Unit - 5 Rotational Motion

Transcription

1 Unit - otational otion 07

2 * mpotant Fomula, Facts and Tems. Cente of mass of system of paticles SUAY cm m m mn m m m fo igid body. cm m fo Two body System n n n n n n mn m n n. m O m m + m +m += + O = m m m +m m m = = O = m m +m m +m m v + m v m v V cm = P vcm p p... pn similaly n n a cm ma ma mn a n F a cm F F F n. Toque = T = F = F sin = poduct of foce and pependicula distance between point of otation and line of action. Angula momentum = L P L = psin = poduct of linea momentum and pependicula distance between point of otation and line of action. oment of inetia = = n m = m n n n m m nn 08

3 4. Low of consevation of angula momentum d L d As L p dt dt d L when = then L emains constant dt ts geometical epesentation in planetay motion Let da is an aeal velocity dt da L da L Then m = O = dt dt m. adius of gyation {K} As = m m m nn f all paticles have same mass then = m n n nm Hee (nm = ) n mk k ( n n 6. Some elations between linea and otational motion. v = w v w Hee ) w = d dt d w d dt dt a a at whee a wv a a a a v 7. Equilibium of a igid body. t T 4 When F F F Fn 0 it is in linea equilibium When P P P Pn it is in otational equilibium 0 8. Two theom fo moment of inetia Z Theom of pependicula axis. X Y cm d Theom of Paallel axis. 09

4 9. olling down of body on an inclined plane. gh sin, a V K K Condition fo olling without sliding s tan, K Fo ing s tan (K = ) disc s tan K solid sphee s tan 7 K 0. otational Kinetic Enegy..K.E = V V K K & L Total kinetic Enegy = otational K.E. + Linea K.E. V K K V V.K.E Now () K LineaK.E () K K otational K. E K Total K. E K K Pecentage otational K.E. = 00% K 0

5 () Tanslational K.E Total K.E K K Compaison between physical quantities of linea motion and otational motion Tanslational motion otational motion Linea displacement, d Angula displacement, Linea velocity, V Angula velocity, w Linea acceleation, ass, m Linea omentum, dv a dt Angula acceleation, oment of inetia, dw dt P mv Angula momentum, L w Foce, F ma Newton's Second Law of otion, dp F dt Toque, dl dt A esult simila to newtown's Second Law, dl dt Tanslational kinetic enegy K = mv otational kinetic enegy K = p m L Wok, W = Fd Wok, W = Powe, P = Fv Powe, P = w Equations of linea motion taking place Equations of otational motion taking place with constant linea acceleation with constant angula acceleation : v = V O + at w = t W O d = v O t+ at W0 t at ad = v -v a = 0 W W0 Law of consevation of linea momentum Law of consevation of angula momentum when F 0 then P is constant mpulse when P 0 then L is constant mpulse linea F t P P otational t L L

6 Value of V, a, t fo some olling Bodies Shape of Body velocity velocity acceleation Time K ing/hollow cylinde gh sin gl g sin gsin Disc/Solid cylinde 4 gh 4 glsin g sin gsin Solid Sphee 0 0 gh glsin g sin 4 gsin Shell/Hollow sphe 6 gh 6 glsin g sin 0 gsin oment of inetia an adius of gyation fo some symmetic bodies Body Axis Figue K Fo olling body K Thin od of Length L Passing though its cente and pe pendicula to its length L L - ing of adius Any diamete - ing of Passing though its adius cente and pependicula to its plane Cicula disc adius Passing though its cente and pependicula to its plane of Cicula disc Any diamete of adius 4 -

7 Body Axis Figue K Fo olling body K Hollow Geometical cylinde of axis of the adius cylinde Solid cylinde Geometical axis of adius of the cylinde Solid sphee Any diamete of adius Hollow sphee of adius Any diamete CQ Fo the answe of the following questions choose the coect altenative fom among the given ones.. The cente of mass of a systems of two paticles is (A) on the line joining them and midway between them (B) on the line joining them at a point whose distance fom each paticle is popotional to the squae of the mass of that paticle. (C) on the line joining them at a point whose distance fom each paticle invesely popotional to the mass of that paticle. (D) On the line joining them at a point whose distance fom each paticle is popotional to the mass of that paticle.. Paticles of gm, gm, gm, gm ae placed at the cones A, B, C, D, espectively of a squae of side 6 cm as shown in figue. Find the distance of cente of mass of the system fom geometical cente of squae. {A} cm {B} {C} cm cm {D} 4 cm. Thee paticles of the same mass lie in the (X, Y) plane, The (X, Y) coodinates of thei positions ae (, ), (, ) and (, ) espectively. The (X,Y) coodinates of the cente of mass ae {A} (, ) {B} (, ) {C} (., ) {D} (,.)

8 4. Conside a two-paticle system with the paticles having masses, and. f the fist paticle is pushed towads the cente of mass though a distance d, by what distance should the second paticle be moved so as to keep the cente of mass at the same position? {A} d {B} d {C} d. Fou paticles A, B, C and D of masses m, m, m and m espectively ae placed at cones of a squae of side x as shown in figue find the coodinate of cente of mass take A at oigine of x-y plane. {D} d {A} 7x 0x x, {B} x, 0 7 {C} x 0x, 7 {D} x 7x, 0 6. Fom a unifom cicula disc of adius, a cicula disc of adius 6 and having cente at a distance + fom the cente of the disc is emoved. Detemine the cente of mass of emaining potion of the disc. {A} 70 {B} A cicula plate of unifom thickness has a diamete of 6 cm. A cicula potion of diamete 4 cm. is emoved fom +ve x edge of the plate. Find the position of cente of mass of the emaining potion with espect to cente of mass of whole plate. {A} - 7 cm {B} + 9 cm {C} - 9 cm {D} + 7 cm 8. Two blocks of masses 0 kg an 4 kg ae connected by a sping of negligible mass and placed on a fictionless hoizontal suface. An impulse gives velocity of 4 m/s to the heavie block in the diection of the lighte block. The velocity of the cente of mass is : {A} 0 m/s {B} 0 m/s {C} 0 m./s {D} m/s 9. A paticle pefoming unifom cicula motion has angula momentum L., its angula fequency is doubled and its K.E. halved, then the new angula momentum is : {A} ½ {B} ¼ {C} L {D} 4L 0. A cicula disc of adius is emoved fom a bigge disc of adius. such that the cicumfeences of the disc coincide. The cente of mass of the emaining potion is fom the cente of mass of the bigge disc. The value of is. {A} ½ {B} /6 {C} ¼ {D} /. Thee point masses, and ae located at the vetices of an equilateal tiangle of side 'a'. what is the moment of inetia of the system about an axis along the attitude of the tiangle passing though,? a a {C} {A} 4 {B} 4 {C} 4 {D} 4 7 a {D} 7 a 4

