Concepts

## Gravitation

- All important concepts of this chapter, in one place
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## Characteristics of Gravitational Force

1. It is a universal attractive force.
2. It is directly proportional to the product of the masses of the two bodies.
3. It obeys inverse square law. It is a long range force and does not need any intervening medium for its operation. Gravitational force between two bodies does not depend upon the presence of other bodies.
4. It is the weakest force known in nature. It is a central force (i.e., it acts along the line joining the centres of the two bodies). It is a conservative force (i.e., work done in moving a body against the gravitational force is path independent). Gravitational- force between two bodies is thought to be caused by an exchange of a particle called graviton.

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## Universal Law of Gravitation

Newton's law of universal gravitation states that any two bodies in the universe attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Mathematically,
where, = Gravitational Force
= Universal Gravitational Constant
are masses of bodies
= distance between their centre
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## Inertial and Gravitational mass

1) Inertial mass

This is defined by Newton's 2nd law-  F = ma, which states that when a force F is applied to an object, it will accelerate proportionally, and that constant of proportion is the mass of that object. In very concrete terms, to determine the inertial mass, you apply a force of F Newtons to an object, measure the acceleration in m/s2, and F/a will give you the inertial mass m in kilograms.

2) Gravitational mass

This is defined by the force of gravitation, which states that there is a gravitational force between any pair of objects, which is given by

where G is the universal gravitational constant, and are the masses of the two objects, and r is the distance between them. This, in effect defines the gravitational mass of an object.

Gravitational mass is measured by comparing the force of gravity of an unknown mass to the force of gravity of a known mass. This is typically done by balance scale.

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## Relationship between acceleration due to gravity and gravitational constant

By newton's second law and universal law of gravitation on the surface of earth,

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## Approximate variation of acceleration due to gravity with height and depth

At a height , approximate expression for acceleration due to gravity is given by:

At a depth , expression for acceleration due to gravity is given by:

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## Gravitational Field due to solid sphere

Gravitational Field at an external point :

Gravitational Field at an internal point :
(Where 'R' is radius of earth.)
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## Gravitational Potential Energy between two point masses

Definition: Gravitational potential energy is energy an object possesses because of its position in a gravitational field. The most common use of gravitational potential energy is for an object near the surface of the Earth where the gravitational acceleration can be assumed to be constant at about . Since the zero of gravitational potential energy can be chosen at any point (like the choice of the zero of a coordinate system), the potential energy at a height h above that point is equal to the work which would be required to lift the object to that height with no net change in kinetic energy.

Two masses and separated at a distance has potential energy:

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## Gravitational Potential (V)

The gravitational potential (V) is the gravitational potential energy (U) per unit mass: where m is the mass of the object. Potential energy is equal (in magnitude, but negative) to the work done by the gravitational field moving a body to its given position in space from infinity.
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## Newton's Conclusions From Kepler's Laws

Kepler's laws and Newton's laws taken together imply that the force that holds the planets in their orbits by continuously changing the planet's velocity so that it follows an elliptical path is (1) directed toward the Sun from the planet, (2) is proportional to the product of masses for the Sun and planet, and (3) is inversely proportional to the square of the planet-Sun separation. This is precisely the form of the gravitational force, with the universal gravitational constant G as the constant of proportionality. Thus, Newton's laws of motion, with a gravitational force used in the 2nd Law, imply Kepler's Laws, and the planets obey the same laws of motion as objects on the surface of the Earth.
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