Question

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

• A. half the present year
• B. one-eighth the present year
• C. one-fourth the present year
• D. one-sixth the present year

Answer 1

Answer: (B)

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, $R_1=R,R_2=\frac{R}{4},T_1=T$
$\therefore \frac{T_2}{T}={\left(\frac{R/4}{R}\right)}^{3/2}$
$={\left(\frac{1}{4}\right)}^{3/2}=\frac{1}{8}$
or $T_2=\frac{T}{8}$
Hence, the duration of the year will be one-eighth the present year.