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ROTATIONAL MECHANICS MOTION OF RIGID BODY ABOUT A FIXED AXIS ) IntRopuction : When a rigid body rotates about a fixed axis, all points of the body meve in civeular paths centred on the axis. Rotational motion is completely described by angular quantities. Key ANGULAR QUANTITIES _3D_DIAGRAM : ROTATION ABOUT FIXED AXiS (i) Angular Displacement (8): ¢ ' 1 ' 1 Angle turned by the body. aia a [ST unit : radian (rad)] circle L to the axis. Angular Velocity (w): Linear 5 P: Rate of change of angular ee displacement. = 2 ti ws S| [rad s*] (iii) Angular Acceleration (a) : Rate of change of angular velocity. dv -2 “ a [rad s*] RELATION BETWEEN LINEAR & ANGULAR QUANTITIES v (tangent) Linear displacement : $= rT, @ (for small 8 in rad) a Linear velocity 9: y=r,w P Linear acceleration : a, =r, & (tangential) Total acceleration a, = 7, Ww (centripetal) Te a @ Moment oF Inertia (I) 3D Diagram (Mass Evement) Tt is the rotational analogue of mass. ® DEPENDENCE : Tt depends on the distribution of mass er + Depends on axis of rotation and the axis of rotation i i * Same body can have different Definition + ei £ For diftent ess T= Zmri, (dieerete) T= [rd dm (continous) ee ty Je : 2 where, Tis perpendicular distance of — yo-” ULE Tha: mass element from the axis * MOMENT OF INERTIA OF COMMON BODIES (About central / symmetry axis ) Thin Rod Thin Rod Solid Dise Hollow Ri Solid Sphere | Thin Spherical Shell Clength L) | (about one end) (radius R) (radius 5 (radius R) Gradius 8) ‘ ' e == i= =>) $ i Sri fore 7 | FE r =I t = 4 mt =4 mt ay Tei Me [ae dm I= 4 mr* | I= 2 mr* * Az [>] A thin vod of length L and mass M is rotated about an axis passing through | | @ print at distance 1/3 from one end and perpendicular to the red [3€€ Main] Find its moment of inertia about this axis Hint : Use T= I,,, + Md® @ Moment of Inertia (IZ) Theorems of Perpendicular Axes (For Plane Lamina) Tf a body lies in xy-plane, then moment of inertia about z-axis (perpendicular to plane) is equal to the sum of moments of inertia about two mutually perpendicular axes in the plane passing through the same point. Moment of inertia about any axis is equal to the sum of moment of i inertia about a parallel axis passing H through the centre of mass and the product of mass and square of distance between the two axes i (d = perpendicular distance between the two parallel axes) @ KINETIC ENERGY oF ROTATION The kinetic energy of a rigid body rotating about a fixed axis with angular speed w is given by 2 T= moment of inertia about the axis iG) ANGULAR Momentum (L) @, Fae i particle Ta txpem(#xF) a fixed axis : |L| = mrvsin® L=-Iw (direction by right hand rule) 3D Diagram ThE (ii) For a rigid body about (vector along the axis) * Rectangular lamina (a xb) about z-axis (through centre) + f i i 1 ‘ - ty Bt ! T= MG! + b*) | ¢ Circular lamina (radius R) | about centre (z-axis) = ' I, = £ MR? |e Ring Cinmer Ry, outer Ry) : | p= $M (Re +R?) I + Thin red (length L) about axis through one end (4 to red): Ly 2 2. Lt dat Tet, +M($) = M+ gta se + Gircular disc about tangent in its plane: # nl ipint ¢ fant aS pa T= Tog + MR* = 5 MR® + MR = SMR Special Case Tf bedy rolls without slipping on a. surface, Kestat = $MVeq + $ Tet Chere Vom = Rua) Conseration of Angular Hamentum | # Tf no external torque acts 7 i on a system, its total angular es momentum remains constant aha es Teg 0 > op i) ’ “ey = T = constant © Torque (7) r | Definikion : The turning effect produced by a force ! - 2 ase | about a point or an axis is called torque. H 5 as ' ane H “FI? + r=0 ' H ees (no tuming effect) | [et] = rF sin® (direction by right hand re) | HF |. Pat + corr ' H | SE Unit Nim (Neuston metre) ee ft Hand Rule (maximum torque) 7 * t po ana Se) : A uniform disc of mass M and radius R is rotating about its vig [Jee Main/Adv.] QUESTION | smmetry axis with angular speed 12, Find : pos: Gy km ¢ (i) Tits moment of inertia about the axis. iba 2. Gi) Tis kinetic energy of rotation. C+ BS Gi) mrt i Gil) Tks angular momentum. SSF Gil) tmp es Page 4 @ MomENT oF INERTIA oF CoMMON BobiES (About Different Axes) | Body Axis 3D Diagram ex 4] Body Asis I 3D Diagram Thin Rod Through 2 Hollow Cylinder | Symmetry ai (ingh ba #) ae fey Lew Te imu eer (roa nin) te rod height hk) b 4 L Solid Sphere Through cantre Gradius R, mass M) (any axis) Thin Rod Through one Gength L, mass M) end to rod Solid Dise | Thragh cone | (radius R, mass M) 1. te plane L Hellow Sphere Through centre (thin phen (any axis) shell, radius R, Hollow Ring Through contre ma (radias R, mass M) to plane E Through centre one Lamina. L te plore cs —— + qi ‘a, breadth b, M(a* + b*) Sed Cinder Sy a0 } (radins R, mass M, (trough contre) 4 eight) Thin Cireular | py | b. Ring (radius R, is «= 4 mp? ng (im plane) r+ 4 mr L 1 | L Cl = = % Note: For any axis, dimensions are in metres (m), mass in kg, so I in kgm* (QQ) Theorem oF PERPENDICULAR AXES (For Plane Lamina) a 7 For @ plane lamina lying in xy-plane, the moment of inertia X about an axis perpendicular to its plane (z-axis) and passing through a point O is equal to the sum of moments of inertia, { i | * Rectangular lamina (ax) about centre : f about tuo mutually perpendicular axes in the plane passing y! \ ' ' \ i | T= 4 M(a*+b*) = I, + Ty \ \ ‘through the same point ' I, = I, + Ty ' (Ty about x-axis, Ty about y-axis, all through 0) 3) Theorem oF PARALLEL Axes ‘Axis through Palla Faas fami dabond wd rT at ie. cm “hain | ®.Thin rod (length one end : Tre moment of inertia of a body about ary axis is equal 1 ' Ta temd® = Laut em(4)t = Lut to the moment of inertia about a parallel axis passing i H t through the centre of mast plus the product of mass * Circular disc about tangent in its plane = and square of distance bebiwer the two parallel axes. ( T= $e + Me® = Sma? +d in sa we shared.» perpendiniler distanea bebsten: ass. See i a i @ Rowing Motion A bedy rolls on a surface without slipping. Let Vem * linear speed of CM, w= angular speed. For ralling without slipping ? Vom = Rw Kinetic Energy in Rolling Ke devine $2. 0% (translation + rotation) Example : Solid cylinder reling ? Tom = 4 RY CRrint of contact inttantanenitly at vest wink. ground) ripe ) a Keds dvd, Ee meas ee. % Forces like static do no work in pure rolling ee A Mechanical energy is conserved (if no external work) eh 3. A thin ciaular ring of vadias R and mass M rotates in a vertical plane about a diameter with angular speed w. Find its kinetic erergy: 4. A uniform square Lamina of side and mast Mis rotating abmut fan eis passing Yrrough a comer and perpendicular te the plane Find ite moment of inertia. 1A uniform red of length Land mass Mit rotating abut an axit Yhrough one end perpendicular to the ved with angular speed. 