(Objective Questions)

Transcription

1 1 A thin od of leth 4 l, ass 4 is bent at the points as shown in the fig. What is the oent of inetia of the od about the axis passi point O & pependicula to the plane of the pap. 10 M M 1 M 4 Moent of inetia of a disc about O O is: The oent of inetia of a doo of ass, leth l and width l about its loe side is M 1. (Obctive Questions) none of these A syetic laina of ass M nsists of a squae shape with a seicicula section ove each of the edge of the squae as in figu The side of the squae is a. The oent of inetia of the laina about an axis though its cente of ass and pependicula to the plane is 1.6 Ma. The oent of inetia of the laina about the taent AB in the plane of laina is. 5. Thee identical ods, each of leth l, ae joined to fo a igid equilateal tial Its adius of gyation about an axis passi though a ne and pependicula to the plane of the tiale is. 6. A squae plate of edge a/ is cut out fo a unifo squae plate of edge 'a' as shown in figu The ass of the eaini potion is M. The oent of inetia of the shaded potion about an axis passi though 'O' (cente of the squae of side a) and pependicula to plane of the plate is : 7. a 5 Ma 1 Ma 16 Ma Thee is each of ass and adius ae so placed that they touch each oth The adius of gyation of the syste about the axis as shown in the figue is : The oent of inetia of a hollow cubical box of ass M and side a about an axis passi though the centes of two opposite faces is equal to 9. 9 Ma Ma 5 Ma 6 5 Ma 1 5 Ma 18 A hollow cylinde has ass M, outside adius R and inside adius R1. Its oent of inetia about an axis

2 paallel to its syety axis and taential to the oute suface is equal to : M (R + R1) M (R R1) M (R + R1) 4 M (R + R1) 10. Let I be the oent of inetia of a unifo squae plate about an axis AB that passes though its cente and is paallel to two of its sides. CD is a line in the plane of the plate that passes though the cente of the plate and akes an ale with AB. The oent of inetia of the plate about the axis CD is then equal to : sin s s( /) 11. A unifo od of ass and leth L is suspended with two assless stis as shown in the figu If the od is at est in a hoizontal position the atio of tension in the two stis t 1/T is: 1: 1 1: : 1 4: 1. Two unifo ods of equal leth but diffeent asses ae igidly joined to fo an L shaped body, which is then pivoted about O as shown. If in equilibiu the body is in the shown nfiguation, atio M/ will be: O 90 0 M Two pesons of equal height ae cayi a lo unifo wooden bea of leth. They ae at distance /4 and /6 fo neaest ends of the od. The atio of noal eactions at thei heads is: : 1 : 4 : 1 : 14. A unifo od of leth is placed syetically on two walls as shown in figu The od is in equilibiu. If N 1 and N ae the noal foces exeted by the walls on the od then N1 > N N1 > N N1 = N N1 and N would be in the vetical diections. N 1 N 15. Fou foces taent to the cicle of adius R ae acti on a wheel as shown in the figu The esultant equivalent one foce syste will be : a In the pulley syste shown, if adii of the bigge and lle pulley ae and 1 espectively and the acceleation of block A is 5 /s in the downwad diection, then the acceleation of block B will be: 0 /s 5 /s 10 /s 5/ /s

3 17. A unifo thin od of ass and leth L is held hoizontally by two vetical stis attached to the two ends. One of the sti is cut. Find the aula acceleation soon afte it is cut : L g L g L g L g L 18. A igid body is in pue otation. You can find two points in the body in a plane pependicula to the axis of otation havi e velocity. You can find two points in the body in a plane pependicula to the axis of otation havi e acceleation. Speed of all the paticles lyi on the cuved suface of a cylinde whose axis nsides with the axis of otation is Aula speed of the body is e as seen fo any point in the body. Two points A & B on a disc have velocities v1 & v at soe oent. Thei diections ake ales 60 and 0 espectively with the line of sepaation as shown in figu The aula velocity of disc is : v1 d d v v1 d v d In the figue shown a i A is olli without slidi with a velocity v on the hoizontal suface of the body B (of e ass as A). All sufaces ae sooth. B has no initial velocity. What will be the axiu height (fo initial position) eached by A on B. v 4g v 4g v g v g A wheel (to be nsideed as a i) of ass and adius R olls without slidi on a hoizontal suface with nstant velocity v. It enuntes a step of height R/ at which it ascends without slidi. a 1. v the aula velocity of the i just afte it es in ntact with the step is v/4r 9 v g the noal eaction due to the step on the wheel just afte the ipact is + 16 R the noal eaction due to the step on the wheel inceases as the wheel ascends the fiction will be absent dui the ascent.. L = nstant. If L = 0 fo a igid body, whee = esultant toque & L = aula oentu about a point and both ae non - zeo. Then: L = nstant L will incease A unifo od of ass, leth is placed ove a sooth hoizontal suface alo y-axis and is at est as shown in figu An ipulsive foce F is applied fo a ll tie t alo x-diection at point A. The x odinate of end A of the od when the od bees paallel to x axis fo the fist tie is (initially the odinate of cente of ass of the od is (0, 0)) : L ay inceas

