question_answer

Two satellites of a planet have periods 32 days and 256 days respectively. If the radius of the orbit of the one is R, find the orbital radius of the other.

Answer:

Here, \[{{R}_{1}}=R\,\,{{R}_{2}}=?\,\,\,{{T}_{1}}=32\,\,days\] \[{{T}_{2}}=256\,days\] We know, \[{{T}^{2}}\propto {{R}^{3}}\,\,or\,\,R\propto {{T}^{2/3}}\] \[\therefore \frac{{{R}_{2}}}{{{R}_{1}}}={{\left( \frac{{{T}_{2}}}{{{T}_{1}}} \right)}^{{\scriptstyle{}^{2}/{}_{3}}}}\,\,or\,\,{{R}_{2}}=R{{\left( \frac{256}{32} \right)}^{{\scriptstyle{}^{2}/{}_{3}}}}=4R\]