question_answer

Assuming density d of a planet to be uniform, we can say that the time period of its artificial satellite is proportional to

A)  \[d\]      

B) \[\sqrt{d}\]

C) \[1/\sqrt{d}\]   

D) \[1/d\]

Correct Answer: C

Solution :

Density of the Planet can be written as \[\rho =\frac{m}{v}=\frac{m}{\frac{4}{3}\pi {{a}^{3}}}\Rightarrow \rho \propto \frac{1}{{{a}^{3}}}\] According to Kepler's law, \[{{T}^{2}}\propto {{a}^{3}}\] \[\Rightarrow \] \[{{T}^{2}}\propto \frac{1}{\rho }\] or \[T\propto \frac{1}{\sqrt{\rho }}=\frac{1}{\sqrt{d}}\] \[[\because \,\,\rho =d]\]