medo12341
2016-04-12 00:41:12
Saturn has an orbital period of 29.46 years. In two or more complete sentences, explain how to calculate the average distance from Saturn to the sun and then calculate it.
$Given:\T=29.46yapprox 9.29cdot 10^8s\M_Sapprox2.0cdot 10^{30}kg\G=6.67cdot 10^{-11} frac{m^3}{kgcdot s^2} \\Find:\R=?\\Solution:\\F_g= Gfrac{mM_s}{R^2} \\F_c= frac{mv^2}{R} \\F_g=F_c\\Gfrac{mM_s}{R^2}=frac{mv^2}{R} Rightarrow Gfrac{M_s}{R^2}=frac{v^2}{R}\\v=omega r\\Gfrac{M_s}{R^2}= frac{omega^2R^2}{R}Rightarrow Gfrac{M_s}{R^2}=omega^2R \\omega= frac{2 pi }{T} \\Gfrac{M_s}{R^3}= frac{4 pi ^2}{T^2}$ $GM_ST^2=4 pi ^2R^3Rightarrow R= sqrt[3]{ frac{GM_ST^2}{4 pi ^2} }\\\R= sqrt[3]{ frac{6.67cdot 10^{-11} frac{m^3}{kgcdot s^2}cdot2.0cdot 10^{30}kg( 9.29cdot 10^8s)^2}{4cdot 3.14^2} } approx 1.42cdot 10^{12}m$