07 Gravitation Tutorial

A 1 . 8 ¡¿ 1 0 ' J B 5 . 5 ¡¿ 1 0 7 J C 1 . 1 ¡¿ 1 0 8 J D 2 . 2 ¡¿ 1 0 8 J to lift a m ass of 2 . o kg from its surface into outer space is m . T he energy needed f(r+ h) 2 - (, + h) " ― ― «ê «Õ ー W hat is the gravitational field strength at a height h above the ground? uniform sphere of radius r. T he gravitationa©¥ field strength at its surface is g . that of a point m ass M at the centre of the sphere. T he E ailh m ay be taken to be a O utside a uniform sphere of m ass M , the gravitational field strength is the sam e asS P 2 A m S - 2 B N m - 2 kg 2 m a kg ' s - Z D m 2 kg - 2 S P 1 T he S l base units of the gravitational constant G is r. f ¡¤ ( U ¡¤ R \ S elf - P ractice Q ueş. Tions the tw o expressions? the orbital velocity of the sam e object around the E arth? W hat are the differences in S tate the expressions for the escape ve©¥ocity of an object from the surface E arth and5 7 5 6 W nat is a geostationary sate©¥lite? W hy is the geostationary orbit unique? 5 5 W hat can you say about the total energy of a satellite that is bound to planet? 5 4 W hat is gravitational potential? S tate an expression for the gravitational potential. strength. 5 3 W nat is gravitationa©¥ fie©¥d strength? S tale an expression for the gravitational fie©¥d to? 5 2 W nat does N ewton * s ©¥aw of gravitalion states? W tral kind of objecta can it be app©¥ied 5 1 W hat l8 a gravitational fie©¥d? self - c w 1 n s T uto"

D orbital period C kinetic energy B centripetal acce©¥eration A angular velocity orbits around the E arth? S P B W hich quantity is not necessarily the sam e for satellites that are in geostationary 2R A JgŘ B m gR C D m g least T o escape * om the gravitational field of the E arth, the speed of the body m ust be at acceleration of * ee fall is g and the radius of the E arth is R . S P 7 A body of m ass m is projected from the E arth ' s s u rface. A t the point of launch, the A T oc R į B T oc R Ĵ C T oc s į D T oc p 3 W nich one of the follow ing correctly show s how T depends on P , R S ? period T . S P 6 A p©¥anet of m ass P m oves in a circular orbit of radius R round a sun or m ass S w ith E g ravitational potentia©¥ energy D angular velocity C linear speed in the orbit B centripeta©¥ acce©¥eration A gravitationa©¥ force change? W nich one of the follow ing quanliliea Increases for the satellite aa a result of the orbot at a greater dlslance from the E arth . S P 6 A n E arth eatellite ©¥©¥ m oved rrom one a©¥able circular orbit Io another atable clrcul©¥r A - 400 kJ B 2 0 0 kJ C ¡ß 2 0 0 kJ D ¡ß 4 0 0 kJ ntlal it X ©¥©¥ - 800 H re of the E arth, W nen a 1 kg m ass l8 laken from X Io Y , the w ork done on the m a©¥©¥ l©¥ kJ kg - 1 . where R II greater ©¥han the radrua of the E arth T he gravilaliona©¥ poler X and Y are tw o polnla al respective di©¥lanco©¥ R and 2 R from the con

«c) C a©¥cu©¥ate the force on each star due to the other star. (b) D eterm ine the gravitationa©¥ potential at X . fie©¥d strength is zero. E xplain w hy you have chosen this point. (a) L abe©¥ on the above diagram , w ith a ©¥etter X , a point w here the gravitationa©¥ ª¹ ー 玛 ¡£ìéÙ£ m ass of the system . m ass 4 . 0 ¡¿ 1 0 " kg , s e parated by 2 . 0 ¡¿ 1 0 " m . T he stars rotate about the centre of T he figure below shows a binary star system w hich consists of ©¥w o stars, e a ch of S P 10 orbit of the S un. C alcu©¥ate the radius of J upiter ' s o rbit. year to com p©¥ete one orbit. T he planet J upiter takes 1 1 . 9 years to com plete an (d) T he earth ls 1 . 50 ¡¿ 1 0 " m from the centre of the S un and lakes exactly one (c) D educe the express©¥on for the lim e taken to com plete one orbit of the S un. (b) U se this expression to find the angu©¥ar ve©¥oclty of the planet ©¥n the orbit. the S un on the p©¥anet. (a) W rite dow n an expression, In term s of G , m , M and r, for the force F exerted by A p©¥anet of m ass m orb©¥te the S un of m ass A￠ In a c©¥rcular path of radius r S P 9 0 9 0M 12 - m o dl©¥led] rt e d from D iscuss©¥on g gm itrE B [G ue¨¡ t©¥m e B ©¡n ! m

