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
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 N 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 : 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.
; 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
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
, 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
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
; and it is directed towards the mass M.