4 5. 6. A nical pendulu nsists of a siple pendulu ovi in a hoizontal cicle as shown. C is the pivot, O the cente of the cicle in which the pendulu bob oves and the nstant aula velocity of the bob. If L is the aula oentu about point C, then L is nstant only diection of L is nstant only agnitude of L is nstant none of the abov In the above poble if L is the aula oentu about the axis CO, then, L is nstant only diection of L is nstant only agnitude of L is nstant none of the abov A unifo ectaula plate of ass which is fee to otate about the sooth vetical hie passi though the cente and pependicula to the plate, is lyi on a sooth hoizontal sufac A paticle of ass ovi with speed 'u' llides with the plate and sticks to it as shown in figu The aula velocity of the plate afte llision will be : 7. 1 u 5 a 1 u 19 a u a u 5a The aula oentu of a paticle about oigin is vayi as L = 4t + 8 (S units) when it oves alo a staight line y = x 4 (x,y in etes). The foce acti on the paticle would be : 1 N N N 4. N A od can otate about a fixed vetical axis. The ass is non-unifoly distibuted alo the leth of the od. A hoizontal foce of nstant agnitude and always pependicula to the od is applied at the end. Which of the followi quantity (afte one otation) will not depend on the infoation that though which end the axis passes? aula oentu kinetic enegy aula velocity none of these 9. A paticle is attached to the lowe end of a unifo od which is hied at its othe end as shown in the figu The iniu speed given to the paticle so that the od pefos cicula otion in a vetical plane will be: 5g 4g a g none of these A body of ass and adius is otated with aula velocity as shown in the figue & kept on a suface that has sufficient fiction then the body will ove : backwad fist and then ove fowad fowad fist and then ove backwad will always ove fowad none of these 1. A sphee of ass ' ' is given soe aula velocity about a hoizontal axis though its cente and gently placed on a plank of ass ' '. The -efficient of fiction between the two is. The plank ests on a sooth hoizontal sufac The initial acceleation of the cente of sphee elative to the plank will be : zeo g (7/5) g g. A unifo cylinde of ass M and adius R olls without slippi down a slope of ale to the hoizontal. The cylinde is nnected to a spi nstant K while the othe end of the spi is nnected to a igid suppot at P. The cylinde is eleased when the spi is unstetched. The axiu distance that the cylinde tavels is 0. Mgsin 4 K Mg tan K Mg sin K 4 Mg sin K

5 . When a peson thows a ete stick it is found that the cente of the stick is ovi with speed 10 / s & left end of stick with speed 0 /s. Both points ove vetically upwads at that oent. Then aula speed of the stick is: 0 ad/ sec 10 ad/sec 0 ad/sec none of these 4. A hollow sooth unifo sphee A of ass olls without slidi on a sooth hoizontal sufac It llides elastically and headon with anothe stationay sooth solid sphee B of the e ass and e adius. The atio of kinetic enegy of B to that of A just afte the llision is : 5 : 1 : 1 : : As shown in the figue, a disc of ass is olli without slippi a aula velocity. When it sses point B disc will be in : tanslational otion only pue olli otion otational otion only none of these 6. A disc of adius R olls on a ough hoizontal sufac The linea distance veed by the point A in one plete evolution is: R none of these A lage spool of ope lies on the gound as shown in the fig. The end, labelled X is pulled a distance S in the hoizontal diection. The spool olls without slippi. The cente of ass of the spool oves a distance a S/4 a a 1 a A ll block of ass is igidly attached at P to a i of ass and adius. The syste is eleased fo est at = 90 and olls without slidi. The aula acceleation of i just afte elease is 9. S/ A solid unifo disk of ass olls without slippi sown an inclined plane with an acceleation a. The fictional foce on the disk due to suface of the plane is 8. S a S 7. 4 R 8 R 5. g 4 40 g 8 g A disc is pefoi pue olli on a sooth stationay suface with nstant aula velocity as shown in figu At any instant, fo the lowe ost point of the disc Velocity is v, acceleation is zeo Velocity is zeo, acceleation is zeo velocity is v, acceleation is 41. g v. R velocity is zeo, acceleation is A syste of unifo cylindes and plates is shown in figu All the cylindes ae identical and thee is no slippi at any ntact. Velocity of lowe & uppe plate is V and V espectively as shown in figu Then the atio of aula speed of the uppe cylindes to lowe cylindes is 1/ 1 none of these v R

6 4. A box of diensions l and b is kept on a tuck ovi with an acceleation a. If box does not slide, axiu acceleation fo it to eain in equilibiu (w..t.tuck) is : 4. g b gb g none of these If the positions of two like paallel foces ae intechaed, thei esultant shifts by one fouth of the distance between the then the atio of thei agnitude is: 1: : : 4 : 5 (SubctiveQ uestions) SECTION : KINEMATICS A body otati at 0 ad/s is acted upon by a nstant toque povidi it a deceleation of ad/s. At what tie will the body have kinetic enegy e as the initial value if the toque ntinues to act? A. A solid body otates about a stationay axis acdi to the law = at bt, whee a = 6.0ad/s and b =.0 ad/s. Find: the ean values of the aula velocity and aula acceleation aveaged ove the tie inteval between t = 0 and the plete stop; the aula acceleation at the oent when the body stops. A solid body stats otati about a stationay axis with an aula acceleation = at, whee a =.0 10 ad/s. How soon afte the beginni of otation will the total acceleation vecto of an abitay point of the body fo an ale = 60º with its velocity vecto.. A. A 1. SECTION : MOMENT OF INERTIA Find the M.I. of a od about (i) an axis pependicula to the od and passi though left end. (ii) An axis though its cente of ass and pependicula to the leth whose linea density vaies as = ax whee a is a positive nstant and 'x' is the position of an eleent of the od elative to its left end. The leth of the od is. B. Thee paticles, each of ass 00 g, ae kept at the nes of an equilateal tiale of side 10 c. Find the oent of inetia of the syste about an axis joini two of the paticles and passi though one of the paticles and pependicula to the plane of the paticles. B. Paticles of asses 1 g, g, g, g ae kept at the aks 1 c, c, c, c espectively on a ete scal Find the oent of inetia of the syste of paticles about a pependicula bisecto of the ete scal B 4. Find the oent of inetia of a unifo half-disc about an axis pependicula to the plane and passi though its cente of ass. Mass of this disc is M and adius is R. B 5. A unifo tiaula plate of ass whose vetices a B 1. ae ABC has leths, and. Find the oent of inetia of this plate about an axis passi though point B and pependicula to the plane of the plat B 6. Find the oent of inetia of a pai of sphees, each havi a ass and adius, kept in ntact about the taent passi though the point of ntact. B 7. The oent of inetia of a unifo od of ass = 0.50 kg and leth = 1 is I = 0.10 kg - about a line pependicula to the od. Find the distance of this line fo the iddle point of the od.

