion: energy and linear momentum. Use your solutions to show that these are both indeed constants.

2.         [4pts] In the Brachistochrone problem, the bead travels a distance B in the horizontal direction and C in the vertical direction. Assume that B=C.
(a) Compare the distance traveled for the Brachistochrone path with the quarter-circle path.
(b) How much much faster is the Brachistochrone path than an alternative path defined by a quarter of a circle. You will need to use some numerical method (e.g. Mathematica, or at least Wolfram alpha) to obtain the answer.

3.         [4pts] 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.