“Date: Qa) 1alaoae Cuaprer System of Particles AND ROTATORY MOTION fs arigid body? What I+ Vs the body whose Shape and sfz2e@ does no change ondey the actfon of force 1 TYPES OF MOTION OF RIGID BODY WN EO =7 Transtatoxy motion =P Ratatfonal motton wIRANSLATOTY Motion ee T+ Ps the medfon PA conch all the parifcles of a ytd body moves wiht @ same velocthy at any Fat Instant of tme- Ex i Motfon of a stone dvopped from a paxtPeular hetght> @ ROTATIONAL MOTION Re I+ ?s the motfon Fn which ail the porifetes of a v¥gtd body moves fn a cPycle about a fPxed axfs. Exi Rototfon of Earthy about Pts ousn axts Rotatfon of a wheel @ DieFeRENCES BETWEEN CENTRE OF MASS AND —eo€_—errr CENTRE oF GRAVITY ~~—~~~ ~~ Lt Centre of MASS ee ee eee CentRE OF GeauitY T+ Ps the point at cht fhe whole mass of He body %s Supposed to be Conten tratea: Tt Ps the pont at tohfch the ushole wefght of the b i eee ody fs soppy.t+ depends onty on ° T+ depends on . " gett buon of mass ALskP LAN OF Mass \ OANA gravPtatPonal Freid. s PostTiON OF CENTRE OF MASS OF SUSTEM OF TWO PARTICLES, OP TWA DIMENSIONAL An ~~ > — me 0 7 re = een Cc OG < Ky INTRODUCTION 2 Inthe above dlagram m, and mM) ave the masses Of hoo particles K,and x, ave the posfifon of hoo parifcles from the Ovfgtn- “is the. postiPon of centye of mass of System of pawtfeles - Along ~« axfs the postion of 1% of centre of mass Ts given by Pg te Mm, wH, tomy, wx, ey mt MENTION AN EXPRESSION For POSITION COORDINATE PARTICLE SYSTEM IN SPACE OR SO Ree Pe Ee, Along %-axPs tne postiPon 1% of centre of Mass Ps gtven by we Uw EK Oy tas Distance Pron ortg in. Along y-ads postifon y of centre of mass fy gtven by Myre yy mtn, f mass 7. * Along 2 axfs postifon 2 of centre © Fs given by MrZy,t ON Zr mMmtmMs Alo enevral CR? posttfPon R of centre of mass oo Fr vs given by Mee Myre mam, @ Position Coorninates oF Ns PARTICLES SYSTEM IN| Fedo SPACE. ee ee ee ee ‘ Along mu ax?s WF CW Mp TOD Wat MV AMY + PHS. TMM ner Mita tema t eee M9 Along neers gem + eye t ria yy be... tmagn . ‘Along Zaxtfs .- : A= Bh Za Ma Zag $M 3 Doe weet penn PU ta eee lion,. in General A R= my) t+ Marit M37v, bmn CATT AM aA eee OD Re eme wp M Me Mtr tater! MA wp TERMS IN RELATION WITH ROTATIONAL NOTIons en Angp lax alsplacement Ce) Angolar VelocPhy Cus) Angvlar accelex~atfon Cx) Tow qrpe Cw Angvlay momento GL). Kfneh?c energy Moment of Tnevtfa Cr) =7 Radfos of gyratfPon Cred m Angucar Dispracenentr Ce) ee ee T+ %s the ORALe descw? bed by a vodfus vector In a gtven +me- O= Ave Length = 40 vadfus + S-T unt+ of angular d&splacement Ps Radfan Radfan CRad) Cradd w Amqueae Vevoeity Cus) omega Rate of change of angvlay alsplacement per ont time Fs caued angolay vetocthy- w= Ao 3 S-D unt 2 sad =2 vad [se es at sec vad sec!° Angular velocthy fs a Vector quan tty - " Anqucar Acceleration Ons ‘i | T+ ts deffred as thevatfo of the change fn | angular velockty of the body and the +fme taken Angular accelevadeon = change Pr angvtar veloc See ee Time taken = FPnal Av- IntHal Av + cL = ud. - “Oe +t * ST ontk = vada /sec® Retaktfon betwoeen Ifneax and Angular ace elevate LB marks =? Ve=~us ie ies nacail ' DiFfeventPating™ both sPaes US¥ +o 4+re. dv 2 ASCrus) | at at 7 aA = YAW | AL | A=yu gm Moment of Tnertia _—eS Moment of Inewt?a of & bedy aboot a giver ©; axis %s deffned as Sum of product of moss | of each part¥cle and squave of MHS pe | _ulay dfstance from Me axts oF xStatfen "| : a_— ~~ 2 ~— w Expresston fox moment of Inby+fa of the eee ae a body TH Su, Mz -Ts Sts ome Me masses of part&les of O Pg id body strated at a alstance of Y1,¥%2,43 From whe axts vespectPvely ,thren moment of PnertPat 2 2 2 TamMM TAM vo t+ O73 Va eee my,y¥,t T= Emv?