> Q. Assertion: The square of the period of revolution of a planet is proportional to the cube of the semi-major axis of the ellipse Reason: Sun’s gravitational field is inversely proportional to the square of its distance from the planet. – LIVE ANSWER TODAY

Q. Assertion: The square of the period of revolution of a planet is proportional to the cube of the semi-major axis of the ellipse Reason: Sun’s gravitational field is inversely proportional to the square of its distance from the planet.

In This post you will get answer of Q.
Assertion: The square of the period of revolution of a planet is proportional to the cube of the semi-major axis of the ellipse Reason: Sun’s gravitational field is inversely proportional to the square of its distance from the planet.

Solution:

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To make our calculations easy, let’s that the semi major axis of the ellipse be equal to the average distance of the sun from the planet. By applying Newton’s law,

$frac{Gmm}{a^{2}} = mleft(omega^{2}aright)$

$omega$ = angular velocity of the planet

$= frac{2pi}{T}$ (T = time period of the planet)

$therefore,,,,frac{Gmm}{a^{2}} = m frac{left(2piright)^{2}}{T^{2}} a$

$Rightarrow,,,,,T^{2} = left(frac{4pi^{2}}{Gm}right)a^{3}$

$Rightarrow,,,,,T^{2} ,propto,a^{3}$

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