ROTATIONAL MOTION PR 1

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1 Eistei Classes, Uit No.,, Vadhma Rig Road Plaza, Vikas Pui Ext., Oute Rig Road New Delhi 8, Ph. : 969, 87 PR ROTATIONAL MOTION Syllabus : Cete of mass of a two-paticles system, Cete of mass of a igid body; Basic cocepts of otatioal motio; momet of a foce, toque, agula mometum, cosevatio of agula mometum ad its applicatios; momet of ietia, adius of gyatio. Values of momets of ietia fo simple geometical objects, paallel ad pepedicula axes theoem ad thei applicatios. Rigid body otatio, equatios of otatioal motio.

2 Eistei Classes, Uit No.,, Vadhma Rig Road Plaza, Vikas Pui Ext., Oute Rig Road New Delhi 8, Ph. : 969, 87 PR CONCEPTS C Cete of Mass Fo a system of paticles, that the distibuted i thee dimesios show i the figue The cete of mass of the system of paticles is give by x M i m x i i y M i m y The positio vecto of cete of mass of the system of paticles is give by R i.e., R M i m Cete of Mass of Solid Bodies i i i i z M i m z i i x î y Solid bodies ae teated as cotiuous distibutio of matte ad the cete of mass fo these bodies is give by x xdm dm y ydm dm Hee x, y, z ae the cete of mass of the diffeetial elemets of the solid bodies ad dm is the mass of the diffeetial elemets. Pactice Poblems :. Fou paticles of masses m, m, m, m ae placed at (l, l), ( l, l), ( l, l) ad (l, l) espectively. The cete of mass will lie i z zdm Fist quadat Secod quadat Thid quadat Fouth quadat. A uifom half cicula ig of adius is placed o the x-y plae as show i figue. The cete of mass of uifom half cicula ig is dm ĵ z kˆ,,,, [Aswes : () c () a]

3 Eistei Classes, Uit No.,, Vadhma Rig Road Plaza, Vikas Pui Ext., Oute Rig Road New Delhi 8, Ph. : 969, 87 PR C Newto s Secod Law Fo System of Paticles Fo a system of paticles, m m Diffeetiatig with espect to time, MV m v... m m v... m v whee V is the velocity of cete of mass of the system of paticles. Diffeetiatig with espect to time, Ma m a m a...m a whee a is the acceleatio of the cete of mass of the system of paticles. Fom Newto s secod law Ma F F... F Amog the foces that cotibute to the ight side of the above equatio will be foces that the paticles of the system exet o each othe (iteal foces) ad foces exeted o the paticles fom outside the system (exteal foces). By Newto s thid law, the iteal foces cacel out i the sum that appeas o the ight side of the above equatio, what emais is the vecto sum of all the exteal foces that act o the system. Hece F Ma. et Pactice Poblems :. Two sphees of masses M ad M ae iitially at est at a distace R apat. Due to mutual foce of attactio they appoach each othe.whe they ae at sepaatio R/, the acceleatio of thei cete of mass would be g m/s g m/s g m/s. A isolated paticle of mass m is movig i a hoizotal plae (x y) alog the x-axis, at a cetai height above the goud. It suddely explodes ito two fagmets of masses m/ ad m/. A istat late, the smalle fagmet is at y = + cm. The lage fagmet at this istat is at y = cm y = + cm y = + cm y = cm [Aswes : () a () a] C Rotatioal Motio Hee we examie the otatio of a igid body (a body with a defiite ad uchagig shape ad size) about a fixed axis (a axis that does ot move), called the axis of otatio o the otatioal axis. Evey poit of the body moves i a cicle whose cete lies o the axis of otatio, ad evey poit moves though the same agle duig a paticula time. I pue taslatio, evey poit of the body moves though the same liea distace duig a paticula time iteval i a staight lie. Hece we ca see the agula equivalet of the liea quatities positio, displacemet, velocity ad acceleatio. Agula displacemet : =. Agula velocity : Aveage agula velocity, <> = t t. t Istat agula velocity, d lim. t t dt Both <> ad ae vectos, with the diectio give by the ight had ule. The magitude of the body s agula velocity is the agula speed. Agula acceleatio : Aveage agula acceleatio, <> = t t. t

