Physics. Rotational Motion.

Transcription

1 Physics otational Motion

2 Table of Content. Intoduction.. Cente of Mass.. Angula Displacement. 4. Angula Velocity.. Angula Acceleation. 6. Equations of Linea Motion and otational Motion. 7. Moment of Inetia. 8. adius of Gyation. 9. Theoem of Paallel Axes. 0. Theoem of Pependicula Axes.. Moment of Inetia of Two Point Masses about Thei Cente of Mass.. Analogy between Tanslatoy Motion and otational Motion.. Moment of Inetia of Some Standad Bodies about Diffeent Axes. 4. Toque.. Couple. 6. Tanslatoy and otatoy Equilibium. 7. Angula Momentum.

3 8. Law of Consevation of Angula Momentum. 9. Wo, Enegy and Powe fo otating Body. 0. Slipping, Spinning and olling.. olling Without Slipping.. olling on an Inclined Plane.. olling Sliding and Falling of a Body. 4. Velocity, Acceleation and Time fo Diffeent Bodies.. Motion of Connected Mass. 6. Time Peiod of Compound Pendulum.

4 . Intoduction Tanslation is motion along a staight line but otation is the motion of wheels, geas, motos, planets, and the hands of a cloc, the oto of jet engines and the blades of helicoptes. Fist figue shows a sate gliding acoss the ice in a staight line with constant speed. He motion is called tanslation but second figue shows he spinning at a constant ate about a vetical axis. Hee motion is called otation. Up to now we have studied tanslatoy motion of a point mass. In this chapte we will study the otatoy motion of igid body about a fixed axis. () igid body: A igid body is a body that can otate with all the pats loced togethe and without any change in its shape. () System: A collection of any numbe of paticles inteacting with one anothe and ae unde consideation duing analysis of a situation ae said to fom a system. () Intenal foces: All the foces exeted by vaious paticles of the system on one anothe ae called intenal foces. These foces ae alone enable the paticles to fom a well-defined system. Intenal foces between two paticles ae mutual (equal and opposite). (4) Extenal foces: To move o stop an object of finite size, we have to apply a foce on the object fom outside. This foce exeted on a given system is called an extenal foce.

5 . Cente of Mass. Cente of mass of a system (body) is a point that moves as though all the mass wee concentated thee and all extenal foces wee applied thee. () Position vecto of cente of mass fo n paticle system: If a system consists of n paticles of masses m, m, m... mn, whose positions vectos ae,, n... espectively then position vecto of cente of mass m m m m m m... m... m Hence the cente of mass of n paticles is a weighted aveage of the position vectos of n paticles maing up the system. n n n y m C.M. m m x m m () Position vecto of cente of mass fo two paticle system: m m and the cente of mass lies between the paticles on the line joining them. If two masses ae equal i.e. m m () Impotant points about cente of mass, then position vecto of cente of mass (i) The position of cente of mass is independent of the co-odinate system chosen. (ii) The position of cente of mass depends upon the shape of the body and distibution of mass. Example: The cente of mass of a cicula disc is within the mateial of the body while that of a cicula ing is outside the mateial of the body. (iii) In symmetical bodies in which the distibution of mass is homogenous, the cente of mass coincides with the geometical cente o cente of symmety of the body. (iv) Position of cente of mass fo diffeent bodies 4

6 S.No. Body Position of cente of mass (a) Unifom hollow sphee Cente of sphee (b) Unifom solid sphee Cente of sphee (c) Unifom cicula ing Cente of ing (d) Unifom cicula disc Cente of disc (e) Unifom od Cente of od (f) A plane lamina (Squae, ectangle, Paallelogam) Point of inte section of diagonals (g) Tiangula plane lamina Point of inte section of medians (h) ectangula o cubical bloc Points of inte section of diagonals (i) Hollow cylinde Middle point of the axis of cylinde (j) Solid cylinde Middle point of the axis of cylinde () Cone o pyamid On the axis of the cone at point h distance 4 h is the height of cone fom the vetex whee (v) The cente of mass changes its position only unde the tanslatoy motion. Thee is no effect of otatoy motion on cente of mass of the body. (vi) If the oigin is at the cente of mass, then the sum of the moments of the masses of the system about the cente of mass is zeo i.e. (vii) If a system of paticles of masses m v Then the velocity of cente of mass cm m m i i 0., m,,... move with velocities v m iv m i i., v,,... v (viii) If a system of paticles of masses m, m, m,... move with acceleations a, a, a,... Then the acceleation of cente of mass A cm m ia m i i

