Einstein Classes, Unit No. 0, 0, Vahman Ring Roa Plaza, Vikas Pui Extn., New Delhi -8 Ph. : 96905, 857, E-mail einsteinclasses00@gmail.com, PG GRAVITATION
Einstein Classes, Unit No. 0, 0, Vahman Ring Roa Plaza, Vikas Pui Extn., New Delhi -8 Ph. : 96905, 857, E-mail einsteinclasses00@gmail.com, C O N C E P T S PG C NEWTON S LAW OF GRAVITATION Evey paticle of matte in the univese attacts evey othe paticle with a foce, known as gavitational foce. Newton s Law of Gavitation states that two paticles with masses m an m, a istance apat, Gmm attact each othe with gavitational foces of magnitue F. Pinciple of Supeposition : If n paticles inteact, the net foce F, net fom all the othe paticles taken one at a time. on a paticle labele as paticle is the sum of the foces on it n F F,net In which the sum is a vecto sum of the foces F on paticle fom paticles,,..., n. i i Gavitation Behaviou of spheically symmetic soli : The gavitational effect outsie any spheical symmetic mass istibution is the same as though all the mass of the sphee wee concentate at its cente. Pactice Poblems :. A point mass of mass M ae boken into two pats an sepaate by cetain istance. The maximum foce between the two pats is given by 4 i 8. Thee point masses each of mass m ae place at the cone of an equilateal tiangle of sie length. The foce on one of the mass is C C [Answes : () b () c] GRAVITATION FIELD AND INTENSITY Gm The magnitue gavitational intensity ue to a mass m at point P is E, iecte towa the mass m. Pinciple of Supeposition : In the pesence of n paticles, the net gavitational intensity E at a paticula point is the vecto sum of the intensities ue to iniviual paticles at that paticula point : E GRAVITATION FIELD INTENSITY DUE TO EARTH AND ACCELERATION DUE TO GRAVITY The gavitational fiel intensity ue to eath an acceleation ue to gavity ae the same. n i E. Outsie o on the Eath : The value of g outsie the eath at the istance fom the cente is g. On the suface of eath, g0 whee R is the aius of eath. R
Einstein Classes, Unit No. 0, 0, Vahman Ring Roa Plaza, Vikas Pui Extn., New Delhi -8 Ph. : 96905, 857, E-mail einsteinclasses00@gmail.com, Above the suface of height h, g is given by, g g 0 R (R h) h g0 R PG Using the appoximation ( + x) n + nx fo x < <, we get g g 0 h if h < < R. R Insie the Eath : The value of g insie the eath at the istance is given by g = Vaiation of g with positon : g 0 R Effect of g ue to Eath s otation : g = g 0 Rcos, whee is the angle of latitue an is the angula velocity of the eath about its axis. case I At the equato, = 0, g = g 0 R case II At the pole, =, g = g0. Hence thee is no effect of eath s otation on g at pole. Pactice Poblems :. If the aius of the eath wee to shink by one pe cent, its mass emaining the same, the value of g on the eath s suface woul incease by 0.5% incease by % ecease by 0.5% ecease by %. Two planets have aii R an R an ensities an espectively. The atio of the acceleation ue to gavity at thei suface is R R R R R R R R. If the value of g at the suface of the eath is 9.8 m/s, then the value of g at a place 480 km above the suface of the eath will be (aius of eath = 6400 km) 4. m/s 7. m/s 8.5 m/s 9.8 m/s 4. If R is the aius of the eath then the altitue at which the acceleation ue to gavity will be 5% of its value at the eath s suface is R/4 R R/8 R/ 5. The aius of the eath is 6400 km an g = 0 m/s. In oe that a boy of 5 kg weight zeo at the equato, the angula spee of the eath shoul be a / s 80 a / s 400 6. As we go fom the equato to the poles, the value of g a / s 800 emains the same inceases a / s 600 eceases ecease up to a latitue of 45 0. [Answes : () b () a () c (4) b (5) c (6) b]
