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Let the period of revolution of a planet at a distance R from a star be T. Prove that if it was at a distance of 2R from the star, its period of revolution will be 8 T . - Science and Technology 1

Short Note

Let the period of revolution of a planet at a distance R from a star be T. Prove that if it was at a distance of 2R from the star, its period of revolution will be \[\sqrt{8}\] T.    

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Solution

From Kepler's third law of planetary motion, we have

\[T^2 \propto r^3\] ..........(i)
Thus, when the period of revolution of the planet at a distance R from a star is T, then from
(i), we have
\[T^2 \propto R^3\] ..............(ii)
Now, when the distance of the planet from the star is 2R, then its period of revolution becomes
\[T_1^2 \propto (2R )^3 \] 
or
\[ T_1^2 \propto 8 R^3 . . . . . \](iii)
Dividing (iii) by (ii), we get
\[\frac{T_1^2}{T^2} = \frac{8 R^3}{R^3}\]
\[ \Rightarrow T_1 = \sqrt{8}T\]
Concept: Kepler’s Laws
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