7 6 (ix) If is a position vecto of cente of mass of a system Then velocity of cente of mass m m m m m m dt d dt d v cm (x) Acceleation of cente of mass m m m m m dt d dt d dt v d A cm cm (xi) Foce on a igid body dt d M M A F cm (xii) Fo an isolated system extenal foce on the body is zeo 0 v cm dt d M F constant cm v. i.e., cente of mass of an isolated system moves with unifom velocity along a staight-line path.

8 . Angula Displacement. It is the angle descibed by the position vecto about the axis of otation. Linea displaceme nt ( s) ( ) Angula displacement adius ( ) () Unit: adian () Dimension: [ M L T ] () Vecto fom S i.e., angula displacement is a vecto quantity whose diection is given by ight hand ule. It is also nown as axial vecto. Fo anti-clocwise sense of otation diection of is pependicula to the plane, outwad and along the axis of otation and vice-vesa. (4) adian 60 evolution. () If a body otates about a fixed axis then all the paticles will have same angula displacement (although linea displacement will diffe fom paticle to paticle in accodance with the distance of paticles fom the axis of otation). Q S P 4. Angula Velocity. The angula displacement pe unit time is defined as angula velocity. If a paticle moves fom P to Q in time t, t d lim t0 () Instantaneous angula velocity t dt whee is the angula displacement. total angula displaceme nt av () Aveage angula velocity total time t t () Unit: adian/sec 0 0 (4) Dimension: [ M L T ] which is same as that of fequency. Q P 7

9 () Vecto fom v [whee v = linea velocity, = adius vecto] Is an axial vecto, whose diection is nomal to the otational plane and its diection is given by ight hand scew ule. n (6) T [whee T = time peiod, n = fequency] (7) The magnitude of an angula velocity is called the angula speed which is also epesented by.. Angula Acceleation. The ate of change of angula velocity is defined as angula acceleation. If paticle has angula velocity at time t and angula velocity at time t Angula acceleation t t d d lim t0 () Instantaneous angula acceleation t dt dt. () Unit: ad /sec 0 0 () Dimension: [ M L T ]. (4) If 0, cicula o otational motion is said to be unifom. av () Aveage angula acceleation t t. (6) elation between angula acceleation and linea acceleation a. then, (7) It is an axial vecto whose diection is along the change in diection of angula velocity i.e. nomal to the otational plane, outwad o inwad along the axis of otation (depends upon the sense of otation). 8

10 6. Equations of Linea Motion and otational Motion. Linea Motion If linea acceleation is 0, u = constant and s = u t. otational Motion If angula acceleation is 0, = constant and t If linea acceleation a = constant, If angula acceleation = constant then (i) ( u v) s t (i) ( ) t v u a (ii) t (ii) t (iii) v u at (iii) t (iv) s ut at (iv) t t (v) v u as (v) (vi) s nth u a(n ) nth (n ) (vi) If acceleation is not constant, the above equation will not be applicable. In this case (i) (ii) dx v dt a dv dt d x dt vdv a ds (iii) If acceleation is not constant, the above equation will not be applicable. In this case (i) d dt d d (ii) dt dt d d (iii) 9

11 7. Moment of Inetia. Moment of inetia plays the same ole in otational motion as mass plays in linea motion. It is the popety of a body due to which it opposes any change in its state of est o of unifom otation. () Moment of inetia of a paticle I m ; whee is the pependicula distance of paticle fom otational axis. () Moment of inetia of a body made up of numbe of paticles (discete distibution) I m m m... () Moment of inetia of a continuous distibution of mass, teating the element of mass dm at position as paticle di dm i.e. I dm m m m m dm T 0 (4) Dimension: [ ML ] () S.I. unit: gm. (6) Moment of inetia depends on mass, distibution of mass and on the position of axis of otation. (7) Moment of inetia does not depend on angula velocity, angula acceleation, toque, angula momentum and otational inetic enegy. (8) It is not a vecto as diection (clocwise o anti-clocwise) is not to be specified and also not a scala as it has diffeent values in diffeent diections. Actually it is a tenso quantity. (9) In case of a hollow and solid body of same mass, adius and shape fo a given axis, moment of inetia of hollow body is geate than that fo the solid body because it depends upon the mass distibution. 0

