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Equations ot mation - c > ubet lope = (1) ve yt at 2) $= vet Watt gle Tea Undey v-t graph = displocement (Co) ve loucty velouty ¢ =-As(AAGC) + Axv(COIMCDE) 6 = TACK RO + Acx Cb xtix(v-v) + teu pi~ = ixteat + vb 2 from eg” G, v-ve at Crom ea @ ZAee Tee pak” ty dime . G) = dag s veut ve louty Slope za | area_urder_vrt graphs displacement Cs) [vou “) $e an (t74pezivm APCD) * tlhith yd: = it x (Adtac) x cD: LL ge ax(vtvy et from eg” Gt $2 yeu time . 6 viv}f{v-u zyteut f = | 2a6% VU a Pavalte leq ram law of vector Addition. if 2 vectors ane taken as aoljacent sides of a Ilo” with ther tails touching , then the ssultant veetoy is ~ resented in_moagnitude & direction by the diaganad foom the point of contact, In SQRT, Rh: A QRt+ RTt In DQRS, sin = QR => GRE BSHnd ie. QRz |Plsna —O Rs I'd e030 BE REE ASeane Ht RSs 13|enb-_ S [ sts RI —@ subshtute @G@8@ ™O = Rl: Tarts Cass)? (21+ Vso + Ary atentod 1ABeWO = ATH AAR cosO & (RI = (a2 +074 2Ancos6 Projectile motion the trajectory ig parabolic” 4 (6,50) fon O, k7 = --@ inban ire ----@ mies rrr) fogs qtand- se rs substihAe G) in @: veor8 = WA tha, Inthe form of _parabole 4 Mgt ¥ hy Pad 2 CEy I Dl Ane BE ‘) = sine mist ==-=O nl wrosO ae y, = tan —_ axe 3 wore UF UC x Mar. height, rarge g Kime of Flight of a projechle- y J = max height (4) Bays ¥5-M5 ies) 2Eq) H = 0- weinte & He cage 5 as ¥ Tina of fight (7) | 7 4 = Myelly 2 Do vsinds | _ysind. 4y Aecant = tage “ 4 2. T= 2usine cy DAeelly, Aecaat = Mesent “tft Ts = dusice a nn R aia F Ranqe (R mat CR) ust + hart? Con 20 \ yes Revet ppedhie Re ulosOd = veosb x Quin = wiindew® - = aes K) 3 h) Uniform reuler motion FOV = Viave Resultant velocihy —>bowards entre 9 "eumbipetel" (ae = Lak) — v cad wD. av Ges t,-t)) > “certnpetal atcelembion' a ary At ee aim > festoget force! |agl = lggl = lath Ben oad + Wag | “6 os Be = = hin, ier = yx Viol Pn) w o£, Expression for Kinetic Ereray SF ge vaeuP web? = Feeose [atsuming, cosOz 4} Za = Ma Xx $ = my(tee) Cassyming vio) we E> imvt me 2 Work Enevauy theorem. total work olone on a particle by nek fore is equal -to if change in Kinetic ener We FP 2 Feeose Cassming cose +t] We Fe - WwW, = SE, We mas) 2 ma (4) =) dmv = tmut 2 Ee, Ey = AE tote if ov z 2 f + Relation blw Kinetic energy % Lineay momentum Exs amv? ; = fo Exc 1Cryyt = _ pr 2b = p% eam 7 Pew ws Ce) = dey Gravitational fotertial Energy Work. done ty raise mass ‘mi toa height ‘W': 2 E, = mah h C We Food , tsbt0) = Wemag [a sg asth) *. we magh which is |ctored og potential energy omy Elastic Potowtial Erevan Fem % 7 Fox = Free wo Frey <= % osnrrera Bee .Ay Hoake's Lowy; Fees + “kx dws fern a> so We faws J Repl C050 (ose =) & . r =») x . . [ ay | Energy vS dicplacement LonrenniAm— Work done by Spring fora, fete r CL i ial qrah for spring MOSS sys pt ! 