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

According to Newton,

Force of gravitation, F µ product of masses

directly proportional to 1/square of the separation between them

Thus, F= G(m1m2/r² )

where m₁ and m₂ are the masses of the particles, r is the distance of separation between them and G in Universal Gravitational Constant.

Magnitude (and unit) of G : 6.67 * 10^–11 N m² / kg²

Dimension of G         : M^–1 L³ T².

Characteristics of the Gravitational Force:

(a)   Gravitational force is always attractive and directed along the line joining the particles.

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

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

(d)   Interaction means that, both the particles experience forces of equal magnitude in opposite directions. If  F1, F2 are the forces exerted on particle 1 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)   It is a conservation force. Therefore, the work done by the gravitational force on a particle is independent of the path described by the particle. It depends upon the initial and final position of the particle. Therefore no work is done by the gravity if a particle moves in a closed path.

(f)    If a particle 1 is acted by n particles, say, the net force F1  exerted on it must be equal to the vector sum of the forces due to surrounding particles.

Þ F = ΣF

where  F1= force acted on the particle 1, by the i^th particle

Hence, gravitational force between any two particles does not depend upon the presence or absence of other particles (bodies).

Illustrations

1. Three identical particles each of mass mare placed at the vertices of an equilateral triangle of side a. Find the forcer 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.

Sol. 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.

(b)       At the centre of the triangle O, the forces FA, FB, Fc will be in magnitude and will subtend 120º with each other. Hence the resultant force on P at O is

F= FA + FB + FC = 0

You must have seen the night-sky composed of stars, planets, moon and you could have also been lucky enough to see a shooting star or meteor as it is termed. If you make a careful study, you could also observe, that the pattern of night-sky keeps on changing with time as well as with seasons in a seemingly complex manner. Our ancient Indian scientists had made careful observation about this changing pattern and had deduced that planets revolve around the sun, a fact which was later rediscovered by Copernicus.

It took the genius of Newton to reduce this complicated motion into a very simple universal law – the law of Gravitation, a law, which not only applies to celestial bodies but also applies to that famous apple which, Newton saw falling.

OBJECTIVE

After learning this chapter we will be able to understand the law of gravitational force, gravitational potential energy. We will also be able to learn the concept of planetary motion of different planets and other heavenly bodies. In this chapter we will revolve around the Universal Law of Gravitation and its applications and try to gain an insight into one of the basic forces in the Universe.

PRE-REQUISITE

Þ            Concept of acceleration

Þ            Earth’s gravity

Þ            Concept of circular motion

Þ      Concept of centripetal and centrifugal forces

Þ      Newton’s third law of motion (Action and Reaction)

CORE CONCEPTS

When you throw a ball up, the ball goes up, its velocity is retarded and it finally comes to rest. Henceforth, it gains acceleration and returns back to the ground. This retardation and subsequent acceleration is a consequence of the universal gravitational force. There is a specific law, which guides the above phenomena, known as Newton’s Law of Gravitation. It states:

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

Let m1 and m2 be the masses of the two particles and r be the separation between them (see figure given below).

“The space around a body within which its force of gravitational attraction is perceptible (by any other body in this space) is called its gravitational field.”

The intensity E, of the gravitational field of a mass ‘m‘ at a point at distance ‘r‘ from it is the force experienced by a unit mass placed at this point in the field. (Assuming that the presence of unit mass does not affect the gravitational field of the mass m)