Newton’s Law of Gravitation

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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.
  • The mathematical expression for the law of gravitation is sometimes written as

The negative sign indicates the force of attraction.



Newton’s Law of Gravitation in Vector Form:

Force of Gravitation:

  • The force of attraction between two material bodies in the universe is known as the force of gravitation.
  • If one of the body is the earth or some other planet or natural satellite then the force of gravitation is called the force of gravity.

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. Hence it is a central force.
  • The gravitational force between two masses is independent of the medium between the two masses. It means the gravitational force between two masses is same when they are kept in a vacuum or in water or in the air. This fact rules out the possibility of making gravity screens.
  • The gravitational force between two masses is independent of their sizes or distribution of mass of the bodies.
  • The gravitational force between two bodies does not depend upon the presence or the absence of other bodies.
  • If the masses of the body are small, the gravitational force between them is negligible. If the masses are large like that of the sun and the earth, the gravitational force of attraction is considerable.
  • The gravitational force between two bodies is called action – at – a distance type of interaction, because the two particles interact even though they are not in contact with each other. Thus gravitational force is a non-contact force.
  • Gravitational force is a conservative force because the work done by the gravitational force is independent of the path between initial and final position.

The universality of Newton’s Law of Gravitation:

  • Newton’s law of gravitation is also called as the universal law of gravitation because
  • It is applicable to all material bodies irrespective of their sizes. It is applicable to very minute particles like atoms, electrons at the same time it is applicable to heavenly bodies like planets, stars etc.
  • The law is applicable to all material bodies irrespective of the distance between them. It is applicable to interatomic distances at the same time it is applicable to stellar distances i.e. the distance between stars.

Evidence Supporting Newton’s Law of Gravitation:

  • The Earth moves around the Sun under the gravitational influence of the Sun on the Earth.
  • The Moon moves around the Earth under the gravitational influence of the Earth on the Moon.
  • The high tide and low tide are caused due to the gravitational influence of the Moon on the Earth.
  • The times of Lunar eclipses and Solar eclipses calculated on the basis of Newton’s law of gravitation are found to be approximately correct.


S.I. Unit of G:

By Newton’s law of gravitation

The SI unit of constant of gravitation is N m2 kg-2 and c.g.s. unit is dyne cm2 g-2 .

Dimensions of G:

By Newton’s law of gravitation

Hence the dimensions of universal gravitation constant are [M-1 L3 T-2]

Definition of G:

  • The gravitational force between two point masses ‘m1’ and ‘m2’ separated by distance ‘r; is given by

Let r = 1 unit, m1 = m2 = 1 unit, then G = F

  • Hence, the universal gravitational constant is the numerical value of the force between two unit masses kept at a unit distance from each other.
  • The value of G is very small and gravitational forces are small unless the masses of the two attracting bodies are large.
  • If the value of G becomes 100 times its present value, then the earth’s attraction would be so large that we would be crushed to the earth.
  • If the value of G becomes 1/100 times its present value, then we would be able to jump from a multi-story building.


Principle of Superposition of Forces:

Newtons law of gravitation

Problems Based on Newton’s Law of Gravitation:

Example – 1:

  • Calculate the force of attraction between two metal spheres each of mass 90 kg, if the distance between their centres is 40 cm. Given G = 6.67 x 10-11 N m2/kg2 . Will the force of attraction be different if the same bodies are taken on the moon, their separation remaining the same?
  • Solution:
  • Given: Mass of first body = m1 = 90 kg, mass of second body = m2 = 90 kg, Distance between masses = r = 40 cm = 40 x 10-2 m, G = 6.67 x 10-11 N m2/kg2 .
  • To Find: Force of attraction = F =?

By Newton’s law of gravitation

  • If the same bodies are taken on the moon, their separation remaining the same, the force of attraction between the two bodies will remain the same, because the force of attraction between two bodies is unaffected by the presence of the third body and medium between the two bodies.

Ans: The force of attraction between two metal spheres is 3.377 x 10-6 N

The force of attraction between two bodies remains the same

Example – 2:

  • Find the gravitational force of attraction between the moon and the earth if the mass of the moon is 1/81 times the mass of earth. G = 6.67 x 10-11 N m2/kg2 , radius of moon’s orbit is 3.58 x 105 km. Mass of the earth = 6 x 1024 Kg.
  • Solution:
  • Given: Mass of Moon =  1/81 times the mass of earth, m= 1/81 me ,Distance between the moon and earth  = r = 3.58 x 105 km = 3.58 x 108 m, G = 6.67 x 10-11 N m2/kg2 . Mass of earth = Me = 6 x 1024 Kg
  • To Find: Force of attraction = F =?

