GRAVITATION 4) R. max. 2 ..(1) ...(2)

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Kelly Higgins
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1 GAVITATION PVIOUS AMCT QUSTIONS NGINING. A body is pojctd vtically upwads fom th sufac of th ath with a vlocity qual to half th scap vlocity. If is th adius of th ath, maximum hight attaind by th body fom th sufac of th ath is: ) 6 ) ) ) (009 ) Ans : Sol: Accoding to th law of consvation of ngy, Total ngy of th body on th sufac of ath Total ngy of th body at a hight h achd by it ( P. + K. ) ( P. + K. ) sufac max. GMm V + m GMm m( 0 ) h..() + GM But V....() Sub () in () w gt h /. Th obit of go-stationay satllit is cicula, th tim piod of satllit dpnds on (008 ) ) mass of th ath ) adius of th obit ) hight of th satllit fom th sufac of ath ) all th abov Ans : Sol: Th tim piod of go-stationay satllit is T π GM Tim piod dpnds on i) mass of ath ii) adius of th obit iii) hight of th satllit fom th sufac of ath and is indpndnt of mass of th satllit. Th mass of a plant is half that of th ath and th adius of th plant is on fouth that of ath. If w plan to snd an atificial satllit fom th plant, th scap vlocity will b, ( V kms ) (007 ) ) kms ) 5.5 kms ) hight 5.55 kms ) 7.78 kms Ans: Sol : V scap GM V M V M plant p ath p

2 Gavitation Givn V ath kms - M M p, p Vplant M Vplant kms 5.55kms M. Asstion (A) : A paticl of mass m doppd into a hol mad along th diamt of th ath paticls is invsly popotional to th squa of th distanc btwn thm fom on nd to th oth nd posssss simpl hamonic motion. ason () : Gavitational foc btwn any two ( 006 ) ) Both A and a tu and is th coct xplanation of A ) Both A and a tu and is not th coct xplanation of A ) A is tu but is fals ) A is fals but is tu Ans: Sol : A) As th paticl movs along to and fo motion it xcuts S.H.M Gmm B) Fom th lation F w can conclud F 5. Dgnat lcton pssu will not b sufficint to pvnt co collaps of whit dwaf if its mass bcoms n tims of ou sola mass. Valu of n is ( 005 ) ) 0.5 ) 0.8 ) ). Ans: Sol : Whit dwaf. sola mass 6. Th scap vlocity of a body on th ath s sufac is V A body is thown up with a spd 5V. Assuming that th sun and plants do not influnc th motion of th body, vlocity of th body at infinit distanc is (00 ) ) 0 ) V ) V ) V Ans: Sol : Fom th law of consvation of ngy (T.) sufac of ath (T.) infinit distanc K. + P. K. + P. ( ) ( ) sufac inf inity GMm GMm mv + mv + but GM GMm m 5 mv 0 + 5GMm GMm mv 5

3 6 Gavitation V Gm V 7. A satllit is launchd into a cicula obit of adius aound th ath whil a scond satllit is launchd into an obit of adius.0. Th pcntag diffnc in th tim piods of th two satllits is : (00 ) ) 0.7 ).0 ).5 ) Ans: Sol : Fom kpls law T By applying small appoximation mthod ΔT Δ T ΔT T ΔT T ΔT 00 T Δ T 00 Pcntag diffnc in th tim piod T 8. A body is pojctd up with a vlocity qual to / th of th scap vlocity fom th sufac of th ath. Th hight it achs is (adius of th ath is ) (00 ) ) 0/9 ) 7/9 ) 9/8 ) 0/ Ans: Sol : Fom law of consvation of ngy T. T. ( ) ( ) at sufac at hight ( K. + P. ) ( K. + P. ) sufac max. hight GMm GMm mv h GM but V V 9 GM GMm GMm m h 9GMm GMm GMm 6 + h 7GMm GMm 6 + h h 7 + 7h 6

4 h 9 7 Gavitation 9. Mass M is dividd into two pats Xm and ( X)m. Fo a givn spaation th valu of X fo which th gavitational attaction btwn th two pics bcoms maximum is (00 ) ) / ) /5 ) ) Ans: Sol : Fom th Nwton s law of gavitation Gmm F G( xm)( x) m Substituting all th valus of x w gt th maximum foc if x 0. and a th adii of th ath and moon spctivly. ρ and ρm a th dnsitis of ath and moon spctivly. Th atio of th acclations du to gavity on th sufac of ath to moon is ρ ρ ρm ρm ) ) ) ) ρm ρ m ρ ρ Ans: Sol : Fom th lation GM G g ( /π ) ρ [ sinc mass volum dnsity] g π Gρ g ρ g ρ m m. Th hight of th point vtically abov th ath s sufac at which acclation du to gavity bcoms % of its valu at th sufac is : [000 ] ) 8 ) 9 ) 0 ) 0 Ans: Sol : W know that GM g.() GM Gavity at a hight g h Dividing () & () gh g + h ( ) ( + h).() 7