4 atm osphere. (c) In the light of your answ er exp©¥ain w hy m any sm all planets do not have gaseous the planet w ere 1 0 3 sm al©¥er, its m ean density rem aining unchanged? (b) B y w hat factor w ould the escape velocity be reduced if the linear dim ensions of (ii) the escape ve©¥ocity from the surface of the planet. (i) the orbital speed of the satellite, m ass 5 . 7 ¡¿ 1 0 24 kg and radius 6 5 0 0 km . C alculate (a) A sm all satel©¥ite is in a stable circular orbit of radius 7 0 0 0 km around a planet of (c» L ist the characteri stics of a geostationary sate©¥lite. (b» F ind the altitude of the satellite ©¥ s o rbit. answ er in rad s - 1 . «a» W hat is the angular velocity of the E arth ' s ro ta tion about its axis? G ive your of the E arth is 5 . 98 ¡¿ 1 0 z4 kg a n d its ra d iu s is 6 . 37 ¡¿ 1 0 e m . Sate©¥litřiŝ aboveí the sam e po©¥nl on the E arth ©¥ s s u rface. A ssum e that the m ass w hich it has an 6 0 1 a7 . V dy aiůät ' tĞ thàr at iivhïch - ttië ' E aiñ fi rota©¥©¥* , s o that the A com m unications satellite of m ass 3 0 0 0 kg is to be put into an equatorial orbit in acceleration is the ' a c c e leration due to gravity ' . C om m ent on his statem ent. m easures its acceleration tow ards the E arth ' s s u l «¤ ace. H e then states that this (c) A student, s ituated al the E quator, re leases a ba©¥©¥ from rest In a vacuum and (©¥©¥) the force as m easured on the N ew ton - m eier. (©¥} the gravilationa©¥ force alone, the E arth ' s s u rface due Io (b) U sing your answ ers Io «©¥), s ta le w hat would be the acceleration of the m aaa al supporllng the m aas. (©¥©¥©¥}deduce the reading on an accurate nerM on - m e ter (©¥prlng ba©¥ance) (©¥©¥} determ ©¥ne the force required Io m ainlaln the circular path of the m ar©¥, the m ar©¥, (©¥) ca©¥cu©¥ale, u s ing N M on ' ©¥ law of G ravi©¥alion. The gr ¡¤ vllalion©¥l force on centre. F or a 1 . 00 kg m ass ©¥ituated al the E qua©¥or, E arth ' a a u rface ©¥©¥ ©¥denlica©¥ ©¥o ©¥hal of a point m ass of 5 . E8 ¡¿ 10 0 kg M the E arth ' ©¥ spinning on i©¥a axi©¥ with a period of 2 4 hours. T he gravi©¥a©¥iona©¥ field it the (a) T he E arth m ay be considered Io be 1 uniform ©¥p©©tore of rad©¥u©¥ 6 3 7 0 km , D 1 0 96©¥©¥1©¥©¥2 - p©¥rq gign btBt©¥ons Y E A R ØßÙ£ P H Y S ©¥C S D E P A R T M E N T R A F F L E S ©¥N S T H U Öõ ©¥O N

2 . The speed of the satel©¥ite. 1 . The radius of the orbit, (ii) H ence, state and exp©¥ain the effect of this change on S tate w hether the tota©¥ energy E t becom es m ore or less negative . resistance. (c) A s the sate©¥lite orbits the E arth, it gradually loses energy because of air 2R E , = - . G M m (ii» H ence, s how that the total energY E t of the sate©¥lite is given by potentia©¥ energy E p of the satel©¥ite . (b) (i) S tate an expression, in term s of G , M , m a n d R for the gravitational energy E k of the sate©¥lite in term s of G , M , m a n d R . (©¥©¥) T he m ass of the satellite is m . D eterm ine the expression for the kinetic R w here M is the m ass of the E arth and G is the gravitational constant, 2 G M (a) (©¥) S how that the speed v ls given by the express©¥on orbit about the E arth ls R . m asses concentrated at their centres, T he salel©¥ile has speed v and the radius of lta B oth the E arth and the sale©¥©¥ile m ay be considered to be point m asses w ith their E ar©¥ A sale©¥©¥ile orblla the E arth ©¥n a circular path, a r l©¥lualraled be©¥ow . D 4 Y E A R S - O pH Y S ©¥C S D E pA R T M E N Öõ R A F FL E S ©¥N S T ©¥T U T ©¥O N üÞ 国

0 . 12 ¡¿ 1 0 6 m above the E arth ' s s u rface. sate©¥lite of m ass 3 0 0 0 kg betw een its launch and w hen it is at a height of U se the expression given in (c) to calculate the gain in the potential energy of a(d) (ii) w hy the