Gravitation

Newton's Law of gravitation states that every particle in the universe attracts every other particle with a force which is directly proportional to the mass of each particle and inversely proportional to the square of the distance between them. If two particles 1 and 2 have masses m₁ and m₂, and, the distance between them is r, the magnitude of the gravitational force between them is:

Fg = Gm1m2 /r2

The constant of proportionality G is called universal gravitational constant, and its value is 6.67 × 10^-11 N-m²/ kg².

Force of gravity is the special name given to the gravitational force exerted by the earth on any body on or near its surface. This force produces an acceleration due to gravity, g, in the body. The vector g is always directed towards the centre of the earth and its standard value at sea level at 45° latitude is 9.81 m/s² . Variation of acceleration due to gravity happens due to a number of factors such as altitude, depth, earth's rotation and its non-spherical shape.

A body possesses two types of mass: inertial mass and gravitational mass. The concept of inertial mass comes from Newton's second law of motion. It gives a measure of resistance to any change of state of rest or motion under an external force. The concept of gravitational mass arises from Newton's Law of gravitation. It gives a measure of the force experienced by the body in a given gravitational field. Experiment shows inertial mass and gravitational mass of a body are the same.

Kepler's laws of planetary motion, as published by Johannes Kepler between 1609 and 1619, are as follows:

1. The planets move in planar elliptical orbits with the sun situated at one focus.
2. The radius vector from the sun to the planet sweeps equal areas in equal times.
3. The square of the period of revolution of any planet about the sun is proportional to the cube of the length of the semimajor axis.

The gravitational field at a point in space is the gravitational force experienced by a unit mass placed at that point. The dimensions of gravitational field are [LT^-2], and its SI unit is newton per kilogram (N/kg).

The gravitational potential energy of a system comprising two particles of masses m₁ and m₂ separated by distance r is:

U = - G m1m2 / r

This is a negative quantity, which becomes zero if the particles are at infinite separation.

The gravitational potential at a point due to a particle is the gravitational potential energy of a system comprising the particle in question and a unit mass placed at the given point. The dimensions of gravitational potential are [L²T^-2], and its SI unit is joule per kilogram (J/kg).

The minimum speed with which a particle must be projected so that it may escape the gravitational attraction of the earth is called the escape speed for the earth. It is given by the formula:

vesc = √(2GME /RE)

where M_E and R_E are the mass and radius of the earth respectively. Its value is 11.2 km/s, independent of the angle of projection of the particle. Other heavenly bodies such as the sun, other planets and their moons also have their respective escape speeds. Escape speed assigned with a specific direction is called escape velocity.

Just like the planets revolve round the sun, many of the planets have one or more natural satellites revolving round them. A natural satellite is also called a "moon". The earth's moon is its only natural satellite. However, man has launched many artificial satellites in orbits round the earth. The orbital speed of a satellite in a circular orbit round the earth is given by:

vorb = √(GME / r)

Here r = R_E + h is the orbit radius, h being the height of orbit above earth's surface. Orbital velocity at a given point on the orbit has magnitude equal to orbital speed and direction along the tangent to the orbit at the given point.

A geostationary satellite is an artificial satellite which always remains above the same spot on the equator. Its orbit is called geosynchronous orbit or parking orbit.

The total mechanical energy of satellite-earth system is given by:

E = - GME m / 2r

where M_E is earth's mass, m is satellite's mass, and r is orbit radius.

Since a satellite revolving round the earth is in a state of free fall, the satellite and its occupants must also be weightless. An astronaut, like any other human, feels comfortable with the sensation of weight. Therefore, weightlessness of an astronaut during their long stay in an orbiting satellite is an unpleasant feeling and may cause medical symptoms.