4 Eistei Classes, Uit No.,, Vadhma Rig Road Plaza, Vikas Pui Ext., Oute Rig Road New Delhi 8, Ph. : 969, 87 PR C Istat agula acceleatio, Both <> ad ae vectos. d d lim t dt d t. Rotatioal Motio with Costat Agula Acceleatio The kiematics equatios fo costat agula acceleatio = + (t t ) = + (t t) + (/) (t t ) = + ( ). Hee the symbols have the followig meaig : Agula posito at t Agula positio at t Agula velocity at t Agula velocity at t Pactice Poblems : Agula acceleatio.. A wheel otates with a costat acceleatio of. ad/s. If the wheel stats fom est, the umbe of evolutios it makes i the fist te secods will be appoximately 8 6 [Aswes : () b] C RELATION BETWEEN LINEAR AND ANGULAR VARIABLES A poit i a igid otatig body at a pepedicula distace fom the otatio axis moves i a cicle with adius. If the body otates though a agle, the poit moves alog the ac with legth s is give by s = whee is i adias. The liea velocity v of the poit is taget to the cicle ad the poit s liea speed v is give by v =, whee is the agula speed of the body. The liea acceleatio a of the poit has both tagetial ad adial compoets. The tagetial compoet is a t =, whee is the magitude of the agula acceleatio of the body. The tagetial dv acceleatio a t epesets oly the pat of liea acceleatio that is esposible fo chage i the dt magitude of the liea velocity v. Like v, that pat of the liea acceleatio is taget to the path of the poit i questio. The adial compoet, esposible to chage the diectio of v, is diected adially iwad. a v. This compoet is Fo a body havig costat velocity, a t = ad a =. Fo a body havig costat agula speed o liea speed, a t = ad a Remembe the followig vecto elatio v a at a a ( ) v. Body havig vaiable agula speed o liea speed, dv a t ad dt v a.

5 Eistei Classes, Uit No.,, Vadhma Rig Road Plaza, Vikas Pui Ext., Oute Rig Road New Delhi 8, Ph. : 969, 87 PR C6 ROTATIONAL KINETIC ENERGY AND ROTATIONAL INERTIA I tems of momet of ietia I, the otatioal kietic eegy K of a igid body is K I. Remembe the followig : Momet of ietia of poit mass I = m Momet of ietia of a system of discet paticles I i i i. Hee ad i i these expessios epesets the pepedicula distace fom the axis of otatio of each mass elemet i the body. This quatity has the same sigificace i otatioal motio as that of mass i liea motio. It is a measue of the esistace offeed by a body to a chage i its otatioal motio. Pactice Poblems :. The momet of ietia of a body about a axis is. kg-m. Iitially the body is at est. I ode to poduce a otatioal kietic eegy of J, a agula acceleatio of ad/s must be applied about the axis fo a duatio of s s 8 s s. Two poits masses m ad m ae joied by a massless od of legth. The momet of ietia of the system about a axis passig though the cete of mass ad pepedicula to the od is m (m m) (m m) [Aswes : () a () c] mm m m mm m m C7 THEOREM OF MOMENT OF INERTIA :. The paallel axis theoem The paallel axis theoem elates the momet of ietia I of a body about ay axis to that of the same body about a paallel axis though the cete of mass (also kow as cetoid axis) as I = I com + M h whee I com is the momet of ietia of the body about the cetoidal axis ad h is the pepedicula distace betwee the two axes. Note that the paallel axis may lie withi o iside the body.. The pepedicula axis theoem This theoem is valid fo a plaa o lamia body (body i two dimesios like a thi disc, ig o thi plate etc.). Let x ad y be the two axes which lie i the plae of the body ad pass though the poit O, as show i figue. The the momet of ietia about a axis (called z-axis) passig though O ad pepedicula to the plae cotaiig x ad y axes is give by I z = I x + I y, whee I x, I y ad I z ae the espective momet of ietia of the body about x, y ad z axes.

