Universal Law Of Gravitation

Gravitation, also known as gravity, is a force that exists among all material objects in the universe. Gravity acts on objects of all sizes ranging from subatomic particles to cluster of galaxies. Sir Issac Newton studied its behaviour with his famous law of gravitation. In physics, gravitation is defined as the force that attracts every object to the centre of gravity. In general, gravitation is the force exerted by the body due to the virtue of its mass.

The Universal Law of Gravitation can be stated as:

“Every object of mass in the Universe attracts every other object of mass with a force which is directly proportional to the product of their masses and inversely proportional to the square of the separation between their centres.”

In this article, let us learn more about the universal law of gravitation.

Newton’s Law of Universal Gravitation

According to the Universal law of gravitation, the force between two bodies is directly proportional to their masses and inversely proportional to the square of the distance between them. Mathematically, it is represented as follows:

\(F \propto \frac{m_{1}m_{2}}{r^{2}}\\ \\ \Rightarrow F=G\frac{m_{1}m_{2}}{r^{2}}\)

where,

F is the gravitational force between two bodies

m1 is the mass of one object

m2 is the mass of the second object

r is the distance between the centers of two objects

G is the Universal Gravitation Constant.

Henry Cavendish, with careful experiments, found the value of Gravitational Constant to be 6.67 x 10−11 m3⋅kg−1⋅s−2N. The value of ‘G’ remains constant throughout the universe.

Universal Gravitation Constant Derivation

Now, the one question to be answered here is how Newton was able to predict that the force is inversely proportional to the square of the distance? For this, he utilized Kepler’s law according to which square of the time period is directly proportional to cube times the distance between the centre and the orbiting body. So we know since the body is in a circular motion so,

\(F \propto \frac{v^{2}}{r}\)……………………….(1)

Also we can say, \(T =\frac{2\pi r}{v}\\ \\ \Rightarrow v=\frac{2\pi r}{T}\)

Now, putting the value of v in equation (1) we get,

\(F \propto \frac{r}{T^{2}}\)……………………….(2)

Now from Kepler’s Law,

\(T^{2}\propto r^{3}\)

Hence, putting the value of \(T^{2}\) in equation(2), we get,

\(F\propto \frac{1}{r^{2}}\)

Now the question is if gravitation exists between any two masses then why we are attracted towards the earth but not towards each other. To realize this let’s take an example, let there be two bodies A (60 kg) and B (80 kg) at a distance of 1m from one another. So the gravitational pull between the two bodies is:

\(F=G\frac{m_{1}m_{2}}{r^{2}}\\ \\ \Rightarrow F=\frac{6.67\times 10^{-11}\times 60\times 80}{1^{2}}N\\ \\ \Rightarrow F=3.2\times 10^{-7}N\)………………….(3)

Now consider the force between body A and the earth.

Mass = \(6\times 10^{24}\) Kg

Radius of earth = \(6.3\times 10^{6}\) m

\(F=\frac{6.67\times 10^{-11}\times 60\times 6\times 10^{24}}{(6.3\times 10^{6})^{2}}N\\ \\ \Rightarrow F=577.4N\)

By looking at (3) and (4), we can say that even though every bit of mass in the universe attracts every other bit, we don’t feel it because, under normal heights, the attraction is far too low to be felt. The Earth, on the other hand, is massive and hence exerts a non-zero force on us. The larger the planet, larger is its force of gravity.

Example of Newton’s Gravitational Law

Question: 

Calculate the gravitational force of attraction between the Earth and a 70kg man if he is standing at a sea level, a distance of 6.38 x 106 m from the earth’s center

Answer:

m1 is the mass of the Earth which is equal to 5.98 x 1024 kg

m2 is = 70 kg

d = 6.38 x 106 m

value of G = 6.673 x 10-11 N m2/kg2

Now According to Law of Gravitation,

\(F=\frac{(6.673\times 10^{-11}Nm^{2}/kg^{2}).(5.98\times 10^{24}kg).(70kg)}{(6.38\times 10^{6}m)^{2}}\)

F = 686 N

Questions To Ponder

Why doesn’t the Moon crash into the Earth?

Ans:

The forces of speed and gravity are what keeps the moon in constant orbit around the earth. The Moon seems to hover around in the sky, unaffected by gravity. However, the reason the Moon stays in orbit is precisely because of gravity. In this video, clearly, understand why the moon doesn’t fall into the earth


Is the force of Gravity same all over the Earth?

Ans: Gravity isn’t the same everywhere on earth. Gravity is slightly stronger over the places with more underground mass than places with less mass. NASA uses two spacecraft to measure the variation in the Earth’s gravity. These spacecraft are a part of Gravity Recovery and Climate Experiment (GRACE) mission.
Given below is the map of gravitation variation on earth. Areas in blue have weaker gravity while areas in red have slightly stronger gravity.

Gravity Map

Stay tuned to Byju’s to know more about gravitation and much more.

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