Theory of Rotation. Chapter 8+9 of the textbook Elementary Physics 2. This is an article from my home page:

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1 Theory of Rotton Chpter 8+9 of the textbook Eeentry Physcs Ths s n rtce fro y hoe pge: Oe Wtt-Hnsen 977 (6)

2 Acknowedgent Ths pper s dgtzed trnston fro Dnsh of chpter n textbook on physcs wrtten n 976, tht ws conceved nd wrtten for the second yer of the Dnsh 3-yer hgh schoo (ced the gynsu, for 6-9 yer-ods ). Reredng t, hve found tht t s st n eeentry yet rgorous presentton of the theory of rotton, nd t covers ost of the spects of the theory. Therefore t cn st be used s n undergrdute text on the subject of echncs. Aso hve not found sr strght nd thorough presentton on the nternet, free fro gtterng pctures nd wth the conteporry c for usng educton thetc progrs, nsted of dong the nytc ccutons by hnd. n the st yers, however, there hs n Derk eerged rsng dend for ths knd of textbook of physcs, sody founded on thetcs, nd where the bsc understndng s the ssue. Whether t s so the cse n other prts of the word, do not know, but cn gther t fro the htherto nterest n y hoe pge: Even n 97 s t ws soewht bove the stndrd eve both n thetcs nd n bstrct coprehenson for use n the second yer of techng physcs, nd tody t woud be entrey out of queston to try to use t n the Dnsh gynsu.

3 Contents. Cross product of two vectors...5. Torque. Moent of force Angur oentu. The equton of rotton oton for prtce Are-veocty nd Keper s second w Torque nd ngur oentu for syste of prtces Torque nd ngur oentu wth respect to n xs of rotton Rotton bout n xs. Moent of nert Rotton energy. Ccuton of oents of nert Theores concernng the centre of ss, oent of nert nd rotton energy Centre of ss Stener s theore Köng s theore Centre of grvty equs centre of ss The physc penduu...39

4 Torque nd ngur oentu 5. Cross product of two vectors n order to defne the physc qunttes Angur Moentu nd Torque, t s necessry to ntroduce the concept of cross product fro vector gebr. f nd b re two vectors n spce the cross product (or the vector product) b s defned s vector, whch hs ength: b b sn v, (where v s the nuerc est nge between the vectors nd b ), nd where the drecton s perpendcur to s we s b, so tht (,b, b ) for rght hnd screw. The tter prt of the defnton requres perhps bt of expnton. There re evdenty two vectors wth the gven ength whch re perpendcur to nd b. To dstngush between the the rght hnd screw rue s nvented. Wth your rght hnd nd your fngers pontng n the drecton of you ke twst to b n the postve drecton. The drecton of b sh then be n the drecton of your thub. You shoud notce tht, wth the gven defnton then the ength of the cross product: b s equ to the nuerc vue of the deternnt of the pr of vectors (, b ), snce det(, b) b sn v, nd fro ths, t foows tht b s the re of the preogr spnned by the two vectors nd b. Fro the defnton foows tht: b when b, ( s pre tob ). Fro the rght hnd rue, t further foows tht the cross product chnges sgn, when nd b re peruted, (s t s so the cse for the deternnt). Fny we enton tht the dstrbutve w s vd for the cross product, such tht c ( b) c c b (The order of the fctors preserved). Torque. Moent of force n the fgure s shown prtce wth ss, where ts poston s gven by the poston vector r OP. The prtce s ffected by the force F. We dssove the force n coponent F pre to r, nd coponent F perpendcur to r. Cery F w seek to drw the prtce rdy wy fro the pont O, whe F seeks to turn the prtce round n xs through O. How uch the prtce w be turned depends of course of F, but so on r r. Such consdertons ed to the concept of torque, so denoted s the oent of force.

