Question

(a) Show that the rotation speed, v(R), of a particle which is moving in a circular orbit of radius R within a spherical mass distribution is given by: whexe M< R) i s the mass W1 hin radins R

Answer 1

$Gravitational \,\,force\,\,of\,\,attraction\,\,F_g=\frac{GMm}{R^2} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,[where\,\,G\,\,is\,\,gravitational\,\,constant,\,\,m\,\,is\,\,mass\,\,of\,\,the\,\,rotating\,\,particle] \\\\ Centrifugal \,\,force\,\,due\,\,to\,\,rotation\,\,F_r=\frac{mv(R)^2}{R}\\\\\\ By\,\,force\,\,balance,\\\\ .\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,F_r=F_g\\\\ \Rightarrow \,\,\frac{mv(R)^2}{R}=\frac{GMm}{R^2}\\\\ \Rightarrow \,\,v(R)=\sqrt{\frac{GM}{R}}$

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