Gravitation

1. Gravitation – The force of attraction between any two bodies in this universe is called Gravitation. Universal Law of Gravitation: According to this law ‘Every particle attracts every other particle in this universe. The force of attraction between them is directly proportional to the product of their masses and inversely proportional to square of the distance between them. F = G (m1m2 / r2), where G is the universal gravitation constant, m1 and m2 are the masses of the two bodies; r is the distance between them. Important characteristics of Gravitational Force: (a). Gravitational force between two bodies form an action and reaction pair, the forces are equal in magnitude but opposite in direction. (b). Gravitational force is a central force, i.e. it acts along the line joining the centers of the two interacting bodies. (c). Gravitational force between two bodies is independent of the nature of intervening medium. (d). Gravitational force between two bodies does not depend upon the presence of other bodies. (e). Gravitational force is negligible in case of light bodies but becomes appreciable in case of heavenly bodies. (f). Gravitational force is a long range force. It is effective even if their distance of separation is very large.

2. Gravity – For the force of attraction between a body and the earth or any other planet (or satellite) instead of Gravitation we use the word gravity. 3. Weight – Force with which a body is attracted towards the centre of the earth. It is given as mg. It is a vector quantity. 4. Weight (mg) of a body changes from one place to another but mass of a body does not change, provided the speed of the body is small as compared to speed of light. 5. Acceleration due to gravity is denoted by ‘g’. The value of g at sea level; is 9.8ms-2. g = GM / R2. 6. The value of ‘g’ depends upon (a). mass of the planet (b). size or radius of the planet. 7. The value of g at height h is gh gh = g[1 + (h/R)]-2 = g[1

– (2h/ R)]

8. Weight of a body at height ‘h’ = mgh. 9. The value of g at depth‘d’ is gd = g [1 – (d / R)]. 10. Value of ‘g’ at the centre of the earth is zero. Value of g at the pole is more then that at the equator of the earth. 11. Weight of the body at depth‘d’ = mgd. Weight of the body at the centre of the earth is zero.

12. The value of ‘g’at latitude θ is g’ = g [1 – (Rω2cos2θ/ g)] At equator θ = 0, so ‘g’ at equator = g [1 – Rω2 / g] At poles θ = 900, so g = 0 at poles. 13. Inertial mass is defined as ratio of the force applied to the magnitude of the acceleration produced in the body. Inertial mass of the body = F / a. Properties of Inertial mass: (a). Inertial mass of a body is directly proportional to the quantity of matter contained in it. (b). Inertial mass of the body does not change due to presence of other bodies near it. (c). Inertial mass of a body is independent of the shape, size and the state of the body. (d). Inertial mass of a body changes with the velocity of the body, provided the velocity of the body is comparable with the velocity of light. (e). Inertial masses are added algebraically. 14. Gravitational mass of the body is defined as the ratio of the magnitude of the gravitational force on the body due to earth to the magnitude of the acceleration due to gravity. Gravitational mass = F / g. 15. Inertial and gravitational masses of the body are equivalent. 16. The space around a body within which its gravitational force of attraction is experienced by other bodies is called gravitational field.

Intensity of gravitational field at point in the gravitational field is defined as the force experienced by a unit mass placed at that point, It is denoted by E. E = GM / r2 = g

17. Gravitational potential energy of a body at a point is defined as the work done in bringing a body from infinity to that point. It is denoted by Ug. Ug = - (GMm / r) 18. Gravitational P.E at ∞ = 0 (maximum). 19. Change in gravitational P.E when body is moved from r1 to r2 where r2 >r1, is dU = - GMm [(1/r2) – (1/r1)] 20. Change in gravitational P.E, when a body is moved from surface of earth to height h is dU = - GMm [(1 / {R+ h}) – (1/R)] = mgh. 21. Gravitational potential at a point in the gravitational field of a body is defined as the amount of work done in bringing a body of unit mass from infinity to that point in the field. It is denoted by V. The unit of gravitational potential is J Kg -1. V = - GM / r 22. Gravitational field intensity and gravitational potential; are related as E = - (dV / dx), where dV / dx = potential gradient. 23. Escape velocity is defined as minimum velocity required to project a body from the surface of earth so that it never returns to the surface of earth is called escape velocity. It is denoted by ve. ve = (2GM / R) 1/2 = √2gR = 11.2Kms -1 24. Escape velocity depends upon (a). mass of the planet. (b). Size of the Planet.

25. Escape velocity does not depend on the mass of the body.

26. Orbital velocity of satellite (close to the earth) is defined as velocity with which a satellite moves in its orbit around the earth. It is denoted by v0, v0 = √gR = 8 Kms-1 ve = √2v0. 27. Time period of a satellite is time taken by a satellite to complete one orbit around the earth. T = circumference of the orbit / orbital velocity = 2π(R + h) / v0 As, v0 = [gR2 / (R + h)]1/2, So, T= (2π / R) [(R + h)3 / g]1/2 28. Height of the satellite, h = [(T2R2g)/ 4π2]1/3 – R 29. Energy of satellite U = - GMm/ 2R 30. Time period of geo stationary satellite = 24hrs. 31. Height of geostationary satellite is approximately equal to 36000 Km. 32. Motion of the planets can be explained using Kepler’s Law. (a). Law of orbits – Each planet moves around the sun in an elliptical orbit with sun at one of the focus. (b). Law of areas – The line joining the sun and a planet sweeps out equal area in equal intervals of time. (c). Law of periods – The Square of time period of a planet is proportional to the cube of semi major axis of the elliptical orbit. T2 α R3 T2 / R3 = constant

33. Mass of earth - We can calculate the mass of earth using Newton’s law of gravitation. F = GMm / R2 Where, M is the mass of earth, R is the radius of the earth, Also, F = mg From the above two equations, we get Mg = GMm / R2 Or, M = (gR2) / G Putting the values, g = 9.8ms-2, R = 6400Km = 6.4 x 106 m G = 6.67 x 10-11 Nm2Kg-2 We get, M = 5.98 x 1024Kg. _______________________________________