5 Torque nd ngur oentu 6 The torque H, wth respect to the pont O (n two denson context), s defned by the expresson: (.) H rf or H rf sn Where φ s the nuercy est nge between the poston vector r nd the force F. Thus: F F cos nd F sn F The torque s vector. Tht the torque ust hve drecton, you y understnd, when you hve n nd tht to twst there wys beong n xs. The drecton of the torque H shoud therefore be ong tht xs. A twst nduces, however, so n orentton, correspondng to eft turn or rght turn. f we express the torque s cross product of vectors, t copes wth these propertes. The torque H, wth respect to the orgn O, on prtce, hvng the poston vector ffected by the force F s gven by the cross product of r wth F. (.3) H r F r OP, nd f r s ced the r, then the torque cn be oosey foruted s: Force tes r. Wth the defnton (.3), nd the defnton of the cross product, one cn reze tht the torque gets the rght vue, ccordng to (.), s we s the correct drecton, snce t s drected ong the xs of the twst, s copyng wth the rght hnd rue. Fro (.) we hve seen tht the torque cn be found s the r r tes F, the coponent of the force perpendcur to r. But the torque y s we be found s the force F tes r beng the coponent of the r perpendcur to the drecton of the force, s shown n fgure (.). vng the condton for equbru: H r g r g, The fgure to the rght shows ever, beng be to tp round O, nd two weghts, tht cn be suspended t dfferent dstnces fro the xs of rotton of the ever. n the present postons of the weghts nd, the torque wth respect to O s zero. H O H H r g r g As the fgure shows, both torques re drected ong the se xs of rotton, but hvng opposte drectons. So we hve H =. Expressng tht the r tes the force, shoud be the se on both sdes of the ttng pont. Ths s (n ore popur context) known s the ever rue.

6 Torque nd ngur oentu 7.6 Expe. Equbru condton for box on sope f the ever s tted, hvng n nge φ wth the horzont, the vector expresson of the torque ren the se s before. H O H H r g r g And tkng the ength: H O r g sn r g sn t s seen, tht the condton for equbru rens the se. f r g sn r g sn, so tht the ever s unbnced, t cn ony be n bnce wth the ever n vertc poston, snce then: H H, becuse r g nd r g The fgure shows two boxes pced on sope, hvng frcton tht prevents the n sdng down the sope. An observer w (by experence) probby c tht the one box w ren where t s, but tht the second box w be overthrown. We sh now ppy the defnton of the torques ctng on the boxes to deterne the condton tht the box tts or not. The grvty s ctng t, the centre of ss (geoetrc centre) of the box. At the frst box t s seen fro the fgure tht the torque fro grvty, H O g w turn the box counter cockwse, towrds the undery, whe the se torque on the second box, w turn the box cockwse nd overthrow t. The t between the two cses, s of course when H O g O g. n ths cse t s seen, tht the ncnton of the sope s equ to the nge between O nd the front sde of the box. Ths nge y however be ccuted fro: tn b /. We thus fnd tht the condton tht the box s overthrown s tht: tn b /. 3. Angur oentu. The equton of rotton oton for prtce We sh now proceed to defne new fundent concept retng to the theory of rottons, ney ngur oentu, the portnce of whch w be crfed n the foowng. n the fgure s shown the trjectory for prtce wth ss. The prtce s deterned fro the poston vector O, s we s ts veocty v. Fro the oentu vector p v, we now defne the ngur oentu vector s:

7 Torque nd ngur oentu 8 (3.) L r p or L r v The veocty v of the prtce cn be resoved n two coponents, one v pre wth r, nd one v perpendcur to r. f φ denotes the est nge between r nd v, we hve: v vcos nd v vsn. Then we re be to express the ength of the ngur oentu. (3.3) L rp sn vr sn or L rp v r Fro the defnton: L s vector, whch s perpendcur to r nd to v, nd we notce tht (3.) L r p v v On the other hnd, f r p then L = rp =vr. Sr to Newton s second w for the oton of prtce, we sh deonstrte tht, regrdng rotton correspondng w s vd, f the force F s repced by the torque H, nd the oentu p s repced wth the ngur oentu L. When dfferenttng the cross product, the se rues ppy, s to dfferenttng the product of two functons, provded tht the orders of the fctors re kept. dl d (3.3) r p p r r F H dr The frst of the two ters becoes zero, becuse (3.4) dl H dp dr v nd p dr v, so tht p, nd thus: Especy t ppes tht, when the torque s zero, the ngur oentu s constnt. dl (3.5) H L r p constnt vector The content of equton (3.7) s of equ portnce s the conservton of oentu nd s ced: The conservton of ngur oentu for prtce. We notce further tht for r ppes: H r F F r F The ngur oentu s constnt, ether f the force on the prtce s zero or the force s drected rdy.

