Question

derive a relation between g nd G​

Answer 1

Relationship between g and G

Let $\rm g$ be the acceleration due to gravity at a planet ( or satellite) of mass $\rm M$ and radius $\rm R$ . By Newton's law of motion, the force $\rm F$due to gravity on a body of mass $\rm m$on its surface will be

$\quad \rm F = mass\times acceleration\ due \ to \ gravity$

$\rm or \qquad F = mg\qquad ...[1]$

By Newton's gravitational law, this attractive force is given by,

$\qquad \rm F = \dfrac{GMm}{R^2} \qquad ...[2]$

From equations [1] and [2]

$\qquad \rm mg=\dfrac{GMm}{R^2}$

or

$\bullet\qquad \rm g= \dfrac{GM}{R^2}\qquad ...[3]$

The above equation [3] relates the acceleration due to gravity g with the gravitational force constant G. Obviously, the value of g on a planet (or satellite) depends on the mass and radius of that planet (or satellite).

Example:-

Taking the mass of earth $\rm M = 5.96 \times 10^{24} kg$ and the radius of earth $\rm R=6.37\times 10^6 \ m,$ the acceleration due to gravity at a place on the surface of the earth comes out to be

$\rm g_{earth} = \dfrac{(6.67\times 10^{-11}) \times (5.96 \times 10^{24})}{(6.37\times 10^6)^2} = 9.8 \ m \ s^{-2}$

Note:- The value of $\rm g$ varies from place to place.

Hope it helps...

Answer 2

Answer:

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