9 . A body of mass m is tied to one end of sping and whiled ound in a hoizontal plane with a anstant angula velocity. The elongation in the sping is one centimete. f the angula velocity is doubted, the elongation in the sping is cm. The oiginal length of sping is {A} 6 cm {B} cm {C} 4 cm {D} cm. A cylinde of mass kg and adius 0 cm, and fee to otate about its axis, eceives an angula impulse of kg S - initially followed by a simila impulse afte evey 4 sec. what is the angula speed of the cylinde 0 sec afte initial imulse? The cylinde is at est initially. {A} 06.7 ad S {B} 06.7 ad S {C} 07.6 ad S - {D} 07.6 ad S - 4. Two cicula loop A & B of adi a and b espectively ae made fom a unifom wie. The atio of thei moment of inetia about axes passing though thei centes and pependicula to thei planes B is 8 b then a is equal to A a {A} {B} 4 {C} 6 {D} 8. f the eath wee to suddenly contact so that its adius become half of it pesent adius, without any change in its mass, the duation of the new day will be {A} 6 h {B} h {C} 8 h {D} 0 h 0 6. n HC molecule the sepaation between the nuclei of the two atoms is about.7 A A 0 m. The appoximate location of the cente of mass of the molecule is A î with espect of Hydogen atom ( mass of CL is. times of mass of Hydogen) {A} {B}. {C}.4 {D}. 7. Two bodies of mass kg and kg have position vecto î ĵ kˆ the cente of mass of this system has a position vecto and (-i-j+k) espectively {A} î kˆ {B} î ĵ kˆ {C} î ĵ kˆ {D} î ĵ kˆ 8. dentify the coect statement fo the otational motion of a igid body {A} ndividual paticles of the body do not undego acceleated motion {B} The cente of mass of the body emains unchanged. {C} The cente of mass of the body moves unifomly in a cicula path {D} ndividual paticle and cente of mass of the body undego an acceleated motion. 9. A ca is moving at a speed of 7 km/h the adius of its wheel is 0.m. f the wheels ae stopped in 0 otations afte applying beaks then angula etadation poduced by the beaks is {A} -. ad s {B} -9. ad s {C} -. ad s {D} -4. ad s 0. A wheel otates with a constant acceleation of.0 ad sec f the wheel stat fom est. The numbe of evolution it makes in the fist ten seconds will be appoximately. {A} 8 {B} 6 {C} 4 {D}. Two discs of the same mateial and thickness have adii 0. m and 0.6 m thei moment of inetia about thei axes will be in the atio {A} : 8 {B} : 7 {C} : 9 {D} :

10 . A wheel of mass 0 kg has a moment of inetia of 60 kg m about its own axis. The adius of gyation will be m. {A} 0 {B} 8 {C} 6 {D} 4. One cicula ig and one cicula disc both ae having the same mass and adius. The atio of thei moment of inetia about the axes passing though thei centes and pependicula to thei planes, will be {A} : {B} : {C} : {D} 4 : 4. One solid sphee A and anothe hollow sphee B ae of the same mass and same oute adii. The moment of inetia about thei diametes ae espectively and A such that B {A} A {B} B A {C} B A {D} B A da db (adio of thei densities). A ing of mass and adius is melted and then molded in to a sphee then the moment of inetia of the sphee will be.. {A} moe than that of the ing {B} Less than that of the ing {C} Equal to that of the ing {D} None of these 6. A cicula disc of adius and thickness /6 has moment of inetia about an axis passing though its cente and pependicula to its plane. t is melted and ecasted in to a solid sphee. The moment of inetia of the sphee about its diamete as axis of otation is {A} {B} 8 {C} 7. One quate secto is cut fom a unifom cicula disc of adius. This secto has mass. t is made to otate about a line pependicula to its plane and passing though the cente of the oiginal disc. ts moment of inetia about the axis of otation is {A} {B} {C} 4 {D} 8 8. A thin wie of length L and unifom linea mass density is bent in to a cicula loop with cente at O as shown in figue. The moment of inetia of the loop about the axis xx' is. B {D} 0 {A} L 8 {B} L 6 L L {C} {D} Two disc of same thickness but of diffeent adii ae made of two diffeent mateials such that thei masses ae same. The densities of the mateials ae in the atio :. The moment of inetia of these disc about the espective axes passing though thei centes and pependicula to thei planes will be in the atio. {A} : {B} : {C} : 9 {D} 9 : 6

