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Derive the Universal law of Gravitation $F = \dfrac{{GMm}}{{{d^2}}}$ .

Last updated date: 23rd Jul 2024
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Hint: Sir Isaac Newton discovered that every object in the universe attract with each other with a force which depends only upon the masses of two bodies and the distance between them and he formulated a mathematical description of this theoretical concept which is known as Universal law of Gravitation.

Complete step-by-step solution:
So, According to the Sir Isaac Newton, Universal law of gravitation is defined as:
The force $F$ with which all bodies in the universe attracts with each other is:
$F$ Is directly proportional to the first body of mass say $M$
And this force $F$ is also directly proportional to the second body of mass say $m$
If the distance between these two bodies is denoted by $d$ then, it’s inversely proportional to the square of distance between them which can be written as:
$F \propto \dfrac{1}{{{d^2}}}$ For distance
$F \propto M$
$F \propto m$
Combining all the proportionality parameter we get
$F \propto \dfrac{{Mm}}{{{d^2}}}$
In order to remove the proportionality constant, Newton pits a constant value which is known as Gravitational constant and it’s denoted by $G$ .
$F = \dfrac{{GMm}}{{{d^2}}}$
Where, G is called gravitational constant and its magnitude is fixed in whole universe and taken as
$G = 6.67 \times {10^{ - 11}}N{m^2}K{g^{ - 2}}$
$F = \dfrac{{GMm}}{{{d^2}}}$ This is called the universal law of gravitation.

Note: It’s important to remember, On $5July,1687$ Sir Isaac Newton discovered this universal law of gravitation which he published on Principia of Mathematic, and Law of Gravitation is valid in whole universe but unfortunately Newton didn’t tells about the how this force works Later, Einstein In $1916$ discover General theory of relativity which tells us about the curved space-time fabric and it shows the reason of gravity.