0 Find (i) its moment of inertia, (ii) kine. energy A sold dite of radian Rand mast M rolls down en inckined plane of height h without slipping. Find the speed of its centre of mass ab the Botton (Practice More —> Understand Concepts —> Apply Formulas —> Crack JEE! * H i I * i | i i Page 5 @) ROLLING MOTION (with Friction) Pare Relling Condition A body rolls on a surface without slipping due to static friction. Vem = Row Static friction provides the required torque. =" {Examples for Pore Rol [OP et apeatir tigo0 —w Veg * Rew H . # tae Re Von | Tem EMRE ‘ dk Frichion is static Cf.) and acts in the direction } From (1) & (2) with Fes 0 that provides torque. ' ated ton. sal Sees) Direction of friction is opposite to Voy during P } (For horizontal, Gq * 0 if no extemal eccelaration down on incling = - Divection (rick) ) Solid, Sphara valing an —* Translational + Rotational Equations H H | If bly tends to ip found | ‘Meng horizontal: Fe~ fy * Maem —(1) | (rite to mf), fin | Himaceeae® feeaca Solve Numericals —> Get Strong in Rotational Mechanics! 3 4 ANGULAR QUANTITIES [2. ROTATIONAL DYNAMICS [3. Moment oF Inearia (1) + Angular displacement: @ = fr (rod) 6 Tore: F=F#xF [t= rf sind] + Angular velocity: wo = $2 + Equation of Rotational Dynamics : + Angular scilration + oh « S [fa = Ta] © Work done: dW ta8, W ls Baki exe paeee ae O + sats tat® «Rina Energy) Ks f To® [wr = we + 20(6- 6) | Angular Momenburn : Cate Peaieiemee] | [eae ia] N vey = 20N vad pee re eee + Depends on mass distribution & axis. * Unit: kgm — Dimension : [ML*T°] + Theorems # G@) Parallel Anis Theorem : I= Toy + Ma* (G) Perpendicular Asis (Lamina): Iq = Ty +Iy Gi) Additivity Tuas © E Ty (iv) Subtraction t+ I= Iiig - Zrmmned (w) Radius ef Gyration: I = Mk* Fa nelF 4. Moment oF INERTIA OF COMMON BoDiES (About Shown Axis) Thin Red (length L) Thin Red (length L) Thin Grodan Ring (radias 8) Selih ise radius R) Hollews Cylinder (vadise R, Temsfcolee nd | Ree oied Lied | Anteaks Chk pe) | Aad ede Ch Opie) a) Pe is j a | Ast syngas ; —— cle Tak G3! r-gm regu I= met I+ 4 mat Perret s Solid Cinder (radian Bhd ) Solid Sphere (rasa R) | Hella Sphara (thin, rdisn R) Rectangular Lamina (axb) Solid Sphere Rolling (no sip) About syrmatry oxi About any diameter Abit any diameter Nod cortre 1 to plane 4; Yom | Oo © x | T+ $ mr* 1-3mM = 2 me? PTA | Bata nore (Pure Reling) || 6. COMBINED TRANSLATION + RoTATION | 7. PoweR & ENERGY Velocity of any point P+ Power: Pata | ol tal Bp = Un t Ox | Kinetic Eneny Ke £ Toot Asante ify oe Ps Wok: w= [rao [dp = dem + Ext + Gx(Gxr) a= ee t 4+ T/me*) due te contripetal, fa= Ji Gem Casts op the plane) | Atrondlation =. Coane contre) 8. ImportaNT Conprmions 9. Sian CONVENTIONS (Right Hand Rule) Dimensions & UNITS ne V Riee eling > Yan = RUD at Re © © 0 Goad) + Dimensiontans eae sling +4 + onclecknise (AW) gy EA] es (TI ¥ Dineckion ef friction opposes slong t+ lockeine (CW) to: [MLtT] = Nm vrlalive mais +2 i _ dae —_ oT: [ML] = kgm (r= Ta) ca) ob: Catt) Aegdar mereka envared § vies f 3) = ve [eK w,U : [MLET“A] = Toute 14. Jee (Apvanced) LeveL Practice QueEsTIONS [let a seg mea ey we tie | A itm ef | One St ge ok | OE Ant gt ep tee OTA ee pie] bin of ong Bathinda at ond nd and” | Bw cm yw mand age sr ngs dee Tweed tev tid Sepeg. Rod rr scent | ft Ketel potion P| kof Wngh 1K Rod nee of pet | aad 8 aes Fo vat. Fed ne 4h an of copter of Be tate | bn chk on els peg Brea S| Sica apt geden parenting eid of ol ond pp ~Q ap + —) t SH H ce mR ©) F) % = r= | | Ane con® $ snd fos. 32 | ans = Brae? Ane. v2 |B ge cin fre, t= 22 | 
