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Gravitation Gravitation is the study of force between the earth ( gravity ) and other bodies Newton’s law of gravitation Statement : Every body in the universe attract other body with the force directly proportional to product of their masses and inversely proportional to square of the distance between them. Force acts along the line joining the centers of two masses. Vikasana – Bridge Course 2012 Explanation Consider two bodies of masses m1 and m2 separated by the distance d. Then from Newton’s law of gravitation, F ∝ m1 m2 ---- ( 1 ) and d F ∝ 1 /d2 ---- (2 ) From ( 1 ) and ( 2 ) m1 m2 F ∝ d2 Vikasana – Bridge Course 2012 m1 m2 F= G d2 Where G is the constant of proportionality called gravitational constant. 11 N m2 /kg It value Its l is i 6.67 6 67 x 10 -11 /k 2 Vikasana – Bridge Course 2012 Gravitational constant Consider m1 m2 F = G d2 F d2 G = m1 m2 Vikasana – Bridge Course 2012 If m1 = m2 = 1 Kg and d = 1m then G = F G it ti Gravitational l constant t t is i numerically i ll equal to force of attraction between two bodies of unit nit masses separated b by unit nit distance Vikasana – Bridge Course 2012 y Newton’s law of gravitation is called universal law, because the law is not only applicable to the bodies on earth but also to the heavenly bodies. The gravitational force between two masses is independent of any other external conditions like temperature, pressure, magnetic field, material medium etc. y ‘G’ is not a mere number. It has dimension of 3 in length -1 length, 1 in mass mass, -2 2 in time time. Vikasana – Bridge Course 2012 y Gravitational force is one of the basic forces of nature and it is the weakest force. f y First accurate measurement of G was made by Cavendish in 1798. 1798 y Newton’s law of gravitation is applicable for both macroscopic bodies ( planets, stars etc) and microscopic bodies ( electrons, protons etc ). Vikasana – Bridge Course 2012 y Newton’s law of gravitation does not violate Newton’s third law of motion y If F12 is the force exerted by body 1 on body 2 and F21 is the force exerted by body 2 on body 1 then, F21 = - F12 They form action – reaction pair - ve sign i shows h th thatt F12 is i opposite it iin direction to F21 Vikasana – Bridge Course 2012 Relation between g and G G M g = R2 g is independent of mass of the body Vikasana – Bridge Course 2012 y g ∝ 1/ R2 . Acceleration due to gravity is inversely proportional to square off radius off planet. y Equatorial E t i l radius di off earth th iis more th than th the polar l radius. Hence g is more at poles than the equator. y Body is carried from equator to pole, then its mass remains same but its weight g increases. Vikasana – Bridge Course 2012 y Acceleration due to gravity is independent of mass, size or shape off the falling f body. y Above Ab th the surface f off earth th g varies i (d decrease ) inversely as the square of distance from centre of earth. But inside the earth it decrease linearly with distance from centre of earth. Vikasana – Bridge Course 2012 y Galileo showed that, gravitation accelerates all objects at the same rate. y This was a major departure from Aristotle’s belief that heavier objects accelerate faster and lighter that, objects accelerate slower. y Galileo correctly postulated that “air air resistance resistance” as reason that lighter objects may fall more slowly in an atmosphere. Vikasana – Bridge Course 2012 y Rate of decrease of acceleration due to gravity with height is twice as compared to that with depth. y Value V l off acceleration l ti d due tto gravity it iis minimum i i ffor the planet mercury and maximum for the planet Jupiter. Vikasana – Bridge Course 2012 Variation of g with altitude ( height ) g1 = g ( 1- 2h / R ) From above equation, it is found that acceleration due to gravity decreases with increase of altitude. Vikasana – Bridge Course 2012 Variation of g with depth g1 = g ( 1- h / R ) From above equation, it is found that acceleration due to gravity decreases with increase of depth. Vikasana – Bridge Course 2012 Variation of g with latitude g1 = g - ω2R Cos2θ At equator θ = 0° g1 = g - ω2R. Hence g is minimum Vikasana – Bridge Course 2012 At poles θ = 90˚ g1 = g - ω2R Cos2 90˚ g1 = g - ω2R . 