♦ Learn about :Newton’s Law of Gravitation ,Characteristics of the Gravitational force , Numerical problems ♦

Newton’s Law of Gravitation :

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

The gravitational force acting between two particles

F ∝ product of masses

F ∝ 1/ (the separation between two particles)2

Thus, F = G m1m/r2

where m1 and m2 are the masses of the particles, r is the distance of separation between them and G is the Universal Gravitational Constant. The value of G was first-experimentally measured by Cavendish in 1798 by using a torsion balance.

Magnitude of G = 6.67 × 10-11 Newton. m2/kg2.

Characteristics of the Gravitational force:

(a) Attractive Force : Gravitational force between two particles is always attractive and directed along the line joining the particles.

(b) Independent of Medium: It is independent of the nature of the medium surrounding the particles.

(c) Universal : It holds good for long distances like inter-planetary distances and also for short distances like inter-atomic distances.

(d) Action – reaction: Both the particles experience forces of equal magnitude in opposite directions. If F1 , F2 are the forces exerted on particle 1 by particle 2 and on particle 2 by particle 1 respectively,
then F1 = – F2 Since the forces F1 and F2 are exerted on different bodies, they are known as action-reaction pair.

(e) Gravitation is conservative: The work done by the gravitational force acting on a particle is independent of the path described by the particle. It depends upon the initial and final positions of the particle. Work done by gravity on a particle moving in a closed path is zero

(f) Superposition principle: If a particle is attracted by n particles, the net force exerted on it must be equal to the vector sum of the forces due to all the n particles.

Illustration 1: Three identical particles, each of mass m, are placed at the vertices of an equilateral triangle of side a. Find the force exerted by this system on a particle P of mass m placed at the

(a) the mid point of a side

(b) centre of the triangle.

Solution: Using the superposition principle, the net gravitational force on P is

F = FA + FB + FC

(a) As shown in the figure, when P is at the mid-point of a side, FA and FB-> will be equal in magnitude but opposite in direction. Hence they will cancel each other. So the net force on the particle P will be the force due to the particle placed at C only.
=> F = FC

along PC

(b) At the centre of the triangle O, the forces FA , FB and FC will be equal in magnitude and will be at 120° with each other.
Hence the resultant force on P at O is F = FA + FB + FC = O

Gravitational Field

The space around a material body, where it exerts a gravitational force on other bodies, is known as the gravitational field.

The gravitational force field is a vector field because a particle placed at any point P within the field experiences a force which depends on the coordinates of the point P.

Gravitational Field Strength (Intensity ) :

The intensity of the gravitational field at a point P is the gravitational force per unit mass exerted on a test particle placed at point P.

The strength of the gravitational field g(P) = (Fg/m) , where Fg is the net force acting on a test particle of mass m kept at the point P.

Its SI unit is N/kg and dimensions are M0LT-2

For earth, the gravitational field g = W/m

The above expression is equal to the acceleration due to gravity g

=> Gravitational field strength at a point on the earth is equal to the acceleration due to gravity at that point.

To find g, due to a point mass M kept at a point O (the origin), we place a point test mass m at P, the observation point, and measure the force exerted by M on the test mass m

It is equal to Fg = G M m/r2

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