Gravitational Constant – Formula, and Newton’s Law.

In physics, Gravitational constant you must be aware that, unlike on Earth, astronomers on the moon float into the atmosphere. You must be curious as to why this is the case. It is gravity. The pull between two bodies is caused by gravity, often known as gravitation. The attraction of things to the Earth is only one aspect of gravity.

There is an attraction between everything in the universe. He discovered gravity when Isaac Newton was sitting under a tree, and an apple fell on his head. He began to speculate as to why the apple had been drawn to the ground in the first place.

Newton’s Law of Gravitation was established due to his discovery of the force of attraction in 1680.

Examples –

1. For instance, a ball put on a table begins to slide down when the table is slanted.
2. If a coin is dropped from a height, it will fall.
3. A body hurled into the air rises reaches heights, and finally falls to the ground.

Newton’s Law of Gravitation

According to Newton’s Law of Gravitation, every particle in the universe is attracted to every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of their distance. Along the line connecting the particles is where the force is applied.

Therefore, from Newton’s Law of Gravitation

F = G m1 m2/ r²

where

• F is the force
• G is the gravitational constant
• m1 is the mass of object 1
• m2  is the mass of the object 2
• R is the distance between the centres of the object

What is the Gravitational Constant Universal Law of Gravitation?

To calculate the gravitational force of attraction, physics introduces the gravitational constant. A significant advancement in the study of physics is the discovery of the Universal Law of Gravitation. It reveals information about how mass and force interact.

According to the law of gravity, everything in the universe is attracted to every other object, so the force applied is proportional to the product of the masses and inversely proportional to the square of the distance between them.

The proportionality constant inherent in the universal law of gravitation is known as the universal gravitational constant. The force of attraction between two unit masses placed at a unit distance apart is mathematically equivalent to the universal gravitational constant value.

The Gravitational constant plays a vital part in defining the force between two objects with maximum distance. Without breaking the rules of physics, the constant gravitation balances the gravitational force.

 Let’s take a closer look at the gravitation law’s mathematical component.

The statement of the law of gravity claims that

The product of the masses of the interacting bodies directly proportionally determines the magnitude of the force exerting on the body. Therefore, we obtain:

⇒ F ∝ m₁ m₂ …….(1)

Newton established a standard value that the force is inversely proportional to the square of the distance between two things, i.e., the magnitude of the force operating between two objects changes fast with increasing distance.

⇒F α 1/r2……….(2)

He then combined (1) and (2) to make both claims more general:

⇒F α m1 m2/ r²………(3)

Where,

m₁ – The first object’s weight

m₂ – The second object’s weight

r – The distance between two objects

The proportionality in Equation (3) is removed and replaced with a constant known as the gravitational constant.

⇒F = G m1 m2/ r²……….(4)

Where,

m₁ – The first object’s weight

m2 – The second object’s weight

r – The distance between two objects

G – The universal Gravitational constant

It has been determined that G = 6.673 x 10^-11 Nm²/kg² is the value of the proportionality constant, which is also the value of the universal gravitational constant.

Newton’s law of gravity, also called the law of gravitational force, is represented mathematically in equation (4). The force exerted by each point mass on the other will be directly proportional to the product of their mass and inversely proportional to the square of their distance, according to equation (4). It is often referred to as the inverse square law.

Only due to their masses can two objects interact gravitationally. One of physics’ four fundamental forces is gravitational attraction. The universe is subject to gravitational pull. One of the two items needs to be larger than the other for a noticeable gravitational force.

Let’s examine the universal gravitational constant now that we have a better understanding of what the gravitational constant is and how it works.

The Universal Gravitational Constant 

The gravitational constant is the constant that correlates the force acting on an object with its mass and its distance from another object. When a unit distance separates two unit masses, the gravitational constant is equal to the numerical representation of the attractive force.

G = 6.673 x 10^-11 Nm²/kg² is the figure determined to be the universal gravitational constant. A proportionality constant to solve the equation is what we refer to as the universal gravitational constant. The gravitational constant dimensions are [M -1L 3T -2]. Nm² kg² represents the SI unit of G.

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. 

We have, according to the law of universal gravitation,

=> F = G m1 m2/  r²

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 distance is measured in metres, masses in kilogrammes, and forces in Newtons. As a result, the units on the right and left are not equal. To achieve equilibrium, the gravitational constant is added. The possible unit of the universal gravitational constant, then, according to our study, is,

⇒ G = Nm²/ kg²

Nm²/ kg² is the unit for the gravitational constant. 

Earth gravitational constant

From the law of universal gravitation, the force on a body acted upon by Earth’s gravitational force is given by

F = G m1 m2/ r² = G M m1/ r²

Where r is the distance between the centre of the Earth and the body (see below), and here we take M to be the mass of the Earth and m to be the mass of the body. Additionally, Newton’s second law, F = ma, where m is mass and a is acceleration, here tells us that

F = mg

Comparing the two formulas it is seen that:

g = G M/ r²

So, at sea level to find the acceleration due to gravity, put the values of the gravitational constant, G, the Earth’s mass m (in kilograms), and the Earth’s radius (in metres), r, to obtain the value of g:

g = G M/ r²

g = 6.67 x 10^-11m³kg^-1s^-2 x 6 x 10 ²⁴ kg / (6.4 x10 ⁶ m) ²

g = 9.77m.s ^-2 

g= 9.81 m s ^-2

Difference between ‘g’ and ‘G’

The main difference between ‘g’ and ‘G’ is that g stands for gravitational acceleration and ‘G’ stands for gravitational constant. The gravitational acceleration ‘g’ varies with altitude, whereas the gravity constant value of ‘G’ remains constant. Gravitational acceleration is a vector quantity, whereas the gravitational constant is a scalar number.

Principle of Superposition of Gravitational Forces

Only the interaction between two particles is addressed by Newton’s law of gravity, and there are n(n – 1)/2 such interactions for systems with ‘n’ particles.

The results may be stated as the vector summing of these interactions; F = F12 + F13 + F14….. + F1n, in accordance with the concept of superposition, assuming each of these interactions acts independently and unaffected by the other entities.

It states that:

“The vector sum of the forces imposed by the individual masses on the given particle is equal to the cumulative gravitational force F acting on a particle owing to the number of point masses.”

Frequently Asked Questions

1. Explain the meaning of the following statement ‘ 1kgf = 9.8N’

The M.K.S system uses kilogram-force (kgf) as the gravitational unit of force 

One kilogram force is the force due to gravity on a mass of 1kg.

1kg = force due to gravity on a mass of 1kg

= mass of 1kg x acceleration due to gravity g m/s^2

= g newton

Given that the average value of g  ( earth gravitational constant ) is 9.8 m/s^2

1kgf = 9.8 newton or 9.8N

2. State two applications of the universal law of gravitation.

Application of the universal law of gravitation:

1. We use this to calculate the force or pull of gravity of the planets of the earth, earth included.
2. This law also comes in handy when calculating the trajectory of astronomical bodies and predicting their motion.

3. When the glass vessels fall on a hard floor, they break, but they do not break when they fall on a carpet (or sand). Why?

When a glass vessel falls on a hard floor from a certain height, it comes to rest within a fraction of a second. So the floor exerts a large force on the vessel and it breaks.

But if it falls on a carpet (or similar soft-cushioned material), the time duration in which the vessel comes to rest increases, and so the carpet (or sand) exerts less force on the vessel and it does not break.