12 8. adius of Gyation. adius of gyation of a body about a given axis is the pependicula distance of a point fom the axis, whee if whole mass of the body wee concentated, the body shall have the same moment of inetia as it has with the actual distibution of mass. When squae of adius of gyation is multiplied with the mass of the body gives the moment of inetia of the body about the given axis. I M O M Hee is called adius of gyation. Fom the fomula of discete distibution I. m m m 4 m m I m m m... mn If m = m = m =... = m then I m (... n )...(i) Fom the definition of adius of gyation, I M...(ii) M By equating (i) and (ii) M m(... n ) nm m(... n ) [As M nm ]... n n Hence adius of gyation of a body about a given axis is equal to oot mean squae distance of the constituent paticles of the body fom the given axis. () adius of gyation () depends on shape and size of the body, position and configuation of the axis of otation, distibution of mass of the body w..t. the axis of otation. () adius of gyation () does not depends on the mass of body. () Dimension [ M L T ]. (4) S.I. unit: Mete. 0 0 () Significance of adius of gyation: Though this concept a eal body (paticulaly iegula) is eplaced by a point mass fo dealing its otational motion.

13 Example: In case of a disc otating about an axis though its cente of mass and pependicula to its plane I M ( ) M M So instead of disc we can assume a point mass M at a distance ( / ) fom the axis of otation fo dealing the otational motion of the disc. Note: Fo a given body inetia is constant wheeas moment of inetia is vaiable. 9. Theoem of Paallel Axes. Moment of inetia of a body about a given axis I is equal to the sum of moment of inetia of the body about an axis paallel to given axis and passing though cente of mass of the body I g and Ma whee M is the mass of the body and a is the pependicula distance between the two axes. I IG I I g Ma a G Example: Moment of inetia of a disc about an axis though its cente and M pependicula to the plane is, so moment of inetia about an axis though its tangent and pependicula to the plane will be I I g Ma I M M I IG G I M

14 0. Theoem of Pependicula Axes. Accoding to this theoem the sum of moment of inetia of a plane lamina about two mutually pependicula axes lying in its plane is equal to its moment of inetia about an axis pependicula to the plane of lamina and passing though the point of intesection of fist two axes. I z I x I y Z X Y Example: Moment of inetia of a disc about an axis though its cente of mass and M pependicula to its plane is, so if the disc is in x y plane then by theoem of pependicula axes i.e. I z M I D ID I x I y I [As ing is symmetical body so x M 4 Note: In case of symmetical two-dimensional bodies as moment of inetia fo all axes passing though the cente of mass and in the plane of body will be same so the two axes in the plane of body need not be pependicula to each othe. I y I D ] ID Y O Z ID X

15 . Moment of Inetia of Two Point Masses about Thei Cente of Mass. Let m and m be two masses distant fom each-othe and and of thei cente of mass fom m and m () () m m m and () m m m m m espectively, then be the distances m Cente of mass m (4) I m m m m I m m () m.] (6) In diatomic molecules lie is deived fom above fomula. mm [whee m m is nown as educed mass m and H, HCl etc. moment of inetia about thei cente of mass 4

16 . Analogy between Tanslatoy Motion and otational Motion. Tanslatoy motion otatoy motion Mass (m) Moment of Inetia (I) Linea momentum P mv P me Angula Momentum L I L IE Foce F ma Toque I Kinetic enegy E mv P E m E I L E I

17 . Moment of Inetia of Some Standad Bodies about Diffeent Axes. Body Axis of otation Figue Moment of inetia / ing About an axis passing though C.G. and pependicula to its plane M ing About its diamete M ing About a tangential axis in its own plane M ing About a tangential axis pependicula to its own plane M 6

18 Body Axis of otation Figue Moment of inetia / Disc About an axis passing though C.G. and pependicula to its M plane Disc About its Diamete M 4 4 Disc About a tangential axis in its own plane M 4 4 Disc About a tangential axis pependicula to its M own plane Annula disc inne adius = and oute adius = Passing though the cente and pependicula to the plane M [ ] 7

