GravitationMedium
Question
The radius of Martian orbit around the Sun is about 4 times the radius of the orbit of Mercury. The Martian year is 687 Earth days. Then which of the following is the length of 1 year on Mercury?
Options
A.88 earth days
B.225 earth days
C.172 earth days
D.124 earth days
Solution
Applying Kepler's $3^{\text{rd}\text{~}}$ law : $T^{2} \propto R^{3}$
Radius of Martian orbit, $R^{'} = 4R$
$$\left( \frac{T^{'}}{T} \right)^{2} = \left( \frac{R^{'}}{R} \right)^{3} = \left( \frac{4R}{R} \right)^{3} = 4^{3} = 64 \Rightarrow \frac{T^{'}}{T} = 8 $$$\therefore\ $ Length of 1 year on Mercury $= T = \frac{T^{'}}{8} = \frac{687}{8} = 85.88$ days
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