5 givn g h % g 00 + h 00 + h 0 h 9 MDICAL 8 Gavitation. Th acclation du to gavity at a hight h abov th ath s sufac is 9ms -. If g 0 - on th ath s sufac, its valu at a point at an qual distanc h blow th sufac of th ath is: (009 M) ) 9ms - ) 8.5ms - ) 0ms - ) 9.5ms - Ans: Sol : If h<< h gh g h 9 0 h 0 At a dpth h blow th sufac of ath h gd g ms -. If th ath shinks such that its dnsity bcoms 8 tims to th psnt valu thn th nw duation of th day in hous will b ( 008 M) ) ) ) 6 ) Ans: Sol : As th mass mains constant m m vd vd πd πd volum of sph π d d But givn d 8d Accoding to th law of consvation of angula momntum Iω Iω π π M M 5 T 5 T

6 Gavitation T T T T T 6hous. A body of mass m is aisd fom th sufac of th ath to a hight n ( -adius of ath). Magnitud of th chang in th gavitational potntial ngy of th body is (g - acclation du to gavity on th sufac of ath) ( 007 M) n n ) mg ) mg ) mg mg n + n n n Ans: Sol : gavitational potntial ngy on th sufac of ath GMm G GMm GMm Gavitational potntial ngy at a hight n G + n ( n+ ) Δ G G G GMm GMm n+ ( ) ) ( ) GMm n+ GMm n n+ n n + GM [ sinc g ] 5. How many tims mo, th mass of th oiginal sta is to b lag than that of th sun fo th fomation of Black Hol? (006 M) ) ) 6 ) 8 ) 0 Ans: Sol : Mass of oiginal sta 0 (mass of black hol) 6. Accoding to th siz, idntify th coct dcasing od in (005 M) a) Oiginal sta b) d giant c) Whit Dwaf ) a,b,c ) b,c,a ) c,a,b ) b,a,c Ans: Sol : d giant > oiginal sta > whit dwaf 9

7 Gavitation 7. A spacship is launchd into a cicula obit of adius clos to th sufac of ath. Th additional vlocity to b impatd to th spacship in th obit to ovcom th ath s gavitational pull is : (g acclation du to gavity) (00 M) ).g ). g ) 0.g ) 0. g Ans: Sol : W know that V V0 Additional vlocity V V0 V V 0 0 V0 ( ) But V0 g 0. V 0 Δ V 0. g 8. Th adius in kilomts, to which th psnt adius of th ath ( 600 km) is to b compssd so that th scap vlocity is incasd tn tims is : (00 M) ) 6. ) 6 ) 60 ) 800 Ans: GM Sol : V.() GM 0V.() Dividing () & () 0 Squaing on both sids and simplifying 6Km Two satllits and a volving ound a plant in coplana and concntic cicula obits of adii and in th sam diction spctivly. Thi spctiv piods of volution a h. and 8 h. Th adius of th obit of satllit is qual to. Thi lativ spd whn thy a closst, in kmph is : (00 M) ) π 0 ) π 0 ) π 0 ) π 0 Ans: Sol : Fom Kpl s law T T T 50

8 Gavitation km Spd of th satllit S π V π 0 kmph T Spd of th satllit S π π 0 V π 0 kmph T 8 lativ spd V V π 0 kmph 0. Whn a satllit going aound th ath in a cicula obit of adius and spd v loss som of its ngy, thn (00 M) ) and v both incas ) and v both dcas ) will incas and v will dcas ) will dcas and v will incas Ans: Sol Accoding to law of consvation of ngy will dcas and V will incas. Th angula vlocity to th ath with which it has to otat so that acclation du to gavity on 600 latitud bcoms zo is (000 M) ).5 0 ad / s ) 5 0 ad/s ) 0 0 ad / s ) ad / s Ans : Sol: gavity at latitudφ g φ gφ g ω cos φ As g φ is zo g ω cos φ g ω cos φ ads - 5

GRAVITATION. (d) If a spring balance having frequency f is taken on moon (having g = g / 6) it will have a frequency of (a) 6f (b) f / 6

GRAVITATION. (d) If a spring balance having frequency f is taken on moon (having g = g / 6) it will have a frequency of (a) 6f (b) f / 6 GVITTION 1. Two satllits and o ound a plant P in cicula obits havin adii 4 and spctivly. If th spd of th satllit is V, th spd of th satllit will b 1 V 6 V 4V V. Th scap vlocity on th sufac of th ath is