® Kg om? w Factoxs on eah?ch moment of a evtfo, depends eg ee e Mass of the body + BYstance Prom axfs of xo tat Pon + Sfze and shape of the body ©¢ S-I untt+ 2 Kg ro @ Toraue AARRE B_LMOMENT oF Fonce: Terqgve acta ON A body about anaxfs of YototPon %s Gefed as the vS Product of moagothde of Rxce actif ON A part¥ele and Pevpend2culax a&stance behoseen ‘tae Pre of ACHPeN of force and axfs of yotatfon BExpression For TORQUE SSGGERSIEN, ‘ Tel ET yay wheve F Ps force acl¥rg on the partfele anda ~ Fs 4he radfus vector or pev pen d&eulay stance behoeen IWNe of actPon OF hoo forces. Eeeee e Torque Ps a vector quanti hy * SeL untt of Torque Ps Newton Metre Nn + Torque fs the vector product of hoo vectors — ae = ET oye = FY Sind “T8s max = 8 =9Q0° TT Fs mPas @ = 0° w Relat elation behoeen Torque and OLPh a oc —T =e. _ Pe est = fey = Maer (f&=may = wer ane Gino ees Total torque = & mun Crav2=2) Vs De "Easterns on conten tore ‘depend s so as “Magnitude of force * Perpendfevtay afstance behoscen ne of Act¥ON of hoo forces? = Awnqucar Momentun AAS PME TOM Angrlox Momentum of a pavifcie '%s dened as the product of tPrneax Omen tom and per pendtevlar d&stance of the Parteele from the axis of votatPon « oR The moment Of near Mementom eof a xAecle aboot AN axls of votaton. PaLt - = ir Ps senesced by the letter L L= Pix ¥7 , | = py sine TE =O 3 angelay Mementem Fs maAfmom = L=OoO Te @=90°5 angvlay momento Ts max = = Eaw © .$L untt of angolay momentum 7s Kg m/sec L=~*P = GO ™ (vere) = ym = maw? L. L = €& my?49 ae a) AR MOMENTUM AND TORE Tox gee Retatrion BETWEEN AN GLU aA SA EOI Cs oe Differe nifoting both sides wth yespect to +#me ustng a aU! _ a. CPR) Ke Tak = (Fe BE) A 87, ) =(pt 1s ACHR ET) fi = (eS TS CED Soot CURED) CTS Zl=0) A*/ go4y Fo OT : Teraqye = rate of change oF angula® raoment™BW Comstavation oF ANGULAR MOMENTUM OER TF the total externat torque actPag on the SyStern of paxitcles ox +Pgtd body “fs ze ¥o 5 then the 4otal angulay momentum of the system fs Conserved o¥ constant We hove Angutaw momentum L = Tvs 5 where T= moment of PntextPa and 6d = ANngvler velockh, According te. Laus of consevuatfon of angular Momentr Es of Tus =constant Therefore PF TL decreases the 09) wit Pacrease and vice versa. mw Examples Nae ZA dlver Tempting from ao Spying board folds hes oxms and legs fn wards As ayesutt of | MTS Moment oF 4 Previta Aeleveases 45 conserve angulay mementum h?s angular velocthy Treveases Therefore, duxtng Jumptng hee can votate Pa 8s sth imaxtmom ang uta Neloctty ; = ; 7 A Bote+ dancey uses the prfnc&ple.o f conservation of angular! moment tp Tacvease. thet speed- They Staxt sptaning estik wet axms Stretched Hen forv toe Wey Orms closer as a yvesgolt moment of sero decreaces and hence angular velocity yieas\s m Cov PLE ‘ ‘ ee 2: ©. ° £ equal and opposite forces wfth . ceed ae of action act€rg on an onfect : E ° 2 eee eord body conshhte o coupte } °gt untt of Couple NaC Newton Meter) | ~~ —_—_— 8 perpendiev lan fc ligt auce A Couple prodvces onty yotadfonal motton - we Momen* of a Couple s- Ps defPned as the i product of eosattee of any of the forces and perpendtcolar dfstance between them, (. re fs the votating abrity of a Couple: Example ~ =7 A Couple fs applfed vsing ovr feagers +o ~votate Ifd of a bottle (=T A Covuete Ps applfed usfrg