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 Keplers law, \[{{T}^{2}}\propto {{a}^{3}}\] \[\Rightarrow \] \[{{T}^{2}}\propto \frac{1}{\rho }\] Or \[T\propto \frac{1}{\sqrt{\rho }}=\frac{1}{\sqrt{d}}\] \[[\because \rho =d]\]