8 Torque nd ngur oentu 9 4. Are-veocty nd Keper s second w The fgure shows prtce, whch perfors oton n the x y pne. We wsh to deterne n expresson for the reveocty, tht s, the re swept by the prtce per te unt. The re da tht the poston vector sweeps n the te, s equ to hf the re of the preogr expnded by the vectors r r (t) nd r dr r ( t ). da det( r, r dr) ( r ( r d r ) da ( r r r dr ) r dr ( r r ) da dr (4.) r r v (= Are veocty) Fro (4.) t ppers ntur to nvent n re veocty vector, by: d A (4.3) r v da where r v s the re veocty. Hodng (4.3) together wth the defnton of the ngur oentu L r v, we see tht da L, fro whch foows the gnfcent theore: f the torque H r F, ether becuse H =, or becuse the prtce s ovng n centr fed, such tht r F, then the ngur oentu L s constnt vector, nd the prtce perfors pne oton wth constnt re-veocty. For the oton of prtce en centr fed e.g. the oton of pnet round the sun, the grvty force F s wys drected opposte to the poston vector r OP, nd therefore, (s deonstrted bove) H r F, nd the ngur oentu of the prtce (the pnet) s constnt, nd the prtce w perfor oton wth constnt re-veocty. Ths s the content of the second w of Keper, whch then trvy foows fro conservton of ngur oentu. 5. Torque nd ngur oentu for syste of prtces For syste of prtces or sod body, the ngur oentu s defned s the vector su of the ngur oentu of the ndvdu prtces. (5.) L L r p r s the poston vector for the th prtce, hvng the oentu p. n the se nner the torque s s the vector su of torques ctng on the ndvdu prtces. f F denotes the torque on the th prtce the tot torque s: (5.) H H r F

9 Torque nd ngur oentu 3 As we hve shown when we ccuted the resutng force on syste of prtces, t s ony the extern torques tht devers contrbuton to the resutng torque. Let F j be the force on whch the 'th prtce cts on the j th prtce. r j beng the vector fro (j) to (). Wth the notton n the fgure we then hve r r r. The torque on whch the j th prtce cts on the j j 'th prtce s then ccuted s: (5.3) H j r F j ( rj rj ) ( Fj ) rj Fj rj Fj We hve pped Newton s 3. w F j Fj, nd the st ter becoes zero becuse r j Fj. The net resut s: (5.4) H j r F j rj Fj H j The ters n the su (5.), cong fro ntern forces thus cnce ech other n prs, nd ony the extern forces gves contrbuton to the resutng torque. We re then be to forute the equton of oton for rotton for syste of prtces: dl (5.5) H ext H d dl L The reton (5.5) expresses the fct tht the extern force s equ to the dfferent quotent of the tot ngur oentu. Especy: dl (5.6) H ext L L constnt vector Ths s the contents of the w of conservton of ngur oentu for syste of prtces. For n soted syste of prtces the ngur oentu of the syste s conserved. One shoud ephsze the vector chrcter of the ngur oentu. t s rther obvous tht there ust be torque to ntte the rotton of body, but perhps ess obvous tht t so needs torque to chnge the drecton of the rotton xs. However, ths s n fu ccordnce wth the fct, tht force s needed to chnge the drecton of the oentu, even f ts nuerc vue s unchnged. (For expe the centrpet force n unfor crcur oton). Experence tes us, however the forer, for expe when rdng bke. The ngur oentu of the whees s drected ong the xs of rotton, nd t heps us to keep the bnce. To keep the bnce on bke t rest for onger perod, s sk reserved for rtsts n crcus. Expe 5.7 The fgure shows n expe of ngur oentu conservton. The person, hodng hevy rottng whee, s stndng on dsc, beng be to rotte wthout frcton ong vertc xs, such tht there s no extern torque ong ths xs. When the person chnges the xs of the rottng whee fro horzont to vertc, he ust suppy torque (drected ong vertc xs, not horzont one), nd fro Newton s w of cton nd recton, he w receve n opposte torque, whch kes h turn the opposte wy of the whee. The ngur veocty of the whee w not chnge, but hs rotton w hve the effect, tht the ngur oentu s conserved to zero ong the vertc xs.

