### 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 ‘m1’ and ‘m2’ 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 m2 kg-2 and in c.g.s. system is 6.673 10-8 dyne cm2 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 L3 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

gh = 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.

### 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.

gh = 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 gat 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 gd and gh:

• 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 ‘gΦ’ 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 mrωwhich acts radially outward. The component of centrifugal force along the radius of the earth is mrωcosΦ.
• Now the body is acted upon by two forces its weight mg acting towards the centre of the earth and the component mrωcosΦ acting radially outward. The difference between the two forces gives the weight of that body at that point.

mgΦ = mg – mrω2cosΦ ………….. (1)

Now cos Φ = O’P / OP = r/R

∴  r  = R cos Φ

Substituting in equation (1)

mgΦ = mg – m(R cos Φ)ω2cos Φ

∴   gΦ = g – R ω2cos2 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.