Physics 422 -- Fall 2006

Homework #2, due Wednesday Sep. 13 at beginning of class

1.         [5pts] Marion & Thornton: problem 7.4. Use polar coordinates.
 
First find the Lagrange equations of motion for r and theta. Notice that the Lagrangian does not depend on theta, which gives a constant of the motion (angular momentum).
 
Then using the two constants of the motion -- angular momentum and total energy -- derive an equation for dr/dt as a function of r, and use it to write the solution to the problem in the form t-t0 = integral of a function of r. You do not need to carry out that integral.

2.         [5pts] Marion & Thornton: problem 7.6. As generalized coordinates, use S1 = distance from a fixed point on the x axis to the pointy end of the ramp (where the angle alpha is), and S2 = the distance measured along the ramp from that pointy end to the point where the hoop touches the ramp. Find the Lagrangian and derive the equations of motion from it.
 
From your past experience in these matters, you know that the total energy and the horizontal component of the momentum of the center of mass must be constants. Use the equations of motion you derived above to verify that these quantities are indeed constant.
 
Solve the equations of motion to find the most general possible solution for S1 and S2 as a function of time, assuming that both velocities are zero at t=0.

3.         [5pts] Marion & Thornton: problem 7.7. Use the angles of the strings with respect to the vertical direction as the two generalized coordinates. Find the equations of motion without assuming the angles are small. (You may find the trig identity cos(theta1 - theta2) = ... useful.)
 
Find the oscillation frequencies in the limit of small oscillations.

4.         [5pts] Marion & Thornton: problem 7.12. Also find the constraint force supplied by the plane. Use the radius r as generalized coordinate. Since the particle starts at rest, you want the solution for which dr/dt = 0 at t=0.
 
Expand your solution for r as a function of t, keeping terms up to t cubed. Do the same for x = r*cos(theta).
 
Make graphs numerically for r(t) for various values of the dimensionless constant g/(r0*alpha^2).