Question

If the earth is $\frac{1}{4}th$ of its present distance from the sun, the duration of the year would be:

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

Answer 1

The correct option is C $\frac{1}{8}th$ of the present year

According kepler's $3$ rd law,

$P^2=a^3\Rightarrow P=\sqrt{a^3}$

[$a=$Radius of earth]

Now if $a\to \frac{a}{4}$

${{\therefore P}^{\prime }}^2=\frac{a^3}{64}$

$\therefore P^{\prime }=\sqrt{\frac{a^3}{64}}=\frac{1}{8}\sqrt{a^3}=\frac{P}{8}$

$\therefore$ If the earth is $\frac{1}{4}th$ of its present distance from the sun the year would be $\frac{1}{8}th$ of the present year.