Einstein Classes, Unit No. 0, 0, Vahman Ring Roa Plaza, Vikas Pui Extn., New Delhi -8 Ph. : 96905, 857, E-mail einsteinclasses00@gmail.com, C4 GRAVITATIONAL POTENTIAL ENERGY : The gavitational potential enegy U() of a system of two paticles, with masses M an m an sepaate by m a istance, is given by U. Potential enegy of the system of paticles : G mm U Gmm Gm m PG 4 U Potential enegy an foce : F ˆ Pactice Poblems :. A boy of mass m is taken fom the eath s suface to a height equal to the aius R of the eath. If g is the acceleation ue to gavity at the suface of the eath, then the change in the potential enegy of the boy is mgr 4 [Answes : () b] C5 GRAVITATION POTENTIAL (V) : mgr mgr mgr It is efine as the Gavitational Potential Enegy pe unit mass. The Gavitational Potential (V) ue to Gm mass m at point P is. v Gavitational Potential (V) an intensity ( E) : E ˆ Pactice Poblems :. The gavitational fiel ue to a mass istibution is E = K/x in the x-iection, whee K is a constant. Taking the gavitational potential to be zeo at infinity, its value at a istance x is C6 K/x K/x K/x K/x [Answes : () ] ESCAPE SPEED An object will escape fom the gavitational pull of an astonomical boy if the object is pojecte with a cetain minimum spee fom the boy is suface an this minimum spee is known as escape spee. Escape spee fom the suface of the eath is V e Let the astonomical boy is of unifom ensity then M =, whee M is the mass of the eath an R is the aius. R 4 R an hence V e 8 GR. The escape spee of an object at a given point in the fiel is inepenent of its mass an state of its motion, but is position-epenent. Fo eath the escape spee fom the suface V e is. km/s. Pactice Poblems :. The escape velocity fom the eath is km/s. The escape velocity fom a planet having twice the aius an the same mean ensity as those of the eath is 5.5 km/5 km/s km/s none of these
Einstein Classes, Unit No. 0, 0, Vahman Ring Roa Plaza, Vikas Pui Extn., New Delhi -8 Ph. : 96905, 857, E-mail einsteinclasses00@gmail.com,. The mass of moon is of eath s mass an its aius is of that of the eath. If the escape 8 4 velocity fom the eath s suface is. km/s, its value fo the moon is 0.4 km/s 0.5 km/s.5 km/s 5.0 km/s. The escape velocity of a paticle of mass m vaies as Km, whee K is a constant. The value of is 0 ½ ½ 4. If a ocket is to be pojecte vetically upwas fom the suface of the eath, it equies an escape velocity of km/s. If the ocket is to be pojecte at an angle of 60 0 with the vetical, the escape velocity equie will be about 5.5 km/s km/s km/s 5.5 km/s [Answes : () c () c () a (4) c] PG 5 C7 SATELLITES : ORBITS AND ENERGY A satellite is any boy evolving aoun a lage boy, une the influence of the latte. Fo example the moon is a gavitational satellite of the eath. Thee ae vaious atificial satellite of the eath, they ae known as geo-static o geo stationay o geo-synchonous satellite. Fo this type of satellites, the conitions ae the obit must be cicula the obit must be in equatoial plane of the eath the peio of evolution of the satellites is 4 hou the angula velocity of evolution of the satellite must be in the same iection as the angula velocity of otation of the eath. If a satellite is to be always seen ovehea, the obseve shoul be on the equato of the Eath. Fo a satellite in cicula obit : Obital spee v 0 R h whee = R + h, R is the aius of plane an h is height fom the suface of planet at which a satellite is evolving. Time Peio T v 0 4 m, Kinetic enegy K mv 0, m m Potential enegy U, Total mechanical enegy = E = K + U = The negative sign inicates that the satellite is boun. Relation between E, U an K : U E K,E,K U Bining enegy of the satellite = m Vaiation of K, U an E with is shown in figue.