2. Elacke potential energy; Cp = tet Cin berms of reGho rine, fou) Slopesk Geren udey the graph = Elestic Ep v Gp: dext Cin terme of external force) é Spring force V6 displace ment araph iva Conservation of mechanical Cnevay Of o freely falling body Em = Ex+tp then; @A: Ear mgh ) Ex=0 ee Em = mgh —O @B: Ep=malh-x) » xe Emvi Am[rx+04 = max to Em= (mah = ape) + gx |= mgh —G) Ce: bexo , Ex= tmvi = Lmclagn)” =mgh —® from O,@ +0, Total mechanical energy tEm) @ A= OB =@¢ = mah Hence, tata mechanical energy of » Freely falling baly ic conserved Collicions in 4 dimension DX Perfect elashe cwllision A ry As A 2 coc @ @» CH GR @ before callision head on collision after collicion To denve for va * Vy. Ace CLM; MYat MBVa > MpVa TMB, ——CO Cr Cun-va) 5 mp (5-48) a Acc: 40 LCKE, x : £ a F aye Fy Legun + Legis = tmava + Lat, e-|_mal ua -va) = mg (vg —u5) My Cun-vp) Cunt va) = mg Va-ua) (vyt Un) | from ©; ™a(varva) = r*y(Ve-¥5) 5 Uns vq °,0 : zsh 7% Variation of pelonticl & kinetic energy 4 height Cua> ug} = Vat Vs oe Up Un = vg va — ©) Maes Ma ee (coett of aestitvbin st fer perfectly elactic collision) Up-45, Van» relative vel--of sepanadim «Pter collision Vy 5 Myst Uy Up —& Trelakive vel. of approach. before ulligion substitute in eq" Oo: MaMa tg Ug = My (Vet Mata) + Va, HOt mya + gta = mV + aUn—™Mata + Mg Va FP Rerpun + UeCMmy-mpn) = Vala tims) ae) “a = Ya (Mig-mp) + 2mava mary 19 Vgc Yalra- mg) +2 gue Matos 0 Mac Yalrra-mg) + 2rpUy % Ye, = Ya (ma-ma) + 2mava_ Mating mat™p astol's law of flvidg : A change in pressure applied w an enclosed incompressible Fluid is transmitted undiminished te wery Porton of the MIvid & to the wally of the container, Appl” . Hydraubie Litt : A, SAr Fi me @A) Pourcune ps ss which iy trangmitted unttorelg 1 ba, Oy Pre Pe a then fore produud @@ 2 PKA, ie. FAL a ve Eye Fao = hor ' t Preswre difference Consider & cylindweal clement of Hhe static fluid. Sina Fluids @ vest; Fret 0 =>-Fj +ma= Fe FL -F, =m FAP Ba Ep 2 9 tt sili such that ike top touches the surface, ‘a (Pa=P,) = p Can) eT = Non No os Pg P= Apy 4 Pressure diff, RPE hea then P—P, = poh => “Gasmge pressure” Equation of continuity 4 Ay 1 Lek AV volume of watr pass trom OD +o @- in ome 't' {0) i” : v fH ie _Ays Aan = Avat ! p) sina volume is wnst. 5 (aV,= dv) e_az—> Av 4b = Ayv, at 4 Ayy FAgva | MW? Ave AQ ZAgv, } t { fo Ay > Ayy, ie Aveconst, “volume flow rate) Bernovil's prnuple : Sum of kinetics potonbal k pressure energy of &n incompressible fluid in ¢ dreambred flow ts Ccengtant. ies P+ tov? d | ier? 4pv + poh const Bret work done & move the Fluid From O t O: AW= W2-W, = Pv-bvis v(R-8) = leh) —O HW Rec- to werk energy theorem, AW= ALK + Ep —O Oe, = Lm(vi-vi) —© eps mg Cha=h,) —@ subshitute values of AEe & AEp into ©) aw am(vin Mi) + malhi-h)) —@ # Or companng OFO, swe de (oh) = gO ow) + agg (hah) ve Poe stew tpw + pgha —pghy Rearrange the terms, Prt seve t poh, = Pt reyr+ egh, | Vemtunmeker. : To measure speed of How oF