5.Gravitation

Gravitation

*Imp- The value of acceleration due to gravity is maximum at the pole of the earth.

Kepler’s Laws: Kepler’s laws of planetary motion describe the orbits of the planets around the Sun. He  published first two laws in 1609 and the third  law in 1619.

1.Law of orbit  

All planets move in elliptical orbits 

around the Sun with the Sun at one of the foci of the ellipse.

2.Law of areas

The line that joins a planet and the Sun 

sweeps equal areas in equal intervals of time.

3. Law of periods

The square of the time period of 

revolution of a planet around the Sun is 

proportional to the cube of the

semimajor axis of the ellipse traced by the planet.

Universal Law of Gravitation:

When objects are released near the surface 

of the Earth, they always fall down to the 

ground, i.e., the Earth attracts objects towards itself. Galileo (1564-1642) pointed out that heavy and light objects, when released from the same height, fall towards the Earth at the same speed, i.e., they have the same acceleration. Newton went beyond (the Earth and objects falling on it) and proposed that the force of attraction between masses is universal. Newton 

stated the universal law of gravitation which led to an explanation of terrestrial gravitation.

*Binding Energy of an orbiting satellite:

The minimum energy required by a satellite  to escape from Earth’s gravitational influence is the binding energy of the satellite.

Escape velocity: The minimum velocity with which a body should be thrown vertically upwards from the surface of the Earth so that it escapes the Earth’s gravitational field, is called the escape velocity of the body.

Projection of Satellite: For the projection of an artificial satellite,  it is necessary for the satellite to have a certain velocity and a minimum two stage rocket. 

Case (I) : 

If tangential velocity of projection vh

is less than the critical velocity, the orbit of satellite is  an ellipse with point of projection as apogee  (farthest from the Earth) and Earth at one of the foci.

Case (II):

If the horizontal velocity is exactly equal

to the critical velocity, the satellite moves in a stable circular orbit round the Earth.

Case (III) :

If horizontal velocity is greater than

the critical velocity and less than the escape

velocity at that height, the satellite again moves in an elliptical orbit round the Earth with the  point of projection as perigee (point closest to  the Earth).

Case (IV):

If horizontal speed of projection is equal

to the escape speed at that height, the satellite travels along parabolic path and never returns  to the point of projection. Its speed will be zero  at infinity. 

Case (V):

If horizontal velocity is greater than the

escape velocity, the satellite escapes

from gravitational influence of Earth transversing a hyperbolic path.

Earth Satellites:

The objects which revolve around the

Earth are called Earth satellites. moon is the

only natural satellite of the Earth. It revolves in almost a circular orbit around the Earth  

with period of revolution of nearly 27.3 days

Communication Satellites: These are

geostationary satellites. They revolve around the Earth in equatorial plane. They have same sense  of rotation as that of the Earth and the same period of rotation as that of the Earth, i. e., one day or 24 hours. Due to this,they appear stationary from the Earth’s surface. Hence they are called geostationary satellites or geosynchronous satellites. These are used for communication, television transmission, telephones and radiowave signal transmission.

Polar Satellites: These satellites are placed

in lower polar orbits. They are at low altitude 500 km to 800 km. Polar satellites are used for weather forecasting and meteorological purpose. They are also used for astronomical observations and study of Solar radiations.Period of revolution of polar satellite is nearly 85 minutes, so it can orbit the Earth16 time per day.

Time Period of a Satellite:

The time taken by a satellite to complete

one revolution round the Earth is its time period.

Weightlessness in a Satellite:

Case I:

Lift having zero acceleration

This happens when the lift is at rest or is

moving upwards or downwards with constant velocity:

The net force F = 0 = mg - N ∴ mg =N

Hence in this case we feel our normal

weight mg .

Case II: 

Lift having net upward acceleration au

This happens when the lift just starts

moving upwards or is about to stop at a lower floor during its downward motion (remember,  while stopping during downward motion, the acceleration must be upwards).

As the net acceleration is upwards, the

upward force must be greater.

∴ F = mau = N - mg ∴ N = mg + mau

, i.e., N > mg, hence, we feel heavier.  

It should also be remembered that this is

not an apparent feeling. The weighing machine really records a reading greater than mg.

Case III (a): 

Lifthaving net downward

acceleration ad This happens when the lift just starts  moving downwards or is about to stop at a higher  floor during its upward motion (remember,  while stopping during upward motion, the  acceleration must be downwards). 

As the net acceleration is downwards, the

downward force must be greater. 

∴ F = mad = mg -N ∴ N = mg - mad , i.e.,

N < mg, hence, we feel lighter.

It should be remembered that this is not an

apparent feeling. The weighing machine really records a reading less than mg.

Case III (b): State of free fall: This will be

possible if the cables of the lift are cut. In this case, the downward acceleration

ad  = g. 

If the downward acceleration becomes

equal to the gravitational acceleration g, we get,

N = mg - mad

 = 0.

Thus, there will not be any feeling of

weight. This is the state of total weightlessness and the weighing machine will record zero.

Critical velocity :

The exact horizontal velocity of projection that must be given to a satellite at a certain height so that it can revolve in a circular orbits around the earth is called as critical velocity or orbital velocity.