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.