A tube of mass M and length ? is free to swing by a pivot at one end. A mass m is positioned inside the tube at this end. The tube is held horizontal and then released (see Fig). Let ? be the angle of the tube with respect to the horizontal, and let x be the distance the mass has traveled along the tube. Find the Euler-Lagrange equations for ? and x, and then write them in terms of ? and ? ? x/? (the fraction of the distance along the tube). These equations can only be solved numerically, and you must pick a numerical value for the ratio r ? m/M in order to do this. Write a program (see Appendix D) that produces the value of ? when the tube is vertical (? = ?/2). Give this value of ? for a few values of r.

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