06 - ROTATIONAL MOTION Page 1 ( Answers at the end of all questions )

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1 06 - ROTATIONAL MOTION Page 1 1 ) A body A of mass M while falling vetically downwads unde gavity beaks into two pats, a body B of mass ( 1 / ) M and a body C of mass ( / ) M. The cente of mass of bodies B and C taken togethe shifts compaed to body A towads does not shift depends on height of beaking body B ( d ) body C [ AIEEE 005 ] ) The moment of inetia of a unifom semicicula disc of mass M and adius about a line pependicula to the plane of the disc though the cente is ( / 5 ) M ( 1 / 4 ) M ( 1 / ) M ( d ) M [ AIEEE 005 ] ) A T shaped object with dimensions shown in the figue, is lying on a smooth floo. A foce F is applied at the point P paallel to AB, such that the object has only tans ational motion without otation. Find the location of P with espec to C. ( / ) l ( / ) l l ( d ) ( 4 / ) l [ AIEEE 005 ] 4 ) A solid sphee is otating in fee space. If the adius of the sphee is inceased keeping mass same, which one of the follow ng will not be affected? Moment of inetia A gula momentum Angula velocity ( d ) Rota ional kinetic enegy [ AIEEE 004 ] 5 ) One solid sphee A and anot e hollow sphee B ae of same mass and same oute adii. Thei moment of inetia about thei diametes ae espectively Ι A and Ι B such that Ι A = Ι B Ι A > Ι B Ι A < Ι B ( d ) Ι A / Ι B = d A / d B [ AIEEE 004 ] 6 ) A paticle pefo ming unifom cicula motion has angula momentum L. If its angula fequency is doub ed and its kinetic enegy halved, then the new angula momentum is L / L / 4 L ( d ) 4 L [ AIEEE 00 ] 7 ) Let F be the foce acting on a paticle having position vecto, and τ be the toque of this foce about the oigin. Then. τ = 0 and F. τ = 0. τ = 0 and F. τ 0. τ 0 and F. τ = 0 ( d ). τ 0 and F. τ 0 [ AIEEE 00 ] 8 A cicula disc X of adius R is made fom an ion plate of thickness t, and anothe disc Y of adius 4R is made fom an ion plate of thickness t / 4. The elation between the moments of inetia Ι x and Ι y is Ι y = 64Ι x Ι y = Ι x Ι y = 16Ι x ( d ) Ι y = Ι x [ AIEEE 00 ] 9 ) The moment of inetia of a igid body depends upon mass of the body angula acceleation of the body angula velocity of the body ( d ) distibution of mass fom the axis of otation [ AIEEE 00 ]

2 06 - ROTATIONAL MOTION Page 10 ) If moment of inetia of a wheel, having adius of gyation 60 cm, is 60 kg-m, then mass of the wheel is 00 kg 500 kg 800 kg ( d ) 1000 kg [ AIEEE 00 ] 11 ) A paticle moves on a cicula path with deceasing speed. Choose t e coect statement. Angula momentum emains constant. Acceleation is towads the cente. Paticle moves on a spial path with deceasing adius. ( d ) The diection of angula momentum emains constant. [ IIT 005 ] 1 ) Mass and adius of a cicula disc is 9M and R espectively Moment of inetia of the disc about an axis passing though point O and pependicula to it plane afte emoval of a disc of adius R / touching its cicumfeence is 7 MR 40 9 MR MR ( d ) 4 MR [ IIT 005 ] V P 1 ) A disc is olling ( without slipping ) on a hoiz ntal suface. C is the cente and P and Q ae two paticles of the disc equidistant fom C. Let V P, V Q and V C be the magnitudes of velocities of the paticles P and Q and the paticle at C esp ctively. Then V Q > V C > V P V Q < V C < V P V Q = V P, V C = ( d ) V Q < V C > V P [ IIT 004 ] 14 ) A child is standing with folded hands at the cente of a platfom otating about its cental axis. The kinetic ene gy of he system is K. The child now stetches his ams so that the moment of inetia of he system doubles. The kinetic enegy of the system now is K K / K / 4 ( d ) 4 K [ IIT 004 ] 15 ) Two identical bodies of mass m ae connected by a light od of length l and placed on hoizontal smooth uface. An impulse mv is given at one end pependicula to the length of the od. The angula velocity of the system is v / l v / l v / l ( d ) v / 4l [ IIT 00 ] 16 ) A paticl undegoes unifom cicula motion. About which point on the plane of the cicle will he angula momentum of the paticle emain conseved? ( ) cente of the cicle on the cicumfeence of the cicle inside the cicle ( d ) outside the cicle [ IIT 00 ] 17 ) Two blocks of masses 10 kg and 4 kg ae connected by a sping of negligible mass and placed on a fictionalless hoizontal suface. An impulse gives a velocity of 14 m/s to the heavie block in the diection of the lighte block. The velocity of the cente of mass is 0 m/s 0 m/s 10 m/s ( d ) 5 m/s [ IIT 00 ] 18 ) A cylinde olls up an inclined plane, eaches some height, and then olls down ( without slipping thoughout these motions ). The diections of the fictional foce acting on the cylinde ae up the incline while ascending and down the incline while descending up the incline while ascending as well as descending down the incline while ascending and up the incline while descending ( d ) down the incline while ascending as well as descending [ IIT 00 ]