7 B 8. Calculate the oent of inetia. of a ppe unifo disc elative to the syety axis pependicula to the plane of the disc, if its thickness is equal to b =.0 and its adius to R = 100 ; B 9. Find the oent of inetia of a thin unifo ectaula plate elative to the axis passi pependicula to the plane of the plate though one of its vetices. If the sides of the plate ae equal to a and b, and its ass is. B 10. Find the oent of inetia of a unifo squae plate of ass and edge a about one of its diagonals. B 11. Find the adius of gyation of a cicula i of adius about a line pependicula to the plane of this line fo the iddle point of the od. B 1. The adius of gyation of a unifo disc about a line pependicula to the disc equals its adius. Find the distance of the line fo the cent y B 1. y=x Find the oent of inetia about x axis of unifo thin plate of kg/ as shown in the Figu d e n s i t y a O y= x x Calculate the oent of inetia of a unifo od of ass & leth about an axis passi though one end & aki ale = 45º with its leth. B 15. The suface density (ass/aea) of a cicula disc of adius a depends on the distance fo the cente of () = A + B. Find its oent of inetia about the line pependicula to the plane of the disc though its cent B 16. Calculate the oent of inetia of a unifo solid ne elative to its syety axis, if the ass of the ne is equal to and the adius of its base to R. B 14. A foce F = A i + B j is applied to a point whose adius vecto elative to the oigin of odinates O is equal C 1. SECTION : TORQUE A foce F1 A j is applied to a point whose adius vecto 1 a i, while a foce F B i is applied to the point whose adius vecto b j. Both adius vectos ae deteined elative to the oigin of odinates O, i and C. a to = a i + b j, whee a, b &, A, B ae nstants, and i, j ae the unit vectos of the x and y axes. Find the oent N and the a of the foce elative to the point O. C. j ae the unit vectos of the x and y axes, a, b, A. B ae nstants. Find the a of the esultant foce elative to the point O. When a sphee of adius 1 = 1. oves in glycein the laina flow is obseved if the velocity of the sphee does not exceed 1 = c/s. At what iniu velocity of a sphee of adius = 5.5 c will the flow in wate bee tubulent? The visities of glycein and wate ae equal to 1 = 1.9 P and = P espectively. SECTION : ROTATIONAL EQUILIBRIUM D 1. Assui fictionless ntacts, deteine the agnitude of extenal hoizontal foce P applied at the lowe end fo equilibiu of the od. The od is unifo and its ass is ' '. D. A unifo ladde of leth 10.0 and ass 16.0 kg is esti against a vetical wall aki an ale of 7 with it. The vetical wall is fictionless but the gound is ough. An electician weighi 60.0 kg clibs up the ladd If he stays on the ladde at a point 8.00 fo the lowe end, what will be the noal foce and the foce of fiction on the ladde by the gound? What should be the iniu of fiction fo the electician to wok fely?

8 D. A unifo ete stick of ass 00 g is suspended fo the ceili though two vetical stis of equal leths fixed at the ends. A ll obct of ass 0 g is placed on the stick at a distance of 70 c fo the left end. Find the tensions in the two stis. D 4. The doo of an aliah is 6 ft high, 1.5 ft wide and weighs 8 kg. The doo is suppoted by two hies situated at a distance of 1 ft fo the ends. If the agnitudes of the foces exeted by the hies on the doo ae equal, find this agnitud c a b le In figue the unifo gate weighs 00 N and is wide and high. It is suppoted by a hie at the botto left ne and a hoizontal cable at the top left ne, as shown. Find : The tension in the cable h in g e The foce that the hie exets on the gate (agnitude and diection). D 5. A unifo od of leth L ests against a sooth wall as shown in figu Find the fiction efficient between the gound and the lowe end if the iniu ale that the od can ake with the hoizontal is. D 7. Figue shows a vetical foce F that is applied taentially to a unifo cylinde of weight W. The efficient of static fiction between the cylinde and all sufaces is 0.5. Find in tes of W, the axiu foce that can be applied without causi the cylinde to otat SECTION (E) : ROTATION ABOUT FIXED AXIS ( H = H ) D 6. A od of ass and leth L, lyi hoizontally, is fee to otate about a vetical axis though its cent A hoizontal foce of nstant agnitude F acts on the od at a distance of L/4 fo the cent The foce is always pependicula to the od. Find the ale otated by the od dui the tie t afte the otion stats. E. A light od of leth 1 is pivoted at its cente and two asses of 5 kg and kg ae hu fo the ends as shown in figu Find the initial aula acceleation of the od assui that= kg it was hoizontal in the beginni. If the od has a ass of 1 kg distibuted unifoly ove its leth. (i) Find the initial aula acceleation of the od. (ii) Find the tension in the suppots to the blocks of ass kg and 5 kg. E 1. a 1=5 kg The unifo od AB of ass is eleased fo est when = 60º. Assui that the fiction foce between end A and the suface is lage enough to pevent slidi, deteine (fo the instant just afte elease) The aula acceleation of the od The noal eaction and the fiction foce at A. (c) The iniu value of patible with the descibed otion. A (c) µ E 4. A unifo squae plate of ass is suppoted as shown. If the cable suddenly beaks, deteine just afte that oentu; The aula acceleation of the plat The acceleation of ne C. (c) The eaction at A. C (c) A E. B A C

