Reference: "An Introduction to Mechanics" by D. Kleppner and R. J. Kolenkow, 2nd Edition. Note: This problem is not from the textbook, as it's from class homework.

Link: https://bayanbox.ir/view/7764531208313247331/Kleppner-D.-Kolenkow-R.J.-Introduction-to-Mechanics-2014.pdf

An object of mass m moves in a central force F(r) = arⁿ where n -1 and the constant a < 0 (i.e., the force is attractive) and has the appropriate units. (a) Find the period T of a circular orbit as a function of the orbit's radius. (b) Show that such an orbit is stable only for n > -3. (c) For n > -3, find the period t of oscillations for the radial coordinate. Assume

that the amplitude of the oscillations is very small. (d) Now assume that n is an integer. i) Calculate the ratio (T / t) and use your result to argue that non-circular

orbits are closed only for some choices of n. (Note that an orbit is defined to be closed if and only if it eventually

retraces itself.) ii) Give the lowest four integer values of n that give closed orbits. Verify that

this includes n = -2 (Universal Gravitation Law) and n = 1 (harmonic

oscillator).