3 06 - ROTATIONAL MOTION Page 19 ) A cicula platfom is fee to otate in a hoizontal plane about a vetical axis passing though its cente. A totoise is sitting at the edge of the platfom. Now the platfom is given an angula velocity ω 0. When the totoise moves along a chod of the platfom with a constant velocity ( with espect to the platfom ), the angula velocity of the platfom ω t will vay with time as 0 ) One quate section is cut fom a unifom cicul disc of adius R. This section has a mass M. It is made to otate about a line pependicula to its plane and passing though the cente of the oiginal disc. Its moment of inetia about the axis of otation is 1 1 MR MR 1 MR ( d ) MR 4 8 [ IIT 001 ] [ IIT 00 ] 1 ) A small block is shot into ea h of the fou tacks as shown below. Each of the tack ises to the same height. The sp ed with which the block entes the tack is the same in all cases. At the highest point of the tack, the nomal eaction is maximum in [ IIT 001 ] ) A equilateal tiangle ABC fomed fom a vetical wie has two small dentical beads initially located at A. The tiangle is set ota ing about the vetical axis AO. Then the beads ae eleased fom est simultaneously and allowed to slide down; one along AB and the othe along AC as shown. Neglecting fictional effects, the quantities that ae conseved as beads slide down ae angula velocity and total enegy ( kinetic and potential ) total angula momentum and total enegy angula velocity and moment of inetia about the axis of otation ( d ) total angula momentum and moment of inetia about the axis of otation [ IIT 000 ]

4 06 - ROTATIONAL MOTION Page 4 ) A long hoizontal od has a bead which can slide along its length and is initially placed at a distance L fom one end A of the od. The od is set in angula motion about A with a constant angula acceleation, α. If the coefficient of fiction between the od and the bead is µ, and gavity is neglected, then the time afte which the b ad stats slipping is µ µ 1 ( d ) infinitesimal [ IIT 000 ] α α µα 4 ) A cubical block of side L ests on a ough hoizontal suface with coefficient of fiction µ. A hoizontal foce F is applied on the block as shown. If the coefficient of fiction is sufficiently high so that the block does not slide befoe toppling, the minimum foce equied to topple the block is infinitesimal mg mg 4 ( d ) mg ( 1 - µ ) [ IIT 000 ] 5 ) A thin wie of length L and unifom linea mass density ρ is bent into a cicula loop with cente at O as shown. The moment of inetia of the loop about the axis XX is ρ L ρ L 5 ρ L ρ L 8 π 16 π 16 π ( d ) 8 π [ IIT 000 ] 6 ) A cubical block of side a s moving with velocity V on a hoizontal smooth plane as s own. It has a idge at point O. The angula speed of th block afte it hits O is V V V ( d ) zeo [ IIT 1999 ] 4 a a a 7 ) A smooth sphee A is moving on a fictionless hoizontal plane with angula velocity ω and cente of mass velocity v. It collides elastically and head on with an identical sphee B a est. Neglect fiction eveywhee. Afte the collision, thei angula speeds ae ω A and ω B espectively. Then ω A < ω B ω A = ω B ω A = ω ( d ) ω B = ω [ IIT 1999 ] 8 ) A disc of mass M and adius R is olling with angula speed ω on a hoizontal plane as shown. The magnitude of angula momentum of the disc about the oigin O is 1 MR ω MR ω MR ω ( d ) MR ω [ IIT 1999 ] 9 ) Let I be the moment of inetia of a unifom 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 makes an angle θ with AB. The moment of inetia of the plate about the axis CD is then equal to I I sin θ I cos θ ( d ) I cos ( θ / ) [ IIT 1998 ]