11 0. Let 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 of Q with AB. The moment of inetia of the plate about the axis CD is then equal to. {A} {B} sin {C} cos 7 {D} cos. A small disc of adius cm is cut fom a disc of adius 6 cm. f the distance between thei centes is. cm, what is the shift in the cente of mass of the disc {A} -0.4 cm {B} -.4 cm {C} -.8 cm {D}. cm. A staight od of length L has one of its ends at the oigin and the othe end at x=l f the mass pe unit length of od is given by Ax whee A is constant whee is its cente of mass. {A} L/ {B} L/ {C} L / {D} L / 4. A unifom od of length L is placed with one end in contact with hoizontal and is then inclined at an angle to the hoizontal and allowed to fall without slipping at contact point. When it becomes hoizontal, its angula velocity will be.. {A} w = g sin L {B} w = L g sin {C} w = 4. A cubical block of side a is moving with velocity V on a hoizontal smooth plane as shown in figue. t hits a idge at point O. The angula speed of the block afte it hits O is. {A} v 4a {B} v a {C} v a {D} zeo 6gsin L {D} w = L gsin. Conside a body as shown in figue, consisting of two identical bulls, each of mass connected by a light igid od. f an impulse J = V is impated to the body at one of its ends, what would be its angula velocity. What is V? {A} V / L {B} V / L {C} V / L {D} V / 4L 6. A thin cicula ing of mass and adius is otating about its axis with a constant angula velocity w. Two objects each of mass m ae attached gently to the opposite ends of a diamete of the ing. The ing will now otate with an angula velocity. {A} ( m) m {B} m {C} m {D} m 7. A smooth sphee A is moving on a fictionless hoizontal plane with angula speed and cente of mass velocity v. t collides elastically and head on with an identical sphee B at est. Neglect fiction eveywhee. Afte the collision, thei angula speeds ae A and B espectively, Then {A} A < B {B} A = B {C} A = {D} = B 8. Two point masses of 0. kg and 0.7 kg ae fixed at the ends of a od of length.4 m and of negligible mass. The od is set otating about an axis pependicula to its length with a unifom angula speed. The point on the od though which the axis should pass in ode that the wok equied fo otation of the od is minimum, is located at a distance of.. {A} 0.4 m fom mass of 0. kg {B} 0.98 m fom mass of 0. kg {C} 0.7 m fom mass of 0.7 kg {D} 0.98 m fom mass of 0.7 kg

12 9. n a bicycle the adius of ea wheel is twice the adius of font wheel. f and F the adius, v F and v ae speed of top most points of wheel espectively then... {A} v = v F {B} v F = v {C} v F = v {D} v F > v 40. Fom a cicula disc of adius and mass 9, a small disc of adius / is emoved fom the disc. The moment of inetia of the emaining potion about an axis pependicula to the plane of the disc and passing though O is. {A} 4 {B} 40 9 {C} 0 {D} 7 9 ae 4. A child is standing with folded hands at the cente of a platfom otating about its cental axis the kinetic enegy of the system is K. The child now stetches his ams so that the moment of inetia of the system doubles. The kinetic enegy of the system now is {A} K {B} K/ {C} K/4 {D} 4K 4. f the eath is teated as a sphee of adius and mass. ts angula momentum about the axis of otation with peiod T is.. {A} T {B} T {C} T {D} 4 T 4. f the angula momentum of any otating body inceases by 00%, then the incease in its kinetic enegy will be.. {A} 400% {B} 800% {C} 00% {D} 00% 44. The.. of a body about the given axis is. kgm initially the body is at est. n ode to poduce a otational kinetic enegy of 00 J. an angula acceleation of ad sec must be applied about that axis fo duation of. {A} 4 sec {B} sec {C} 8 sec {D} 0 sec 4. An automobile engine develops 00kw when otating at a speed of 800.p.m. what toque does it delive? {A} 0 Nm {B} 440 Nm {C} Nm {D} 68 Nm 46. The moment of inetia of two otating bodies A and B ae A and B A B and thei angula momentum ae equal. f thei K.E. be K and A K espectively then. B {A} K A, K B {B} KB KA {C} KA KB {D} KB KA 47. The cente of mass of the disc undegoes S.H.. with angula fequency equal to.. {A} k m {B} k m {C} 48. Thee ings, each of mass P and adius ae aanged as shown in the figue the moment of inetia of the aangement about YY' axis will be. k m {D} 4k m {A} 7 {B} P {C} P 7 {D} P P 8

13 49. f distance of the eath becomes thee times that of the pesent distance fom the sun then numbe of days in one yea will be. {A} 6 {B} 6 7 {C} 6 9 {D} 6 0. A solid sphee and a solid cylinde having same mass and adius oll down the same incline the atio of thei acceleation will be. {A} : 4 {B} 4 : {C} : {D} :. The atio of angula momentum of the electon in the fist allowed obit to that in the second allowed obit of hydogen atom is {A} {B} {C} ½ {D}. A playe caught a cicket ball of mass 0 gm moving at a ate of 0 m/s f the catching pocess is Comlitad in 0. sec the foce of the flow exeted by the ball on the hand of the playe.. N {A} {B} 0 {C} 0 {D} 00. Two disc one of the density 7. gm/cc and othe of density 8.9 gm/cc ae of the same mass and thickness thei moment of inetia ae in the atio of {A} {B} {C} {D} Two identical hollow sphees of mass and adius ae joined togethe and the combination is otated about an axis tangential to one sphee and pependicula to the line connecting thei centes. The moment of inetia of the combination is. {A} 0 {B} 4 {C} {D} 4. A od of length L otate about an axis passing though its cente and nomal to its length with an angula velocity. f A is the coss-section and D is the density of mateial of od. Find its otational K.E. {A} AL D {B} 6 AL D {C} 4 AL D {D} AL D 6. nitial angula velocity of a cicula disc of mass is w Then two sphees of mass m ae attached gently two diametically oppsite points on the edge of the disc what is the final angula velocity of the disc? m 4m {A} w {B} w {C} w {D} w 4m m 7. A cicula disc x of adius is made fom an ion plate of thickness t. and anothe disc Y of adius 4 is made fom an ion plate of thickness t/4 then the otation between the moment of inetia and X y is {A} y 64 x {B} y x {C} y 6 x {D} y x 8. A Pulley of adius m is otated about its axis by a foce F 0t t N whee t is in sec applied tangentially. f the moment of inetia of the Pulley about its axis of otation is 0 Kg,