19 Body Axis of otation Figue Moment of inetia / Annula disc Diamete M [ 4 ] Annula disc Tangential and Paallel to the diamete M [ 4 ] Annula disc Tangential and pependicula to the M [ ] plane Solid cylinde About its own axis L M Solid cylinde Tangential (Geneato) M 8

20 Body Axis of otation Figue Moment of inetia / Solid cylinde About an axis passing though its C.G. and pependicula to its L M 4 L 4 own axis Solid cylinde About the diamete of one of faces of the cylinde L M 4 L 4 Cylindical shell About its own axis M Cylindical shell Tangential (Geneato) M 9

21 Body Axis of otation Figue Moment of inetia / Cylindical shell About an axis passing though its C.G. and pependicula to its own axis L L M Cylindical shell About the diamete of one of faces of the cylinde L L M Hollow cylinde with inne adius = and oute adius M Axis of cylinde ( ) = Hollow cylinde with inne adius = and oute adius M Tangential ( ) = 0

22 Body Axis of otation Figue Moment of inetia / Solid Sphee About its diametic axis M Solid sphee About a tangential axis 7 7 M 7 Spheical shell About its diametic axis M Spheical shell About a tangential axis M Hollow sphee of inne adius and oute adius About its diametic axis M

23 Body Axis of otation Figue Moment of inetia / Hollow sphee Tangential ( M[ ) ] M Long thin od About on axis passing though its cente of mass and pependicula L ML L to the od. Long thin od About an axis passing though its edge and pependicula to the L ML L od ectangula Passing though the lamina of length l and beadth b cente of mass and pependicula to the plane b l M [ l b ] Tangential ectangula lamina pependicula to the plane and at the mid- M [4l b ] point of beadth ectangula lamina Tangential pependicula to the plane and at the midpoint of length t b l ii iii i M [ l 4b ]

24 Body Axis of otation Figue Moment of inetia / ectangula paallelopipe d length l, beadth b, thicness t Passing though cente of mass and paallel to (i) Length (x) (ii) beadth (z) (iii) thicness (y) iii i ii M[ b t ] (i) M[ l t ] (ii) M[ b l ] (iii) (i) ectangula paallelepipe d length l, beath b, thicness t Tangential and paallel to (i) length (x) (ii) beadth (y) (iii) thicness(z) M [l (ii) M [ l (iii) b b t t ] ] M [ l b t ] Elliptical disc of semimajo axis = a and semimino Passing though CM and pependicula to the plane M [ a 4 b ] axis = b Solid cone of adius and height h Axis joining the vetex and cente of the base a a M 0 a

25 Body Axis of otation Figue Moment of inetia / Equilateal tiangula lamina with side a Passing though CM and pependicula to the plane Ma 6 ight angled tiangula lamina of sides a, b, c Along the edges Mb () 6 Ma () 6 () M a b 6 a b 4. Toque. If a pivoted, hinged o suspended body tends to otate unde the action of a foce, it is said to be acted upon by a toque. o the tuning effect of a foce about the axis of otation is called moment of foce o toque due to the foce. otation otation O P F O F (A) (B) 4

26 If the paticle otating in xy plane about the oigin unde the effect of foce F and at any instant the position vecto of the paticle is then, Y F sin F cos Toque = F sin F P [Whee is the angle between the diection of and F ] d 90o X () Toque is an axial vecto. i.e., its diection is always pependicula to the plane containing vecto and F in accodance with ight hand scew ule. Fo a given figue the sense of otation is anti-clocwise so the diection of toque is pependicula to the plane, outwad though the axis of otation. () ectangula components of foce As o F F cos adial component of foce, F F sin tansvese component of foce F sin F Thus the magnitude of toque is given by the poduct of tansvese component of foce and its pependicula distance fom the axis of otation i.e., Toque is due to tansvese component of foce only. () As F sin o F( sin ) Fd [As d sin fom the figue ] Toque is also called as moment of foce and d is called moment o leve am. (4) Maximum and minimum toque: As F o F sin maximum F When sin max minimum 0 When sin min 0. 0 o 80 i.e., 90 F is pependicula to i.e F is collinea to

