Gravitation

Gravitation


Newton's law of Gravitation :

According to this law, the force of attraction between two points of masses m₁ and m₂ is directly proportional to the product of two masses and inversely

proportional to the square of the distance r between them

F = .

Here G is universal gravitational constant it value is 6.67 × 10^-11 Nm² kg² .

Gravitational field According to the field concept, a mass particle modifies the space around it and sets up what is called, a Gravitational field. This field acts

upon other particles and exerts forces upon them.


Keplers laws of Planetary motion :

Ist Law : (Law of orbits)

Every planet revolves around the sun in an elliptical orbit, with sun at one of foci.

IInd Law (Law of areas)

The line joining the sun and the earth sweeps out equal areas is equal intervals of time i.e. areal velocity is constant.

Note : According to second law a planet moves faster when it is nearer to sun and moves slower when it is far away from the sun.


III law : Law of Periods :

Square of the period of any planet (T²) about the sun is proportional to cube of the mean distance (R³)of the planet from the Sun.

T²a R³or T²/R³= constant.





Gravitational field strength :
It is the Gravitational force experienced by a unit mass kept at a point in a gravitational field.

If a mass m experiences a force F at a point in a Gravitational field, the Gravitational field strength at that point is g = F/m
Gravitational field strength of a mass M at a distance r is g =

(Relation between G and g)

Gravitational field strength at a point is the acceleration due to gravity at that point.


Variation of g with Altitude :

As the altitude increases, 'g' decreases.

If R is the radius of the earth, M its mass and h is the altitude of the given point from the surface of the earth. The acceleration due to gravity at the given altitude is



r = R + h

(r = distance, from centre of the earth)



For very small altitudes (h < < < < R)




Variation of g with depth :

If r is the mean density of a planet the acceleration due to gravity on the surface of the planet is



At a depth 'd' it is





If Altitude = depth, then g_h< g_d

In order to produce the same change in the value of g, then d = 2h (Small distances)

If a tunnel is dug along the diameter of the earth and a body is released into it, the body executes SHM.

Note : Due to spin of earth, the acceleration due to gravity changes with latitude.

Gravitational potential :at a point in a gravitational field, is the work done in moving a unit mass from infinity to that point. The work done is stored in the form

of potential energy.
Gravitation potential near a mass M is , where r is the distance of the given point from the centre of the given planet. M is the mass of the planet.

Orbital velocity :

The orbital velocity a satellite is defined as the velocity required to satellite into given orbit around earth.

If the orbit radius is r, orbital velocity is



But the time period of a satellite is T =

The orbital angular velocity of a satellite revolving very close to a planet is .

For an orbit close to earth surface orbital velocity is

The period of a satellite in a circular orbit close to the surface of the earth T = =84.6 min


Escape velocity :

The minimum velocity required to product a body vertically upwards from the surface of the earth so that it may not return to the earth.

The escape velocity of a body on earth or on any planet is

or =

Orbital velocity and escape velocity are related as



The escape velocity depends upon the mass and radius of the planet from the surface of which the body is projected. Its value on earth surface is 11.2 kms^-1.