question_answer

Two planets from the sun are\[\text{1}{{0}^{\text{13}}}\]and \[\text{1}{{0}^{12}}\text{m}\] respectively. Find the ratio of the time periods of the two planets.

Answer:

               

Planet-A	Planet-B
\[{{R}_{A}}={{10}^{13}}m\]	\[{{R}_{B}}={{10}^{12}}m\]
\[{{T}_{A}};{{T}_{B}}=?\]

We know \[\frac{{{T}^{2}}}{{{R}^{3}}}=const\] \[\Rightarrow \frac{T_{A}^{2}}{T_{B}^{2}}=\frac{R_{A}^{3}}{R_{B}^{3}}={{\left[ \frac{{{R}_{A}}}{{{R}_{B}}} \right]}^{3}}\] \[\frac{{{T}_{A}}}{{{T}_{B}}}={{\left[ \frac{{{R}_{A}}}{{{R}_{B}}} \right]}^{\frac{3}{2}}}\] \[\frac{{{T}_{A}}}{{{T}_{B}}}={{\left[ \frac{{{10}^{13}}}{{{10}^{12}}} \right]}^{\frac{3}{2}}}\] \[=\frac{10\sqrt{10}}{1}\] \[{{T}_{A}}:{{T}_{B}}=10\sqrt{10}:1\]