6 Eistei Classes, Uit No.,, Vadhma Rig Road Plaza, Vikas Pui Ext., Oute Rig Road New Delhi 8, Ph. : 969, 87 C8 MOMENT OF INERTIA OF IMPOTANT BODIES :. A thi od PR 6 About a axis though cete of mass pepedicula to legth, I Ml About a axis though oe of the ed ad pepedicula to legth, I ML. A ig o Hoop About a cetal axis ad pepedicula to the plae I = About ay diamete I About a taget i the plae of ig I About a taget pepedicula to the plae of ig I =. Hollow cylide About cetal axis I =. Solid cylide ML I About cetal axis o axis of symmety I About cetal diamete ML I

7 Eistei Classes, Uit No.,, Vadhma Rig Road Plaza, Vikas Pui Ext., Oute Rig Road New Delhi 8, Ph. : 969, 87. Disc PR 7 About a axis though the cete ad pepedicula to the plae I. About the diamete I About the taget pepedicula to the plae of disc I 6. A ectagula plate About the taget i the plae of disc About pepedicula axis though cete I M(a b ) I 7. Solid sphee About its diamete I 8. Hollow sphee About its taget 7 I About its diamete I About its taget 9. Aula cylide (o ig) I About cetal axis I M(R R ), hee R is the oute adius ad R is the ie adius. Pactice Poblems :. Which of the followig has the highest momet of ietia if each has the same mass ad the same adius? A ig about its axis pepedicula to the plae of the ig A solid sphee about oe of its diametes A spheical shell about oe of its diametes A disc about its axis pepedicula to the plae of its disc.. The momet of ietia of a uifom cicula disc about a diamete is I. Its mometum of ietia about a axis pepedicula to its plae ad passig though a poit o its im is I I I 6 I

8 Eistei Classes, Uit No.,, Vadhma Rig Road Plaza, Vikas Pui Ext., Oute Rig Road New Delhi 8, Ph. : 969, 87 PR 8. The adius of gyatio of a od of mass m ad legth L about a axis of otatio pepedicula to its legth ad passig though the cete is L L. A wie of mass pe uit legth ad legth L is used to fom a cicula loop. The momet of ietia about the xx is L L L 8 [Aswes : () a () d () a () c] L 8 L 8 L 8 C9 Momet of Foce o Toque Toque is a tuig o twistig actio o a body about a otatio axis due to a foce F. It has the same ole i otatioal motio as that of foce i liea motio. Coside a foce F is exeted at a poit give by the positio vecto elative to the axis, as show i figue. Its toque about O is give by F. Newto s Secod Law fo Rotatio The otatioal aalog of ewto s secod law is et = I whee et is the et toque actig o a paticle o igid body, I is the otatioal ietia of the paticle o body about the otatio axis, ad is the esultig agula acceleatio about that axis. Hee et ad I ae take with espect to the same otatio axis. Note the followig poits fo et = I Equilibium : A igid body is said to be i equilibium if Net exteal foce equal to zeo. This is the coditio of tasiatioal equilibium F. Net exteal toque equal to zeo. This is the coditio of otatioal equilibium. Pactice Poblems :. A wheel of adius cm ad mass. kg otates feely about a axis passig though the cete ad pepedicula to the plae of the wheel by applyig a costat foce F ad it is foud that its agula speed iceases fom zeo to ad/s i s. The foce F actig o the wheel to do so. N. N. N 6. N [Aswes : () a] C WORK IN ROTATIONAL MOTION Whe a toque acts o a igid body that udegoes a agula displacemet fom i to f the wok W doe by the toque is W t i d. If the toque is costat, the W = ( f i ) =. Gaphical itepeatio of otatioal wok doe is show i figue.