27 Toque () Fo a given foce and angle, magnitude of toque depends on. The moe is the value of, the moe will be the toque and easie to otate the body. Example: (i) Handles ae povided nea the fee edge of the Planc of the doo. (ii) The handle of scew dive is taen thic. (iii) In villages handle of floumill is placed nea the cicumfeence. (iv) The handle of hand-pump is ept long. (v) The am of wench used fo opening the tap, is ept long. (6) Unit: Newton-mete (M.K.S.) and Dyne-cm (C.G.S.) (7) Dimension: [ ML T ]. (8) If a body is acted upon by moe than one foce, the total toque is the vecto sum of each toque.... (9) A body is said to be in otational equilibium if esultant toque acting on it is zeo i.e. 0. (0) In case of beam balance o see-saw the system will be in otational equilibium if, 0 o F l F l 0 F l F l Howeve if, L.H.S. will move downwads and if..h.s. will move downwad. and the system will not be in otational equilibium. F l l F () On tilting, a body will estoe its initial position due to toque of weight about the point O till the line of action of weight passes though its base on tilting, a body will topple due to toque of weight about O, if the line of action of weight does not pass though the base. G Tilt G Tilt Toque O O W W 6

28 () Toque is the cause of otatoy motion and in otational motion it plays same ole as foce plays in tanslatoy motion i.e., toque is otational analogue of foce. This all is evident fom the following coespondences between otatoy and tanslatoy motion. otatoy Motion Tanslatoy Motion I F m a W d W F ds P P F v dl dp F dt dt. Couple. A special combination of foces even when the entie body is fee to move can otate it. This combination of foces is called a couple. () A couple is defined as combination of two equal but oppositely diected foce not acting along the same line. The effect of couple is nown by its moment of couple o toque by a couple τ F. F F () Geneally both couple and toque cay equal meaning. The basic diffeence between toque and couple is the fact that in case of couple both the foces ae extenally applied while in case of toque one foce is extenally applied and the othe is eactionay. () Wo done by toque in twisting the wiew C. Whee C ; C is nown as twisting coefficient o couple pe unit twist. 7

29 6. Tanslatoy and otatoy Equilibium. Foces ae equal and act along the same line. F F F 0 and 0 Body will emain stationay if initially it was at est. Foces ae equal and F F 0 and otation i.e. spinning. does not act along the 0 same line. F Foces ae unequal and F 0 and Tanslation i.e. slipping o act along the same line. F F 0 sidding. Foces ae unequal and F F 0 and otation and tanslation does not act along the 0 both i.e. olling. same line. F 7. Angula Momentum. The tuning momentum of paticle about the axis of otation is called the angula momentum of the paticle. O The moment of linea momentum of a body with espect to any axis of otation is nown as angula momentum. If P is the linea momentum of paticle and its position vecto fom the point of otation then angula momentum. L P L P sin nˆ L Angula momentum is an axial vecto i.e. always diected pependicula to the plane of otation and along the axis of otation. 8

30 () S.I. Unit: g-m -s o J-sec. T - () Dimension: [ ML ] and it is simila to Planc s constant (h). () In Catesian co-odinates if xi ˆ yj ˆ zˆ and P P ˆi P ˆj P ˆ x y z Then L P ˆi x P x ˆj y P y ˆ z P z = (yp zp )ˆi ( xp zp )ˆj ( xp yp ) ˆ z y z x y x (4) As it is clea fom the figue adial component of momentum P P cos Tansvese component of momentum So magnitude of angula momentum P P sin L P sin Y P sin P cos L P P d X i.e., The adial component of linea momentum has no ole to play in angula momentum. () Magnitude of angula momentum L P ( sin ) L Pd [As d sin fom the figue.] Angula momentum = (Linea momentum) (Pependicula distance of line of action of foce fom the axis of otation) (6) Maximum and minimum angula momentum: We now L P L m[ v] m v sin P sin [As P m v ] L m mv aximum When sin max L minimum 0 When sin min 0. 0 o 80 i.e., 90 v is pependicula to i.e v is paallel o anti-paallel to 9