By Newton’s law of gravitation

Ans: The gravitational force of attraction between the moon and the earth is 2.213 x 1020 N

F = Antilog (2.3805 – (3 + 0.0163)) x 1021

F = Antilog (2.3805 + 3 – 0.0163)) x 1021

F = Antilog (2.3642 + 3) x 1021  = Antilog (1.3642) x 1021

F = 2.314 x 10-1 x 1021 = 2.314 x 1020

Example – 3:

  • Two bodies of masses 5 kg and 6 x 1024 kg are placed with their centres 6.4 x 106 m apart. Calculate the force of attraction between the two masses. Also find the initial acceleration of two masses assuming no other forces act on them.
  • Solution:
  • Given: Mass of first body = m1 = 5 kg, mass of second body = m2 = 6 x 1024 kg, Distance between masses = r =  6.4 x 106 m, G = 6.67 x 10-11 N m2/kg2 .
  • To Find: Initial accelerations of the two masses =?

By Newton’s law of gravitation

  • Initial acceleration of body of mass 5 kg

By Newton’s second law of motion  F = ma

Thus a = F/m = 48.85 / 5 = 9.77 m/s2

  • Initial acceleration of body of mass 6 x 1024 kg

By Newton’s second law of motion  F = ma

Thus a = F/m = 48.85 / 6 x 1024  = 8.142 x 10-24 m/s2

Ans: The Initial acceleartion of body of mass 5 kg is 9.77 m/sand

That of body of mass 6 x 1024 kg is 8.142 x 10-24 m/s2.

Example – 4:

  • A sphere of mass 40 kg is attracted by another spherical mass of 15 kg by a force of 9.8 x 10-7 N when the distance between their centres is 0.2 m. Find G.
  • Solution:
  • Given: Mass of first body = m1 = 40 kg, mass of second body = m2 = 15 kg, force between them = F =  9.8 x 10-7 N, Distance between the masses = r = 0.2 m.
  • To Find: Universal gravitation constant = G =?

By Newton’s law of gravitation

Ans: The value of universal gravitation constant is 6.533 x 10-11 N m2/kg2 .

Example – 5:

  • A sphere of mass 100 kg is attracted by another spherical mass of 11.75 kg by a force of 19.6 x 10-7 N when the distance between their centres is 0.2 m. Find G.
  • Solution:
  • Given: Mass of first body = m1 = 100 kg, mass of second body = m2 = 11.75 kg,  distance between masses = r = 0.2 m, force between them = F =  19.6 x 10-7 N,
  • To Find: Universal gravitation constant = G =?

By Newton’s law of gravitation

Ans: The value of universal gravitation constant is 6.672 x 10-11 N m2/kg2 .

Example – 6:

  • The distance of a planet from the earth is 2.5 x 107 km and the gravitational force between them is 3.82 x 1018 N. Mass of the planet and earth are equal, each being 5.98 x 1024 kg. Calculate the universal gravitation constant.
  • Solution:
  • Given: Mass of Planet = m1 = 5.98 x 1024 kg , mass of earth = m2 = 5.98 x 1024 kg, distance between them = r = 2.5 x 107 km = 2.5 x 1010 m, force between them = F =  19.6 x 10-7 N,
  • To Find: Universal gravitation constant = G =?

By Newton’s law of gravitation

Ans: The value of universal gravitation constant is 6.676 x 10-11 N m2/kg2 .

Example – 7:

  • Three 5 kg masses are kept at the vertices of an equilateral triangle each of side of 0.25 m. Find the resultant gravitational force on any one mass. G = 6.67 x 10-11 S.I. units.
  • Given:  m15 kg, m2 = 5 kg, m3 = 5 kg, r = 0.25 m, G = 6.67  x 10-11 N m2/kg2 .
  • To find: Force on m= ?

By Newton’s law of gravitation, The force on mass m1 due to mass m2.

By Newton’s law of gravitation, The force on mass m1 due to mass m3.

The angle between F12 and F13 is 60°. (Angle of equilateral triangle). The net force on m1 is given by

Newtons Law of GravitationThe two forces are equal, hence their resultant act along angle bisector towards centroid.

Ans:  Force on any mass is 4.621 x 10-8 N towards the centroid



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