9 Eistei Classes, Uit No.,, Vadhma Rig Road Plaza, Vikas Pui Ext., Oute Rig Road New Delhi 8, Ph. : 969, 87 Wok - Eegy Theoem fo Rotatioal Motio Wok eegy theoem fo otatioal motio of a igid body is Pactice Poblems : W K Kf Ki If Ii. PR 9. A uifom cylide of adius R ad mass M is spied about its axis to the agula velocity ad the placed ito a coe. The coefficiet of fictio betwee the walls ad the cylide is equal to µ. The total wok doe is [Aswes : () d] C POWER IN ROTATIONAL MOTION Whe the body otates with agula velocity, the powe P (ate at which the toque does wok) is dw P. dt Pactice Poblems :. A electic moto exets a costat toque of = N-m o a gidstoe mouted o it shaft. The momet of ietia of the gidstoe is kg-m. If the system stats fom est, the kietic eegy at the ed of 8s is J 8 J 6 J J. I the above poblem, the istat powe at t = 8s deliveed by the moto is W W W 8 W [Aswes : () c () c] C ANGULAR MOMENTUM. Agula mometum of a paticle The agula mometum L of a paticle, with liea mometum p, mass m ad liea velocity v is a vecto quatity defied elative to a fixed poit (usually a oigi). It is L p m( v). The magitude of L is give by L mv si p mv p mv. whee is the agle betwee pepedicula to ad adp p ad v ae the compoets of p ad v espectively,, is the pepedicula distace betwee the fixed poit ad the lie of extesio of p. The diectio of L is give by the ight had ule fo coss poducts.. Agula mometum of a system of paticles The agula mometum L of a system of paticles is the vecto sum of the agula mometum of idividual paticles, L i. Agula Mometum of a Rigid Body L i. Whe a symmetic igid body with momet of ietia I otates with agula velocity about a statioay axis of symmety, its agula mometum is give by L I. If the body is ot symmetic o the otatio axis is ot a axis of symmety, the compoet of agula mometum alog the axis of otatio is equal to I.

10 Eistei Classes, Uit No.,, Vadhma Rig Road Plaza, Vikas Pui Ext., Oute Rig Road New Delhi 8, Ph. : 969, 87 PR. A mass M is movig with a costat velocity paallel to the x-axis. Its agula mometum with espect to the oigi is zeo emais costat goes o iceasig goes o deceasig. Whe a mass is otated i a plae about a fixed poit, its agula mometum is diected alog the adius the taget to the obit a lie at a agle of to the plae of otatio the axis of otatio. [Aswes : () b () d] C RELATION BETWEEN TORQUE AND ANGULAR MOMENTUM The ate of chage of agula mometum of a igid body equals the et toque actig o it i.e., Pactice Poblems : dl et. dt. A costat toque actig o a uifom cicula wheel chages its agula mometum fom A to A i secods. The magitude of this toque is [Aswes : () a] A A A A C CONSERVATION OF ANGULAR MOMENTUM The agula mometum L of a system emais costat if the et exteal toque actig o the system is dl zeo i.e. L cos tat. dt This is a law of cosevatio of agula mometum. It is oe of the fudametal cosevatio laws of atue, havig bee veified eve i situatio (ivolvig high speed paticles o subatomic dimesio) i which ewto s laws ae ot applicable. Pactice Poblems :. A thi cicula ig of mass M is otatig about its axis with a costat agula velocity. Two objects, each of mass m, ae attached getly to the opposite eds of a diamete of the ig. The ig ow otates with a agula velocity. M M m (M m) M m M M m ( M m) M. A smooth uifom od of legth L ad mass M has two idetical beads of egligible size, each of mass m, which ca slide feely alog the od. Iitially the two beads ae at the cete of the od ad the system is otatig with a agula velocity about a axis pepedicula to the od ad passig though the mid-poit of the od. Thee ae o exteal foces. Whe the beads each the eds of the od, the agula velocity of the system is

