Newton's universal law of Gravitation

Newton's Universal Law of Gravitation :
Every object in the universe attracts the other with a force of attraction. This is by virtue of the mass of the objects. This force of attraction was supposed to be thought upon by Newton while contemplating on the free fall of an apple towards the ground.
The force of attraction, which is the gravitational pull due to mass of objects, exists universally. The factors that affect gravitational force were studied and a law was put forth which is known as 'Newton's Universal Law of Gravitation'.
Universal Law of Gravitation
The universal law of gravitation states that every object in the universe attracts every other object with a force called the gravitational force. The force acting between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.

Derivation
For two objects of masses m and m and the distance between them r, the force (F) of attraction acting between them is given by the universal law of gravitation as:
F ∝ m ------------------------(i)
F ∝ m -----------------------(ii)
F ∝ (1/r ) ------------------(iii)

From the above equations we can rewrite them as the following
F ∝ m × m × (1/r )
F ∝ (m × m ) / ( r )
F = G(m × m ) / ( r ) , where, G is the proportionality constant called universal gravitational constant.
F = Gm m / r -----------------------(iv)
G stands for Newton's universal gravitational constant, whereas g stands for the acceleration due to gravity at a certain point.
G = 6.67300 × 10 N.m kg , G is a constant throughout space and time and it is a scalar quantity.
Equation (iv) is the mathematical expressiton for Newton's universal law of gravitation.
The second part of the law is called the 'inverse square rule' or 'inverse square law'. The force with which earth attracts any object on its surface is the weight (W) of the object, which is the product of the mass (m) of the object and its acceleration due to gravity (g).
Weight changes from place to place on the earth on account of variation in 'g'. Thus the mass of an object remains the same throughout the universe where the weight of an object changes from place to place.
Mass
One way of defining mass is on the basis of the fact that the mass of an object is the measure of its inertia. Such a mass is known as “inertial mass”. If a force “F” acting on a body produces an acceleration “a” in the body, then, its inertial mass is defined as the ratio of “F” and “a”.
The SI unit for mass is the “kilogram”. When a body is placed in the earth’s gravitational field, the body is attracted by the earth. The force with which the earth attracts a body is known as the “weight of the body”.
Internal mass = F/a
If the acceleration gained by a body due to the earth’s gravitational attraction is “g,” then its weight is equal to “mg”.
Weight of a body = mg
Since the “weight” of an object is a forcee, its SI unit is the “newton”.
The weight of a body is a vector quantity, and always acts towards the centre of the earth. If a body is taken from the “earth” to the “moon”, then there will be no change in its mass, but its weight will decrease. This is because the moon attracts the body with less force than that exerted by the earth. In fact, the weight of a body on the surface of the moon is only one-sixth of its weight on the surface of the earth.
Relation Between g and G
G stands for Newton's universal gravitational constant, whereas g stands for the acceleration due to gravity at a certain point.
G = 6.67300 × 10 N.m kg , G is a constant throughout space and time and it is a scalar quantity .
g = 9.8 m.s , g is acceleration due to gravity which is a variable quantity and a
vector qualtity .
According to Newton's law of universal gravitation the force of attraction between two bodies is given by
F = GMm/r ---------- (i)
From Newton's second law of motion the weight of a body of mass m is
F = mg -----------------(ii)
From (i) and (ii)
mg = GMm/r
or
g = GM/r
Note :
g is a constant at a given location, which depends upon M and r.

Note : g is a constant at a given location, which depends upon M and r.