9 E 5. The pulley (unifo disc) shown in figue has ass. Assui the inclined planes to be fictionless, calculate the acceleation of the ass. If the fiction efficient between the block A and the plane below it is = 0.5 and the plane below the block is fictionless. 0 0 A sti is wapped on a wheel of oent of inetia 0.0 kg - and adius 10 c and goes though a light pulley to suppot a block of ass.0 kg as shown in figu 0.0 kg - 10 c.0 kg If the lle pulley is assue to be light, find the acceleation of the block. The lle pulley has its adius 5.0 c and oent of inetia 0.10 kg-. Find the tension in the pat of the sti joini the pulleys. E 7. The ends of thin theads tightly wound on the axle of adius of the Maxwell disc ae attached to a hoizontal ba. When the disc unwinds, the ba is aised to keep the disc at the e height. The ass of the disc at the axle is equal to, the oent of inetia of the aaeent elative to its axis is I. Find the tension of each thead and the acceleation of the ba. E 8. Each of the double pulleys shown has a centoidal ass oent of inetial of 0.5 kg, an inne adius of 100 and an oute adius of 150. Assui that the beai fiction at A and at B is equivalent to toque of agnitude 0.45 N., deteine ; The velocity of the cylinde sec afte the syste is eleased fo est. The tension in the d nnecti the pulleys. E 9. A unifo disc of adius R is spinned to the aula velocity and then caefully placed on a hoizontal suface such that its axis is vetical. How lo will the disc be otati on the suface if the fiction efficient is equal to k? The pessue exeted by the disc on the suface can be egaded as unifo E 10. A flywheel with the initial aula velocity 0 deceleates due to the foces whose oent elative to the axis is popotional,to the squae oot of its aula velocity.find the ean aula velocity of the flywheel aveaged ove the total deceleation ti E 11. One end of a unifo od of ass M and leth L is hied to the ceili such that it can otate feely about the hie in vetical plan The aaeent is shown in the figu A ball of ass ovi hoizontally with speed u hits the od at point P sepaated by a distance x fo hi Fo what value of x eaction in hoizontal diection of the hie on the od is iniu and what is it? xx E 1. A od of leth R and ass M is fee to otate about a hoizontal axis passi though hie P as in figu Fist it is taken aside such that it bees hoizontal and then eleased. At the lowest point the od hits the block B of ass and stops. Find the atio of asses such that the block B pletes the cicula tack of adius R. Neglect any fiction. a E 6. M P B SECTION (F) : ROTATION ABOUT FIXED AXIS (ENERGY CONSERVATION) F 1. A ete stick is held vetically with one end on a ough hoizontal floo. It is gently allowed to fall on the floo. Assui that the end at the floo does not slip, find the aula speed of the od when it hits the floo. F. A solid cylinde of ass M = 1kg & adius R = 0.5 is pivoted at its cente & has thee paticles of ass = 0.1kg ounted at its peiete as shown in the figu The syste is oiginally at est. Find the aula speed of the cylinde, when it has swu though 900 in anticlockwise diection.

10 F. Two paticles of asses 7 and ae fastened to the ends, A, B espectively of a weightless igid od, 15 feet lo, which is feely hied at a point O, 5 feet fo A. If the od is just distubed fo its position of unstable equilibiu. Find the velocity with which A will pass though its position of stable equilibiu. F 4. A igid body is ade of thee identical thin ods each of leth L fastened togethe in the fo of lette H. The body is fee to otate about a fixed hoizontal axis AB that passes though one of the legs of the H. The body is allowed to fall fo est fo a position in which the plane of H is hoizontal. What is the aula speed of the body, when the plane of H is vetical. B A The pulleys shown in figue has a adius of 0 c and oent of inetia 0. kg-. The sti goi ove it attached at one end to a vetical spi of spi nstant 50 N/ fixed fo below, and suppots a 1 kg ass at the othe end. The syste is eleased fo est with the spi at its natual leth. Find the speed of the block when it has descended though 10 c. Take g = 10 /s. F 6. A unifo od of leth ' ' is kept as shown in the figu H is a hoizontal sooth suface and W is a vetical sooth wall. The od is eleased fo this position. Find the aula acceleation of the od just afte the eleas F 5. SECTION (G) : ROTATION ABOUT FIXED AXIS ( H = H & ENERGY CONSERVATION) A unifo od pivoted at its uppe end has vetically. It is displaced though an ale of 60 C and then eleased. Find the agnitude of the foce acti on a paticle of ass d at the tip of the od when the od akes an ale of 7 with the vetical. G. A ba of leth L and ass has a fictionless pivot though its id point. Thee is an additional point ass on the ight end of the ba and an additional point ass on the left end of the ba. The ba is held in hoizontal position by a vetical d attached at L/4 fo the left end as shown in the Figu The additional asses & eain fixed on the od. The od is initially hoizontal. Find the tension in the d. Find the foce that the pivot exets on the ba. If the d is cut, what is the aula acceleation of the ba iediately afte the d is cut. When the ba has otated though 900 and is vetical, what is the linea velocity of the ass. (c) (d) a G 1. SECTION (H) : ANGULAR MOMENTUM H 1. A paticle havi ass kg is ovi alo staight line x + 4y = 5 with speed 8/s. Find aula oentu of the paticle about oigin. x and y ae in etes. H. A paticle havi ass kg is ovi with velocity ( i j )/s. Find aula oentu of the paticle about oigin when it is at (1, 1, 0). H. A unifo squae plate of ass.0 kg and edge 10 c otates about one of its diagonals unde the action of a nstant toque of 0.10 N.. Calculate the aula oentu and the kinetic enegy of the plate at the end of the fifth send afte the stat. H 4. A wheel of oent of inetia kg- and adius 0.0 c is otati about its axis at an aula speed of 0.0 ad/s. It picks up a stationay paticle of ass 00 g at its edg Find the new aula speed of the wheel. H 5. A dive havi a oent of inetia of 6.0 kg- about an axis though its cente of ass otates at an aula speed of ad/s about this axis. If he folds his hands and feet to decease the oent of inetia to 5.0 kg-, what will be the new aula speed?

