The Universal Law of Gravitation

Introduction

The force of attraction with which an object attracts another object in the universe is known as Gravitational Force. Newton, from his famous experience of ‘an apple falling from a tree towards the ground’ could derive the equation behind the Universal Law of Gravitation. There are several inherent forces of nature prevalent in this universe, and Gravitational Force is one of them. The different forces of nature can be classified as (i) gravitational Force, ii) electromagnetic force, iii) strong nuclear force, and iv) weak nuclear force. The force of gravity plays a significant role in holding objects together and thus binding the universe together.  

Newton’s Universal Law of Gravitation :

Let consider there are two objects of mass, m₁, and m₂, be separated by distance r.

Gravitational Force existing between two objects will be directly proportional to the product of their masses and inversely proportional to the square of the distance between them,

I.e., F=Gm1m2/r² ……………(1) 

 where G is the universal constant, ‘r’ is the distance between two objects.

Universal constant G has a value of  6.67 X 10-11 Nm2kg-2 and is the same everywhere in the universe. 

The value of G is a constant and does not depend on the physical size and nature of the body or the medium existing between them. 

G has a dimensional formula of  [M-1L3T-2 ].

Gravitational Force: Salient Features

1. i) Gravitational force is an attractive force between two objects existing in this universe irrespective of the medium between the two bodies.
2. ii) It is a conservative force by nature

iii) It is a weak force

1. iv) The strength of the force is 1036 times smaller than electrostatic force and 1018 times smaller than the nuclear force.
2. v) It exists between all bodies irrespective of their size, shape, and position
3. vi) Gravitational force acting between the sun and planets is responsible for the centripetal force for orbital motion

vii) The earth’s gravitational pull is also knowns as gravity

viii) It also explains Newton’s third law of motion

Limitations to Universal Law of gravitation:

While the law is applicable even for very large and very small distances, it does not hold well when the bodies are in order of intermolecular distances, of the order 10⁻⁹. 

Weight and the Gravitational Force

We define mass as the amount of matter contained by a particular body. Weight is the force of gravity exerted on this object with a certain mass. 

Therefore for an object with a mass ‘m’ 

Its weight ‘w’ =mg………………………………(2)

 where ‘g’ is the acceleration due to gravity

‘g’ is a constant, and its value is 9.8m/sec² 

From the equation, we derive that mass and weight are directly proportional to each other and ‘g’ is the acceleration due to gravity.

Hence from this, we also understand that while mass is a scalar quantity, weight is a vector quantity with magnitude and direction.

This can also be explained that mass is independent of direction and position, but weight depends on the position and location of the object. 

Precisely, the same object would weigh differently in the Moon compared to the earth as the gravitational force on the object is different in the Moon compared to the earth.

From the above equation, it is evident that objects with greater mass will attract each other with more gravitational force. An increase in mass content of an object will increase the gravitational force it exerts on another object, e.g., if the mass of one of the objects is doubled, the gravitational force between them will also double. 

According to the law above, the gravitational force between the objects is inversely proportional to the distance between them. Therefore as the objects move away from one another, the force of gravity between them decreases proportionally, and the nearer they get, the force of gravity increases following equation …(1). 

Universality of Gravity

Gravitational force is a universal concept. It exists not only between the earth and other objects on the earth but between objects themselves. The force so applied is directly proportional to the mass of the objects. This is extremely important to study the planetary movements of the solar system. The elliptical orbital pathways of the solar system are also an effect due to the gravitational pull between the planets. 

Acceleration due to Gravity

The force of gravity exerted on a freely falling body causes uniform acceleration of the body. The acceleration acting on the body is caused due to the earth’s gravitational pull and is uniform throughout the fall, provided no other force is acting on the body. It is denoted by ‘g,’ and its unit is m/s². Acceleration due to gravity is a vector quantity with direction towards the earth’s center. However, ‘g’ does not depend on the mass of the body, and the fall is a free fall under the force of gravity. The value of ‘g’ is 9.8m/s² at earth surface.

Solved Numericals:

 Example 1) The mass of the earth is 6×1024 kg, and that of the moon is 7.4 × 1022 kg. Given the distance between the earth and the moon is 3.84×105 km, measure the force exerted by the earth on the moon. ( G = 6.7 × 10-11 N m2 kg-2)

Ans)

The mass of the earth, M=6×1024 kg 

The mass of the moon,

m=7.4×1022 kg

The distance between the earth and the moon,

d = 3.84×105km

= 3.84×105×1000m = 3.84×108m

G = 6.7×10-11 Nm2 kg-2

 From Eq. (1), the force exerted by the earth on the moon is

 ​​F = GMm/ d2

=( 6.7×10-11 Nm2 kg-2 ×6×1024 kg×7.4×1022 kg)/  (3.84×108 m)2

Thus, the force exerted by the earth on

= 2.02 × 1020 N.

So the force exerted by the earth on the moon is 2.02 × 1020 N.

Example 2) Mass of an object is 10 kg. What is the weight of the object on the earth?

Ans)

Mass, m = 10 kg

Given acceleration due to gravity   ‘g’ is a constant  = 9.8 m/s2

W=m×g

W= 10kg×9.8m/s2 = 98N 

Thus, the weight of the object is 98 N.

Conclusion

The gravitational force is a universal force, as this is what manages the planetary motion. Every object on Earth is subjected to this force to stay grounded overwise without gravity. All of us would have been floating. The force of gravity decreases concerning the altitude. It also decreases while going from the poles to the equator. The weight of a body is the force that is attracted by the Earth. To understand this concept, Sir Issac Newton gave the Universal Law of Gravitation which applies to every object in the universe. The law states that the force of attraction between any two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.