14 the numbe of otations made by the pulley befoe its diection of motion is evesed is : {A} moe than but less then 6 {B}moe than 6 but less then 9 {C} moe than 9 {D} Less then 9. Two sphees each of mass and adius / ae connected with a mass less od of length as shown in figue. What will be moment of inetia of the system about an axis passing though cente of one of the sphees and pependicula to the od? {A} {B} {C} 60. Fou paticles each of mass 'm' ae lying symmetically on the im of the disc of mass and adius moment of inetia of this system about an axis passing though one of the paticles and pependicula to plane of disc is {A} 6 {B} 6 {C} {D} Zeo {D} 6. A mass is suppoted by a mass less sting wound aound a unifom cylinde of mass and adius as in figue. With what acceleation will the mass fall on elease? {A} /g {B} g/ {C} g {D} 4g/ 6. Calculate otational K.E. of eath due to its otation about its own axis. 4 e =6 0 kg e =6400 Km {A} Joule {B} Joule {C} Joule {D} Joule 6. A cod is wound ound the cicumfeence of wheel of adius. the axis of the wheel is hoizontal and moment of inetia about it is A weight mg is attached to the end of the cod and falls fom the est. Afte falling though the distance h. the angula velocity of the wheel will be. {A} gh m {B} mgh m {C} mgh m 64. f otational K.E. is 0% of tanslational K.E. then the body is.. {A} ing {B} solid cylinde {C} Hollow sphee {D} Solid sphee 6. A mete stick of mass 400 gm is pivoted at one end and displaced though an angle 60 0 the incease in its P.E. is J. {A} {B} {C} Zeo {D} 66. Tow unifom od of equal length but diffeent masses ae igidly joined to fom L shaped body which is then pivoted as shown. f in equilibium the body is in the shown configuation atio /m will be. {A} {B} {C} {D} {D} gh 0

15 67. A light od caies thee equal masses A, B and C as shown in the figue the velocity of B in vetical position of od if it is eleased fom hoizontal position as shown in the figue is. {A} g {B} 8g 4g 8g {C} {D} A gamophone ecod of mass and adius is otating with angula speed W. f two pieces of wax each of mass ae kept on it at a distance of / fom the cente on opposite side then the new angula velocity will be.. {A} {B} m m {C} m {D} m 69. A solid cylinde olls down a smooth inclined plane 4.8m high without slipping what is its linea speed at the bottom of the plane, if it stats olling fom the top of the plane? (take g = 0 m/s ) {A} 4 m/s {B} m/s {C} 0 m/s {D} 8 m/s 70. The. of a disc of mass and adius about an axis passing though the cente O and pependicula to the plane of disc is. f one quate of the disc is emoved the new moment of inetia of disc will be.. {A} {B} 4 {C} 8 {D} 7. The moment of inetia of a unifom od about a pependicula axis passing though one of its ends is. The same od is bent in to a ing and its moment of inetia about a diamete is, Then {A} is. {B} 4 {C} 8 {D} 6 7. A molecule consist of two atoms each of mass 'm' and sepaated by a distance of 'd' f 'K' is the aveage otational K.E. of the molecule at paticula tempeatue then its angula fequency is. {A} d m {B} d k m {C} m d d k {D} 4 m k 7. A ca is moving with a constant speed the wheels of the ca make 0 otations pe minute the beaks ae applied and the ca comes to est in 8 sec how many otation ae completed by the wheels befoe the ca is bought to est. {A} 4 {B} 6 {C} 8 {D} The angula momentum of a wheel changes fom L to L in seconds what is the magnitudes of toque acting on it? {A} L {B} L/ {C} L/ {D} L/

16 7. A unifom disc of mass 00kg and adius m is otating at the ate of 600.p.m. what is the toque equied to otate the disc in the opposite diection with the same angula speed in a time of 00 sec? {A} 600 Nm {B} 00 Nm {C} 400 Nm {D} 00 Nm 76. The moment of inetia of a mete scale of mass 0.6kg about an axis pependicula to the scale and passing though 0 cm position on the scale is given by (Beath of scale is negligible). {A} 0.04 kg m {B} 0.08 kg m {C} kg m {D} 0.48 kg m 77. How much constant foce should be applied tangential to equato of the eath to stop its otation in one day? {A} 8. 0 N {B} N {C}. 0 N {D} None of these 78. A constant toque of 00 Nm tuns a wheel of moment of inetia 00 kg m about an axis passing though its cente the angula velocity of the wheel afte sec will be... ad/sec {A} {B} 0 {C} {D} A disc of mass and adius is olling with angula speed w on a hoizontal plane, as shown in figue. The magnitude of angula momentum of the disc about the oigin O is {A} {B} {C} {D} 80. A mass m is moving with a constant velocity along the line paallel to the x-axis, away fom the oigin. ts angula momentum with espect to the oigin {A} Zeo {B} emains constant {C} goes on inceasing {D} goes on deceasing 8. A body is olling down an incline plane. f the otational K.E. of the body is 40% of its tanslational K.E. then the body is. {A} ing {B} Cylinde {C} solid sphee {D} hollow sphee 8. A spheical ball olls on a table without slipping, then the faction of its total enegy associated with otation is. {A} / {B} / {C} /7 {D} /7 8. A binay sta consist of two stas A (. s) and B(mass s) whee s is the mass of sun. They ae sepaated by distance d and ae otating about thei cente of mass, which is stationay. The atio of the total angula momentum of the binay sta to the angula momentum of sta B. about the cente of mass is. {A} 6 {B} ¼ {C} {D} ½ 84. A small object of unifom density olls up a cuved suface with initial velocity 'u'. t eaches up to maximum height of v 4g with espect to initial position then the object is. {A} ing {B} solid sphee {C} disc {D} hollow sphee 8. A paticle of mass m slides down on inclined plane and eaches the bottom with linea velocity V. f the same mass is in the fom of ing and olls without slipping down the same inclined plane. ts velocity will be. {A} V {B} V {C} V {D} V

17 GAPHCAL QUESTONS. 86. oment of inetia of a sphee of mass and adius is. keeping mass constant if gaph is plotted between and then its fom would be. {A} {B} {C} {D} 87. Accoding to the theoem of paallel axis cm md the gaph between d will be {A} {B} {C} {D} 88. The gaphs between angula momentum L and angula velocity w will be. {A} {B} {C} {D} 89. The gaphs between loge L and loge P is whee L is angula momentum and P is linea momentum {A} {B} {C} {D} 90. Let E is the otational kinetic enegy and L is angula momentum then the gaph between Loge E and log e L can be {A} {B} {C} {D}

