A) half the present year
B) one-eighth the present year
C) one-fourth the present year
D) one-sixth the present year
Correct Answer: B
Solution :
The square of the planets time period is proportional to the cube of the semi-major axis of its orbit. \[{{T}^{2}}\propto {{R}^{3}}\] where R is the distance of earth from the sun. or \[{{\left( \frac{{{T}_{2}}}{{{T}_{1}}} \right)}^{2}}={{\left( \frac{{{R}_{2}}}{{{R}_{1}}} \right)}^{3}}\] Given, \[{{R}_{1}}=R,\,{{R}_{2}}=\frac{R}{4},\,{{T}_{1}}=T\] \[\therefore \] \[\frac{{{T}_{2}}}{{{T}_{1}}}={{\left( \frac{R/4}{R} \right)}^{3/2}}={{\left( \frac{1}{4} \right)}^{3/2}}=\frac{1}{8}\] or \[{{T}_{2}}=\frac{T}{8}\] Hence, the duration of the year will be one-eighth the present year.
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