11 A wheel of oent of inetia 0.10 kg- is otati about a shaft at an aula speed of 160 ev/inut A send wheel is set into otation at 00 ev/inute and is upled to the e shaft so that both the wheels finally otate with a on aula speed of 0 ev/inut Find the oent of inetia of the send wheel. Show that the kinetic enegy of the bined syste is less than the su of initial kinetic enegeis of the two disc. How do you acunt fo this loss? H 7. A dub-bell nsists of two identical ll balls of ass 1/ kg each nnected to the two ends of a 50 c lo light od. The dub-bell is otati about a fixed axis though the cente of the od and pependicula to it at an aula speed of 10 ad/s. An ipulsive foce of aveage agnitude 5.0 N acts on one of the asses in the diection of its velocity fo 0.10 s. Find the new aula velocity of the syst H 8. A kg sphee ovi hoizontally to the ight with an initial velocity of 5 /s stikes the lowe end of an 8 kg igid od AB. The od is suspended fo a hie at A and is initially at est. Knowi that the efficient of estitution between the od and sphee is 0.80, deteine the aula velocity of the od and the velocity of the sphee iediately afte the ipact. H 9. A ound boad of ass M and adius R is placed on a fixed sooth hoizontal plane and is fee to otate about an axis which passes though its cent A an is standi on the point aked A on the cicufeence of the boad. At fist the boad & the an ae at est. The an stats ovi alo the i of the boad at nstant speed v o elative to the boad. Find the ale of boad s otation when the an passes his stati point on the disc fist ti H 10. Two unifo thin ods A & B of leth 0.6 each and of asses 0.01 kg & 0.0 kg espectively ae igidly joined, end to end. The bination is pivoted at the lighte end P as shown in figue such that it can feely otate about the point P in a vetical plan A ll obct of ass 0.05 kg ovi hoizontally hits the lowe end of the bination and sticks to it. What should be the velocity of the obct so that the syste uld just be aised to the hoizontal position? [JEE - 94] H 11. A ete stick weighi 40 g is pivoted at its uppe end in such a way that it can feely otate in a vetical plane though this end (figue). A paticle of ass 100 g is attached to the uppe end of the stick though a light sti of leth 1. Initially, the od is kept vetical and the spi hoizontal when the syste is eleased fo est. The paticle llides with the lowe end of the stick and sticks the Find the axiu ale though which the stick will is H 1. A hoogeneous od AB of leth L = 1.8 and ass M is pivoted at the cente O in such a way that it can otate feely in the vetical plane as shown (fig). The od is initially in the hoizontal position. An insect S of the e ass M falls vetically with speed V on the point C, idway between the point O & B. Iediately afte falli, the insect oves towads the end B such that the od otates with nstant aula velocity. Deteine the velocity in tes of V & L. If the insect eaches the end B when the od has tuned though an ale of 90º, deteine V. H 1. A boy is standi on a platfo which is fee to otate about its axis. The boy holds an open ubella in his hand. The axis of the ubella incides with that of the platfo. The oent of inetia of the the platfo plus the boy syste is.0 10 kg - and that of the ubella is.0 10 kg-. The boy stats spinni the ubella about the axis at an aula speed of.0 ev/s with espect to hiself. Find the aula velocity ipated to the platfo. a H 6. SECTION ( ) : COMBINED TRANSLATIONAL + ROTATION MOTION (KINEMATICS) 1. The end B of od AB which akes ale with the floo is bei pulled with a velocity v0 as shown. Taki the leth of the od as, calculate the followi at the instant when = 7º The velocity of end A The aula velocity of od (c) Velocity of CM of the od.

12 A point A is located on the i of a wheel of adius R = 0.50 which olls without slippi alo a hoizontal suface with velocity v = 1.00 /s. Find: the odulus and the diection of the acceleation vecto of the point A ; the total distance s tavesed by the point A between the two successive oents at which it touches the sufac I. A sphee of ass olls on a plane sufac Find its kinetic enegy at an instant when its cente oves with speed v. 4. A ball of adius R =10.0 c olls without slippi down an inclined plane so that its cente oves with nstant acceleation =.50 c/s; t =.00 s afte the beginni of otion its position esponds to that shown in Fig. Find : the velocities of the points A, B and O ; the acceleations of these points. 5. A cylinde olls without slippi ove a hoizontal plan The adius of the cylinde is equal to. Find the cuvatue adii of tactoies taced out by the points A and B in Fig 6. A otati disc (figue) oves in the positive diection of the x axis. Find the equation y (x) descibi the position of the instantaneous axis of otation, if at the initial oent the axis C of the disc was located at the point O afte which it oved With a nstant velocity v, while the disc stated otati unte clockwise with a nstant aula acceleation (the initial aula velocity is equal to zeo) ; With a nstnat acceleation (and the zeo initial velocity), while the disc otates unteclockwise with a nstnat aula velocity. I 7. A dinki staw of ass is placed on a table othogonally to the edge such that half of it extends beyond the tabl A fly with ass lands on the inne end of the staw and walks alo the staw untill it eaches the oute end. It does not tip even when anothe fly lands on the top of the fist on Find the lagest ass that the send fly can hav (Neglect the fiction between staw and table). (dinki staw) () a. SECTION (J) : COMBINED TRANSLATIONAL & ROTATIONAL MOTION (DYNAMICS) A cylinde olls on a hoizontal plane sufac If the speed of the cente is 5 /s, what is the speed of the highest point? J. Deteine the point of the od AB of leth 0.9 at which the foce P should be applied if the acceleation of point B is to be zeo. Knowi that the agnitude of P is.70 N, deteine the espondi aula acceleation of the od and the acceleation of the cente of the od. ( Mass of the od is 1.5 kg. Foce P is to be applied pependicula to the od and the od is in hoizontal sooth plane). J. A unifo od 1.0 lo is initially standi vetically on a sooth hoizontal sufac It is stuck a shap hoizontal blow at the top end, with the blow diected at ight ales to the od axis. As a esult, the od acquied an aula velocity of ad/sec. What is the tanslational velocity of the cente of ass of the od afte the blow. Which point on the od is stationay on the od just afte the blow. J 4. A ll spheical ball is eleased fo a point at a height h on a ough tack shown in figu Assui that it does not slip anywhee, find its linea speed when it olls on the hoizontal pat of the tack. J 1.

13 J 5. A ll disc is set olli with a speed v on the hoizontal pat of the tack of the pevious poble fo ight to left. Two what height will it clib up the cuved pat? J 6. A sphee stats olli down an incline of inclination. Find the speed of its cente when it has veed a distance. J 7. A solid sphee of ass is eleased fo est fo the i of a heispheical cup so that it olls alo the sufac If the i of the heisphee is kept hoizontal, find the noal foce exeted by the cup on the ball when the ball eaches the botto of the cup. J 8. A capet of ass 'M' ade of inextensible ateial is olled alo its leth in the fo of a cylinde of adius 'R' and is kept on a ough floo. The capet stats unolli without slidi on the floo when a negligibly ll push is given to it. Calculate the hoizontal velocity of the axis of the cylindical pat of the capet when its adius educes to R/. [JEE - 90] R A block X of ass 0.5 kg is held by a lo assless sti on a fictionless inclined plane inclined at 0º to the hoizontal. The sti is wound on a unifo solid cylindical du Y of ass kg and adius 0. as shown in figu The du is given an initial aula velocity such that block X stats ovi up the X plan ) 0º Find the tension in the sti dui otion. At a cetain instant of tie the agnitude of the aula velocity of Y is 10 ad s 1. Calculate the distance tavelled by X fo that instant of tie until it es to est. Y J 9. [JEE - 94] By pulli on the d of a yo-yo just fast enough, a peson anages to ake the yo-yo spin unteclockwise, while eaini at a nstant height above the floo. Denoti the weight of the yo-yo by W, the adius of the inne du on which the d is wound by, and the adius of gyation of the yo-yo byk, deteine ; The tension in the d The aula acceleation of the yo-yo. J 11. A sti is wapped ove the edge of a unifo disc and the fee end is fixed with the ceili. The disc oves down, unwindi the sti. Find the downwad acceleation of the disc. J 1. A hollow sphee is eleased fo the top of an inclined plane of inclination. What should be the iniu efficient of fiction between the sphee and the plane to pevent slidi. Find the kinetic enegy of the ball as it oves down a leth l on the incline if the fiction efficient is half the value calculated in pat. J 1. A solid sphee of ass 0.50 kg is kept on a hoizontal sufac The efficient of static fiction between the sufaces in ntact is /7. What axiu foce can be applied at the highest point in the hoizontal diection so that the sphee does not slip on the suface? J 14. The 6 kg caiage is suppoted as shown by two unifo disks, each havi a ass of 4 kg and a adius of 75. Knowi that the caiage is initially at est, deteine the velocity of the caiage.5 s afte the 10 N foce has been applied. Assue that the disks oll without slidi. J 15. A thin unifo od AB of ass = 1.0 kg oves tanstionally with acceleation w =.0 /s due to two antipaallel foces F1 and F (Fig.). The distance between the points at which these foces ae applied is equal to a = 0 c. Besides, it is known that F = 5.0.N. Find the leth of the od. a J 10. F1 a F B