18 ACHNG COLUN TYPE 9. atch list with list and select the coect answe. List - System (x) A ing about it axis (y) A unifom cicula disc about it axis (z) A solid sphee about any diamete List - oment of inetia () () () 7 (w) A solid sphee about any tangent ( 4) s Select coect option () Option? X Y Z W {A} 4 {B} 4 {C} 4 {D} atch the shape of gaph with given pai of physical quantities. Physical Quantities {X} oment of inetiadistance {Z}Angula momentum(l) letagen paallel axis angula geloaty (w) {Y} log E log L e e {W} log e L log e P Shape of gaph {P} {Q} {} {S} Select Coect Option Option? X Y Z W {A} S P Q {B} Q P S {C} S Q P {D} P Q S 4

19 ASSETON - EASONNG TYPE 9. n the following questions statement - (Assetion) is followed by statement - (eason). Each question has the following fou choices out of which only one choice is coect. {A} Statement - is coect (tue), Statement - is tue and Statement- is coect explanation fo Statement - {B} Statement - is tue, statement - is tue but statement- is not the coect explanation fou statement -. {C} Statement - is tue, statement- is false {D} Statement- is false, statement - is tue 9. Statement - The angula momentum of a paticle moving in a cicula obit with a constant speed emains conseved about any point on the cicumfeence of the cicle. Statement - f no net toque outs, the angula momentum of a system is conseved. 94. Statement - A sphee and a cylinde slide without olling fom est fom the top of an inclined plane. They will each the bottom with the same speed. Statement - Bodies of all shapes, masses and sides slide down a plane with the same acceleation. 9. Statement - Fiction is necessay fo a body to oll on suface Statement - Fiction povides the necessay tangential foce and toque. 96. Statement - A body tied to a sting is moved in a cicle with a unifom speed. f the sting suddenly beaks the angula momentum of the body becomes zeo. Statement - The toque on the body equals to the ate of change of angula momentum. 97. Statement - f thee is no extenal toque on a body about its cente of mass, then the velocity of the cente of mass emains constant. Statement - The Linea momentum of an isolated system emains constant. 98. Statement - Two cylinde one hollow and othe solid (wood) with the same mass and identical dimensions ae simultaneously allowed to oll without slipping down an inclined plane fom the same height. The hollow will each the bottom of inclined plane fist. Statement - By the pinciple of consevation of enegy, the total kinetic enegies of both the cylindes ae identical when they each the bottom of the incline. 99. Statement - A thin unifom od AB of mass and length L is hinged at one end A to the hoizontal floo initially it stands vetically. t is allowed to fall feely on the floo in the vetical plane, The angula velocity of the od when its ends B stikes the floo Statement - The angula momentum of the od about the hinge emains constant though out its fall to the floo. 00.Statement - f the cylinde olling with angula speed- w. suddenly beaks up in to two equal halves of the same adius. The angula speed of each piece becomes w. Statement - f no extenal toque outs, the angula momentum of the system is conseved. g L

20 PASSAGE BASED QUESTONS Passage - A Solid sphee of mass and adius is eleased fom est at the top of a fictionless inclined plane of length 'd' and inclination?. n case (a) it olls down the plane without slipping and in case (b) it slides down the plane 0. The atio of the acceleation of the sphee in case (a) to that in case (b) is [A] [B] / [C] /7 [D] 7/9 0. The atio of the velocity of the sphees when it eaches the bottom of the plane in case (a) to that in case (b) is [A] [B] 0. The atio of time taken by the sphee to each the bottom in case (a) to that in case (b) as [A] [B] [C] 7 [D] [C] [D] Passage - A unifom disc of mass and adius olls without slipping down a plane inclined at an angle with the hoizontal. 04. The acceleation of the cente of mass of the disc is [A] g sin [B] g sin 0. The fictional foce on the disc is [C] gsin g sin [A] [B] [C] 06. The magnitude of toque acting on the disc is g sin g sin [D] 7 7 g cos [D] None g sin g sin [A] g [B] g sin [C] [D] 07. f the disc is eplaced by a ing of the same mass and the same adius, the atio of the fictional foce on the ing to that on the disc will be [A] / [B] [C] [D] Passage - A solid cylinde of mass and is mounted on a fictionless hoizontal axle so that it can feely otate about this axis. A sting of negligible mass is wapped ound the cylinde and a body of mass m is hung fom the sting as shown in figue the mass is eleased fom est then 08. The acceleation with which the mass falls is [A] g [B] mg [C] mg m ng [D] m 6

21 09. The tension in sting is mg [A] mg [B] m mg [C] m g [D] m 0. The angula speed of cylinde is popotional to h n, whee h is the height though which mass falls, Then the value of n is [A] zeo [B] [C] ½ [D]. The moment of inetia of a unifom cicula disc of mass and adius about any of its diamete is ¼, what is the moment of inetia of the disc about an axis passing though its cente and nomal to the disc? [A] [B] [C] [D]. A solid cylinde of mass and adius olls down an inclined plane of height h. The angula velocity of the cylinde when it eaches the bottom of the plane will be. [A] gh [B] gh. A cylinde of mass m and adius is otating about its axis with constant speed v ts kinetic enegy is [A] mv [B] mv [C] mv [D] mv 4. A cicula disc of mass m and adius is olling on a smooth hoizontal suface with a constant speed v. ts kinetic enegy is [A] 4 mv [B] mv [C] 4 mv [D] mv. A solid sphee is otating about a diamete at an angula velocity w. if it cools so that its adius educes to /n of its oiginal value. ts angula velocity becomes [A] n [B] [C] n [D] n n 6. n above question () f the oiginal otational K.E. of the sphee is K, ts new value will be [A] K n [B] K 4 n [C] n K [D] n 4 K 7. A solid sphee is otating about a diamete due to incease in oom tempeatue, its volume inceases by %, f no extenal toque acts. The angula speed of the sphee will. [A] incease by nealy / % [B] decease by nealy / % [C] incease by nealy ½ % [D] decease nealy by ½ % 8. A cylinde of mass has length L that is times its adius what is the atio of its moment [C] of inetia about its own axis and that about an axis passing though its cente and pependicula to its axis? [A] [B] gh [D] [C] [D] gh 7