Einstein Classes, Unit No. 0, 0, Vahman Ring Roa Plaza, Vikas Pui Extn., New Delhi -8 Ph. : 96905, 857, E-mail einsteinclasses00@gmail.com, PG 6 (e) Spees an natue of obits : If V < V 0 ; the obit is elliptical with the cente of the eath as the futhe focus, E is negative. If V = V 0 ; the obit is cicula, E is negative If V 0 < V < V e : the obit is elliptical, E is negative If V = V e : the boy escapes, E is zeo. If V > V e : the boy escapes along a hypebolic Path, E is positive. Hee E stans fo total mechanical enegy of the boy. Pactice Poblems :. A satellite of mass m is evolving aoun the eath at a height R above the suface of the eath. If g is the gavitational fiel intensity at the eath s suface an R is the aius of the eath, the kinetic enegy of the satellite is mgr 4 mgr mgr mgr. The obital velocity of an atificial satellite in a cicula obit just above the eath s suface is v. Fo a satellite obiting at an altitue of half the eath s aius, the obital velocity is v v. Let the total enegy, kinetic enegy an potential enegy of the atificial satellite ae E, E an E espectively. Then E : E : E is : : : : : : : : 4. Time peio of evolution of a satellite close to the suface of a spheical planet of aius R is T. The peio of evolution close to the suface of anothe planet of aius R an same ensity is C8 (i) (ii) (iii) T T T 9T [Answes : () a () c () b (4) a] KEPLER S LAW Fist Law (law of obit) v The planets move aoun the sun in elliptical obits with the sum at one focus. Secon Law (law of aea) The line joining the sun to a planet sweeps out equal aeas in equal times. This law is base on consevation of angula momentum. Thi Law (law of peio) The squae of the peio of planet is popotional to the cube of its mean istance fom the sun. The mean istance tuns out to be the semi-majo axis, a, i.e., T a v
Einstein Classes, Unit No. 0, 0, Vahman Ring Roa Plaza, Vikas Pui Extn., New Delhi -8 Ph. : 96905, 857, E-mail einsteinclasses00@gmail.com, Pactice Poblems :. A planet moves aoun the sun. At a point P it is close to the sun at a istance an has a spee v. At anothe point Q, when it is fathest fom the sun at a istance, its spee will be v v v v. The figue shows the motion of a planet aoun the sun in an elliptic obit with the sun at one focus. The shae aeas A an B can be assume to be equal. If t an t epesent the times taken by the planet to move fom a to b an fom c to espectively, then PG 7 t < t t > t t = t fom the given infomation the elation between t an t cannot be etemine. [Answes : () c () c]. A missile is launche with a velocity less than the escape velocity. The sum of its kinetic an potential enegies i.e. mechanical enegy is positive negative zeo may be positive o negative epening on its initial velocity. A planet is moving in an elliptic obit. If T, V, E an L stan, espectively, fo its kinetic enegy, gavitational potential enegy, total enegy an angula momentum about the cente of foce, then T is conseve V is always positive INITIAL STEP EXERCISE E is always negative magnitue of L is conseve but its iection changes continuously.. The escape velocity fom the suface of a planet is 0 4 m/s. If a mass of kg falls fom infinity to the suface of the planet, its kinetic enegy an the magnitue of its potential enegy on eaching the suface will be 0 8 J, zeo 0 8 J, 0 8 J 0.5 0 8 J, 0.5 0 8 J 0 8 J, 0 8 J 4. Gas escapes fom the suface of a planet because it acquies an escape velocity. The escape velocity epens on (i) (ii) (iii) (iv) mass of the planet mass of the paticle escaping tempeatue of the planet aius of the planet. Select the coect answe fom the coes given below (i) an (ii) (ii) an (iv) (i) an (iv) (i), (iii) an (iv) 5. A penulum clock is set to give coect time at the sea level. This clock is move to a hill station at an altitue of 500 m above the sea level. In oe to keep coect time on the hill station, the length of the penulum has to be euce has to be incease nees no ajustment nees no ajustment but its mass has to be incease.