14 J 16. A unifo slende ba AB of ass is suspended fo two spis as shown. If spi beaks, deteine at that instant ; J 17. The aula acceleation of the ba. The acceleation of point A. (c) The acceleation of point B. A ectaula igid fixed block has a lo hoizontal edg A solid hoogeneous cylinde of adius is placed hoizontally at est with its leth paallel to the edge such that the axis of the cylinde and the edge of the block ae in the e vetical plan Thee is sufficient fiction pesent at the edge so that a vey ll displaceent causes the cylinde to oll of the edge without slippi. Deteine : : The ale c though which the cylinde otates befoe it leave ntact with the edg The speed of the cente of ass of the cylinde befoe leavi ntact with the edg (c) The atio of tanslational to otational kinetic enegies of the cylinde when its cente of ass is in hoizontal line with the edg [JEE - 95] Figue shows a ough tack, a potion of which is in fo of a cylinde of adius R. With what iniu linea speed should a sphee of adius be set olli on the hoizontal pat so that it pletely goes ound the cicle on the cylindical pat. J 19. Figue shows a ll spheical ball of ass olli down the loop tack. The ball is eleased on the linea potion at a vetical height H fo the lowest point. The cicula pat shown has a adius R. Find the kinetic enegy of the ball when it is at a point A whee the adius ake an ale with the hoizontal Find the adial and the taential acceleations of the cente when the ball is at A. (c) Find the noal foce and the fictional foce acti on the ball if H = 60 c, R = 10 c, = 0 and = 70 g. J 0. A unifo plate of ass '' is suspended in each of the ways shown. Fo each case deteine iediately afte the nnection at B has been eleased ; a J 18. A (1/) Pin Suppots B Spi Wies A B (1/) (i) (ii) The aula acceleation of the plat The acceleation of its cente of ass. A B (1/) (iii)

15 J 1. A unifo disc olls without slidi on a hoizontal sufac Find the atio of total kinetic enegy of uppe half pat to the total kinetic enegy of the disc. J. A unifo solid cylinde of ass ests on two hoizontal planks. A thead is wound on the cylind The hai end of the thead is pulled vetically down with a nstant foce F. Find the axiu agnitude of the foce F which still does not bi about any slidi of the cylinde, if the efficient of fiction between the cylinde and the planks is equal to k. What is the acceleation wax of the axis of the cylinde olli down the inclined plan J. In the aaeent shown in the figue weight A possesses ass, a pulley B possesses ass M. Also known ae the oent of inetia I of the pulley elative to its axis and the adii of the pulley ae R and R espectively. Conside the ass of the theads is negligibl Find the acceleation of weight A afte the syste is set fe (Assue no slippi takes place anywhee) B B R R A A unifo ba of leth 6 a & ass 8 lies on a sooth hoizontal tabl Two point asses '' & ovi in the e hoizontal plane with speeds v & 'v' espectively stike the ba as shown & stick to the ba afte llision. Calculate ; Velocity of the cente of ass. Aula velocity about cente of ass (c) Total kinetic enegy, just afte llision. Two ll sphees A & B espectively of ass & ae nnected by a igid od of leth & negligible ass. The two sphees ae esti on a hoizontal, fictionless sufac When A is suddenly given the velocity v 0 as shown. Find velocities of A & B afte the od has otated though 180º. [JEE - 91] v0 A B A unifo od of ass and leth is stuck at an end by a foce F pependicula to the od fo a shot tie inteval t. Calculate the speed of the cente of ass, the aula speed of the od about the cente of ass, (c) the kinetic enegy of the od and (d) the aula oentu of the od about the cente of ass afte the foce has stopped to act. Assue that t is so ll that the od does not appeciably chae its diection while the foce acts. A unifo od of leth L lies on a sooth hoizontal tabl A paticle ovi on the table stikes the od pependiculaly at an end and stops. Find the distance tavelled by the cente of the od by the tie it tuns though a ight al Show that if the ass of the od is fou ties that of the paticle the llision is elastic. K. a K. K 1. SECTION (K) : CONSERVATION OF ANGULAR MOMENTUM (COMBINED TRANSLATION & ROTATIONAL MOTION) K 4. K 5. A i olls on a hoizontal suface without slidi. The velocity of the cente is v. It enuntes a step of height 0. R whee R is the adius of the i. Calculate the aula velocity of the i just afte the ipact. Assue that the i does not etun back. (and thee is sufficient fiction to avoid slippi). Find the iniu value of ' v ' so that the i ascends the step.. K 6. A thin spheical shell of adius R lyi on a ough hoizontal suface is hit shaply and hoizontally by a cu Whee should it be hit so that the shell does not slip on the suface? K 7. A unifo wheel of adius R is set into otation about its axis at an aula speed. This otati wheel is now placed on a ough hoizontal suface with its axis hoizontal. Because of fiction at the ntact, the wheel acceleates fowad and its otation deceleates till the wheel stats pue olli on the sufac Find the linea speed of the wheel afte it stats pue olli.