22 9. A unifom od of length L is suspended fom one end such that it is fee to otate about an axis passing though that end and pependicula to the length, what maximum speed must be impated to the lowe end so that the od completes one full evolution? [A] gl [B] gl [C] 6 gl [D] gl 0. The height of a solid cylinde is fou times that of its adius. t is kept vetically at time t=o on a belt which is moving in the hoizontal diection with a velocity v =.4t whee v in m/s and t is in second. f the cylinde does not slip, it will topple ove a time t = [A] second [B] sec. [C] sec. [D] 4 sec.. The moment of inetia of a thin od of mass and length L about an axis passing though the point at a distance L/4 fom one of its ends and pependicula to the od is 7 L L L L [A] [B] [C] [D] A thin unifom od AB of mass and length L is hinged at one end A to the hoizontal floo. nitially it stands vetically. t is allowed to fall feely on the floo in the vetical plane. The angula velocity of the od when its end B stikes the floo is [A] g L [B] g L. A cicula disc of adius is fee to oscillate about an axis passing though a point on its im and pependicula to its plane. The disc is tuned though an angle of 60? and eleased. ts angula velocity when it eaches the equilibium position will be [A] g [B] g 4. The moment of inetia of a hollow sphee of mass and inne and oute adii and about the axis passing though its cente and pependicula to its plane is [A] [B] [C]. f a is aeial velocity of a planet of mass its angula momentum is [A] [B] A [C] 8 [C] [C] g L g [D] [D] [D] g L g 6 A [D] A 6. A wheel having moment of inetia kg about its vetical axis, otates at the ate of 60 pm about this axis. The toque which can stop the wheels otation in one minute will be.. [A] Nm [B] Nm [C] N m [D] Nm 8 7. A wheel is otating at 900 pm about its axis. When powe is cut off it comes to est in minute, the angula etadation in ad / sec is [A] [B] 4 [C] 6 [D] 8 8. What is the moment of inetia of a solid sphee of density and adius about its diamete? [A] 0 76 [B] 76 0 [C] 0 76 [D] 76 0

23 9. A wheel is subjected to unifom angula acceleation about its axis. nitially its angula velocity is zeo. n the fist two second it otate though an angle, in the next sec. it otates though an angle, find the atio = [A] [B] [C] [D] 4 0. A gamophone ecod of mass and adius is otating at an angula velocity w. A win of mass is gently placed on the ecod at a distance /. fom its cente. The new angula velocity of the system is w [A] m w [B] m [C] [D] wm KEYNOTE Q.No. Ans Q.No. Ans Q.No. Ans Q.No. Ans Q.No. Ans C A 6 A 9 D A A C 6 B 9 B C B A 6 B 9 D B 4 C 4 A 64 B 94 A 4 D D A 6 D 9 A B 6 A 6 B 66 D 96 D 6 A 7 C 7 C 67 D 97 D 7 A 8 C 8 B 68 C 98 D 8 B 9 B 9 C 69 D 99 C 9 C 0 D 40 A 70 C 00 A 0 A B 4 B 7 C 0 C B 4 D 7 A 0 C A 4 B 7 C 0 D 4 A 44 B 74 A 04 B A 4 C 7 C 0 A 6 C 46 B 76 C 06 D 7 B 47 D 77 A 07 A 8 B 48 A 78 C 08 D 9 A 49 C 79 C 09 D 0 B 0 A 80 B 0 C A C 8 C B D B 8 C C B C 8 A D 4 C 4 D 84 C 4 C B C 8 C D 6 C 6 D 86 D 6 C 7 A 7 A 87 C 7 B 8 D 8 A 88 A 8 A 9 B 9 A 89 B 9 C 0 A 60 B 90 B 0 A 9

24 Hints. Let cm is at oigin cm m m O = m + m m m m = m m m ve sign ignoe as distance. Hee ^ ^ = 0i + 6J A ^ ^ = 6i + 6J B ^ ^ = 6i + 0J C ^ ^ = 0i + 0J D = = 6 gm cm m A A m B B m C C m D D. The x and y co.odinates of cente of mass ae x = m x + m x + m x m + m + m as m = m = m (x + x + x) = similtaaly fo y = (x, y) = (, ) 4. m x = m x and m (x d) = m (x d ) md = md' md d ' = m. Hee at oigin A (0, 0), B (x, 0), C (x, x), D (0, x) x cm = (m 0) + (mx) + (mx) + (4m 0) m + m + m + 4m y = cm (m 0) + (m 0) + (mx) + (4m x) m + m + m + 4m 0

25 6. Let mass pe unit aea of disc = m ass of disc = = m ass of emoved disc = m ' = m = 6 6 fom figue 00' = 0 = ' + ( ')x 'x = ' + x x = ' ' 7. Let mass pe unit aea of Plote = m ass of whole Plote = 6 = m ass of emoved pat = 4 = m ass of emaining Potion = C. of whole disc = O at oigin C. of emoved Plote = = 8 = 7cm C. of emaining Potion =? O = + i i 8. The Velocity of C.. is given by 9. V cm m v + mv = m + m E = = = L E E' L = L' = ' 0. Let m is the mass of unit aea then mass of big disc = () m = Let m is the mass of unit aea then mass of small disc = m = = 4 ass of emaining Potion = = = 4