31 (7) A paticle in tanslatoy motion always have an angula momentum unless it is a point on the line of motion because L mv sin and L if o 0 o o 80 (8) In case of cicula motion, L P m( v) = mv sin o In vecto fom (9) Fom L mv m [As L I [As m = I] L I d L d L I I = dt dt d I [As v andv ] dt and I ] I.e. the ate of change of angula momentum is equal to the net toque acting on the paticle. [otational analogue of Newton's second law] (0) If a lage toque acts on a paticle fo a small time then 'angula impulse' of toque is given by J dt av t t dt o Angula impulse J av t L Angula impulse = Change in angula momentum () The angula momentum of a system of paticles is equal to the vecto sum of angula momentum n. of each paticle i.e., L L L L... L () Accoding to Boh Theoy angula momentum of an electon in n th obit of atom can be taen as, h L n [Whee n is an intege used fo numbe of obit] 0

32 8. Law of Consevation of Angula Momentum. Newton s second law fo otational motion d L dt d L So if the net extenal toque on a paticle (o system) is zeo then 0 dt i.e. L L L L... = constant. Angula momentum of a system (may be paticle o body) emains constant if esultant toque acting on it zeo. As L I so if 0 then I constant I Since angula momentum I emains constant so when I deceases, angula velocity inceases and vice-vesa. Examples of law of consevation of angula momentum: () The angula velocity of evolution of a planet aound the sun in an elliptical obit inceases when the planet come close to the sun and vice-vesa because when planet comes close to the sun, its moment of inetia I deceases theefoe inceases. () A cicus acobat pefoms feats involving spin by binging his ams and legs close to his body o vice-vesa. On binging the ams and legs close to body, his moment of inetia I deceases. Hence inceases. () A peson-caying heavy weight in his hands and standing on a otating platfom can change the speed of platfom. When the peson suddenly folds his ams. Its moment of inetia deceases and in accodance the angula speed inceases.

33 (4) A dive pefoms somesaults by Jumping fom a high diving boad eeping his legs and ams out stetched fist and then culing his body. () Effect of change in adius of eath on its time peiod Angula momentum of the eath L L M T I constant constant T [If M emains constant] 4 If becomes half then time peiod will become one-fouth i.e. 6hs Wo, Enegy and Powe fo otating Body. () Wo: If the body is initially at est and angula displacement is d due to toque then wo done on the body. W d [Analogue to wo in tanslatoy motionw F dx ] () Kinetic enegy: The enegy, which a body has by vitue of its otational motion is called otational inetic enegy. A body otating about a fixed axis possesses inetic enegy because its constituent paticles ae in motion, even though the body as a whole emains in place. otational inetic enegy Analogue to tanslatoy inetic enegy K I KT mv K L K T Pv L K I P K T m

34 () Powe: ate of change of inetic enegy is defined as powe P In vecto fom Powe d dt d d ( K ) I I dt I I dt [Analogue to powe in tanslatoy motion P F v ] 0. Slipping, Spinning and olling. () Slipping: When the body slides on a suface without otation then its motion is called slipping motion. In this condition fiction between the body and suface F 0. Body possess only tanslatoy inetic enegy K T mv. Example: Motion of a ball on a fictionless suface. = 0 v () Spinning: When the body otates in such a manne that its axis of otation does not move then its motion is called spinning motion. In this condition axis of otation of a body is fixed. Example: Motion of blades of a fan. In spinning, body possess only otatoy inetic enegy K I. o K mk v mv K i.e., otatoy inetic enegy = K times tanslatoy inetic enegy. K K Hee is a constant fo diffeent bodies. Value of sphee) (ing), K (disc) and K (solid

35 () olling: If in case of otational motion of a body about a fixed axis, the axis of otation also moves, the motion is called combined tanslatoy and otatoy. Example: (i) Motion of a wheel of cycle on a oad. (ii) Motion of football olling on a suface. In this condition fiction between the body and suface F 0. Body possesses both tanslational and otational inetic enegy. Net inetic enegy = (Tanslatoy + otatoy) inetic enegy. K N K T K K K N mv mv I mv mv K v. olling Without Slipping. In case of combined tanslatoy and otatoy motion if the object olls acoss a suface in such a way that thee is no elative motion of object and suface at the point of contact, the motion is called olling without slipping. Fiction is esponsible fo this type of motion but wo done o dissipation of enegy against fiction is zeo as thee is no elative motion between body and suface at the point of contact. olling motion of a body may be teated as a pue otation about an axis though point of contact with same angula velocity. By the law of consevation of enegy K N mv I [ Asv ] m I = [ m I] = [ I m ] IP [As I P I m ] By theoem of paallel axis, whee I = moment of inetia of olling body about its cente O and I P = moment of inetia of olling body about point of contact P. O P v 4

