Ais of Rotation Rotational Motion | Centre _of Mass - Centre ob mars of a booly or system da a point which moves such thed all the mars 4s concentyated here ancl all externad forces ane applied Hur. * The position of Centre of mass clipends upon the Shape of the body and cistributfon of mass «Tn Symmetrical bodies in which olishibution of mass ib homegenceus, the centre of mass coincides With the geometrical Centye of the body, # Centre of mars moy be inside or outsile the body. &g Circular dise > beithin the bedy Cirewlon Ring > Outside the body Position vectey of centre of mars for _n particle system Lt a system consist of 0 y Particles of maxes Mp Mma5 .0 Mn Where positon vectors ant ¥], Vay Ye... Fh vespectively: Then position vector of Centre of mess Fo MM tye. Mn my t Matern Mn Position vector of centre of mand for two particle system 7 2 Me + me my +mIMPORTANT POINTS LJ Ih the origin is at the centre of mass, then the sum of the ‘moments of the masses of the system about the centre of mars is cero. Umit =o 2) Tp a system of particles of masses mM 5 May. In move with velociHes Vio Vaz--- Vn then the velocity of centre of mars 4 ZF baad Va = et =m 3} Th a system ob parttdes of massed m5 ™2,... Mn move with accelerations Q,,92,--+An then the acceleration of centre of mass is HI Fore on a rigid body F=M@ eo May NOE easy Coleuledion Se fee EM Gey paint around Cenhe of mans colculote tRAT 2,58 Print Br origin (0,0,0) WE assume x Fe = |Body Position of Centre of Mars Uniform holleus/Selid sphere! Centre of Sphere Uniform circular ving [elise Centre of ring /cline Uniform Red Centre of roc Squane, Rectangle , Paralluegrem Pant of intersection of aliagenls Triangular plane Lamina Point of Intersection of mesicns Rectangular [Cubical bleek Points of intersection of dlagemals Hollow / Solid Cybinder Midldle pein of the axis Solid Gone or Pyramid On the anis of the cone ot Pent Aatance Sh/y from the Vertex » h= height of the Cone Equations of Linear MoHon + Rotational Notion @) visu tar qi) s = ut +Lat® div) v® = ub+ 295 W) S =U + 4a n-1) Wid v= ae wit) a= ay = oe Gi) We = Hat WH) @ = wt + Lae® dv) We = Ww? + 200 &) Om = Ww, + da (@an-1) MW) w= Wil) a= de = aeMoment of Insatia - It Jaa property of @ body due dy which ak Opposes any change in the state of rest or unifirm rotator. Moment of ineha plays the Same rele ‘in rotational motion ab mars plays in Lineax motion The moment of Inertia of es @ partele with respect 40 an axis of retadion is equal fo the product P m of the mass of the particle and Square of distance from yotational axis. Moment of Inertia of system of Particles I= mrPamyis-- axis Ln no Foy Continuous distribution of mass cripanesil Te bat) Note - (i) SZ Umit - kgem™ Gi) Moment of inurtia A y ‘ epenals on mam, shape, size = demtity of the body, I+ also depends pen es Position ob axis of YotaHin and mars clistrbubin. ; os ened Ob inrtia does Not dupencls upon Angular Hy, a Acceleration, torque ana angular momentum of the body, Bimensions- Cai 7°]—_—— Radius of Gyrection (x) The radius ob guration of a body is the perpendiuler distance $rem axis of Totatio, the Squane of this distance When multiplied by the mars of body then it gives the moment: of Inwntia of the bey (T= Mkt) about seme oxic of yotation. T=mMK* > /K From the formula of otgevele olis tribution : 2 2 Toe my t mate too mnt Tg om, ema = um, zm thin a Loe mata ade 8) Hen MK* = m (rtadte.. 13) (im) k* = om (yP4 G4 ) [1 = Totod No. of Pasticea’) +e Ta nm K Hence radius of gyration of a bedy about a given axis 4 equal to the root mean square olistane of the consti~ tuent Partdes of the body from the given aris. Note— | 41 SZ unit - Metre Dimensions - Cm’t?] | 21 Radius eb gyration depends on axis of rotation and distribution of masd of the body (Shape ond size) . 