Imagine that you are living in a universe where gravity acts like a spring, rath

Question

Imagine that you are living in a universe where gravity acts like a spring, rather than obeying a 1/ r^2 force law. This "spravity" force between two masses m1 and m2 is given by F = - Gm1m2r r^ (a) Show that the potential energy of two masses separated by a distance r is U = 0.5G m1m2 r^2 (What do you use as your reference point?) (b) Show that the trajectory of an object moving in a spravitationally-bound orbit takes the form r = C/ (1 + ?cos 2?)^0.5 What are C and ? , in terms of the constants E, l, k = G m1m2, and the reduced mass ?? (Note that this is equivalent to an ellipse with semi-major and semi-minor axes of C/ (1 - ?)^0.5 and C/ (1 + ?)^0.5 respectively. Extra credit: show this.) Draw a diagram of a typical orbit of a small mass m about a much larger mass M. (c) (10) Find the equivalent of Kepler's Third Law. What is the period of an orbit? Answer: P = 2?/ (G(m1 + m2))^0.5

Explanation / Answer

(a) the - of potential gradint is force and hence -dU/dr=- Gm1m2*r........integrating it and taking the reference point as the U=o when R=0 ,we get U = 0.5G m1m2 r^2 ...the reference point is taken to be the centre of m1......there is no restriction on the refence as there is nothing called absolute potential .......what matter most in physics is the deltaU

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