Gravitation

The force of attraction between any two bodies in the universe is called gravitation.

Force is that push or pull which changes or tends to change the state of rest or uniform motion in a straight line.

Newton's law of gravitation

Newton's law of gravitation states that everybody in this universe attracts every other body with a force which is:

1. Directly proportional to the product of their masses i.e. F ∝ m₁m₂...(i) and
2. Inversely proportional to the square of distance between their centers i.e ..(ii)

Where G is the proportionality constant which is known as universal gravitational constant.

Numerically,

G = 6.67 X 10^-11

\( \therefore F\; = \; G \frac{m_1\times \;m_2}{d^2}\) .....(iii)

Universal gravitational constant G

The force of gravitation that is exerted between two bodies of unit mass placed at a unit distance from their centers.

i.e. m₁ =1 kg

m₂ = 1 kg

d = 1m

Then from Eqn (iii), we get

F= G

Or, F = G

∴G = F

Thus, the gravitational constant 'G' is numerically equal to the gravitational force 'F' when two masses of 1kg are separated by 1meter.

This law is called universal law because this law is applicable to all objects of any size (larger or small) all across the universe (terrestrial or celestial)

Value of G on other planets:

The value of G anywhere in the universe is 6.67 x 10^-11. It is because G is universal constant and its value remains same everywhere in the universe.

Q: Two objects with a certain masses placed at a certain distance are ones separated by air medium and then by water medium. In which case is gravitational force greater:

A: Since the gravitation force depends upon mass and the distance between masses and does not depend on the medium where two objects are kept, gravitational force is equal in both air and water medium.

Effect on gravitational force of attraction if:

1. The masses of both objects is doubled keeping distance constant?

   According to the question:

   m₁= 2m₁

   m₂ = 2m₂

   D = d

   ∴ New force of attraction(f_n)

   f_n = From Eqn iii

   Or, f_n = 4

   Or, f_n= 4F

   Hence the new force will be 4 times more than the original force.

2. The masses of both objects is tripled keeping distance constant

   According to the question:

   m₁ = 3m₁

   m₂ = 3m₂

   D = d

   ∴ new force of attraction

   f_n = From Eqn (iii)

   f_n = 9

   Or, f_n = 9F

   Hence the new force will be 9 times more than the original force.
3. The mass of one body is doubled with distance reduced to half

   According to the question:

   m₁ = 2m₁

   m₂ = m₂

   D = d/2

   ∴ new force of attraction

   f_n =

   Or, f_n=

   Or, f_n = 2 ×4

   Or, f_n = 8F From Eqn (iii)

   Hence the new force will be 8 times more than the original force

4. The mass of one body increased fourfold, that of other decreased by one-third with distance increased three times

   Let,

   Two masses are m₁ and m₂

   Distance of separation is d

   ∴ The mutual force of attraction,

   F = ........ (1)

   According to the question:

   m₁ = 4m₁

   m₂= 1/3 x m₂

   D = (3 x d)

   ∴ new force of attraction

   f_n = G From Eqn (1)

   Or, f_n =

   Or, f_n =

   Hence the new force will be 4/27 times less than the original force.

   
5. The masses of both is doubled and distance increased by double?

   According to the question:

   m₁ = 2m₁

   m₂ = 2m₂

   D = 2d

   ∴ new force of attraction

   f_n=

   Or, f_n =

   Or, f_n = F from Eqn (1)

   Hence the new force will be equal to the original force.

Factors Affecting Gravitational force.

The factors affecting gravitational force are as following:-

Since,

F = therefore, the affecting factors are:-

Mass of the two bodies' i.e. F∝M₁M₂ when mass increases gravitational force increases.The distance between their centers

i.e. \(F\propto\frac{1}{d^2}\)when the distance increases gravitational force decreases.