0 g1 = g Hence g is maximum Vikasana – Bridge Course 2012 y If the rate of rotation of earth increases, the value of acceleration due to gravity decreases at all places on the surface of earth except at poles. y If earth stops rotating, at equator acceleration due to gravity increases by 0.034m/s2 . But at poles there is no change in acceleration due to gravity. Vikasana – Bridge Course 2012 Kepler’s laws of planetary motion First law : All planets revolve around the sun in elliptical orbits with sun at one of its focus. Vikasana – Bridge Course 2012 Second law : Planets revolve around the sun in such a way that i imaginary i liline d drawn ffrom sun tto planet l t sweeps outt equal area in equal time intervals. Vikasana – Bridge Course 2012 Third law : Square of the period of rotation of the planet revolving round the sun is directly proportional to cube of their mean distances from the sun. Vikasana – Bridge Course 2012 Orbital velocity : It is the velocity with which a satellite revolve around the earth in a particular orbit. Vikasana – Bridge Course 2012 Expression for orbital velocity is GM GM Vo = or (R+h) r where r = R + h h = height of satellite from the surface of earth Vikasana – Bridge Course 2012 It is also given by g R2 Vo = r Orbital velocity is independent of the mass of body Vikasana – Bridge Course 2012 Expression for period of revolution of satellite is 2πr T= 2π ( R + h ) = Vo Vo Vikasana – Bridge Course 2012 Escape velocity It is the minimum velocity with which an object is projected from the surface of planet so that it escapes from the gravitational force of attraction of the planet. Vikasana – Bridge Course 2012 Expression for escape velocity is 2GM Ve = R or Ve = √ 2 g R Vikasana – Bridge Course 2012 y The value of escape velocity from the surface of earth is 11.2 11 2 Km/s. / y Escape E velocity l it iis iindependent d d t off th the angle l th through h which body is projected. y Escape velocity is independent of mass of the body. Vikasana – Bridge Course 2012 Geo – Stationary satellite y Geo - stationary satellite : Geo stationary satellite appears to be stationary over the same place to an observer on earth. They are also called synchronous satellites. satellites y Geo – stationary satellite is one whose orbital period is same as the p period of rotation of earth about its own axis. Hence satellite appears to be stationary to the observer on earth. Vikasana – Bridge Course 2012 y Geo – stationary satellite has height of about 36,000Km 36 000 above the equator. y It orbits round the earth from west to east ( just like the earth )). y Its orbital velocity is nearly 3.1Km/s. y Plane of orbit of geo stationary satellite is normal to axis of rotation of earth. Vikasana – Bridge Course 2012 y Geo – stationary satellites are communication satellites. They provide relay facilities f for f international communication, internal communication TV communication, TV, radio communication etc etc. y APPLE was the first Indian experimental communication satellite launched in the y year 1981. Vikasana – Bridge Course 2012 y INSAT is the series of Indian communication satellites. y INSAT – 1A, 1A 1B 1B, 1C 1C, 1D 1D, 2A 2A, 2B 2B, 2C and d 2D are Indian other communication satellites. Vikasana – Bridge Course 2012 I R S Satellites y Remote sensing is a process by which objects on earth are sensed ( acquisition of information ) remotely by satellites or rockets rockets. y Remote sensing satellites are used in the filed of Agriculture, Forestry, mineral location, Geology, Meteorology, gy Hydrology y gy etc. Vikasana – Bridge Course 2012 y Bhaskara – 1 was the first Indian remote sensing satellite launched in the year 1979. y ISRO had developed operational remote sensing satellites The first in the series is IRS – 1A. satellites. 1A Vikasana – Bridge Course 2012 y The mass of the body which measure gravitational force of attraction with earth is called gravitational mass. mass y The mass of the body which measure its inertia is called inertial mass. y Gravitational mass and inertial mass are same. Vikasana – Bridge Course 2012 Weightlessness y For a body in an artificial satellite, the gravitational force of attraction is balanced by the centrifugal force due to rotation i.e gravity is not felt. This situation is called weightlessness Vikasana – Bridge Course 2012