Question

If the earth is one-fourth of its present distance from the sun, the duration of the year will be changed to :

• A. half of the present year
• B. $\frac{1}{4}th$ of the present year
• C. $\frac{1}{8}th$ of the present year
• D. $\frac{7}{8}th$ of the present year
• E. $\frac{1}{16}th$ of the present year

Answer 1

Answer: (C)

$\frac{1}{8}th$ of the present year

From Kepler's law,
$T^2\propto r^3$
Hence, ${\left(\begin{array}{c}\frac{T_1}{T_2}\end{array}\right)}^2={\left(\begin{array}{c}\frac{r_1}{r_2}\end{array}\right)}^3={\left(\begin{array}{c}\frac{r}{r/4}\end{array}\right)}^3$
$\frac{T_1}{T_2}=(4{)}^{3/2}$
$T_2=\frac{T_1}{4^{3/2}}$
$T_2=\frac{T_1}{8}$