GRAVITATION Gravitation is a natural phenomenon by which

GRAVITATION

Gravitation is a natural phenomenon by which all objects with mass attract each other. In everyday life, gravitation is most familiar as the agency that endows objects with weight. It is responsible for keeping the Earth and the other planets in their orbits around the Sun; for keeping the Moon in its orbit around the Earth, for the formation of tides;

l Modern physics describes gravitation using the general theory of relativity, but the much simpler Newton's law of universal gravitation provides an excellent approximation in most cases.

Newton's theory of gravitation l In 1687, English mathematician Sir Isaac Newton published Principia, which hypothesizes the inverse-square law of universal gravitation. . ”

Newton's theory of gravitation l In his own words, “I deduced that the forces which keep the planets in their orbs must be reciprocally as the squares of their distances from the centers about which they revolve; and thereby compared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the Earth; and found them answer pretty nearly

Newton's theory of gravitation

Kepler's three laws are: n n Law of Orbits The orbit of every planet is an ellipse with the sun at one of the foci. An ellipse is characterized by its two focal points; see illustration.

n Thus, Kepler rejected the ancient Aristotelian and Ptolemaic and Copernican belief in circular motion.

Law of Areas n A line joining a planet and the sun sweeps out equal areas during equal intervals of time as the planet travels along its orbit.

n This means that the planet travels faster while close to the sun and slows down when it is farther from the sun. With his law, Kepler destroyed the Aristotelean astronomical theory that planets have uniform velocity.

Law of Periods n The squares of the orbital periods of planets are directly proportional to the cubes of the semi-major axes (the "half-length" of the ellipse) of their orbits.

n This means not only that larger orbits have longer periods, but also that the speed of a planet in a larger orbit is lower than in a smaller orbit.

Escape velocity l In physics, escape velocity is the speed where the kinetic energy of an object is equal in magnitude to its potential energy in a gravitational field.

l It is commonly described as the speed needed to "break free" from a gravitational field; however, this is not true for objects under their own power.

Escape velocity l Space Shuttle Atlantis launches on mission STS-71. The need to reach escape velocity does not strictly apply to powered vehicles and those not leaving the Earth's orbit, like the Space Shuttle.

Escape velocity Space Shuttle Atlantis launches on mission STS 71. The need to reach escape velocity does not strictly apply to powered vehicles and those not leaving the Earth's orbit, like the Space Shuttle.

Escape Velocity n The minimum velocity given to the object on the surface of the planet so that it ejects from the gravitational field region & never comes back to the surface of the planet is known as escape velocity.

n For conservation of energy n 1 mve 2 = GMm n n n n 2 R ve 2 = 2 GM R ve = since, gsurface = GM/R 2 GM = gsurface R 2 Thus ve = Ve =

Weightlessness in a satellite n F – N = mv 2 r n N = F – mv 2 r n N = GMm - m. GM r 2 n i. e. a man in a satellite feels weightlessness.

Acceleration due to gravity n The acceleration produced in the body due to the force of gravity is called acceleration due to gravity.

n Suppose an object of mass m placed on the surface of the earth & acceleration to shell theorem suggested by newton the whole mass of the planet & earth is concentrated at its centre. So the earth mass system is treated as the 2 points masses placed as separation equal to the radius.

n n n F = GMm R 2 mg = GMm R 2 g = GM R 2 Since, M = 4 R 3 3 g = G. 4 R 3 R 2 3 g = 4 RG 3

Effect of g: Due to shape of the earth: n ge = GM Re 2 n gp = GM Rp 2 Re > Rp ge < gp

Due to height from the surface • • • g 1 = GM (R + h)2 g 1 = G R 2(1 + h/R)2 g 1 = g (1 + h/R)2 If h << R g 1 = gsurface(1 + h/R) -2 g 1 = gsurface ( 1 – 2 h/R) As height increases g decreases

Due to depth from the surface • g = 4 G R 3 • g 1 = 4 G (R-h) 3 • g 1 = 4/3 G (R-h) g 4/3 G R • g 1 = R - h g R • g 1 = g(R-h) R • g 1 = g(1 – h/R) When we go inside the planet the value of g decreases and at the centre of the planet it becomes zero.

Due to rotation of earth • mg 1 = Fc 2 + (mg)2 + 2 Fcmg cos (180 – λ) • (mg 1)2 = (mω2. Rcos λ)2+ (mg)2 – 2 mω2 Rcosλ. gcosλ • (g 1)2 = ω4 R 2 cos 2λ + g 2 - 2ω2 Rg cos 2λ • (g 1)2 = ω4 R 2 cos 2λ + g (g - 2ω2 R cos 2 λ)

• (g 1)2 = g 2 – 2 ω2 Rg cos 2 λ (since, ω is very small) • g 1 = (g 2 - 2 ω2 Rg cos 2 λ)1/2 • g 1 = g (1 - 2 ω2 R cos 2 λ)1/2 g • g 1 = g – ω2 R cos 2 λ

SATELLITE The heavenly bodies which revolves around the planet is known as satellite.

ORBITAL VELOCITY o The velocity of the satellite by which it revolves in the orbit is called its orbital velocity.

o The gravitational attraction of the planet of the satellite provides the required centripetal force. o F = GMm/r 2 o mv 2/r = GMm/r 2 o V= GM/r

o It is clear from above the orbital velocity does not depend upon the mass of the satellite so each orbit has its own specified speed for a given planet.

ENERGY ASSOCIATED WITH SATELLITE o The satellite has 2 types of energy 1. Kinetic Energy (due to its motion) 2. Potential Energy (due to presence of gravitational force)

Kinetic Energy l KE = ½ mv 2 l Since, v = GM/R l KE = ½ m. GM/R l KE = ½ GMm/R

POTENTIAL ENERGY U = -G Mm/R Total Energy (E) = KE + U GMm/2 R – GMm/R E = -GMm/2 R

BINDING ENERGY l The minimum amount of energy required to the satellite in order to emerge from the gravitational field of the planet is called binding energy. l Ebinding = GMm/2 R

Geo – Stationary Satellite The satellite which appears to be at rest with respect to the earth is known as geostationary satellite.