Question
Derive the relation between acceleration due to gravity (g) and Gravitational constant G.
✓
Answer
When a body is at rests on the surface of the Earth, it is acted upon by the gravitational force of the Earth. Let us compute the magnitude of this force in two ways. Let, $M$ be the mass of the Earth and $m$ be the mass of the body. The entire mass of the Earth is assumed to be concentrated at its centre. The radius of the Earth is $R =6378 km =6400 km$ approximately. By Newton's law of gravitation, the force acting on the body is given by
$F =\frac{G M m}{R^2}$
Here, the radius of the body considered is negligible when compared with the Earth's radius. Now, the same force can be obtained from Newton's second law of motion.
According to this law, the force acting on the body is given by the product of its mass and acceleration (called as weight). Here, acceleration of the body is under the action of gravity hence $a = g$
$F = ma = mg \ldots \ldots . .$
$F =\text { weight }= mg$
Comparing equations (1) and (2), we get
$mg =\frac{G M m}{R^2}$
Acceleration due to gravity $g =\frac{G M}{R^2}$
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