Gravitational Constant and Universal Law of Gravitation

The Gravitational constant is introduced in physics to determine the gravitational force of attraction. The Universal law of gravitation is an important discovery in the field of physics. It gives an insight into the relationship between mass and force. The law of gravitation states that- every object in the universe attracts every other object such that the force exerted will be proportional to the product of the masses and inversely proportional to the square of the distance between them. 

The proportionality constant incorporated in the universal law of gravitation is known as the universal gravitational constant. The universal gravitational constant is numerically equal to the Force of attraction between two unit masses placed at a unit distance apart. The Gravitational constant plays an important role in determining the force between two objects with maximum distance. The gravitational constant balances the gravitational force without violating the laws of physics.

What is the Gravitational Constant Universal Law of Gravitation?

Before understanding what is the gravitational constant in the universal law of gravitation, let us have a look into the mathematical part of the gravitation law in detail. As we know that universal gravitation law states that the force between two point mass objects is directly proportional to the product of masses and inversely proportional to the distance between them. 

Therefore, Newton’s law of gravitation says that,

• The magnitude of the force acting between two point masses is directly proportional to the product of their masses.

• The magnitude of the force acting between two point masses decreases rapidly as distance increases.

Mathematically we write,

Consider two objects having masses m₁ and m₂ separated by a distance r, as shown in the figure. 

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According to the statement of the law of gravitation,

The magnitude of the force acting on the body is directly proportional to the product of the masses of interacting bodies, then we get:

⇒ F ∝ m₁ m₂ …….(1)

Also, the magnitude of the force acting between two objects is changing rapidly with increasing distance, then Newton gave a standard value that, the force is inversely proportional to the square of the distance between them, i.e.,

\[\Rightarrow F\alpha \frac{1}{r^{2}}\]……….(2)

Then, he generalized both statements by combining (1) and (2) :

\[\Rightarrow F\alpha \frac{m_{1}m_{2}}{r^{2}}\]………(3)

Where,

m₁ - The mass of the first object

m₂ - The mass of the second object

r - The distance between two objects

Equation (3) is re-arranged by removing proportionality and replacing it with a constant known as gravitational constant.

\[\Rightarrow F=G\frac{m_{1}m_{2}}{r^{2}}\]……….(4)

Where,

m₁ - The mass of the first object

m₂ - The mass of the second object

r - The distance between two objects

G - The universal Gravitational constant

The value of the proportionality constant that is the value of the universal gravitation constant is found to be G = 6.673 x 10^-11Nm²/kg².

Equation (4) is known as the mathematical form of Newton’s law of gravitation or the law of gravitational force. From equation (4) we find that the force acting on each other will be directly proportional to the product of point masses and inversely proportional square of the distance between them. It is also known as the inverse square law. 

The gravitational force acting between two objects is only due to their masses. The gravitational force is one of the four basic forces of physics. The gravitational force is valid throughout the universe. For significant gravitational force, one among the two objects must be larger than the other.  

Since we have understood what is the gravitational constant universal law of gravitation, let us look at the universal gravitational constant.

Define Universal Gravitational Constant

The gravitational constant can be defined as the constant relating the force exerted on the objects to the mass and distance between the objects. The gravitational constant is equal to the numerical value of the attracting force when two unit masses are separated by a unit distance. 

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The value of the universal gravitation constant is found to be G=6.673 x 10^-11Nm²/kg². We define universal gravitational constant as a constant of proportionality to balance the equation. The dimension of the gravitational constant is [M^-1L³T^-2].SI unit of G is given by, Nm²kg^-2.

The gravitational constant serves a very specific purpose in the law of universal gravitation, it is not just a number but also has units. Let us have a look at the units of the gravitational constant. 

From the law of universal gravitation we have,

\[\Rightarrow F=G\frac{m_{1}m_{2}}{r^{2}}\]

Where,

m₁ - The mass of the first object

m₂ - The mass of the second object

r - The distance between two objects

G - The universal Gravitational constant

We know that force is measured in Newton’s and masses measured in the kilograms and distance in meters. Then the units on the right-hand side are not equal to the left-hand side, we need something that will balance, thus the Gravitational constant is introduced. Then from our analysis the possible unit of the universal gravitational constant is,

⇒ G=Nm²/kg²

Therefore, the unit of gravitational constant is Nm²/kg².

Did You Know

What is interesting here is that, albeit it's Newton’s Universal Law of Gravitation, the worth of G wasn’t given by him. This was calculated by Cavendish in 1798 through a series of experiments and observations. It is hypothesized the influence of the earth’s core on the experiments alters its rotational inertia due to which the worth of G given isn't always constant throughout the globe.

Another theory regarding the universal gravitational constant ( it is also referred to as Big G) is that, if it is true that the universe is expanding since the Big Bang, then the worth of G will keep decreasing.

The universal gravitational constant is employed in Newton’s Universal Law of Gravitation, Einstein’s General Theory of Relativity and also Kepler’s Third Law of Planetary Motion, to calculate the period of time of a planet to finish one full revolution in its orbit.