A small block of mass \(m\) and a weight of mass \(M\) are connected by a string of length \(D\). The string has been threaded through a small hole in a tabletop, so the block can slide without friction on the tabletop, while the weight hangs vertically beneath the tabletop. We can let the hole be the origin of coordinates, and use polar coordinates \(r, \theta\) for the block, where \(r\) is the block's distance from the hole, and \(z\) for the distance of the weight below the tabletop.

(a) Using generalized coordinates \(r\) and \(\theta\), write down the Lagrangian of the system of block plus weight.

(b) Write down a complete set of first integrals of the motion, explaining the physical meaning of each.

(c) Show that the first integrals can be combined to give an equation of the form

and write out an expression for \(U_{\text {eff }}(r)\).

(d) Find the radius of a circular orbit of the block in terms of constants of the motion.

(e) Now suppose the block executes small oscillations about a circular orbit. What is the frequency of these oscillations? Is the resulting orbit of the block open or closed? That is, does the perturbed orbit of the block continually return to its former position or not?

Transcribed Image Text:

E= = (M + m) + Uest(r) Ueff