Question

If the Earth is at $\frac{1}{4}th$ of the present distance from the Sun, then the duration of a year on the Earth would have been ____ of the present duration of a year.

• A. $\frac{1}{8}th$
• B. $\frac{1}{16}th$
• C. $\frac{1}{6}th$
• D. $\frac{1}{4}th$

Answer 1

The correct option is A $\frac{1}{8}th$

From Kepler's third law,
$T_1^2\propto r_1^3$ and given $r_2=\frac{1}{4}{r}_1$

Therefore, $\frac{T_1^2}{T_2^2}=\frac{r_1^3}{r_2^3}$

or, $\frac{T_1^2}{T_2^2}=\frac{(4r_2{)}^3}{r_2^3}$

or, $T_2=\frac{T_1}{8}$