Tiêu đề 8.Gravitation-f.pdf ✅

1 | P a g e NEET-2022 Ultimate Crash Course PHYSICS Gravitation
2 | P a g e POINTS TO REMEMBER AND FORMULAE 1. Kepler’s laws are applicable not only to the solar system but to the moons going around the planets as well as to the artificial satellites 2. Kepler’s laws are valid whenever inverse-square law is involved. 3. Kepler’s laws, which are empirical laws (i.e., laws based on observations and not on theory), sum up nearly how planets of the solar system behave without indicating why they do so
3 | P a g e 4. Newton’s laws are about motion and force in general and as such involve an interaction between objects. Kepler’s laws describe the motion of only a single system, i.e., the planetary system and do not involve interactions. 5. Newton’s laws are dynamic and relate force, mass, distance and time. Kepler’s laws are kinematic and give a relation between distance and time. 6. Since F12 and F21 are directed towards the centre of mass of the two particles, the gravitational force is a central force. 7. Gravitational force is always attractive while electric and magnetic forces can be attractive or repulsive 8. Gravitational force is independent of the medium between the particles whereas electric and magnetic forces depend on the nature of the intervening medium 9. Gravitational force is a conservative force which means that work done by it is independent of path followed. This fact can also be stated by saying that work done in moving a particle round a closed path under the action of gravitational force is zero. 10. Newton’s law of gravitation is valid for objects lying at huge distances (interplanetary distances) and also for very small distances (interactomic distances) i.e., it holds over a wide range of distances. 11. Newton’s law of gravitation is of universal application and it holds irrespective of the state and the nature of the attracting bodies 12. From eqn. (3), 2 1 F r  .........(5) This means that the force exerted on a planet by the sun varies inversely as the square of the distance from the sun, i.e., gravitational force is inverse square force. Though we have taken the help of all the three laws of Kepler to deduce Newton’s law of gravitation, eqn.(5) is a direct outcome of Kepler’s third law. Thus, Kepler’s third law enables us to determine the way in which the gravitational force varies with the distance, i.e., it established the inverse square nature of gravitational force. 13. Although we have bot proved there, Kepler’s first law is also a direct consequence of the fact that the gravitational force varies as 1/r2 . It can be shown that under an inverse square force, the orbit of a planet is a conic section (i.e., circle, ellipse, parabola or hyperbola) with the sun at one focus. 14. From Art. 14.17(Part-I) dA L 2m dt = or dA L dt 2m = According to Kepler’s second law , dA dt = a constant. Hence, this implies that the angular momentum of the planet is constant, i.e., Kepler’s second law follows from conservation of angular momentum. As L is constant,  =   = r F,r F 0 or rf sin  = 0 or  = 00 or 1800 Thus, r and F must act along the same line. Such a force F , which acts along r , is called the central force. Thus, Kepler’s second law established that the gravitational force is central. In fact, this law applies to any situation that involves central force whether inverse square or not. 15. We have derived Newton’s law of gravitation from Kepler’s laws on the assumption that Newton was guided by these laws while formulating the law of gravitation. By comparing the acceleration of Moon (a) with the acceleration due to gravity on the Earth’s surface (g), he only checked the correctness of the inverse square nature of gravitational force on which his law was based. There is another view point according to which it is believed that having discovered the (1/r2 ) nature of gravitational force by comparing (a) and (g), newton formulated his universal law of gravitation, Later on, he was able to derive Kepler’s laws using his laws of motion and universal law of gravitation.
4 | P a g e We have already talked about the derivation of first law (comment 2) and second law [Art.14.17(part-I)] The third law can also be derived as discussed in Q.6(page 42). From there it follows that 2 2 3 3 4 r r Kr GM    = =     Where K is a constant whose value depends on M. In case of planets moving around the sun, M= MS (mass of the sun), 2 19 2 3 S S 4 K K 2.97 10 s / m GM −    = = =      For Moon and other satellites around the Earth, 2 13 2 3 E E 4 K K 10 s / m GM −    = = =     16. The historical connection between Kepler’s laws and Newton’s law of gravitation can best be understood from the following two statements of Newton. i) “If have seen farther from others, it is because I stand on the shoulders of giants” ii) “From Kepler’s third law, I deduced the inverse-square property of gravitational force and thereby compared the force requisite to keep the Moon in her orbit with the force of gravity at the surface of the Earth, and found them answer pretty nearly” (Newton’ s nostalgic look back-a year before his death) But all this is now a part of glorious history of physics. 17. The component 2 mr sin   changes the direction of mg 18. If the Earth were to stop rotating, the weight would increase due to the absence of the centrifugal force. 19. Gravitational intensity is a vector quantity with [LT2 ]as its dimensional formula. Its SI units is N/kg. 20. For a point of mass M, gravitational field at a distance r from it is given by 2 GM I r = 21. For a sphere of radius R, a) 2 GM I (r R, for external point) r =  b) 2 GM I (r R, at the surface) R = = c) 3 GM I r(r R R =  , where r is the distance of the internal point from the centre of the sphere) d) I 0 r 0,at the centre = = ( ) 22. For a spherical shell of radius R, a) ( ) 2 GM I r R r =  b) ( ) 2 GM I r R at the surface R = = c) I 0 r R =  ( ) d) I 0 r 0, at the centre = = ( ) 23. The expression for intensity of gravitational field in case of the Earth is the same as that for a sphere.