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I need d and e only please.
A.2 Orbital Motion a. An object moves under a central force F(r) = -k/r". The radial component of Newton's Second Law in polar coordinates can be written as F(r) L2 der dt2 + m Substituting u = 1/r and changing the independent variable to 0, show that du de2 kmur-2 L2 -U. [5] [5] b. Argue that for F(r) = -k/r3, the motion is that of a simple harmonic oscillator in u(Q), and determine its frequency. c. Starting from the velocity v expressed in polar coordinates, show that (for any central force) the kinetic energy T = my v in polar coordinates can be written as dr 2 1 1 T = -m 2 +mrºw? dt Using this, determine the total energy of an object moving under the force F(r) = [5] [5] d. Show that the effective potential, defined as the components of the energy that depend only on r, is proportional to 1/r2. Furthermore, show that for L = km, the effective potential is identically zero. e. For L = km, show that the frequency of oscillation from part (b) vanishes. Write down the total energy in this case, and describe in words what the motion must look like. [5]