16 K 8. A thin spheical shell lyi on a ough hoizontal suface is hit by a cue in such a way that the line of action passes though the cente of the shell. As a esult, the shell stats ovi with a linea speed v without any initial aula velocity. Find the linea speed of the shell afte it stats pue olli on the sufac K 9. A sphee of ass and adius is pocted alo a ough hoizontal suface with the initial velocities indicated. If the final velocity of the sphee is to be zeo, expess ; K 10. The equied 0 in tes of 0 and.. The tie equied fo the sphee to e to est in tes of 0 & the efficient of fiction k. Thee paticles A, B, C of ass each ae joined to each othe by assless igid ods to fo an equilateal tiale of side a. Anothe paticle of ass hits B with a velocity v 0 diected alo BC as shown. The llidi paticle stops iediately afte ipact. A, B, C 'a' '' v 0 BC 'B' Calculate the tie equied by the tiale ABC to plete half evolution in its subsequent otion. What is the net displaceent of point B dui this inteval? Suppose the od with the balls A and B of the pevious poble is claped at the cente in such a way that is can otate feely about a hoizontal axis though the clap. The syste is kept at est in the hoizontal position. A paticle P of the e ass is dopped fo a height h on the ball B. The paticle llides with B and sticks to it. Find the aula oentu and the aula speed of the syste just afte the llision. What should be the iniu value of h so that the syste akes a full otation afte the llision. K 1. A solid sphee olli on a ough hoizontal suface with a linea speed v llides elastically with a fixed, sooth, vetical wall. Find the speed of the sphee afte it has stated pue olli in the backwad diection. K 1. A slende od of leth L foi an ale with the vetical stikes a fictionless floo K 11. A od AB of leth is eleased fo est with AB inclined at ale with hoizontal. It llides elastically with gound afte falli though a height h. What is the height upto which the cente of ass the od ebounds afte ipact? Take = 60º. Find the height h so that the od is hoizontal fist tie when its cente of ass is at the axiu height. K 14. a at A with a vetical velocity 1 and no aula velocity. Assui that the ipact at A is pefectly elastic, deive an expession fo the aula velocity of the od iediately afte the ipact. SECTION (L) : TOPPLING. L 1. A cubical block of ass and edge a slides down a ough inclined plane of inclination with a unifo speed. Find the toque of the noal foce acti on the block about its cent (JEEQuestionsofPeviousYeas) 1. A unifo disc of ass and adius R is pocted hoizontally with velocity 0 on a ough hoizontal floo so that it stats off with a puely slidi otion at t = 0. Afte t 0 sends, it acquies a puely olli otion as shown in figu Calculate the velocity of the cente of ass of the disc at t 0. Assui the efficient of fiction to be, calculate t 0. Also calculate the wok done by the fictional foce as a function of tie & the total wok done by it ove a tie t uch loe than t0. [JEE - 97]

17 . A unifo disc of ass and adius R is olli up a ough inclined plane, which akes an ale of 0º with the hoizontal. If the efficients of static and kinetic fiction ae each equal to and only the foces acti ae gavitational and fictional, than the agnitude of the fictional foce acti on the disc is and its diection is (wite 'up' o 'down') the inclined plan [JEE - 97]. Two thin cicula disks of ass kg and adius 10 c each ae joined by a igid assless od of leth 0 c. The axis of the od is alo the pependicula to the planes of the disk though thei centes. This obct is kept on a tuck in such a way that the axis of the obct is hoizontal and pependicula to the diection of otion of the tuck. The fiction with the floo of the tuck is lage enough, so that obct can oll on the tuck without slippi. Take x-axis as the diection of otion of the tuck and z-axis as the vetically upwad diection. If the tuck has an acceleation of 9 /s, calculate [JEE - 97] the foce of fiction on each disk. the agnitude & diection of fictional toque acti on each disk about the cente of ass O of the obct. Expess the toque in the vecto fo in tes of unit vectos i, j & k alo x, y & z diection. [JEE /100] A ass is ovi with a nstant velocity alo a line paallel to the, x-axis, away fo the oigin. Its aula oentu with espect to the oigin : [JEE- 97] Is zeo Reains nstant Goes on inceasi Goes on deceasi 5. A syetic laina of ass M nsists of a squae shape with a seicicula section ove each of the edge of the squae as in figue. The side of the squae is a. The oent of inetia of the laina about an axis though its cente of ass and pependicula to the plane is 1.6 M a. The oent of inetia of the laina about the taent AB in the plane of laina is. [JEE - 97 /100] 6. A od of weight 'w' is suppoted by two paallel knife edges 'A' and 'B' is in equilibiu in a hoizontal position. The knives ae at a distance 'd' fo each oth The cente of ass of the od is at a distance 'x' fo 'A'. The noal eaction on 'A' is and on 'B' is [JEE - 97 /100] [JEE - 97 /100] 7. A wedge of ass and tiaula ss section (AB = BC = CA =R) is a 4. ovi with a nstant velocity-v i towads a sphee of adius R fixed z on a sooth hoizontal table as shown in the figu The wedge akes an z C elastic llision with the fixed sphee and etun alo the e path withv y R out any otation. Neglect all fiction and suppose that the wedge eains in ntact with the sphee fo a vey shot tie At, dui which the sphee x A B exets a nstant foce F on the wedg Find the foce F and also the noal foce N exeted by the table on the wedge dui the tie t. Let h denote the pependicula distance between the cente of ass of the wedge and the line of action of foce F. Find the agnitude of the toque due to the noal foce N about the cente of the wedge, dui the tie t. [JEE - 98] 8. A unifo cicula disc has adius R & ass. A paticle also of ass is fixed at a point A on the edge of the disc as shown in the figu The disc can otate feely about a fixed hoizontal chod PQ that is at a distance R/4 fo the cente C of the disc. The line AC is pependicula to PQ. Initially the disc is held vetical with the point A at its highest position. It is then allowed to fall so that it stats otati about PQ. Find the linea speed of the paticle as it eaches its lowest position. [JEE /00] 9. Let be the oent of inetia of a unifo squae plate about an axis AB that passes though its cente and is paallel to two of its sides. CD is a line in the plane of the plate that passes though the cente of the plate and akes an ale with AB. The oent of inetia of the plate about the axis CD is then equal to [JEE - 98 /00] I I sin I s I s ( /)