ovx legs to peddte a bfcycle* Roows of Gussrion toe fate Shore Radtee ihre" Radius of Gyratfon af a body need mn axts Ss the perpen d&evlay Afstance from the ax’s to Re potnt at vohich entre mass of the body Ts assumed fo be concentrated Tr m 2s the naass of the body and k 7s “the d&stance of the polat From the axts of yotodfon at udh?Pch entYre mass of the body Fs co acentrated then? Tamir : 2 Wheve kK os catted vyadlus of gyrvatfon about & Brven ax?s of votatron.Ke a actors on wohfch yadfus of Gyratfon of body depends fe NE et ae | Coo een Mic eedaa % SfTze and shape of the body =T Axis of votatfon. ROTATIONAL KINETIC Creer Oy a= Rotatfon Kinetfe energy Ps the energy possessed by a System of parifcles shea {4+ executes votatfonat motton . Rototfon al KE = 17, 2+ Yen gua? 28 & rete t Yam ny, 2 t=2\ WRT =? v=, U9 at 4 Nav, U9 Ve 2 ae Rot Kee = ve Cea yy,tu9 Se = Mauer Cmte my eyten maAvn2) = VW2awr> Err> es Rotoatfon al KeE = ']2 Tus ® =7 K-E= Nop MV® =—G rans latory mot fon — ° = 2 : : =7 KeE= "/y Tus? 7 Rotatfonal mortfon - {= } DAA 13 MomeAt Of Inevifa plays the Same vole Pr retarPon al mot eon as omass Plays. Pal’ trans latory motPon- Moment of Tnev Pa Pe. the votatfona} analog fox mass PA! Hr Gn Sla tery mMmothton: - ' 9 irLinear Motion RoTATLONAL Motion mz neay as placement | © = Angulay ALSPlacement | ve Final Vanear velocPhy Wes Fenal angular Velo ly ve TaPtfat \Pnear Veloct hy wIF= Tnhthal ange lar velocthy as accelevatton <= Angolay aceelerat Pon, | a 7 EQUATION S — ke Veutat Wer = bogeet ae Ns urt Zar Wpr=09;24 AKO Hever | 2 ° = © = wWrtt Yyne® Prusical Qty Linear ROTATIONAL | Ve loctty Vale w= Olt | ©, | AccetevarPon |. as veru mK = We- De) } | aS a | | =} | : | | | Force (Torqve Fema verte | | | — | ‘foam e Vo rus2 | Work | é | 02> Fac w=Te | Power P= FV p= us | | | Mome ntu = ’ = | ™ P=’ seAnGLLaAR Terns RELATION VRE G8 SumBoc TERMS UmBoL | | Displacement * Angular | © | A&splacement Vetocthy Vo) Angelar ot | Veto ty J | Accertevatton a Angvlar I , Pars | accetevatfon | (eae | ! Mass eal | moments TL laem < Momen torn P Angela | | Momentom Fovee Fe Torqgve | Vsry BW DERIVE AN EXPRESSION FOR VELOCITY OF CENTRE EES oor MAS SOOO ao oe eee cm SF ONT, IN {we ee eee M Eocpressfon For velocthy of cmEe AN EXPRESSION FOR ACCELERATION OF wo Der Centre oF MASS —7 a a C vem) mom —_ at -— & oh Seeieeee eS at Mm [A C “Tat Crow + Bat Crna Te )4--) aot = = —e BE Yy Coma Ra WD where dvi Ss the acceleratten of fie st at powssPcle and of the second paxt®cte- Av2zlat TS the acceleratfon —_+ = —— a, MO ce = ONE Ha Aa torres Cevoss moltipl) Net, Exteanac Force. + —= es =r — —~ Come Mm Cra, engage wemy =r = = Moem = Cmial + ma UL t--- D> Mealem Ys the net external force acting on a xigid bedy = Feu =T 7 oP Fes. = fF, +f. 4-3 where mar 9s the force acting on Fevst partfele and mas Fs the force acting on Second part cle -» Shous that IPreax momentom remains constan when the net external force acting on & partfcte Ts zero MVon Pmt myo 4 lll. —v pager Poem = PTH PL, Po —S If Fexs =O . According +o Neuston 's ad law! —c ap = fae At * fp + ae), =0 C derPuatue of a constan lak = 2 Fs o> ato nalp > e + P Ts a constant =F w Niecoamical Equiugrrin ee eee A AAgrd boay) fs SoPd 46 ' be In mMechantcal eqklPbstum PF both IPneay momentyem and ang vlax (mamentom ave, Kept) constant es?th vespect to42me , = —s * ' @ TRANSLATIONAL EQutuBgRions +) r oso A ~ighd body % sa2d to be Ta Hranstdldoney CqullPer¥um PF PE ~wemnatn s 2 * —It§ —TtR ontts- t