Properties of Circular Orbits Learning Goal: Part A Find the orbital speed v for a satellite in a circular orbit of radius R. Express the orbital speed in terms of G, M, and R. To teach you how to find the parameters characterizing an object in a circular orbit around a much heavier body like the earth. The motivation for Isaac Newton to discover his laws of motion was to explain the properties of planetary orbits that were observed by Tycho Brahe and analyzed by Johannes Kepler. A good starting point for understanding this (as well as the speed of the space shuttle and the height of geostationary satellites) is the simplest orbit--a circular one. This problem concerns the properties of circular orbits for a satellite orbiting a planet of mass M. ► View Available Hint(s) νπΙ ΑΣΦ ? For all parts of this problem, where appropriate, use G for the universal gravitational constant. V =
Part B Find the kinetic energy K of a satellite with mass m in a circular orbit with radius R. Express your answer in terms of m, M, G, and R. ΟΙ ΑΣΦ ? KK =
Part D Find the orbital period T. Express your answer in terms of G, M, R, and 1. ► View Available Hint(s) VO AEO t ? TT =
Part G The quantities v, K, U, and L all represent physical quantities characterizing the orbit that depend on radius R. Indicate the exponent (power) of the radial dependence of the absolute value of each. Express your answer as a comma-separated list of exponents corresponding to v, K, U, and L, in that order. For example, -1,-1/2,-0.5,-3/2 would mean va R-1,K « R-1/2, and so forth. View Available Hint(s) IVO AEO ?
Solution: (A) yuz & fo GMA RA for GM R where, G = Gravitational cont. R = Orbit Radius M = mass of earth KIE I mv² Putting value of w from above, we get KE = I mx GM 2 R K.E=GMM 2R orbital Period (T) AR V TE 27RoR'2 SGM 03/2 T= 2x R² Sam
(G) VaR p/ K&R UXRM, L&R' Can be written -12, -, -1, 0.5