18 10. The toque on a body about a given point is found to be equal to A J whee A is a nstant vecto J and is the aula oentu of the body about that point. Fo this it follows that [JEE - 98 /00] d J is pependicula to J at all instants of tie dt the ponents of J in the diection of A does not chae with tie the agnitude of J does not chae with tie J does not chae with tie A sooth sphee 'A' is ovi on a fictionless hoizontal plane with aula speed and cente of ass velocity 'v'. It llides elastically and head on with an identical sphee 'B' at est. Neglect fiction eveywhe Afte the llision, thei aula speeds ae A and B, espectively. Then [JEE - 99 /00] A< B A = B A= B= 1. A disc of ass M and adius R is olli with aula speed on a hoizontal plane as shown. The agnitude of aula oentu of the disc about the oigin O is [JEE - 99 /00] (/)MR MR A an pushes a cylinde of ass 1 with help of a plank of ass as shown. Thee is no slippi at any ntact. The hoizontal ponent of the foce applied by the an is F. Find : [JEE /00] 1. MR (1/)MR 11. A cubical block of side a is ovi with velocity v on a hoizontal sooth plane, as shown It hits a idge at point O. The aula speed of the block afte it hits O is. [JEE - 99 /00] v a v a zeo A cubical block of side L ests on a ough hoizontal suface with efficient of fiction. A hoizontal foce F is applied on the block as shown If the efficient of fiction is sufficiently high so that the block does not slide befoe toppli, the iniu foce equied to topple the block is L µ F (toppli) [JEE (Sc 1/5] 15. v 4a a 14. The acceleations of the plank and the cente of ass of the cylinde, and The agnitudes and diections of fictional foces at ntact points.) g g/ g (1 ) 4 A thin wie of leth L and unifo density is bent into a cicula loop with cente at O as shown. The oent of inetia of the loop about the axis XX is [JEE Sc 1/5] Infinitesial L 8 L 16 5 L 16 L 8 An equilateal tiale ABC foed fo a unifo wie has two ll identical beads initially located at A. The tiale is set otati about the vetical axis AO. Then the beads ae eleased fo est siultaneously and allowed to slide down, one alo AB and the othe alo AC as shown. Neglecti fictional effects, the quantities that ae nseved as the beads slide down, ae [JEE Sc. 1/5] Aula velocity and total enegy (kinetic enegy and potential enegy) Total aula oentu and total enegy Aula velocity and oent of inetia about the axis of otation Total aula oentu and oent of inetia about

19 A od AB of ass M and leth L is lyi on a hoizontal fictionless sufac A paticle of ass '' tavelli alo the suface hits the end A of the od with a velocity v 0 in the diection pependicula to AB. The llision is pletely elastic. Afte the llision the paticle es to est. Find the atio /M. [JEE (Mains) ++/100] A point P on the od is at est iediately afte the llision. Find the distance AP. (c) Find the linea speed of the point P at a tie L/ (v0 ) afte the llision. 19. A paticle is placed at a ne P of a cube of side 1 et Foces of agnitudes, and 5 kg. wt. act on the paticle lo the diagonals of the faces passi though the point P. Find the oent of these foces about the ne opposite to P. [REE - 000] 0. A unifo od AB of leth 10 etes and weight 6kg.wt. is esti with its end A on a sooth hoizontal plane AD and end B on a sooth plane DB inclined at ale 60 with the hoizontal. The od is kept in equilibiu by typi a sti DP to a point P of the od. If the leth of the sti is equal to AP and AD = BD, find the tension in the sti. [REE - 000] 1. One end of a unifo od of ass M and leth L is suppoted by a fictionless hie which can withstand a tension of 1.75 Mg. The od is fee to otate in a vetical plan To what axiu ale should the od be otated fo the vetical position so that when left, the hie does not beak. [REE - 000]. Two discs of e thickness but of diffeent adii ae ade of two diffeent ateials such that thei asses ae The densities of the ateials ae in the atio 1 :. The oents of inetia of these discs about the espectively axes passi though thei centes and pependicula to thei planes will be in the atio [REE - 000] 1 : : 1 1 : 9 9 : 1. One quate secto is cut fo a unifo cicula disc of adius R. This secto has ass M. It is ade to otate about a line pependicula to its plane and passi though the cente of the oiginal disc. Its oent of inetial about the axis of otation is : [JEE - 01 (Sc.) /105] MR MR MR MR 4 8 Two heavy etallic plates ae joined togethe at 90º to each oth A laina sheet of ass 0 kg is hied at the line AB joini the two heavy etallic plates. The hies ae fictionless. The oent of inetia of the laina sheet about an axis paallel to AB and passi though its cente of ass is 1. kg. Two ubbe obstacles P and Q ae fixed one on each etallic plate at a distance 0.5 fo the line AB. This distance is chosen so that the eaction due to the hies on the laina sheet is zeo dui the ipact. Initially the laina sheet hits one of the obstacles with an aula velocity 1ad/sec and tuns back. If the ipulse on the sheet due to each obstacle is 6 N/s, Find the location of the cente of ass of the laina sheet fo AB At what aula velocity does the laina sheet e back afte the fist ipact (c) Afte how any ipacts, does the laina sheet e to est. [JEE - 01(Mains) 8+1+1/100] a 4. A cylinde olls up an inclined plane, eaches soe height and then olls down (without slippi thoughout these otions). The diections of the fictional foce acti on the cylinde ae : [JEE - 0 (Sc.) /90] Up the incline while ascendi and down the incline while descendi Up the incline while ascendi as well as descendi Down the incline while ascendi and up the incline while descendi Down the incline while ascendi as well as descendi. 6. A cicula platfo is fee to otate in a hoizontal plane about a vetical axis passi though its cent A totoise is sitti at the edge of the platfo. Now the platfo is given an aula velocity 0. When the totoise oves alo a chod of the platfo with a nstant velocity (with espect to the platfo) the aula velocity of the platfo (t) will vay with tie t as : [JEE - 0 (Sc.) /90] 0 0 t > (t) > (t) > (t) > (t) 0 t > > 5. 0 t > t >