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Q. If distance between earth and sun become four times then time period becomes

Delhi UMET/DPMTDelhi UMET/DPMT 2007Gravitation

Solution:

According to Kepler's third law (law of periods),
$T^{2} \propto R^{3}$
where $T$ is time taken by the planet to go once around the sun $R$ is semimajor axis (distance) of the elliptical orbit.
$=T^{2} \propto k R^{3}$
(where $k$ is constant of proportionality)
When $R$ becomes $4$ times let time period be $T$'.
$=T^{2}=k(4 R)^{3}$
$=\frac{T^{2}}{T^{2}}=\frac{1}{64}$
or $\frac{T}{T'}=\frac{1}{8}$
or $T'=8 T$
So, time period becomes $8$ times of previous value.