Einstein Classes, Unit No. 0, 0, Vahman Ring Roa Plaza, Vikas Pui Extn., New Delhi -8 Ph. : 96905, 857, E-mail einsteinclasses00@gmail.com, 6. Accoing to Kepla s secon law, the aius vecto to a planet fom the sun sweep sout equal aeas in equal intevals of time. This law is a consequence of the consevation of linea momentum angula momentum mechanical enegy mass PG 8 7. The pecentage change in aius of the eath is % without change in its mass. The pecentage change in time peio of the simple penulum is ½ % % %. % FINAL STEP EXERCISE. Imagine a light planet evolving aoun a vey massive sta in a cicula obit of aius R with peio T. If the gavitational foce of attaction between the planet an the sta is popotional to R 5/. then T is popotional to R R 7/ R / R.75. The magnitues of gavitational fiel at istances an fom the cente of a unifom sphee of aius R an mass M ae F an F, espectively. Then F if < R an < R F F if > R an < R F F if > R an > R F F if < R an < R F. Fo a planet moving aoun the sun in an elliptical obit of semi-majo an semi-mino axes a an b, espectively, an peio T, then the toque acting on the planet about the sun is zeo the angula momentum of the planet about the sun is constant the aeal velocity is ab/t all ae coect 4. Two objects of masses m an 4 m ae at est at an infinite sepaation. They move towas each othe une mutual gavitational attaction. If G is the univesal gavitational constant, then at a sepaation the total enegy of the two objects is zeo thei elative velocity is (0 Gm/) / in magnitue. the total kinetic enegy of the objects is 4 Gm / all ae coect 5. The masses an aii of the eath an the moon ae M, R an M, R espectively. Thei centes ae at a istance apat. The minimum spee with which a paticle of mass m shoul be pojecte fom a point mi-way between the two centes so as to escape to infinite is G (M M ) G (M M ) G (M M ) G (M M ) 6. A soli sphee of unifom ensity an aius R applies a gavitational foce of attaction equal to F on a paticle place at A, istant R fom the cente of the sphee. A spheical cavity of aius R/ is now mae in the sphee as shown in the figue. The sphee with cavity now applies a gavitational foce F on the same paticle place at A. The atio F /F will be
Einstein Classes, Unit No. 0, 0, Vahman Ring Roa Plaza, Vikas Pui Extn., New Delhi -8 Ph. : 96905, 857, E-mail einsteinclasses00@gmail.com, / 7/6 7/9 7. A paticle is pojecte fom the suface of eath with spee such that it attains a height equals to the times the aius of eath. The minimum value of the spee is PG 9 ( )R ( )R 4 ( )R 5 ( )R ANSWERS (INITIAL STEP EXERCISE). b. c. 4. c 5. a 6. b 7. b ANSWERS (FINAL STEP EXERCISE). b. a. 4. 5. a 6. 7. a
Einstein Classes, Unit No. 0, 0, Vahman Ring Roa Plaza, Vikas Pui Extn., New Delhi -8 Ph. : 96905, 857, E-mail einsteinclasses00@gmail.com, PG 0 TEST YOURSELF. Two satellites of the same mass ae obiting oun the eath at heights of R an 4R above the eath s suface : R being the aius of the eath. Thei kinetic enegies ae in the atio of 4 : : 4 : 5 :. The atio of the escape velocity of an eath satellite to its obital velocity is vey nealy equal to / /. Thee paticles, each of mass m, ae place at the vetices of an equilateal tiangle of sie a. The gavitational fiel intensity at the centoi of the tiangle is Gm zeo a Gm a Gm a 4. A comet is evolving aoun the sun in a highly elliptical obit. Which of the following will emain constant thoughout its obit? Kinetic enegy Angula momentum Total enegy both an ae coect 5. The change in the gavitational potential enegy when a boy of mass m is aise to a height nr above the suface of the eath is (hee R is the aius of the eath) n mgr n nmgr n mgr n mgr n.. a. a 4. 5. a ANSWERS