Show that the radius of an orbit may be expressed as follows: R^3=((GM)/4*pi^2)T^2
Start with Newton's Law of Gravitation: F=(GMm)/R^2 (1) Since orbits are assumed to be circular recall the equation for centripetal force: F=(mv^2)/R (2) We can now equate these 2 forces due to them being action-reaction pairs (Newton's 3rd Law) (GMm)/R^2= (mv^2)/R We notice that small m on both sides cancel and 1/R^2 may be reduced to 1/R on the LHS giving an equation for v^2: v^2=GM/R (3) Since we have a circular orbit we can use the radial velocity equation: v=Rw (4) We then sub (4) into (3) R^2w^2=GM/R (5) Remember w=2pi/T (6) this can be substituted in and the R terms may be collected to give R^3 (4pi^2/T^2)R^3=GM (7) Finally divide by 4pi^2/T^2 to give the correct equation R^3=((GM)/4*pi^2)T^2 (8)
LM
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