11 Eistei Classes, Uit No.,, Vadhma Rig Road Plaza, Vikas Pui Ext., Oute Rig Road New Delhi 8, Ph. : 969, 87 PR M M m M M m M M m M M 6m [Aswes : () c () d] C ROLLING MOTION : (i) The combied motio of taslatio ad otatio is kow as ollig motio. (ii) Coditio of pue ollig motio o fixed suface : v c = whee v c is the velocity of the cete of mass of the ollig body of adius ad is the agula velocity about the cete of mass. (iii) Kietic eegy of ollig body = mv c I. (iv) Fo the body ollig alog the iclie plae of icliatio : a c = gsi( + I/ ), v c Pactice Poblems : gh /( I / ). A solid cylide of mass M ad adius R olls dow a iclied plae fom height h without slippig. The speed of its cete of mass whe it eaches the bottom is gh gh. A thi, uifom, cicula disc is ollig dow a iclied plae of icliatio without slippig. Its liea acceleatio alog the plae is gh g/ g/ g/ g/. A solid sphee, a hollow sphee ad a solid cylide, all of the same adius, oll dow a iclied plae fom the same height, statig fom est. Which of them takes the least time i eachig the bottom of the plae? Solid sphee Hollow sphee Solid cylide All will take the same time. A ig is ollig without slippig o a hoizotal suface. The velocity of cete of mass of the ig is v. The factio of otatioal kietic eegy of the total kietic eegy is / / / / [Aswes : () b () b () a () a] g h

12 Eistei Classes, Uit No.,, Vadhma Rig Road Plaza, Vikas Pui Ext., Oute Rig Road New Delhi 8, Ph. : 969, 87 PR INITIAL STEP EXERCISE. The momet of ietia of a thi squae plate ABCD of uifom thickess about a axis passig though the cete O ad pepedicula to the plae is 6. A uifom cube of side l ad mass m ests o a ough hoizotal table. A hoizotal foce F is applied omal to oe of the faces at a poit that is diectly above the cete of the face, at a height l above the base. The miimum value of F fo which the cube begis to tip about a edge is / mg / mg / mg / mg I + I I + I I + I all the above. A paticle pefoms uifom cicula motio with agula mometum l. If the fequecy of the motio of the paticle is doubled ad its kietic eegy halved, the agula mometum becomes l l l/ l/. Two loops P ad Q ae made fom a uifom wie. The adii of P ad Q ae ad espectively, ad thei momets of ietia ae I ad I espectively. If I /I = the equals / / / /. The otatioal kietic eegy of a body is E ad its momet of ietia is I. The agula mometum of the body is E I E I 7. A stig is wapped aoud a cylide of mass m ad adius. The stig is pulled vetically upwad to pevet the cete of mass to fall as the cylide uwids the stig. The tesio i the stig is mg/ mg/ mg mg 8. Two discs with momet of ietia I ad I iitially they ae otatig with agula velocities ad espectively i aticlockwise diectio, ae pushed with foces actig alog the axis, so as ot to apply ay toque o eithe disks. The disks ub agaist each othe ad evetually each a commo fial agula velocity. Which of the followig quatity will be coseved? Kietic eegy Agula Mometum Liea mometum All the above 9. If the adius of eath cotacts to half of its peset day value, the mass emaiig uchaged, the duatio of the day will be 8 hs hs hs 6 hs E I E/I. A false balace has equal ams. A object weighs x whe placed i oe pa ad y whe placed i the othe pa. The tue weight of the object is equal to xy x y x y x y