36 () Linea velocity of diffeent points in olling: In case of olling, all points of a igid body have same angula speed but diffeent linea speed. Let A, B, C and D ae fou points then thei velocities ae shown in the following figue. B v B v B v C v D v = 0 v + C = D C D A v v A v v = 0 v Tanslation otation olling () Enegy distibution table fo diffeent olling bodies: Body K Tanslato y (K T) mv otato y (K ) mv K Total (K N) mv K K K T N (%) K (%) K N ing Cylindica l shell Disc solid cylinde Solid sphee Hollow sphee mv mv mv mv mv mv 4 4 (66.6%) mv 7 mv mv 7 0 (7.%) (0%) (0%) (.%) 7 (8.%) mv mv mv (60%) (40%) 6

37 . olling on an Inclined Plane. When a body of mass m and adius olls down on inclined plane of height h and angle of inclination, it loses potential enegy. Howeve it acquies both linea and angula speeds and hence, gain inetic enegy of tanslation and that of otation. By consevation of mechanical enegy mgh () Velocity at the lowest point: v gh mv () Acceleation in motion: Fom equation v u as By substituting u 0, S g sin a h and sin v gh () Time of descent: Fom equation v u at we get By substituting u = 0 and value of v and a fom above expessions t sin h g Tanslation otation S B h C Fom the above expessions it is clea that, v Note: Hee facto ; a ; t is a measue of moment of inetia of a body and its value is constant fo given shape of the body and it does not depend on the mass and adius of a body. Velocity, acceleation and time of descent (fo a given inclined plane) all depends on of inetia of the olling body lesse will be the value of lesse will be the time of descent.. Lesse the moment. So geate will be its velocity and acceleation and If a solid and hollow body of same shape ae allowed to oll down on inclined plane then as S H, solid body will each the bottom fist with geate velocity. 6

38 If a ing, cylinde, disc and sphee uns a ace by olling on an inclined plane then as while least velocity. ing sphee minimum maximum, the sphee will each the bottom fist with geatest velocity while ing at last with Angle of inclination has no effect on velocity, but time of descent and acceleation depends on it. Velocity, time of decent and acceleation.. olling Sliding and Falling of a Body. Figue Velocity Acceleation Time olling 0 h gh g sin K sin h g Sliding 0 gh g sin sin h g Falling 0 = 90 o gh g h g 7

39 4. Velocity, Acceleation and Time fo Diffeent Bodies. Body v Velocity gh Acceleation gsin θ a Time of descent t sin θ h g ing o Hollow cylinde Disc o solid cylinde Solid sphee Hollow sphee gh g 4 sin gh sin o o o gh gh gh g sin gh sin g 4 h sin 7 sin g g 0 h sin sin g. Motion of Connected Mass. A point mass is tied to one end of a sting which is wound ound the solid body [cylinde, pulley, disc]. When the mass is eleased, it falls vetically downwads and the solid body otates unwinding the sting m = mass of point-mass, M = mass of a igid body = adius of a igid body, I = moment of inetia of otating body () Downwads acceleation of point mass () Tension in sting () Velocity of point mass T I mg I m v (4) Angula velocity of igid body gh I m mgh I m g a I m T m h 8

40 6. Time Peiod of Compound Pendulum Time peiod of compound pendulum is given by, Hee l = distance of cente of mass fom point of suspension L l T whee L g = adius of gyation about the paallel axis passing though cente of mass. l Body Axis of otation Figue l L l l T π L g ing Tangent passing though the im and pependicula to the plane Tangent paallel to the plane T T g g Tangent, Pependicula to plane T g Disc Tangent paallel to the plane 4 4 T 4 g Spheical shell Tangent T g Solid sphee Tangent 7 T 7 g 9

41 40