# 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.

#### 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
Is there an error in this question or solution?