13 Eistei Classes, Uit No.,, Vadhma Rig Road Plaza, Vikas Pui Ext., Oute Rig Road New Delhi 8, Ph. : 969, 87 PR FINAL STEP EXERCISE. Two skates A ad B, havig masses kg ad 7 kg espectively, stad facig each othe 6 m apat o a hoizotal smooth suface. They pull o a ope stetched betwee them. How fa does each move befoe they meet? both move m A moves. m ad B moves. m A moves. m ad B moves. m oe of the above. At t =, the positio ad velocities of two paticles ae as show i the figue. They ae kept o a smooth suface ad beig mutually attacted by gavitatioal foce. The positio of cete of mass at t = s is x = m x = 7 m x = m x = m. A hollow sphee ad a solid sphee, havig the same mass, ae eleased fom est simultaeously fom the top of a smooth iclied plae. Which of the two will each the bottom fist? solid sphee hollow sphee the oe which has the geate desity both will each the bottom simultaeously. A paticle of mass m is pojected with a velocity v makig a agle of with the hoizotal. The magitude of the agula mometum of the pojectile about the poit of pojectio whe the paticle is at its maximum height h is zeo mv g mv m g gh. Fou sphees, each of mass M ad diamete, ae placed with thei cetes o the fou coes of a squae of side a (> ). The momet of ietia of the system about oe side of the squae is M( a ) M( a ) M( a ) 6. A cod is woud oud the cicumfeece of a wheel of adius. The axis of the wheel is hoizotal ad its momet of ietia about this axis is I. A weight mg is attached to the ed of the cod ad allowed to fall fom est. The agula velocity of the wheel, whe the weight has falle though a distace h, is gh I m mgh I m / / mgh I m (gh) / / 7. A body of mass M ad adius, ollig o a smooth hoizotal floo with velocity v, olls up a iegula iclied plae up to a vetical height v. The body may be g sphee solid cylide disc both ad 8. A thi od of legth L ad mass M is held vetically with oe ed o the floo ad is allowed to fall. The velocity of the othe ed whe it hits the floo, assumig that the ed o the floo does ot slip gl gl gl gl 9. Thee uifom ods each of mass m ad legth L, is used to fom a equilateal tiagle. The momet of ietia of this fame about a axis though the cetoid ad pepedicula to the plae of tiagle is ml ml ml ml M( a )

14 PR. A uifom disk, with mass M ad adius R mouted o a fixed hoizotal axle. A block with mass M hags fom a massless cod that is wapped aoud the im of the disk. The cod does ot slip, ad thee is o fictio at the axle. The acceleatio of the fallig block is mgh h Noe of these. A cicula hole of adius R/ is cut fom a homogeous cicula disc of a adius R. The cete of mass of the emaiig disc is g g/ / g g/. A uifom ladde of mass kg ests agaist a smooth vetical wall makig a agle of with it. The othe ed ests o a ough hoizotal floo. The fictio foce that the floo exets o the ladde is 6 N 98 N 7 N 86 N. Figue shows a mass m placed o a fictioless hoizotal table ad attached to a stig passig though a small hole i the suface. Iitially, the mass moves i a cicle of adius with a speed v ad a peso holds the fee ed of the stig. The peso pulls o the stig slowly to decease the adius of the cicle to. Let the tesio i the stig whe the mass moves depeds o adius as. The value of is. A sphee of mass m ad adius R is ollig without slippig with agula speed o a hoizotal plae. The agula mometum of the sphee about ay poit lyig o the suface is / mr / mr 7/ mr 8/ mr. A wheel of adius ad mass m stads i cotact with step of height h. The least hoizotal foce F which should be applied to the axle of the wheel to foce it climb oto the step is mg [h( h)] h mgh( h) h R/6 towads left R/6 towads ight R/ towads left R/ towads ight ANSWERS (INITIAL STEP EXERCISE). d. d. b. c. b 6. b 7. c. c. b. d. b. d 6. b 7. d 8. a 8. b 9. d ANSWERS (FINAL STEP EXERCISE) 9. b. c. a. c. c. a. a Eistei Classes, Uit No.,, Vadhma Rig Road Plaza, Vikas Pui Ext., Oute Rig Road New Delhi 8, Ph. : 969, 87