5 06 - ROTATIONAL MOTION Page 5 0 ) The toque τ on a body about a given point is found to be equal to Α L whee Α is a constant vecto and L is the angula momentum of the body about that point. Fom this it follows that d L dt is pependicula to L at all instants of time the component of L in the diection of Α does not change with time the magnitude of L does not change with time ( d ) L does not change with time [ IIT 1998 ] 1 ) A mass m is moving with a constant velocity along a line pa llel to X-axis, away fom the oigin. Its angula momentum with espect to the oigin is zeo emains constant goes on inceasing ( d ) goes on deceasing [ IIT 1997, 1985 ] ) A thin cicula ing of mass M and adius is otating about its axis with a constant angula velocity ω. Two objects, each of mass m, ae attached gently to the opposite ends of a diamete of the ing. The wheel now otates with the angula velocity ωm / (M + m) ω(m m) / ( + m) ωm / (M + m) ( d ) ω(m + m) / M [ IIT 198 ] ) The moment of inetia of a thin squae plate ABCD, as shown in the figue, of unifom th ckn ss about an axis passing though the cente O and pe endicula to the plane of the plate is I 1 + I I I I 1 + I ( d ) I 1 + I + I + I 4 [ IIT 199 ] 4 ) A tube of ength L is filled completely with an incompessible liquid of mass M and closed a both the ends. The tube is then otated in a hoizontal plane about one of its ends with a unifom angula velocity ω. The foce exeted by the liquid at the othe end is M ω L / M ω L M ω L / 4 ( d ) M ω L / [ IIT 199 ] 5 ) A unifom ba of length 6a and mass 8m lies on a smooth hoizontal table. Two point masses m and m moving in the same hoizontal plane with speeds v and v espectively, stike the ba as shown in the figue and stick to the ba afte collision. Denoting angula velocity about the cente of mass, total enegy and velocity of cente of mass by ω, E and V c espectively, we have afte collision V c = 0 v v mv ω = ω = ( d ) E = 5 a 5 a 5 [ IIT 1991 ]

6 06 - ROTATIONAL MOTION Page 6 6 ) A paticle of mass m is pojected with a velocity V making an angle 45 with the hoizontal. The magnitude of the angula momentum of the pojectile about the point of pojection when the paticle is at its maximum height h is m V m V zeo ( d ) m g h [ IIT 1990 ] 4 g g 7 ) Two paticles A and B initially at est, move towads each othe unde a mutual foce of attaction. At the instant when the speed of A is V and the speed of B is V, the speed of the cente of mass of the system is zeo V 1.5 V ( d ) V [ IIT 198 ] 8 ) A ifle of mass M shoots a bullet of mass m at a stationay bloc of mass M distant D away fom it. When the bullet has moved distance d towads the block, the cente of mass of the bullet-block system is at a distance of md + MD ( D - d ) m fom the ifle fom the block m + M M + m M DM ( D - d ) fom the block ( d ) fom the ifle M + m M m 9 ) A hollow cylinde and a solid cylinde having diffeent mass and diamete ae eleased fom est simultaneously fom the top of an inclined plane. Which will each the bottom fist? solid cylinde can t b dete mined without knowing thei masses hollow cylinde ( d ) can be detemined without knowing thei diametes 40 ) A ca is moving with a speed of 108 km / h on a cicula path of adius 500 m. Its speed is inceasing at the ate of m / s. What is the acceleation of the ca? 9.8 m / s.7 m / s.6 m / s ( d ) 1.8 m / s 41 ) A spheical ball olls on a table about its diamete without slipping. Then the faction of its total enegy ssoci ted with its otation is ( d ) ) Two discs o same mass and thickness ae made fom two diffeent mateials of dens ties 1 and d espectively. The atio of thei moments of inetia about thei ax s passing though thei centes and pependicula to thei planes is d 1 : d d : d 1 ( d 1 / d ) : 1 ( d ) 1 : d 1 d 4 ) Two point masses M 1 and M ae at a distance apat. The moment of inetia of the system about an axis passing though the cente of mass and pependicula to the line joining the two masses is M1 M M1 + M M1 + M M1 M M1 + M M1 - M ( d ) M1 - M ) A paticle of mass m = 5 is moving with a unifom speed v = 5 in the XOY plane along the line y = x + 4. The magnitude of the angula momentum about the oigin is 40 units 60 units 80 units ( d ) 100 units M M

7 06 - ROTATIONAL MOTION Page 7 45 ) A solid cylinde fist olls without sliding and then slides without olling down the same inclined plane. The atio of velocities in the two cases, at the bottom of the plane is 1 : 1 : 6 : ( d ) : 1 Answes a c d b c b a a d d d d a b a a c b c a a b a c d a c c a a,b,c b c b a,c,d b,d a b a b b b a d c