an incompressible tube. H+|L put + pgh q Pr ba pus tg perrh) 1 (va-¥) Oo Aww, = Fav, Ye Aut —@ sunchtute in; SUR 1 (Apt wt) be AGN) s a(t At-Aa)) on transposing, vies 2¢P,-) Ar _G@) From Ae U-trbe, pattwure diff. P,-P, » yo —© —subohitve © mG, v's _ye'g (zat) oy > foyer en) | t Tres oh FO0t-A3) rs Speed of efflux = Speed of body under free fall just before hitting ground pe, + apei+gghy= Baiguts gh, Ty = Pe) = Pat \=0 & h,20 Then, Be aPOr rash Ror gastegylel > ve agh heli Expsuasion for terminal velooity (y) Fig H la | f, When body reacher terminal) velocity Free 20 Fee mas Fe Fa-O -@- Fep b ¥mg = FQ = Psy —@ - Vy # Fe agg + o8ny —@ sehchitvte in | —) 4nrtpg = 4 rado'g + omyny > eltnry = £879 (9-0) _EE ay) # Figs ORV —CStoke'¢ tow) ~@) then v= 4984 (9-8) save 2nbq (p-0) ray ~ Pmncple vf calorimetry 2 Heat energy loot by tke hotter system = Heak ensrgy geined hy Lolden system [vorsider a no phase transition seenaruo) combined Sustem systems stem My OP + Esl = ray ting Ty Tr T mCi > mC) T = GT mc T) Law of Thewal vonducdion Cconditions: Slab of face aa A > length L whose faces ane mainiained @ Temp. Ta Ty dove to fag Ain tme tJ ga c Tish 2 bystewns mixed te Quelecred erin (Q=TY = mC, (T=) > TONG 20) = =. Bethel mc, 7, tm, GT, Change in temp per wie Rngth -¢ temp- gradien a, Ts ive anew minkire & thermal eqm. mat +m, GT: mie, tly CT >Tr). Constant Meat cupply @ sara A "15 ht ———— Bea Qat then Roe ATLA Ss Qs Karka. Ck: thermal conductivity of materal] Rate of heat flow, Bets zeroth law of thermodynamics + 4 2 syskems are in theumal equa adiabatic walls are in “Heamal egm: with each othr A 8 A is im thermal em wl © =>) The Te isgthermal 6 is in thermal eam wi € => Ty>Te Ta=Th Work done in _an_iso-theamal process Al) Iscthermall expansion of Vg T N. on ideal gas Te anst. => AU=O f PVE conk. i eee ee | y) F2PA T cont. 3 We Snes way AW: Fax fo w= Jaw = fete “we MRT Ln Vp 2 2-303 RT log Me . Work done in an adiahatre process awed aw= Pav — 9. w+ frave vt LSEY) NY Sy tm sia process bys const. rn { Wav res P= ‘, -~@ aha moj ¥, bet ky” Vi ts ah al ryt Ce - with a ad system sLplohatelgt ahah Aha bi vee we |PAdx = [PAV—O ew I x = | ewe nRT.% C= nRT —G) subbtinte in O v Me WRTAWV LS = nel tote =|" i) fi eal rd P= OY aK hen on PE yt Bw = (tag ~ PME) Ne (tM Pave) = CTE Te i et Mayer's Relekion 1 law of thermmpdynamics + O@s su + aw —O c= 48 [_ MST © pt 48 @ cant. wlume in © 4G6,: aU vos Sv * SQ Ay => Gr AQ . AU Se ee r a por AQ e unt. preant in; DR, = Ay + (POV), —> Mvide by ATs BQp 2 OU + PAY aT oT aT ve. Ope © Gps BY + pa —~@) Replace ups hy Cv from _G_: Coz ly +e Av/t —~©) oe Cp-Gys R pY 7 yRT Cyst) => pav= RAT ys, vay 2R —G—> subs bate ia G) : fp=tytR Lreod_gas_eql Waoyle’e law : va @ canst temp & ait. of Substank Combining all 3: Cor am idaa\ as) > Chovle's laws Vex |@_conet— amt. of substance + preeurnt Vor pT > PVR wt = Pye yRT é ? Arvegad