15 Eistei Classes, Uit No.,, Vadhma Rig Road Plaza, Vikas Pui Ext., Oute Rig Road New Delhi 8, Ph. : 969, 87 PR AIEEE ANALYSIS []. Two idetical paticles move towads each othe with velocity v ad v espectively. The velocity of cete of mass is v v/ v/ zeo. A solid sphee, a hollow sphee ad a ig ae eleased fom top of a iclied plae (fictioless) so that they slide dow the plae. The maximum acceleatio dow the plae is fo (o ollig) solid sphee hollow sphee ig all same. Iitial agula velocity of a cicula disc of mass M is. The two small sphees of mass m ae attached getly to two diametically opposite poits o the edge of the disc. What is the fial agula velocity of the disc? M m M M m m. Momet of ietia of a cicula wie of mass M ad adius R about the diamete is / /. A paticle of mass m moves alog lie PC with velocity v as show. What is the agula mometum of the paticle about P? mvl mvl mv zeo M M m M M m AIEEE ANALYSIS [] 6. Let F be the foce actig o a paticle havig positio vecto ad T be the toque of this foce about the oigi. The.T ad F. T.T ad F. T.T ad F. T.T ad F. T 7. A cicula disc X of adius R is made fom a io plate of thickess t, ad aothe disc Y of adius R is made fom a io plate of thickess t/. The the elatio betwee the momet of ietia I X ad I Y is I Y = I X I Y = 6I X I Y = I X I Y = 6I X. A solid sphee is otatig i fee space. If the adius of the sphee is iceased keepig mass same which oe of the followig will ot be affected? momet of ietia agula mometum agula velocity otatioal kietic eegy [] 8. A paticle pefomig uifom cicula motio has agula mometum L. If its agula fequecy is doubled ad its kietic eegy halved, the the ew agula mometum is L L/ L/ L 9. Two spheical bodies of mass M ad M ad adii R ad R espectively ae eleased i fee space with iitial sepaatio betwee thei cetes equal to R. If they attact each othe due to gavitatioal foce oly, the the distace coveed by the smalle body just befoe collsio is 7. R. R. R. R AIEEE ANALYSIS [/]. Oe solid sphee A ad aothe hollow sphee B ae the same mass ad same oute adii. Thei momet of ietia about thei diametes ae espectively I A ad I B such that I A = I B I A > I B I A < I B I A /I B = d A /d B whee d A ad d B ae thei desities []

16 Eistei Classes, Uit No.,, Vadhma Rig Road Plaza, Vikas Pui Ext., Oute Rig Road New Delhi 8, Ph. : 969, 87 PR 6. A aula ig with ie ad oute adii, R ad R is ollig without slippig with a uifom agula speed. The atio of the foces expeieced by the two paticles situated o the ie ad oute pats of the ig, F is F R R R R R R []. A mass m moves with velocity v ad collides ielastically with aothe idetical mass. Afte collisio the st mass moves with velocity v i a diectio pepedicula to the iitial diectio of motio. Fid the speed of the d mass afte collisio. The momet of ietia of a uifom semicicula disc of mass M ad adius about a lie pepedicula to the plae of the disc though the cete is M M M M [] 6. A body A of mass M while fallig vetically dowwads ude gavity beaks ito two pats; a body B of mass M ad a body C of mass M. The cete of mass of bodies B ad C take togethe shifts compaed to that of body A towads body C body B depeds o height of beakig does ot shift [] 7. A T shaped object with dimesios show i the figue, is lyig o a smooth floo. A foce ' F' is applied at the poit P paallel to AB, such that the object has oly the taslatioal motio without otatio. Fid the locatio of P with espect to C v v v v []. A spheical ball of mass kg is statioay at the top of a hill of height m. It olls dow a smooth suface to the goud, the climbs up aothe hill of height m ad fially olls dow to a hoizotal base at a height of m above the goud. The velocity attaied by the ball is m/s m/s m/s m/s [] l l AIEEE ANALYSIS [6] l l [] 8. A foce of F kˆ acts o O, the oigi of the coodiates system. The toque about the poit (, ) is F(î ĵ) F(î ĵ) F(î ĵ) F(î ĵ) 9. A thi cicula ig of mass m ad adius R is otatig about its axis with a costat agula velocity. Two objects each of mass M ae diamete of the ig. The ig ow otates with a agula velocity =

