GRAVITATION

Gravitation is the force of attraction between two objects in the universe.

• Gravitation may be the attraction of objects by the earth.    e.g., 
• a) If a body is dropped from a certain height; it falls downwards due to earth’s gravity.
• b) If a body is thrown upwards, it reaches a certain height and then falls downwards due to the earth’s gravity.
  

• Gravitation may be the attraction between objects in outer space. 

a) Attraction between the earth and moon.
b) Attraction between the sun and planets.

 Newton’s Law of Gravitation: 

Every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them.
Mathematically, this law, and the magnitude of the force due to the gravitational interaction between two particles, is expressed with

                           

 Gravity: 

The gravitational force of earth is called gravity. The acceleration produced in a body due to force of gravity is called acceleration due to gravity. It is denoted by ‘g’ and its value is 9.8 metre/ sq. second.

• It is important to note, that the gravitational acceleration and force exerted by the earth vary with the position of the particle above the surface of the earth; they both decrease as the particle is located further away from the surface (or the centre) of the earth. Hence, ‘g’ is maximum at poles and minimum at equator.
• If angular speed of earth becomes 17 times its present value, a body on the equator becomes weightless.

Weight of a body in a lift:

• If lift is stationary or moving with uniform speed (either upward or downward), the apparent weight of a body is equal to its true weight.
• If lift is going up with acceleration, the apparent weight of the body is more than the true weight.
• If lift is going down with acceleration, the apparent weight of the body is less than the true weight.
• If lift falls freely, the weight of a body in the lift becomes zero. This is the situation of weightlessness.
• While going down, if the acceleration of the lift is more than acceleration due to gravity, a body in the lift goes in contact of the ceiling of lift.

 Kepler’s Laws of Planetary motion:

• The Law of Ellipses: The path of the planets about the sun is elliptical in shape, with the center of the sun being located at one focus. 
• The Law of Equal Areas: An imaginary line drawn from the center of the sun to the center of the planet will sweep out equal areas in equal intervals of time.

It describes the speed at which any given planet will move while orbiting the sun. The speed at which any planet moves through space is constantly changing. A planet moves fastest when it is closest to the sun and slowest when it is furthest from the sun.

For instance, if an imaginary line were drawn from the earth to the sun, then the area swept out by the line in every 31-day month would be the same. This is depicted in the diagram below. As can be observed in the diagram, the areas formed when the earth is closest to the sun can be approximated as a wide but short triangle; whereas the areas formed when the earth is farthest from the sun can be approximated as a narrow but long triangle. These areas are the same size. Since the base of these triangles are shortest when the earth is farthest from the sun, the earth would have to be moving more slowly in order for this imaginary area to be the same size as when the earth is closest to the sun.
                                   

 The Law of Harmonies: 

• The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the sun. In other words, the square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

If ‘P’ is the time taken for a planet to complete an orbit round the sun, and ‘R’ is the mean value between the maximum and minimum distances between the planet and sun, then,
                                                         

• The law says that the expression has approx the same value for all the planets in the solar system.
• The time taken by Mercury to complete an orbit round the sun is 88 days whereas for Neptune it is 164.8 years.

 Satellite: 

Satellites are natural or artificial bodies revolving around a planet under its gravitational attraction. Moon is a natural satellite while INSAT-IB is an artificial satellite of earth.

Orbital Speed of a Satellite:

• Is independent of its mass. Hence satellites of different masses revolving in the orbit of same radius have same orbital speed.
• Depends upon the radius of orbit (height of satellite from the surface of earth). Greater the radius, lesser will be the orbital speed.
• Orbital speed of the satellite revolving near the surface of earth is 7.9 km/ sec.
• The velocity corresponding to the circular orbit is sometimes called the first cosmic velocity.

Period of Revolution of satellite:
Time taken by a satellite to complete one revolution in its orbit is called period of revolution

                  Period of revolution = Circumference of orbit/orbital speed

The period of revolution of satellite revolving near the surface of earth is 1 hour 24 minute (84 minute).

 Geo-stationary satellite: 

A geostationary satellite is an earth-orbiting satellite, placed at an altitude of approximately 35,800 kilometers (22,300 miles) directly over the equator, that revolves in the same direction the earth rotates (west to east).
At this altitude, one orbit takes 24 hours, the same length of time as the earth requires to rotate once on its axis. The term geostationary comes from the fact that such a satellite appears nearly stationary in the sky as seen by a ground-based observer. The orbit of geostationary satellite is called parking orbit.

 Escape Velocity: 

Escape velocity is that minimum velocity with which a body should be projected from the surface of earth so as it goes out of gravitational field of earth and never return back.

• It is independent of the mass, shape and size of the body and its direction of projection.
• It is also called second cosmic velocity.

For earth, Escape velocity = 11.2 km/s.
For moon, Escape velocity = 2.4 km/s.
• If   is the escape velocity, G is the gravitational constant i.e., G = 6.67×10−11 m3 kg−1 s−2 M is the mass of the body being escaped from, r is the radius of earth, g is the gravitational acceleration, and μ is the standard gravitational parameter, then,
                                     

The escape velocity is  2^(1/2) times the orbital velocity. Therefore if the orbital velocity of a satellite is increased to 2^(1/2)  times (increased by 41%), the satellite will leave the orbit and escape.