21 — ob gyration oes not depend on mars of the oly.Moment of Inertia of some standard Bodies ino = eS Shape of Body | _Axisof Rotation Figure Moment of Inertia | — + | ‘About an axis v : ‘i passing through z MR? z the centre and perpendicularto | theplane of dise 1 Cirle ive | oun dianttts MR £ Mass=M | axis Radius= R About an axis SA | tangential tthe Bice = MR’ | | nmand lyingin 4 | Sr | theplane ofthe disc ‘About an axis tangential tothe aa | GD | | perpendicularto | 2 the plane of disc ! About an axis Y | passing through | the centre and 2 | Circular Ring | erpendicularto me | Nass=M | theplane of Ring Rade | : — r= | About a di | | R a diametric | | | axis | 2 MR’ ezShape of Body | Axis of Rotation About its geometricalaxis Solid Cylinder ‘aaah Mass=M | ‘@sgentialto the | cylinder surface Radivs=R | & parallelto its ay | Beometionlaxis Aboutan axis passing through the centre of and || perpendicularto its length|_ Shape of Body | Axisof Rotation Tigure ‘Moment of Inertia x About an axis | the centre and Mp? «Rt Riek . Stat + R81 ‘Anmular Dise perpendicalarto e Mase = M the plane of disc | Internal Redius = R, 7 ~ Outer Radius = fy About a diametric Hae = Sin? «ab Aboutits | Diameterioaxis Bat | Pp 3 Passing through Hollow Sphere | i, centre | Mass =M. Radius=R About tangent 5 to the sphere sum ‘Abomis, 2 met ‘Diameteric axis | Solid Sphere | Passing through | Mass=M | itscentre ibis po About tangent sue to the sphere| Axis or Rotation Moment of nertis | About an axis | passing through 2 centre & a Thin Rod pemendicular to ats Tenath, Mass = M About an axis Length = L. og one end & perpendicular to its lengthTorque (OY Moment of force) The turning edfect ef a force about the axis of rotation Calle torque oy moment of force clue to the given fora: Rotation Rotation Radial Component of force z Rix Fos OC) ‘ Thartverse Component of force F< F sino oO @ As T = YFSiNe = TF CY T= Pesitton Vector X Transverse Componend of force @ Hence Teg ue 44 clue to transverse. Component oy fore only: Tn vector form SI Unit - Nem Bimenions - Cutty Direction - torque is an axial vector. Ths direction Js perpendicwiar. to the plane. containing vector FV ard alu Fe ee direction ly given by ‘Right hand sere Rule’. Gi) For clockwise yototion Ts ve Ud For AwHelverwice vototion Te +veLJ Fev a given force and angle, magnitude ob torque ov v, TRe more the value of r, the more will be the forqut and Us easy to rotate the body. work W Power P Couple - A i = Tt Lb olepined a> the Gmbinoton of two equal bub ebpositely ivected Forad not acting along tne same Line + F w= Teer —— Yr ——H F Note - The basic olippeence betwen torque and Conple is that in Case Couple both the forces are extern: Leal hile in ea eee ae oe ee Yeactionany-rngUuan Momentum — Tre angilan momentum of a body absut a given iis Us the pioduct of its Linear momentum and per— wendiculan olistnte of Line of achon of Linear momorum rector from the axis of yatation. Angular Memontum BS Plame i To wechr fom : or T = mPxv) c ST Unit — Toule-see. Dimensons — Coit P In Cantesion coordinates ity = xt +yf 42k srt +A) + Pk v Pe and Voz Uut+w)4ve® Then ” tar Cele y 2] = (ya-zayt-ur ~ 2AM + (XR, - YAR | Pe | wt tami i T=m : x = = mle ~AM)E-Gee-em P+ (ony)Note - Ad lan momentum is an anial vector Its olirectin is given by ‘Right hand Serew Rule’. Z) Fr 6-0 or eo , Lm =o For @ = 30° > Lime = mvr -BJ- In case of Circular motion T=¥eP = m(PKV) = mvrsine Loe mvry =mrtw CAs FLV J (Aas tem] As a de and T=1e The rode of change of angular momentum is ejual to the net torque acting on the particle, This expression is also Known os Neuston's 2™ Law of rotational moton. BJ In Case of rotational Motion Angular Momentum Lb = Iw Rotational Kinetic Energy Ena = 2 : p 4 = ee -3t(e] 9] & eee Ss) Voge momentum of