Gravitation

Important Points (Part - III)

For the planets orbiting around the sun, the angular speed, the linear speed and kinetic energy change with time but the angular momentum remians constant.

The minimum velocity with which a body must be projected in the atmosphere so as to enable it to just overcome the gravitational attraction of the earth is called escape velocity. i.e. = , where R = radius of earth.

The relation between orbital velocity of satellite ad escape velocity is = . i.e. if the orbital velocity of a satellite revolving close to the earth happens to increase to times, the satellite would escape.

There is no atmosphere on the moon because the escape velocity on the moon is low.

If the orbital radius of the earth around the sun be one fourth of the present value, then the duration of the year will be one eighth of the present value.

Weightlessness is a situtation in which the effective weight of the body becomes zero. Circumstances when this condition arises.

When the body is taken at the centre of the earth.
When a body is lying in a freely falling lift, (a = g).
When the body is inside a space craft or satellite which is orbiting around the earth.

Kepler's first law (law of orbit): Every planet revolves around the sun in an elliptical orbit. The sun is situated at one focus of the ellipse.

Kepler's second law (law of area): The radius vector drawn from the sun to a planet sweeps out equal areas in equal intervals of time. i.e. the areal velocity of planet around the sun is constant.

Kepler's third law (law of period): The square of the time period of revolution of a planet around the sun is directly proportional to the cube of semi-major axis of its elliptical orbit, i.e. , where a = semi-major axis of the elliptical orbit of the planet around the sun.

Shape of orbit of a satellite: The shape of orbit of a satellite depends upon its velocity of projection v from earth.

If v < v₀, the satellite falls back to earth following a spiral path.
If v = v₀, the satellite revolves in circular path/orbit around the earth.
If v > v₀, but less than escape velocity v_e, the satellite shall revolve around earth in elliptical path.
If v = v_e, the satellite will escape away following a parabolic path.
If v > v_e, the satellite will escape away following a hyperbolic path.

Geo-stationary satellite/ Parking orbit

Time period = 24 hour
It is isochronous with earth.
The angular velocity of satellite must be in the same direction as that of the earth. It thus revolves around earth from west to east. Its relative angular velocity with respect to earth in zero.
The orbit of satellite should be circular.
The orbit should be in equatorial plane of earth. It contains centre of earth as well as equator.
It should be at 36000 km from the surface of earth. It is thus (36000 + 6400) km or 42400 km from the centre of earth.
Radius of parking orbit = 42400 km.
A satellite revolving in stable orbit does not require any energy from an external source. The work done by the satellite in completing its orbit is zero.
Acceleration due to gravity is used up in providing centripetal acceleration to the satellite. Hence effective g inside the satellite is zero.
Its orbital velocity = 3.1 km/sec

Its angular velocity = radian/hour.

For a satellite orbiting near earth's surface:

Time peroid = 84 minute approximately
Orbital velocity = 8 km/sec
Angular speed = = 0.00125 .

Inertial mass and gravitational mass

Inertial mass =
Gravitational mass =
They are equal to each other in magnitude.
Gravitational mass of a body is affected by the presence of other bodies near it. Inertial mass of a body remians unaffected by the presence of other bodies near it.