### Newton’s Law of Gravitation

#### Statement:

- Every particle of matter in the universe attracts every other particle of matter with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

#### Explanation :

- Let ‘m
_{1}’ and ‘m_{2}’ be the masses of two particles separated by a distance r as shown. According to Newton’s Law of gravitation, these particles will attract each other by a force ‘F’ such that

- Where ‘G’ is constant of proportionality and known as Universal gravitation constant. The value of ‘G’ in S.I. system is is 6.673 10
^{-11}N m^{2}kg^{-2 }and in c.g.s. system is 6.673 10^{-8}dyne cm^{2}g^{-2}.

#### Characteristics of Gravitational Force:

- The gravitational force between two bodies forms the action-reaction pair. The gravitational force between two masses is always that of attraction. If the first body attracts the second body with force F (direction of force from the second body to the first body), then the second body attracts the first body with equal force F (direction of force from the first body to the second body).
- The gravitational force between two masses is always acting along the line joining the centre of the two masses.
- The gravitational force between two masses is independent of the medium between the two masses.
- The gravitational force between two bodies does not depend upon the presence or the absence of other bodies.
- Gravitational force is a conservative force because the work done by the gravitational force is independent of the path between initial and final position.

#### Dimensions of G:

By Newton’s law of gravitation

Hence the dimensions of universal gravitation constant are [M^{-1} L^{3} T^{-2}]

### Acceleration Due to Gravity

- When a body is released from a height, it gets accelerated towards the earth with constant acceleration, this constant acceleration is called the acceleration due to gravity.

**The Expression for Acceleration Due to Gravity on the Surface of the Earth:**

Let m = mass of body resting on the surface of the earth

M = Mass of the earth

R = Radius of the earth

g = acceleration due to gravity on the surface of the earth.

This is the expression for acceleration due to gravity on the surface of the earth.

**Expression for acceleration due to gravity At a Height ‘h’ From the Surface of the Earth:**

Let m = mass of body resting on the surface of the earth

M = Mass of the earth

R = Radius of the earth

h = Height of the body above the surface of the earth

r = R + h = Distance of the body from the centre of the earth

g_{h} = acceleration due to gravity at height ‘h’

Where r = R + h

This is the expression for the acceleration due to gravity at height h from the surface of the earth.

#### Relation Between g and g_{h}:

**Variation in Acceleration Due to Gravity:**

**Variation in Acceleration Due to Gravity ****Due to Altitude:**

Let m = mass of body resting on the surface of the earth

M = Mass of the earth

R = Radius of the earth

h = Height of the body above the surface of the earth

r = R + h = Distance of the body from the centre of the earth

g = acceleration due to gravity on the surface of the earth.

g_{h} = acceleration due to gravity at height ‘h’

Expanding binomially and neglecting terms of the higher power of (h/R) we get

- This is an expression for the acceleration due to gravity at small height ‘h’ from the surface of the earth.
- The acceleration due to gravity decreases as we move away from the surface of the earth.

**Variation in acceleration due to gravity **** Due to Depth:**

- The acceleration due to gravity on the surface of the earth is given by

Let ‘ρ’ be the density of the material of the earth.

Now, mass = volume x density

Substituting in the equation for g we get

- Now, let the body be taken to the depth, ’d’ below the surface of the earth. Then acceleration due to gravity g
_{d }at the depth, ’d’ below the surface of the earth is given by

Dividing equation (3) by (2) we get

- This is an expression for the acceleration due to gravity at the depth, ’d’ below the surface of the earth.
- The acceleration due to gravity decreases as we move down into the earth.

**Relation between g _{d} and g_{h}:**

- We have

- Thus the acceleration due to gravity at a small height ‘h’ from the surface of the earth is the same as the acceleration due to gravity at the depth ‘d = 2h’ below the surface of the earth. It means that the value of acceleration due to gravity at a small height from the surface of the earth decreases faster than the value of the acceleration due to gravity at the depth below the surface of the earth.

**Variation of g with Altitude and Depth**

**Variation in acceleration due to gravity ****Due to Latitude of the Place:**

- The latitude of a point is the angle Φ between the equatorial plane and the line joining that point to the centre of the earth. Latitude of the equator is 0° and that of poles is 90°.
- Let us consider a body of mass ‘m’ at a point P with latitude ‘Φ’ as shown on the surface of the earth. Let ‘g
_{Φ}’ be the acceleration due to gravity at point P.

OP = Radius of earth = R

O’P = Distance of point P from the axis of earth

- Due to rotational motion of the earth about its axis, the body at P experiences a centrifugal force which is given by mrω
^{2 }which acts radially outward. The component of centrifugal force along the radius of the earth is mrω^{2 }cosΦ. - Now the body is acted upon by two forces its weight mg acting towards the centre of the earth and the component mrω
^{2 }cosΦ acting radially outward. The difference between the two forces gives the weight of that body at that point.

mg_{Φ} = mg – mrω^{2}cosΦ ………….. (1)

Now cos Φ = O’P / OP = r/R

∴ r = R cos Φ

Substituting in equation (1)

mg_{Φ} = mg – m(R cos Φ)ω^{2}cos Φ

∴ g_{Φ} = g – R ω^{2}cos^{2} Φ

- This is an expression for acceleration due to gravity at a point P on the surface of the earth having latitude Φ.
- At the equator Φ = 0°. Hence ‘g’ is minimum on the equator. For the poles Φ = 90°. Hence ‘g’ is maximum on the poles.