ve! law : Vom @ const temp 2 pressure Pressure of Ideal gas ‘oar j favg * Pet ttf —D PV= wet — [R= Rydberg'c unstart = 6-4 T mal K' | Bef (where Fe chit Fax t Bat... a Fy, (for n molecules in x-direction) ry time for simatarecus ealtisions: 2 — Cat) —t, lt consider | particle: S Spy = (Mx = Gmy,,) = ZV —-@ r) ail va 7o Ay 2 nd % Fy Sts = aon = mal ey MW Bs vee (Mine velet 2" poadicle vrtta-direchion) Va then FL 5 Fes Fant s+: Se 2 a ait + ae. ~ tata ae (fe eva ai pt vie) + Pye fee ca (yet a te AP Mean Free path (A) Conditions = 4: Tas molecules —> hard spheres of diameter a 2. Collisions ave perfectly elastic 3. All molecules except the one under ohservation are @ vest. % A moving molecule can be in Ullision with other molecules around if their centyal distance of separation 18 equal to ‘dl! n= moleuake density = _no- oF molewher 2 No of molecules = nV Veluwe for_tylindrical-columnny Ve Tith = nad?) 2, no oF wolemles confined nm the cake? 64) area cweph by eentyal molecules & SUTIOUAR AG woleuules tanderotng collision = nd wghindrical OnL, sue pt. mah ; a 1 5 <% no. oF Collictons = nia? Q ae AF SE - Aistane SIONG = + im 1g - Aigtan Jw_callision aa ana (mean Free pocth) experimen 2 Y Mole = . é. Mate = aaa + PV: F N, . c 6 4 = _k,T Piglet AT = kyMAT es JT Nel = * Node apr Simple Harmonic Motion : Puvedic motion in whith displacement is a sinusoidal funtion of time & alcelevation due to : wsbring fru is directly proportional 4 Aeplacement, such that they bppese an Aiveckion Fag ae Maye, OHH PE Dye Oe =H Foku ¢) MAs—kx (4) We He ! eee (for spring mass syst.) a, x1, Cinttantaneous) So RE TONS - ME Fa A= Whe beam ‘1 CF > restoring fore) aay ad nm # Consider oscillation in a Spana- mass system eka sper linear tiple harmonic oscillation Fema = m(-ot) = -(meot)x ao fe angular Frequency) Ts an (= DFerukial @a” for sim Foc -% 4 For Ring mast system, Free = “kx % S 2 rahi: Sle be: mas —kex 4: hh) = oe Sma = ake bee mad + k= -0 oR ae + bE 20 (eimai) + amt ota 20 m Uniform Cirenlay Motion & SHM; Particle moving in uniform civetlon mole => Ite projections on the divechta oui undergo SHM In Dope’ cos = Fs Me = ReasO [ore oF ot a cs Me nest —(@ Yeo 4a < Al repeat CW es herct it ig a SHM re a= aM = Aroxnaty = = /-nWsinwt — 6) [ Compaung 040; =A eas WE —G) = Aoswt , vy =-nwsinwk 2, O, = ROME = Wr (Kepset) = 0 ay = new cos at = -w eee? i Yon as Ainenz: tn Bor! | sind = ye dive chy't “Ly? nsing [eo] yes nsinct —Q) Us agin ook ve Am = Actaswt _@ at Vee aAweoswt = wht ae Oe = -Awsinok = ~ootyy a= dive -narsinest 2 blr (asinwt) - ~ty, Simple pendulum % SHM T= mages) Frey = MOE Mgsin® we Oye, = 491N8 assuming © vem) Sine 0 0 Ones = 4° —@ 6: OA OA Of ve) OS Ue + oO , zle + v aa bia Mr hye Ayo ye 5? SHM Sneed of Aonagitudinel wave y= .|= Ep edacttity of medium = = iF Pe sensity 2 medium. Newton's formula for speed of sound : ave [8 Air >isothermal = Py = wnst.