SUMMARY OF PRINCIPLES. LINEAR MOTION KINEMATICS (constant acceleration) KE = W net

Transcription

1 SUMMRY OF PRINCIPES INER MOTION KINEMTICS (cstat acceleati) x = v t+ ½at v= v + at v = v + ax T apply i tw dimesis, chse a x-y cdiate system s that the acceleati cicides with eithe the x y diecti. The the mti alg the the cdiate diecti is at cstat velcity (cmpet acceleati i that diecti is ). The cmpets mti i the x ad y diectis ae aalyzed sepaately. The cmpets mti ca be cmbied t give magitude ad diecti by usig Pythagas ad the actaget ucti. NEWTON S FIRST W (EQUIIBRIUM): I a bject is at est mvig at cstat velcity (cstat speed i a staight lie), thee is NET ce actig it. i.e. The vect sum all the ces actig the bject is. F = which meas F = ad F = NEWTON S SECOND W (DYNMICS) I thee is a et ce actig a bject, the it will be acceleatig. The acceleati is give by: F v a = s F = ma ad a, F have the same diecti m Chsig a x-y cdiate system s that the acceleati is i the x- diecti, istace, yields: ΣF x = ma ad ΣF y = UNIFORM CIRCUR MOTION F a bject mvig i a cicula path at cstat speed v thee is a cetipetal acceleati, a c, diected twad the cete the cicle: a ad this acceleati must be pduced by a et cetipetal ce diected adially iwad such that c = v Fc = Fadial = m x v y CONSERVTION OF MECHNIC ENERGY The chage i the kietic eegy a system equals the et wk de the system (Wk-Eegy Theem). The et wk is the algebaic sum the wk de due t all the ces actig the system, equivaletly, the wk de by the et ce actig the system. KE = W et The wk de by a ce is give by W = (F csθ) s whee θ is the agle betwee the ce ad the displacemet the bject. I the et wk de a system is due ly t ces which a ptetial eegy ca be deied (e.g. csevative ces such as gavity, the ideal spig ce, the electic ce), the Mechaical Eegy is Cseved. E = E KE + PE = KE + PE I geeal, t iclude -csevative ces: W c = KE + PE (KE + PE ) = E E whee W c = wk de by the -csevative ces (e.g. icti) I the case icti, W c is egative because the ce icti is diected ppsite t the displacemet (agle betwee ictial ce ad displacemet is 8 ) CONSERVTION OF MOMENTUM I a system is t acted by a et exteal ce, the the ttal mmetum the system is cstat. (The mmetum a bject with mass m mvig with velcity v is p = mv.) P = P i.e. p + p + p +... = p + p + p +... x x 3x x x 3x ad py + py + p3y +... = py + py + p3y +... i.e. the ttal mmetum is cseved i each cmpet diecti.

2 ROTTION KINEMTICS Cside a pit the im a wheel adius, a bject mvig i a cicula path s adius. Recall s = θ (deiiti adia θ agle measue). I the wheel is tuig, the bject mvig (i the cicula path), the v = s/ whee s is the distace tavelled alg the cicle i time. t ay paticula istat the velcity is taget t the cicula path. s θ θ v = = = θ / is the ate tati. valid uit θ / is evlutis pe miute, althugh the SI uit is adias/sec. θ / is give the symbl ω ad is called the agula velcity. v = ω I the wheel ( bject) is spiig aste ad aste slwe ad slwe, the a pit the im is acceleatig. The (tagetial) acceleati is give by v ω a t = = = α whee α is called the agula acceleati. We ca wite the kiematic equatis a pit the im as: s= v t+ a t v= v + at v = v + as ½ t t Hweve, pits at the pats the wheel will have dieet s, v, ad a t depedig thei distaces m the cete the wheel. I we ecall that s = θ, v t = ω, ad a t = α, the we ca wite the kiematic equatis usig the agula vaiables: θ = ω + t αt ω = ω + αt ω = ω + αθ ad these equatis apply t all pits the wheel, t just at the im. ROTTION DYNMICS I a bject is udegig agula acceleati thee must be a et tque actig it, such that Στ ext = Iα whee I = mmet ietia = Σm Tque τ is deied as: t τ = Fl whee the mmet am l is the pepedicula distace betwee the lie acti the ce F ad the axis tati. I a bject is i equilibium, the ΣF ext = ad Στ = ROTTION WORK ND KINETIC ENERGY The tatial wk de by a cstat tque τ actig thugh a agula displacemet θ is W R = τθ. Fm the wk-eegy theem, KE R = Iω is the kietic eegy a bject with agula speed ω ad mmet ietia I. The kietic eegy a igid, exteded bdy is the sum its taslatial kietic eegy ad its tatial kietic eegy. I a bject is llig withut slippig, the taslatial speed ad agula velcity ae elated by NGUR MOMENTUM v = ω. = Iω whee = agula mmetum bject with mmet ietia I tatig abut a ixed axis with agula velcity ω. The ttal agula mmetum a system emais cstat i the et exteal tque actig the system is : I ω = I ω whe Στ ext = SIMPE HRMONIC MOTION ccus due t a Hke s aw Fce (a estig ce), F = cstat displacemet m equilibium F = kx = ma (m Newt s d aw) Kiematic Equatis SHM: x = cs ( ωt) v = ωsi ( ωt) a = ω cs ( ωt) whee = maximum displacemet = amplitude v max a max = = π ω = k = maximum speed T m k = = = = m 4π ω maximum acceleati T Nte that x ad a have maximum magitude whe displacemet is at maximum ad speed is. v has maximum magitude whe bject is

