question_answer

If the Earth is \[\text{1}/\text{4th}\] of its present distance from the Sun, the duration of the year would be (a) 1/4th of present year (b) 1/6th of present year (c) 1/8th of present year (d) 1/16th of present year

Answer:

               

\[{{R}_{1}}=R\]	\[{{T}_{1}}=1\text{ }year\]
\[{{R}_{2}}=R/4\]	\[{{T}_{2}}=?\]

\[{{T}^{2}}\propto {{R}^{3}}\]\[\Rightarrow \]\[T\propto {{R}^{3/2}}\] \[\Rightarrow \]\[\frac{{{T}_{2}}}{{{T}_{1}}}=\frac{R_{2}^{3/2}}{R_{1}^{3/2}}={{\left( \frac{{{R}_{2}}}{{{R}_{1}}} \right)}^{3/2}}={{\left( \frac{R}{4R} \right)}^{3/2}}\] \[\frac{{{T}_{2}}}{1}=\frac{1}{{{4}^{3/2}}}=\frac{1}{{{({{2}^{2}})}^{3/2}}}=\frac{1}{8}\] \[\therefore \]\[{{T}_{2}}=\frac{1}{8}\]of the present year.