a System of particles is equal t vector sum of angulaA momentum of each particle. T =t4+te... Tr Tw ve F.com www. sccsikalAngular Tepulse — Ih a large torque acts on a body for a small hme me Angulon Impulse = TAt Hence angulea Tmpulse 4s ema) to tre change in angela, momentum. It has ame unit, dimension and direction ay angular momentum. Law of Gonsewation of Angular mMementum- Ib te reuttant torque acting on a bey is zew tren total angular momentum of the System remains Constant. By Newton's 2° Law of retational Motion “Es AL re Ai 2 < Ip T=o0 then aE =e > Also Henee change. in angular. momentum =O, or L= Contant or Li = Le > Tw = hee & When a person havi weights im he hands and ‘Standing on @ Yotating platform, suddenly folds his auma, ther its mement 04 inuatia clecreases and in accordance the angulan speed increases. ra 4; Iv wtNote - Ip external torque of the system is zero, Han the Qngular. momentum Jy Conserved» Homer the rotatimal KincHe enrgy ds not Conserwed + Geo then TW, = Tale or Stuy s $B0r > a (SneP) = (2%) teen Tike = Inka. Hene ih > TL ten Key < Kee So ik moment of inetia decreases, the rotational Kinetic energy increases ane vice-versa.Rotational Kinetic Enengy - The energy due so votatonal moHon of a body JA Known ab rotational Kintte enmngy Lat a vigid body is roteding about an axis uth umiform angular velocity W. Ib the becly a Composed of particle of massed my ,m,,-- trun Kinetic energy of Totating. body = dmyhs Luts As tre Linear velouties off He porteles Yr wy , Va eK 5. en Eps eS +myrd4 ese Work in Rotational Motion - Let Q tangential Force Fr acts at the rim ob pivoted Lisk . . The dise totates a small angle d@ during a 5 dime dt. The usnrk done by the force Fy while @ point on the rim moves a distance as iy dw = Frds Ty A6 ney amgulan cisplacement then ols = Rdo dw = RRdb Waris anIne WIyAL GUL aD TRL TOrCe fy Ad TERR dw = tdo swing an an displacement from 8, to 8, +the total ork clone by the torque is 8; Ww = fav >| WwW =: {tae 8, 8, St the torus is cmstant while the angle changes by d) do Ba then e, w= tLe], >| we t le-6) Fen the work done by 9 Constant torqui is the product of torque and the angular disphtement. Let T represents the net torque on the becly se thot t= 1a Gssuming tet the bedy is rigid So that the momect Of ‘media I is Constant, then Tap = (Ta)do = Tdi alo = Iwaw Hen total Work dont on the rotating body is We L[wt -wF] Wr w =filwds > wy) & When a torque cloes work on a rotating body » the Kinetic energy changes by an amount equal -to the work clone,Work Energy theorem in Rotatonal Moton- The change in +he rotational Kinetic energy of a rigid body Us qual 4o the Work done by external tormut. Rotational ower - T+ Js te Powth asodiated with the work done by @ tovqua acting om a rotating body. Work done by the torque T aluring Amal olisplaconint AS Ae dw = tae dividing beth side by the Ame intewal ot curing which the angular displacement cecurss we get aye = tae >[R=te Whe Ro as instantaneons Power Rotational Equilibrium - A body is oid do be in rototimal equilibrium 'b resultant torque acting on it is zewo. xT =0 £3, In case of beam balance the System will be In rotational equilibrium ib op Trt =O or GFoet. or. Fily = FleComparison behowen formin of translator and Retetenal MoHon Translatory Motion RotaHonal Mohon Fore Fs oF sma Torque T= He = 1k Lintan Momentum P's me | Angulen momerdum T= 723 Linear Kinetic Energy K = dev] Retational Kimete Energy Ep $1 Work dene by Consterd Force | Work clone by Contant Torqus wee w: Ue Work clone by variable Force | Work dene by variable Torque Power P= das PV Power p= = ot Work Energy theorem in TM. | Werk Energy treorem jn RM. | | w = SFas w = ft-d8 | | | | | w = dmvd - mye w = 41h -diuh | | Linear. Impulee 2 AB = FAL | Angular Tmpule = AL = Tat