# Newton’s Law of Gravity

## Introduction

In the late 1600s, Sir Isaac Newton came up with the law of gravity which is also known as the universal law of gravitation. Sir Isaac Newton’s inspiration for deducing the revolutionary law of gravity was an apple falling from a tree. We are all pretty familiar with the story of Newton and how he discovered gravity. The falling of an apple made him discover what is Newton's gravity and the law of gravitation. Newton’s law of gravity plays an important role in mechanics

Newton had a simple question out of curiosity is why an apple was falling instead of either sideways or upward!!! Later Newton realized that the earth must be responsible for the apple to fall downwards perpendicular to the ground. This was the major turning point and then he developed the law of gravity.

## Law of Gravitation:

### What is Newton's Law of Gravitation?

The law of gravity 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.

According to Newton’s Law of Gravitation,

• 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.

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:

$\Rightarrow F\alpha m_{1}m_{2}$ …….(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 is found to be G = 6.673 x 10-11 Nm2/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. In some articles, it is also referred to as the first law of gravity.

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. Sometimes it is also referred to as Newton gravity or Newton's gravity. The gravitational force is valid throughout the universe. For significant gravitational force, one among the two objects must be larger than the other.

### Characteristics of Gravitational Force:

• The gravitational force is always attractive and it is directed along with the line joining of two interacting bodies.

• The gravitational force is independent of the medium and the surrounding environment.

• The gravitational force is valid for long distances like the distance between two planets and for short distances like interatomic distances.

• The force of gravitation is conservative. Thus the work done gravitational force will be zero.

• If a particle is acted by n particles then the net force exerted on it will be equal to the vector sum of the forces due to surrounding particles. i.e., $F_{net}=\sum_{i=1}^{n} F_{i}$ where Fi is the force acted on the object by ith particle.

### Weight:

• From the law of gravity or Newton’s law of gravitation, we understood that mass is a crucial entity. There is always a confusion between mass and weight, we consider mass and weight to be the same, but in reality, they are interrelated but are different from each other.

• Weight is the gravitational force exerted on any object of a certain mass. The weight of an object can be estimated by multiplying the mass m of the object by the acceleration due to gravity, g, at the surface of the Earth. The measured acceleration due to gravity at the Earth’s surface is found to be about 9.8 m/s2 or 980 cm/s2

• The measure of how much matter is in an object is known as mass, while weight is the measure of the gravitational force exerted on the material in a given gravitational field; thus, mass and weight are proportional to each other.

⇒ W∝ m

Where,

m - The mass of the object

⇒ W = mg

Where,

g - acceleration due to gravity.

• It is observed that the mass of the given object will be constant, but the weight depends on the position of the object.

### Examples:

#### 1. Calculate the Gravitational Force Between the Earth and a 90 kg Man. If He is Standing at Sea Level at a Distance of 6.38 x 106m from the Earth’s Center.

Sol:

Given that,

Mass of the man = m1= 90kg

Mass of the earth = m2 = 5.98 x 1024 kg

Distance between the man and the earth as measured from the center of the earth

= r = 6.38 x 106

We are asked to determine the force of attraction between the Earth and a man who is standing at sea level. We know that from the law of gravitation, the gravitational force of attraction is given by:

$\Rightarrow F= G \frac{m_{1}m_{2}}{r^{2}}$ …….(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

Substituting the given values in the above expression,

$\Rightarrow F=\left ( 6.673 \times10^{-11} \right )^{\frac{(5.98 \times10^{24})(90)}{(6.38\times10^{6})^{2}}}$

⇒ F = 882.3N

Therefore, the force of attraction between the earth and a man is 882.3N.