# Universal Law of Gravitation

## Gravitation Definition

In 1665, the concept of gravitation was put forth by Sir Isaac Newton when he was sitting under the tree, an apple fell down from that tree on the earth.

This sparked an idea in his mind that all bodies are attracted towards the center of the earth where he said that the gravitation is the force of attraction between any two bodies separated from each other by a distance.

This concept played a major role in the initiating birth of stars, controlling the entire structure of the universe.

At present, this concept has significant applications in advancement of physics.

Gravitational force is the weakest force among all the basic forces of nature.

## Universal Law of Gravitation

In 1687, an English mathematician and physicist, Isaac Newton put forward this law to explain the observed motions of planets and their moons.

Newton’s law of universal gravitation states that any particle of matter in the universe attracts another one with a force varying directly as the product of the masses and inversely as the square of the distance between them.

Consider two bodies A and B of mass m1 and m2  separated by a distance r such that the force of attraction acting on them are represented as shown in the figure below:

In figure.1, the two bodies having forces of attraction F1 and F2 have a tendency to move towards the center of gravity..

Such that F1 = F2.

Therefore, the universal law of gravitation formula is given by,

F ∝ m1 . m2 / $r^{2}$

Or

 F = G (m1 . m2)/ $r^{2}$ ….(1)

Here, G is called the Universal gravitational constant (a scalar quantity).

The value of G remains constant throughout the universe and is independent of the nature and size of the bodies.

Definition of G

When m1 = m2 = 1  and r =1

Then from eq (1)

 F = G

It says that the magnitude of the attractive force F is equal to G, multiplied by the product of the masses and divided by the square of the distance between them.

State Two Applications of Universal Law of Gravitation

Newton’s law of gravitation holds good for the objects lying at very large distances and at short distances as well.

It fails when the distance between the two bodies is less than $10^{-9}$m.

There are various applications where Newton’s law , two of them are discussed below:

• The predictions about the orbits and time period of the modern artificial satellites made on the basis of this law proved to be very accurate.

• The prediction about solar and lunar eclipses, made on the basis of this law came out to be very true.

### Importance of Universal Law of Gravitation

• The gravitational force of earth ties the terrestrial objects to the earth.

• This law explains the attractive force between any two objects having a mass.

• The formation of tides in the ocean is due to the force of attraction between the moon and ocean water.

• All planets make an elliptical revolution with the sun.

• The rotation of the earth around the sun.

• The rotation of the moon around the earth.

### Derivation of Universal Law of Gravitation

This law states that any two objects pull on each other with force gravity.

Newton’s law brought up the new concept where he said:

Total force  acting on an object = object’s mass  x  object’s acceleration

Total force is the force of gravity or Fg.

So,                     Fg  (gravity force pulling on object) ∝ object’s mass (m)

The earth pulls the object towards itself.

The mass of earth  = M and gravity force = Fg

So,                  Fg  (gravity force) ∝ Earth’s mass (M)

Planets move around the sun in an elliptical orbit because gravity force provides the net centripetal force pulling the planet towards the center of its circle given by

Fg = F netc  = m/r $v^{2}$ …(2)

Since moon orbits the circumference of the circle in one period given by

velocity,   v = 2 πr/ T   putting in eq(2)

Fg = m/r. $(2\pi r/ T)^{2}$

On solving,

Fg = 4  $\pi ^{2}$ mr/ $T^{2}$

Multiplying both the sides by  $T^{2}$/r we get

Fg. $T^{2}$/r = 4 $\pi ^{2}$ mr/ $T^{2}$ . $T^{2}$/r

$T^{2}$/r Fg = 4 $\pi ^{2}$ m …(3)

Since Fg $r^{2}$ we get F = k $r^{2}$ putting it in eq(3)

We get that $T^{2}$/r (k/($r^{2}$) = 4 $\pi ^{2}$ m $\pi ^{2}$m equivalent to the equation of kepler’s third law i.e., $T^{2}$/$r^{3}$ = constant   (Newton considered Fg 1/ $r^{2}$

Therefore, we inferred that

Fg

m

M

1/ $r^{2}$

Combining these three terms we get,

Fg mM/ $r^{2}$

Removing this proportionality constant we get

 F = G (mM)/$r^{2}$

and

 G = F . $r^{2}$ / (m1 . m2)

1. What is the value of G?

Ans: According to the universal gravitational law formula:

F = G (mM)/ r2

Where

 G = F . r2/ (m1 . m2)

Value of G

In international systems, SI

 6.67 x 10-11N m2 kg-2

In cgs system

 6.67 x 10-8 dyne cm2 g-2

2. How to prove that the value of g is 9.8 m/s2?

Ans: Since we know that acceleration due to gravity is given by

g = GM/r2  …(a)

(F = G (mM)/ r2 = mg)

The mass of earth M =6 x 1024 Kg , G = 6.67 x 10-11N m2 kg-2 and radius of earth

(r) = 6.4 x 106 m

Putting these values in eq(a) we get,

g  = 6.67 x 10--11N m2 kg-2 x (6 x 1024 Kg)2/ (6.4 x 106 m)2…(b)

On solving, eq(b) we proved the value of gravity

 g   = 9.7705 〜 9.8 m/s2