10 Torque nd ngur oentu 3 6. Torque nd ngur oentu wth respect to n xs of rotton When rgd body s fxed to rotte ong n xs, t s ost sutbe to defne the torque nd ngur oentu wth respect to tht xs of rotton. n the fgure the rgd body rottes ong n xs. For n rbtrry ss eeent of the body, ts oentu: p v s perpendcur to the xs. Let L nd H be the ngur oentu nd the torque of the rottng body wth respect to pont O on the xs. f e denotes unt vector ong the xs of rotton, the coponents of L nd H ong the xs y be wrtten s L L e nd H H e. Furtherore et be the poston vector fro O to the ss eeent, nd et r be the dstnce fro to the xs. Wth the nottons n the fgure t foows tht: r, where s drected ong the xs. We sh then evute L, the ngur oentu wth respect to the xs. (6.) L ( p ) e ( ( r p p ) e ( r p r p ) e rv ( r ) p ) e n the second equton the frst ter vnshes, becuse p nd therefore: p e. The second ter cn be wrtten s the product of the engths of the vectors, becuse r p nd therefore r p e. For the torque we hve sr stuton, snce t s ony forces F pre to v whch contrbutes to the rotton. The other coponents re cnceed by the rectve forces fro the xs. (6.) H ( r F ) e r F dl Fro Newton s second w for rotton: H t edtey foows, when utpyng ths equton by e, berng n nd tht e s constnt vector. (6.3) dl d H e e ( L e) So the equton of oton for rotton dl H rens vd, when the torque nd the ngur oentu s evuted wth respect to fxed xs. H dl

11 Rotton bout n xs. Moent of nert 3 7. Rotton bout n xs. Moent of nert When rottng bout fxed xs, s show n the fgure on the precedng pge, of the bodes prtces perfor crcur oton wth the se ngur veocty ω. nsertng v = ωr n the expresson (6.) for the ngur oentu wth respect to n xs, we fnd: (7.) L rv rr r The su: Where the su s extended to prts of the body s ced the oent of nert of the body wth respect to the xs. The S unt of the oent of nert s kg. The oent of nert depends ony on the ss, the shpe of the body, nd the poston of the xs wthn or wthout the body. So for rotton ong fxed xs the reton: L ppes. f the xs s understood, one cn drop the ndex. Further the rotton nge φ s often used nsted of ω, where ω = dφ/. So we y n gener wrte: r (7.) L or d L where r Usng ths notton the equton of oton for rotton tkes the for: (7.3) dl H or H d d or H The two tter expressons re ony vd for rgd body, where the oent of nert s constnt n te. f the torque wth respect to the xs s zero, the ngur oent s constnt. (7.4) H L const const For rgd body (7.4) ens tht t n ck of ny extern torque, t rottes wth constnt ngur veocty. However, the oent of nert cn be chnged by ntern forces nd snce the ngur oentu s constnt, ths w chnge the ngur veocty. For expe skte prncess preprng prouette, strts out wth oderte rotton speed wth her rs stretched, next when she pus her rs n, gnng ngur veocty becuse her oent of nert s dnshed. The fst rotton stops, s she pus out her rs (or even eg).

12 Rotton bout n xs. Moent of nert Expe The fgure shows test person who hs been persuded to tke set n chr. The set cn rotte freey bout vertc xs. The person s tod to hod two 5 kg weghts out n stretched rs t dstnce.75 nd (rther rresponsbe) the techer puts h n sow rotton. Fro sgn fro the techer he pus hs rs n, so tht the weghts re now hed t dstnce of.5, fro the xs (The techer ust be precutous, tht he w not get hurt, when he fs down fro the wdy rottng chr). usuy referred ths to: Lernng by dong, for the ess cever students. Assung tht the oent of nert of the person s kg, nd tht the nt rotton s round per sec, we sh ke n estte of wht hppens. The oents of nert cn be ccuted s: = person + weghts. before = kg + 5 (.75 ) = 7.6 kg. nd fter = kg + 5 (.5 ) =.6 kg. Accordng to the conservton of ngur oentu: before ω before = fter ω fter, fro whch we ccute ω fter. before fter before rd / sec 8.4 rd /sec. 9.6 fter rps 8. Rotton energy. Ccuton of oents of nert We sh then proceed to deterne the knetc energy of rgd body, rottng wth n ngur veocty ω ong fxed xs. For the prtce wth ss, hvng the dstnce r fro the xs of rotton, ts veocty s: v = ω r. t then foows: E E v r ) (8.) Where s the oent of nert ong the xs. kn rot E rot ( r The oent of nert hs the se sgnfcnce for rotton s the ss hs for body n trnston. The nogy between ner oton nd rotton bout fxed xs s n fct cose nd fr rechng. Beow s schee ustrtng the nogous concepts. Lner oton Rotton bout fxed xs Coordnte s Rotton nge φ Veocty ds Angur veocty d v Mss Moent of nert Moentu p= v Angur oentu L = ω Force dp Torque (Moent of force) dl F H Knetc energy E kn v Rotton energy E rot