=n RT sie; Pv= Cp+aP)(v-Av) Le 2. PaveAPV 3 Po epv cB wove 4v ON = pv - Pave sev—4Pay Lewplace corrected Newton's formula for speed of sound By Newton's formula, speed of sound = 280m/s #332 m)s Cexperimentally } Laplace > "oir is not isothermedl, but rether adiahetic!? be. P¥T= conet (rPv)0 > dvi + vid p so te PIV lV + VAP 20 ah vi at de de py dys -vTdP PYAV = —y!l AP = -VdP > Pr =-v. = -AP 2B 9, Bc PT then v:|[—= ave pee APs oa a fe ak five [et Analytical treatment of interferenie of transverse wAvey 4 “Follows prrncinle | of Superpu sition VS) { i> Sree? Bit Y \4 4, s-asinCkaswt) sas asmCky= | Yre asin(ka=w te avinC kx =ck+ R, och / initial pease: in tig casey phase Adtjuunee O04 Inet-2 yt Bo Cdiplacements ane collintar in dike ase = Bain kx-wt) + atin (kak + f) = A] gin(ka-we) + sinCka= at +6 | Ear’ asin (aha mab) sf ) (4) 2 2 + -2asin(kx-wt + $6) colFA) oe aaeet(£) sin (meat ample In terms of path differene : o¢: max “Be =2ab unctructive interference | # for onst udive inkerferunce C46 22h) > ans 22a, ah Neon At # mar. amplitude c2a (when es Shea ie gs 0,20, yf.) when | Gz ank ¥ nz 0/2 )3-- ¥ min. amplitude = cs afi p Aas Salhi i } destructive interference. | for destructive interferene (AG = Gat 9M) : axe d-bati7 fensiy = =o - 7m N=Oj12--- Latensity of wave reat [a= amplitude) 1,0 fracosep))* > 2, as yates $f. os Lyesvitant = kelp co? fr = 4kIpws Ph + T,,, Crorstrudive interferenn) =t4kTy C $a + on,m...J a when $2 20; nso 2-~ #Inin Cdestrurtive interference) = 0 [FAS MATA J > when P= anti nz 07,7 Retledion of waves X Reflected wave from vigiA boundary —> phase reversal takes place along w/ divection change ; ie crest becomes trough v trovah becomes crest: 4= asin ( ke +a) Phase reversal =y a= 160° ve 4 = asin (kx=wt #1) = =asin (kx -wk) “y= asinC ka +at) § y)=—asin( ke at) ¥ Reflected wave from free boundary —> na phase weversal takes place; ie only divection charge y= asin( ky +t) then y's asinCkx wt) ys asin ka tat) and y's asinCkx-wk) Aa Stationary / tanding waves: Formed due to interference of identical waves from opp. directions. 4 NLA yz asin (kx-ot) ty’ = asin (Kx + cot) AA Yrek = asin (kaw) + asin Cke tot) <= Zasinkxoos(-mt) = zasinkxposwt Ay ce aredet ° peplieake By DS antinodes. # amkinodta 3 max, geaplitude - 2A When sinky =4 fe: kes args Th, Mee. ie Wx’ |. w= Qasr when | x= (msi) (nz 9,12, 3-4) * x z y nodes = min. amp lityde = O when sinke=0- ier kes ame = 071727 — je dex: n& es Dd when-x= bh (nz 072,34) * > a Transverse stationary wave ina ctring fined on both sides: Z| Fundamental / Natural frequency | ya ~~ Ae Les ye 2k Ve a a ‘ * S H ore ez ———— ] -—— I DL second mode2f vibration esses At fe Dae 2X9q => a* overtone (+91) pas = ve C2nd_harmontc + Ao wae eee ) : hb wa = DY Thivl mode of vibration 3 ar, et 932495) 27 overkove ene 420) MMs vs 39, SOS ye Be (FA harmonic) hy Pn ee | 3 oe Mn = (n> pvenkore $ KT hanmonc. 