26 Let G be the C. of emaining Potion (OG) = (OO ) ( ) = 4 4 =. The moment of inetia about AD =? = m (Pependicula distance of m fom AD) + m (Pependicula distance of m fom AD) + m (Pependicula distance of m fom AD) m + m a (m + m ) 4. Let L is oiginal length & K sping anstant then m (L + x ) w = kx & (L + x ) w = kx Taking atio L + x L + x x = x given x = cm, x = cm and =. nital angula momentum = L i = i = x 0 = 0 angula mumentum afte initial inpulse = kgm s angula mumentum afte initial 4 sec = + = 6kgm s angula mumentum afte initial 8 sec = 6 + = 9kgm s angula mumentum afte initial 8 sec = 4 kgm s angula mumentum afte initial 0 sec = 4 kgm s = 4 hee = = (0.) = 0.kgm 4 4 = = = 06.7 ad s A = m a a, B = m b b m B b b = A ma a Let K is the mass of unit length of the wie then m = ( )k and m b = ( b)k a a m m b a = b a m B b b b = 8 = = A ma a a b a =

27 . Let be the mass and the initial adius of the eath is the angula veloalty of the otation of the eath, the duation T of the day is T = and T = Accoding to law of consevation of angula momentum = 6. m = m =. = 0 =.7 i m + m cm = m + m 7. m + m cm = m + m ^ 8. Theoy [B] The cente of mass of lucky emains unchaged. 9. o Go 7 000/600 = = 80ad/sec 0. = O, = n = 0 = 40 ad As = w w o 0.. = wot + t = 00 ad = = ( t ) As t ae same 4 =. = K = K = = = 6 K = 4 m 0 4. ing = = Disc : 4. A = Solid = = 0.4 and B = hollow = = 0.66 A <. ing = As Volume and ass emain same = Solid <<< B

28 6. Volume of disc = Volume of Sphee = =. of disc = V = t = 6 4 = 6 4 V = = 8 = = =. of sphee = = = 7. ass of the ontie disc would be A and its moment of inetia about the given axis would be (A). Fo the given section the moment of inetia about the since axis be one qvate of this is (A). 8. ass pe unit length of the wie = ass of L length = L When it is lent in fom of ciculaing = L L = oment of inetia of ing about given axis = 9.. of disc = = = t t As thei mass & thickness ae some t = = t 0. Let Z be the. of squae plote about the axis passing though the cente and pependicula to the plane of squae, hence accoding to Pefendicula axis theom. Z = AB + A'B' Also Z = CD + C'D' As axis ae symmetic AB = A'B' = Z And CD = C'D' = Z So we can say that AB = A'B' = CD = C'D' =. Let the adius of complete disc is a & that of small disc is b Afte small disc is cut fom complete disc let the C.. shift to O at distance x flem oiginal cente O. The Position of new C.. is givenly let 6 is mass pecunitaea. 6b cm X = 6 a 6 b 4

29 . Let the mass of an element of length dx of the od located at a distance X away fom left and is dx. the x codinate of the C.. is given by. L Total mass of od = 0 AL Ax dx = x cm = xdm = x (Ax dx) AL 0. By Consevation of Enegy P.E. of od = otational K.E. g sind = = ml = g sin L As hee l = L g sin = L 4. Angula momentum of Block w..t. O befoe collision with O = V a on collision the block will otate about the side passing though O. Now its angula momentum = w Acc. to law of consevation of angula momentum V a = a a + 6 v = 4 v hee is the. of block about the axis pependicula to the plane passing though O.. Given system of two putides will otate about its cente of mass. L initial angula momentum = v L Final angula momentum = By law of consevation of angula momentum V L = L V = L 6. nitial angula momentum of ing L = = Final angula momentum of ing and paticles = ( + m ) ' As No extenal foque so Accoding to Law of consevation of angula momentum. = ( + m ) ' w ' = ( + m)

30 7. As it is head-on elastic collision between two idential balls thee foe they will exchange thei linea vecocity is A comes to est and B stats moving with linea velocity V. As thee is no fiction any whee, foque on both the sphees about thei cente of mass is zeo and thei angula velocities emains unchanged. Theefoe A = and B = 0 8. = 0.x (.4 x) Fo minimum wok moment of inetia of the system should be minimum is d = 0 = 0 x 0.7 (.4 x) = 0 dx x = 0.98 m [B] 0.98 m fom mass of 0. kg 9. Angula speed fo both wheels ae diffeent but linca speed fo both same so V F = V 40.. of complete disc about O point Total = (g) adius of emoved disc = As = i.e. ass of emoved potion = g g. of emoved disc about it own axis = = of emoved disc about O. emoved = total = emoved disc + emaining pat E = L = K given cm + x = + = 8 K f L is constant when child stetches his ams the moment of inetia of system get doubled so kinetic enegy will becomes half i.e. K 4. oment of inetia of sphee = L = 4. L E = E L about its axis 44. otational K.E. = = 00 Accoding to w = wo + t 6

31 4. = 800 pm P = 800 = 60 ad/sec As K = K A = A J and J K B = B 47. As is the mass of disc, the foce is poducing angula accoation in the disc, theefoe 4kx 4k x m = m( ) = 48. Fo ing and and about.. of system = + + YY = ' ' and fo ing. about YY = 49. T T = g sin 0. a = ( + k / ) g sin 7 a = and a = g sin. nh L =. mpulse = Change in momentum = F t L F = t. = = 4. Apply theom of paallel axis. otational K.E. = 6. f = 0 w = w 7. ass = Volume = = t. of x is x = m 8. Hee diection of otion will be evesed when foce F = 0 o 0 t = 0 o t = 4sec. f is angula accellaation then foque = 4t t and d d w = Also = t dt dt 7 = = F. O 0 = (0t t ) O

32 9. d =.t dt = (4t t ) t dt = (4t t ) Taking intigation 4 4t t = f n otations ae completed in As then Putting t = = n = 64 = n =.4 which is <.4 < 6 = + () Accoding to the Paallel axis theom. of disc about an axis passing though paticle () and pependicula to plane of disc is 6. n falling though a height h. P.E. = K.E + = 6. mgh = mv + = mv + m Accoding to V v = ad taking v = 0 & d = h, V = ah so = = 6 0 (6.4 0 ) = kgm Angulavelocity of eath T mgh = m ah 4 K.E. of otation of the eath gh 6. We know = k + v V gh mgh = = + k m + mk 64. k = 0% k T 6. Cente of mass of stick is at midpoint when it is displaced by 60 0 ts c.m. ises up to height l l l h fom figue h = cos = ( cos ) l so incease in P.E. = mgh = mg ( cos ) o ( cos60 ) = Joule 8