13 Rotton bout n xs. Moent of nert 34 For ppyng the theory of rotton to the physc word t s of course pertve to be be to ccute the oents of nert of vrous rgd bodes. For extended bodes, the oent of nert y n soe cses be evuted by ntegrton. f the body s hoogenous wth densty ρ, the ss d, beng n the voue eeent: dv = dxdydz s gven by: d = ρ dxdydz. The oent of nert s then ccuted by ntegrton, when repcng the ss eeent by d, nd repcng suton by ntegrton. (8.3) r d r dv 8.4 Expe. The oent of nert of hoogenous rod We sh frst consder rod rottng bout n xs, whch s perpendcur to the rod t ts end pont. The rod hs the ength nd the ss. The ss pr unt ength s /, nd the ss octed t dx, n the dstnce x fro the xs s therefore: nert s then ccuted fro (8.3). d dx. The oent of 3 3 (8.4.) x d x dx x 3 3 f the xs goes through the centre (of ss) of the rod, the oent of nert s found, chngng the ts of ntegrton. 3 3 (8.4.) x dx 3 x 8.5 Expe. The oent of nert of whee wth ss ong the pereter, nd tht of dsc. We sh then ccute the oent of nert fro (bke) whee, where the xs s through the centre of the whee perpendcur to the whee. Snce of the ss s octed t r (the rdus of the whee), the oent of nert s spy: (8.5.) whee r r r We sh next ccute the oent of nert of dsc or cynder, where the xs s through the centre, perpendcur on the dsc. We do ths by dvdng the dsc nto rngs wth rdus r nd thckness dx. We then hve: d xdx r r xdx

14 Rotton bout n xs. Moent of nert 35 x dsc r x dx r 3 4 (8.5.) 4 x r r We sh then ccute the oent of nert of sod b wth respect to n xs through ts centre. The b s ssued to hve the ss nd the rdus r. The ccuton s done by dvdng the b n dscs wth rdus y nd thckness dx. f such dsc hs the ss d, the oent of nert of the dsc s d =½d y, s we found n the prevous expe. A subordnte theore fro eeentry pne geoetry sttes tht the squre of the heght n rght nge trnge s equ to the product of the sdes n whch t dvdes the bse ne. y =x(r x). We y then fnd n expresson for d expressed by x. 3 3 d dvdsc y dx y dx x(r x) dx r r r And we y perfor the ntegrton over x. 3 r r r 3 y d x ( r x dx 8 3 r ) r r 3 4 y d 8 4 r x dx x dx 4r 3 r r 3 x dx b 3 r r r 6 3 r 5 5 r 5 9. Theores concernng the centre of ss, oent of nert nd rotton energy The centre of ss s ctuy defned n nother chpter of the textbook n whch ths chpter beongs to, so sh repet t here. 9. Centre of ss The fgure shows syste of prtces or ssve body. The snge eeent of the syste s descrbed by the poston vector: OP r nd s the ss of the syste. We now defne (fctve) pont, by the equton: (9.) r O r s ced the centre of ss of the syste, nd t s of vt portnce, when sovng the equtons of oton for the syste. Often (for ssve bodes) we sh repce the suton by ntegrton. (9.) r rd r dv where r ( dx, dy, dz), ρ s the densty, nd dv = dxdydz s the voue eeent. The three coordntes of the centre of ss, re then found by:

15 Rotton bout n xs. Moent of nert 36 (9.5) x xdv y ydv z zdv f body hs syetry ne or syetry pne, the centre of ss ust e on tht ne or n tht pne. f the body hs two non pre syetry pnes, the centre of ss es n ther ntersectng ne. f the body hs three non pre syetry pnes, the centre of ss es n ther coon ntersectng pont. The centre of ss for box, therefore es the ntersecton of the cross dgons. The centre of ss of sphere es n ts centre. The centre of ss of trnge es n the coon ntersectng pont of the edns. 9.6 Expe. The centre of ss fro two bodes hvng dfferent sses (e.g. erth nd oon). ( O OP ) As n expe we sh deterne the poston of the centre of ss for syste consstng of two sses nd, pced n the postons P nd P Accordng to the defnton of the centre of ss we hve. O ( O OP ) ( OP OP ) ( O OP OP ) P P P P The ccuton bove shows, tht the centre of ss for syste consstng of two prtces es n ther connectng ne, nd dvdes ths ne n the nverse rto of the two sses. Ths resut cn for expe be pped on the syste consstng of the erth nd the oon. The rto M oon /M erth =.3, nd ther dstnce s roughy 6R, where R = R erth = 6.37 k. f r s the dstnce to the centre of ss fro the centre of the erth, then we hve: r M oon.3 r. 79R 6R r M erth The centre of ss es (surprsngy) wthn the surfce of the erth. As consequence of the utu ttrcton of the erth nd oon, both objects ove n eptc orbts, wth focus n ther coon centre of ss. However, becuse of Newtons 3. w: F = - F = -.The rton between ther cceertons s so the nverse rton between ther sses Stener s theore Becuse of the nce syetry propertes of the centre of ss (CM), t often dvntgeous to ccute the oent of nert wth respect to n xs gong the centre of ss. f the xs s not through the CM, then one y ppy Stener s theore: The oent of nert, wth respect to n xs, whch does not goes through the CM, cn be ccuted s the oent of nert ong n xs through the CM nd pre to the gven xs pus k, where k s the seprton between the two pre xs. (9.7) = + k

16 Rotton bout n xs. Moent of nert 37 Wth the nottons shown n the fgure bove we hve: k k k r ) ( k k k The st ter becoes zero for the foowng resons. k k ) ( Frsty becuse k nd secondy becuse r s the poston vector fro to the ss. So tht ) ( r r so k 9.3 Köng s theore f body perfors cobnton of trnston nd rotton, then the over knetc energy cn be wrtten s: (9.8) (9.6) rot trns kn v E E E Here v s the veocty of the CM, nd the second ter s the rotton energy retvey to the CM. n the fgure to the rght s showed body n oton descrbed fro reference pont O. Let r be the poston vector to the ss prtce, nd O r the poston vector of the CM. s defned by the reton: r r. dr v / s the veocty of the prtce wth ss, retve to O, nd d u / s the veocty of the prtce retve to. Fro the reton: r r we therefore hve: u v v. v v s the veocty of the CM retve to O. u, whch foows fro dfferenttng the reton.. A ccuton now shows: ) ( kn u v v E kn u v u v u v E (9.) ) ( kn v v v E We hve used tht v s constnt vector tht cn be oved n front of the suton.

17 Rotton bout n xs. Moent of nert 38 The frst ter: s the CM energy or the trnston energy. t s the knetc energy retve to other objects, wht we htherto hve ced the knetc energy. The second ter: u s the knetc energy of the sses retve to the centre of ss. t s therefore ced the retve energy, ntern knetc energy or the rotton energy. For rgd body ths oton cn wys be descrbed s n nstnt rotton bout n xs through, wth oent of nert. n the second equton the st ter becoes zero becuse of the reton: u. Thus we hve proved Köng s theore. (9.) E kn v. Centre of grvty equs centre of ss The centre of grvty of rgd body s usuy descrbed s pont of suspenson, where the body s bnced (equbru), wth respect to every dspceent fro ths poston. The condton of equbru s obvousy tht the su of torques ctng on the body s zero. The fgure shows body suspended n pont O (or on rod through O) n the grvty fed. The su of torques fro grvty ctng on the ss eeents stuted t r wth respect to O, s ccuted fro: H r g r g (.) H r g r g Fro (.) t ppers tht the entre torque (oent of force) cn be found, s f the whoe ss of the body s stuted n the CM of the body. Especy f O =, tht s, the body s suspended n ts centre of ss, r nd therefore H, the body w be n equbru turned n ny poston, nd ths w so hod f the body s suspended on n xs through the CM. The centre of grvty s the se s the centre of ss. Fro (.) t s seen tht suspended body s n bnce, tht s, when H, when r s pre wth g. t therefore foows tht the body s bnced, f the pont O of suspenson or support s rght beow or bove the centre of ss. f s beow O, the equbru s stbe, snce the body w ove towrds the equbru poston fro ny nor dspceent. However, f s vertcy bove O, the equbru s unstbe, snce even nor dspceent fro the equbru poston, w generte torque, turnng (wth ncresng strength) the body