3 passig thugh equilibium psiti (whee displacemet ad acceleati ae ). Examples SHM: Mass ideal spig Simple Pedulum k g gula Fequecy ω = ω = m Oscillati Peid Oscillati Fequecy ENERGY RETIONS IN SHM m T = π T = π k k = = π m π E = KE + PE = ½mv + ½kx = ½k = ½mv max KE = maximum whe PE = (at x =, equilibium psiti) PE = maximum whe KE = (at x =, maximum displacemet) PRESSURE ND DEPTH IN STTIC INCOMPRESSIBE FUID P = P + ρgh whee P is the pessue at a depth h belw a pit whee the pessue is P. BUOYNT FORCE: bject whlly patially submeged i a luid eels a upwad buyat ce equal t the weight the vlume luid that has bee displaced: F B = W luid = ρ gv luid whee ρ is the luid desity ad V luid is the vlume luid displaced. I the bject is cmpletely submeged: V luid = V bject I the bject is latig: F B = weight bject = M bject g = ρ bj V bj g INCOMPRESSIBE FUID FOW: Q = vlume lw ate = cstat v = v BERNOUI S EQUTION: POISEUIE S W (VISCOUS FOW) 4 πr ( P P) Q = 8η g P + ½ρv + ρgy = P + ½ρv + ρgy HRMONIC WVES (SINE WVE) pduced by a SHM scillat Wave displacemet as a ucti psiti ad time is give by: g πx yxt (, ) = si πtm λ v = wave speed = λ/t = λ T = peid = time e cmplete scillati λ = wavelegth = distace betwee pits wave that have idetical chaacteistics. e.g. distace betwee csecutive cests I a wave is ppagatig uimly m a pit suce, the itesity, I, (= pwe lwig thugh uit aea) is P P I = = 4 π whee = distace m the suce ad = 4π = aea the sphee thugh which the pwe is passig. I eegy lsses ae egligible, the P is a cstat (eegy is cseved) ad I I = I ; DOPPER EFFECT chage i equecy assciated with the elative mti a wave suce ad bseve v ± v = s vs m v whee = bseved equecy s = suce equecy v = bseve speed v s = suce speed v = wave speed The uppe sig is used whe the bseve mves twad the suce the suce mves twad the bseve, ad the lwe sig is used mti away. CONSTRUCTIVE ND DESTRUCTIVE INTERFERENCE F tw wave suces vibatig i phase, a dieece i path legths that is a itege umbe wavelegths (λ, λ, 3λ,...) leads t cstuctive

4 iteeece; a dieece i path legths that is a dd multiple hal the wavelegth (λ/, 3λ/, 5λ/,...) leads t destuctive iteeece. STNDING WVES ND RESONNCE Whe waves ca ly exist alg a cetai legth (e.g. guita stig, ga pipe) the ly cetai equecies waves ca exist whe the medium is made t vibate by a distubace a scillat. The esat equecies ae detemied by the buday cditis, i.e. by what happes at the eds the medium. N F a stig ixed at bth eds, the eds must be des. λ v v = λ = = = λ = λ = v Pssible esat equecies ae: = v ; = 3,,,K Sud Waves i Pipes: COSED PIPE (clsed at e ed, pe at the the) N OPEN PIPE (pe at bth eds) N N = dd multiple λ /4 + λ = ; =,,,K 4 N + v = 4 = λ = v EECTROSTTICS Eects due t chage distibutis ae detemied by calculatig the eect due t each chage, ad the addig these idividual eects (vect sum ce ad electic ield, algebaic sum ptetial). Electstatic ce betwee chages q ad q sepaated by distace is: F k qq = k = 9. 9 N m /C The ce is diected alg the lie betwee the tw chages; ad is epulsive i the chages ae bth psitive bth egative, ad attactive i e chage is psitive ad the the egative. EECTRIC FIED: The electic ield at a pit i space is the ce that wuld act a uit psitive chage placed at this lcati. (i.e. E = ce pe uit chage) The ce actig a chage q placed whee thee is a electic ield E is F = q E Nte that the ce is i the same diecti as the electic ield i q is psitive, ad is i the ppsite diecti t the electic ield i q is egative. The magitude the electic ield pduced by a pit chage q is kq E = whee is the distace m q E pits away m a psitive chage ad twad a egative chage. EECTRIC POTENTI: The electic ptetial at a pit i space is the ptetial eegy that a uit psitive chage wuld have i placed at this lcati. (V = EPE/q = EPE pe uit chage) chage q placed whee thee is a ptetial V has a electstatic ptetial eegy: EPE = q V I a chage q mves thugh a ptetial dieece V vlts, the the chage i the electstatic ptetial eegy this chage is PE = q V Fm eegy csevati: E = E