33 66. Net foque about O should be zeo Hence o o g sin 60 = g sin Loss in PE = gain in agula kinetic enegy l + mg l g + mg l = l w Hee = oment of inetia about fixed point l l 4 + m m l = ml 9 Fom fig. 68. = '' 4 6g 6g 89l 9 4l 4l 7 mgl = ml = As V l = = + m ' 4 = ( + m) ' gh 69. V = k New ass 4. of disc = ' New = = = 4 [C] 8 7. Fo a od of mass and length L., the about a pependicula axis passing though one and is L = when it is bent to fom a ing, then L = L = The. of the ing about its diamete is L L = = = 4 8 d d 7. The. of the molecule = + m d md = m = 4 The otational K.E. of the moldule ( K) = = K 9

34 o = = 4 ad/sec 60 Now = 0 + t Total angle descited in 8 second is = w ot + t dl L L L = = = = L dt Fo disc = = so angula momentum L = angula speed = final angula momentum in opposite diection = 600 = kgm sec So change in angula momentum = L = kgm /sec dl = = = 400 N.m. dt By Paallel axis theom solved poblem. 77. = ad/day and = 0, t = day w w = t Foque equied to stop the eath = T = = F... F = 78. d = = dt 79. The disc have two types of motion tanslational and otational so thee will be two types of angulamotion thee foe total angula momentum should be total of both L = L T + L 80. L = omentum x pepenlicula distance between point of otation and line of action = m.v.y all emain constant L = emaing constant K.E. = T.K.E 00 4 = mv = mv 0 V mk = mv 40

35 8. Total enegy E = + mv = m + m 8. Hee C.. wt A. d d = +. 6 LA d d =. w = w 6 6 L B d d = = At the highest point the whole enegy is conseted to P.E. of the object. P.E. = mgh, Total K.E. of body = Now, K.E. = P.E. 8. Hee in fist case n second case v h = v mv PE = mg = 4g 4g 4 v ' = V mv + w = mv + mg = mvh v = gh gh + K As fo the ing K = v ' = gh 86. = This elation shows that the gaph between and will be paacola symatic to axis 87. Gaph should be paacola symmetic to - axis, but it should not pass fom oigin because thee is a constant value cm plesent fo d = L = w L w f constant so gaph between L and w will be staight line with constant stop and passing fom oigin. 89. L = P 90. logel = logep + loge f gaph is dawn between log e L and log e P then it will be staight line which will not pass though oigin at log e P = 0 log e L = log e L E = so Log E = log L log () So the gaph log E Log E = log () log L will be staightline with constant stop and when Log L = 0. 4

36 9. The coect choice is [D] since the centifetal foce is adial. Foque is zeo so L = constant. 94. The coect choice is [A] f a body slides down an indined plane its acaleation is a = g (sin cos ) which depends only on g, and. 98. a = g sin + m cm cm of hollow cylinde is loss, so it will have moe accdeation and will take less time to each bottom. 99. Loss in P.E. is equal to gain in otational K.E. As the cente of mass of the od falls though the distance L L L g g = L 00. f is the.. of the complete cylinde, The. of each iece becomes since L =, the angula speed of each Piece becomes. 0. n case (a) accelaation down the plane is a g sin = = g sin 7 + a So = a 7 n ase (b) a = g sin 0. using V = ad we can find. 0. Fom a a v v = = v v 7 d = at 04. g sin f = a , we find that d 7 = d a =. As =, = and = f a Hence f = = a f = a g sin f = g sin f = 4

37 07. Fo a ing = Coect choice is 08. g T = a, T is Tension in sting foque on cylinde = T = a = = = a mg a = ( + m) [D] 09. g T = + [D] 0. Fom consevation of enegy we have gh = V + + ( m + ) 4 4mgh = ( + m) As h. Accoding to pependicula axis theom x = y = 4 So (Consideing two pependicula diamete in x & y diections) x y = c = + = 4 4. Fo solid cylinde = gh = V + As. K.E. in otation w Hee m and V = theefoe v K.E = = m 4 4. otational K.E = K.E = v + K.E = mv & Tanslational 4

38 . A solid sphee. = m As w = = = m m n 6. n = = n K = and K = K = m = m (n ) n n K = n kn 7. V = 4 4 log V = log + log diffeentiating we have dv v = d d dv = = 0.% = % v 6 8. No extenal foque so = constant = constant log + log = log c m = constant d d d d + = 0 = = % = % dt 6 -ve sign fo decensing L =, = n one full evolution the incase in P.E = gl, whee is the mass of od. L gl = = 0. The cylinde will topple when the foque mg equals the foque h so ma = mg Now V =.4t and g g a = = h dv a = dt h ma a = d dt [.4(t) ] = 4.gt 44

39 . Using = cm + d Paallel axis theom L L L d = = 4 4 and L cm = L L 7L = + = Loss in P.E = gain in otational K.E. cente of ass of od is at L. P.E = gl Gain in L.E = = gl L g = = 6 L. P.E. at 60 0 = gh ( cos ) Eqvillibium at K.E = Fo Accoding to Paallel axis theom (as d = ) = cm + = + = Hee = gh ( cos 60 ) o 4. We can obtain hollow sphee as it solid sphee of adius is emoved fom a solid sphee 4 4 of mass of hollow sphee = t is density = ( ) and = 8 = oment of inetia of hollow sphee = () By substituting the values of and. Aeal Velocity A = T and T = A = = = L = A 4

40 6. Hee = k g m n = 60pm = ps T =? n = 0 t = mn = 60 sec ( ) T =. = t 7. Angula etadation = t 8. 4 Fo sphee = = 9. Fom = wt + t = 0 + () = + = 0 + (4) = 8 0. Angula momentum of ecoul L = Whee = Let w' be the angula velocity afte putting coin of mass m at distance fom cente the angula momentum of system L' ( + m ) ' since T = 0 so L' = L ( + m ) ' ' = = = + m + m m + ' = + m but = 46