18 Rotton bout n xs. Moent of nert 39 wy fro the equbru. We known experence fro everyone, ncudng bncng rtsts. Ths s ustrted n fgure (5.3).. The physc penduu A physc penduu s rgd body suspended on horzont rod, nd perforng hronc osctons n the grvty fed. n the fgure to the eft the penduu s t rest. The poston vector fro the xs of rotton to the centre of ss s denoted. n the second fgure the penduu s dspced n nge φ fro the equbru poston. Snce we hve proven tht the ctng torque cn be found s f the whoe ss of the body s stuted n, the torque s: H g, seekng to turn the body bck to equbru poston. Tkng the engths of the vector gves: H gsn g (for s devtons < ) We then ppy Newton s. w for rottons wth respect to the rotton xs through O d H O Aso usng Stener s theore ccordng to whch, the oent of nert O wth respect to n xs t O, y be ccuted s the su of the oent of nert hvng pre xs through, pus, where s the dstnce between the xes. d d H O H g O g d g (.) O The dfferent equton (.) s recognzed s the equton for hronc oscton. t hs therefore the souton: cos( t ). Ney, when dfferenttng twce, we get: d g (.3) cos( t ) cos( t ) O Whch shows tht t s souton, f nd ony f: g g O O snce T

19 Rotton bout n xs. Moent of nert 4 We cn then fnd for the perod of oscton for the physc penduu: (.4) T O g The foru (6.4) s the foru for the perod for the physc penduu. Note tht t cn so be pped for thetc penduu wth penduu ength =, snce n tht cse O =, nd we retreve the fr foru: (.5) T.6 Expe. The oent of nert of bcyce whee. The foru for the physc penduu y for expe be pped to deterne the oent of nert of whee, whch cn turn freey round n xs. The fgure s suppose to show the front whee of bcyce. We sh desgn n outost spe experent deternton of ts oent of nert. On the r of the whee t the dstnce.35, we pce weght wth ss of.5 kg. The whee s put nto osctons round ts horzont xs, nd the perod s esured to T = 3.6 s. Fro ths esureent, t s strghtforwrd to deterne the oent of nert of the whee. Snce the whee s syetrc wth respect to ts xs of rotton the torque on the whee coes excusvey fro the weght. n the foru for the perod: g, s the rdus n the whee. The oent of nert s = whee +, where =.5.35 kg =.53 kg. Fro the foru for the perod of physc penduu we fnd. g T O T O g.4 => kg g whee O.5 kg.7 Expe. Cock frequency for n od ong cse cock. (supposedy sec) Next we sh try to ccute the theoretc perod of penduu n ong cse cock, s shown n the fgure (Dnsh text). The penduu conssts of rod wth ss rod =.5 kg nd ength rod =.7, together wth dsc wth ss dsc =.5 kg nd rdus r dsc =.. Frst we deterne the poston of the CM for the penduu. Accordng to our prevous resuts, the CM cn be deterned s f the whoe ss of the rod nd the dsc were octed n ther respectve CM s rod nd dsc. f the poston of the CM s esured fro O, we hve: dsc ( r ) rod ( ) rod ( ) dsc ( r ). 69 The oent of nert s ccuted s the oent of nert wth respect to the xs t O usng Stener s theore. rod dsc dsc ( r ) r ( r ).5 kg 3 rod dsc dsc The oent of force g = ( dsc + rod )g = 3.5 N, nd when nserted n the foru for the perod of physc penduu t gves: O.5 T s. 75 s g 3.5 (But s physcst t s ore portnt to understnd wht you re dong, thn to obtn the correct resut). rod dsc

20 Rotton bout n xs. Moent of nert 4.8 Expe. The Yo-yo. The fgure shows yo-yo. For convenence we sh ssue tht the yo-yo s hoogenous dsc wth rdus r = 3. c. Further we ssue tht the cord s wound on dsc wth rdus =. c. We wsh to fnd the cceerton of the yo-yo, when t s dropped freey, but hed on the cord. We ppy the equton of oton for rotton, wth respect to n xs through O. The oent of force wth respect to O s H = g. To ccute the oent of nert wth respect to O, we ppy Stener s theore: O = + = ½ r +. Furtherore: v =ω, where ω s the ngur veocty, nd v s the veocty of the CM. We y then wrte the equton of oton: H dv O d g r g ( r ) r g d dv 8 7 d g r g 4.6 / s