* Differences Between g and G

Acceleration due to gravity (g)
Universal Gravitational Constant (G)
1.
The Acceleration produced on a
Freely falling body due to
gravitaional force is known as
Acceleration due to gravity.
1.
The force of attraction between any
two objects of unit masses separated
by unit distance in the universe is
known as Universal Gravitational Constant.
2.It is denoted by g 2.It is denoted by G
3. It changes from place to place 3.
Its value is constant everywhere in the
universe.
4. Its units are m/sec 4. Its units are Nm /Kg
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Summary
Every object in the universe attracts the other with a force of attraction. This is by virtue of the mass of the objects. This force of attraction was supposed to be thought upon by Newton while contemplating on the free fall of an apple towards the ground.
The force of attraction, which is the gravitational pull due to mass of objects, exists universally. The factors that affect gravitational force were studied and a law was put forth which is known as 'Newton's Universal Law of Gravitation'.
Universal Law of Gravitation
The universal law of gravitation states that every object in the universe attracts every other object with a force called the gravitational force. The force acting between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres.
Derivation
For two objects of masses m and m and the distance between them r, the force (F) of attraction acting between them is given by the universal law of gravitation as:
F ∝ m
F ∝ m
F ∝ (1/r ) ---------------(iii)
From the above equations we can rewrite them as the following
F ∝ m × m × (1/r )
F ∝ (m × m ) / ( r )
F = G(m × m ) / ( r ) , where, G is the proportionality constant called universal gravitational constant.
F = Gm m / r -----------------------(iv)
G stands for Newton's universal gravitational constant, whereas g stands for the acceleration due to gravity at a certain point.
G = 6.67300 × 10 N.m kg , G is a constant throughout space and time and it is a scalar quantity.
Equation (iv) is the mathematical expressiton for Newton's universal law of gravitation.
The second part of the law is called the 'inverse square rule' or 'inverse square law'. The force with which earth attracts any object on its surface is the weight (W) of the object, which is the product of the mass (m) of the object and its acceleration due to gravity (g).
Weight changes from place to place on the earth on account of variation in 'g'. Thus the mass of an object remains the same throughout the universe where the weight of an object changes from place to place.
Mass
One way of defining mass is on the basis of the fact that the mass of an object is the measure of its inertia. Such a mass is known as “inertial mass”. If a force “F” acting on a body produces an acceleration “a” in the body, then, its inertial mass is defined as the ratio of “F” and “a”.
The SI unit for mass is the “kilogram”. When a body is placed in the earth’s gravitational field, the body is attracted by the earth. The force with which the earth attracts a body is known as the “weight of the body”.
Internal mass = F/a
If the acceleration gained by a body due to the earth’s gravitational attraction is “g,” then its weight is equal to “mg”.
Weight of a body = mg
Since the “weight” of an object is a forcee, its SI unit is the “newton”.
The weight of a body is a vector quantity, and always acts towards the centre of the earth. If a body is taken from the “earth” to the “moon”, then there will be no change in its mass, but its weight will decrease. This is because the moon attracts the body with less force than that exerted by the earth. In fact, the weight of a body on the surface of the moon is only one-sixth of its weight on the surface of the earth.
Relation Between g and G
G stands for Newton's universal gravitational constant, whereas g stands for the acceleration due to gravity at a certain point.
G = 6.67300 × 10 N.m kg , G is a constant throughout space and time and it is a scalar quantity .
g = 9.8 m.s , g is acceleration due to gravity which is a variable quantity and a
vector qualtity .
According to Newton's law of universal gravitation the force of attraction between two bodies is given by
F = GMm/r ---------- (i)
From Newton's second law of motion the weight of a body of mass m is
F = mg -----------------(ii)
From (i) and (ii)
mg = GMm/r
or
g = GM/r
Note :
g is a constant at a given location, which depends upon M and r.

Differences Between Mass and Weight
Mass Weight
1. Mass is defined as the matter contained in body.
2. Mass of a body is constant throughout the universe,
3. Mass is a scalar quantity .
4.     S.I. Unit: kilo gram.
1. Weight is defined as the gravitational force
(pull) acted by the earth on the body.
2. Weight of the body changes from place to place depending on acceleration due to gravity.
3. Weight is a vector quantity .
4.     S.I. Unit: kilo gram weight.