17 Eistei Classes, Uit No.,, Vadhma Rig Road Plaza, Vikas Pui Ext., Oute Rig Road New Delhi 8, Ph. : 969, 87 PR 7 m (m M) ( m M) m m (m M) (m M) (m M). Fou poit masses, each of value m, ae placed at the coes of a squae ABCD of side l. The momet of ietia of this system about a axis passig though A ad paallel to BD is ml ml ml ml AIEEE ANALYSIS [7]. A cicula disc of adius R is emoved fom a bigge cicula disc of adius R such that the cicumfeece of the discs coicide. The cete of mass of the ew disc is R bigge disc. The value of is fom the cete of the. A oud uifom body of adius R, mass M ad momet of ietia I, olls dow (without slippig) a iclied plae makig a agle with the hoizotal. The its acceleatio is g si / I g si I / 6 g si / I g si I /. Agula mometum of the paticle otatig with a cetal foce is costat due to costat toque costat foce costat liea mometum zeo toque. Fo the give uifom squae lamia ABCD, whose cete is O, I AC = I EF I AD = I EF I AC = I EF I AC = I EF ANSWERS AIEEE ANALYSIS. c. d. c. a. b 6. c 7. b 8. c 9. a. b. c. a. a. c. b 6. d 7. a 8. d 9. b. a. c. c. d. d TEST YOURSELF. I the HCl molecule, the sepaatio betwee the uclei of hydoge ad chloie atoms is.7å. If the mass of a chloie atom is. times that of a hydoge atom, the cete of mass of the HCl molecule is at a distace of..7 Å fom the hydoge atom Å fom the chloie atom 6..7 Å 6. fom the chloie atom both ad ae coect. The atio of the adii of gyatio of a cicula disc ad a cicula ig of the same adii about a tagetial axis is : : 6 : :. A solid sphee is otatig about its diamete. Due to icease i oom tempeatue, its volume iceases by.%. If o exteal toque acts, the agula speed of the sphee will icease by ealy % decease by ealy %

18 Eistei Classes, Uit No.,, Vadhma Rig Road Plaza, Vikas Pui Ext., Oute Rig Road New Delhi 8, Ph. : 969, 87 PR 8 icease by ealy % µg µg decease by ealy %. A pulley of adius m is otated about its axis by a foce F = (t t ) ewto (whee t is measued i secod) applied tagetially. The foce is the withdaw. If the momet of ietia of the pulley about its axis of otatio is kg m, the umbe of otatios made by the pulley befoe its diectio of motio if evesed, is vey ealy equal to 8. Momet of ietia of uifom hoizotal solid cylide of mass M about a axis passig though its edge ad pepedicula to the axis of the cylide whe its legth is 6 times its adius R is A cicula potio of diamete R is cut out fom a uifom cicula disc of mass M ad adius R as show i the figue. The momet of ietia of the emaiig (shaded) potio of the disc about a axis passig though the cete O of the disc ad pepedicula to its plae is µg µg 8. A disc is otatig with agula velocity. A foce F acts at a poit whose positio vecto with espect to the axis of otatio is. The powe associated with the toque due to the foce is give by ( F) ( F) (F ) (F ) 9. Whe a explosive shell, tavellig i a paabolic path ude the effect of gavity explodes, the cete of mass of the fagmets will move fist vetically upwads ad the vetically dowwads vetically dowwads alog the oigial paabolic path fist hoizotally ad the alog a paabolic path.. Two blocks m ad m, havig masses kg ad kg espectively, ae placed o a fictioless hoizotal suface ad ae coected by a light spig of foce costat N/m. m is i cotact with a igid wall. m is pushed though a distace of cm towads m ad the eleased. The velocity of the cete of mass of the system whe m beaks off the wall is / cm/s / cm/s cm/s cm/s A small coi is placed at a distace fom the cete of the gamophoe ecod. The otatioal speed of the ecod is gadually iceased. If the coefficiet of fictio betwee the coi ad the ecod is µ, the miimum agula fequecy of the ecod fo which the coi will fly off is give by. d. b. b. a. d ANSWERS 6. c 7. c 8. a 9. c. b