5 KE + EPE = KE + EPE EPE EPE = KE KE (EPE EPE ) = KE KE ΕPE = KE ad ΕPE = q V i.e. a chage q mvig thugh a ptetial dieece lses EPE ad gais a equivalet amut KE. Uit electvlt (ev): Whe the chage a bject is a small multiple the elemetay chage, e, it is te cveiet t expess the chage i uits e athe tha Culmbs, ad eegy i uits electvlts athe tha Jules. e.g. The chage i ptetial eegy whe a bject with a chage +4e is mved thugh a ptetial dieece 5 Vlts is: ΕPE = q V = (4e)(5V) = ev The ev, called the electvlt, is a valid uit eegy. The cvesi act ev t Jules is btaied by substitutig the value e: ev = (.6 9 C)(V) = (.6 9 C)(J/C) ev =.6 9 J EECTRIC POTENTI DUE TO POINT CHRGES Deiig the electic ptetial t be at a iiite distace m a pit chage q, the electic ptetial at a distace m a pit chage q is kq V = RETION BETWEEN EECTRIC FIED ND CHNGE IN EECTRIC POTENTI Cside a pit chage +q placed i a uim electic ield. The chage is mved a distace s (m pit t pit B) alg a electic ield lie, i the diecti the ield. The chage i ptetial, V (= V B V ) equals the egative the wk de by the electic ce actig the chage divided by the chage (egative wk de pe uit chage). Wel Fel scs qe s V = = = = E s q q q E V = s I the esistace, R, a cmpet is cstat ve a age vltage ad cuet values, the abve elatis ae called Ohm s aw. The pwe dissipated i a cmpet cayig cuet I, acss which thee is a vltage dp V is: P = VI I the cmpet is a esist, the COMPONENTS IN SERIES: P= VI = I R= V R have same cuet ttal vltage dp = sum idividual vltage dps have equivalet esistace R se whee R se = R + R + R COMPONENTS IN PRE: have same vltage dp ttal cuet = sum idividual cuets have equivalet esistace R pa whee = + + +K R R R R pa 3 KIRCHHOFF S WS: Cuet aw Ttal cuet eteig a pit i a cicuit = ttal cuet leavig this pit Vltage aw I a cmplete lp i a cicuit, the sum the applied em s equals the sum the vltage dps. RE VOTGE SOURCE: eal vltage suce (as cmpaed t a ideal e) has a iteal esistace. eal vltage suce is epeseted as a ideal suce i seies with a esistace: EECTRIC CIRCUITS Vltage, Cuet, ad Resistace a cicuit cmpet ae elated by R V V = ; V = IR ; I = I R E

6 MGNETISM I geeal, mvig chages, whethe i the m mvig chaged paticles cuet i a wie, eel a ce whe thee is a exteal magetic ield. This ce is called the etz Fce. The ce a paticle chage q mvig with speed v i a magetic ield B is: F = q vb siθ whee θ is the agle betwee v ad B. F is t v ad Bad its diecti is give by a ight - had ule: Exted the ight had s that the iges the ight had ae pitig i the diecti the magetic ield ad the thumb the ight had is pitig i the diecti the velcity v the chage. The palm the had is w pitig i the diecti the magetic ce that acts a psitive chage. I the chage is egative, the ce is i the ppsite diecti. Whe B v, F = q vb Sice F v, whe a chaged paticle mves it a egi uim magetic ield the i v B, the etz ce causes uim cicula mti. Fm F = ma, F etz = ma c qb v = mv = qb mv The adius the cicula tajecty the paticle depeds its mass, speed, ad chage, ad the magetic ield. The diecti cuvatue, sese, the tajecty depeds the sig q ad is give by the ighthad ule. THE REFECTION OF IGHT MIRRORS Behavi is detemied by aw Relecti: agle icidece = agle electi THE REFRCTION OF IGHT ight tavels slwe i a medium tha i a vacuum. c speed light i vacuum = eactive idex = = v speed light i medium Sell s aw θ θ siθ = siθ I >, the θ < θ Ttal Iteal Relecti: I <, the θ = θ c = citical agle, θ = 9 ( eacti ccus). ENSES behavi detemied by Sell s aw Reacti Paallel light cusses (cvegig les) appeas t igiate (divegig les) m a pit a distace (cal legth) m the les. Picipal Rays (leses). Ray paallel t picipal axis eacts thugh, seems t have cme m, the cus.. Ray thugh twad the cus eacts paallel t the picipal axis. 3. Ray thugh the cete the les is udeviated. Sig Cveti ( sigle les, bject distace is psitive): + cvegig divegig d bject same side les as icidet light bject ppsite side les m icidet light d i image ppsite side les m icidet light image same side les as icidet light m upight image iveted image F multiple les systems, image st les is bject d les, image d les is bject 3d les,... es Equati:

7 = + d d image height h d i magiicati, m = = = bject height h d i Huma Eye ad Optical Istumets The mal ea pit is the smallest bject distace that the eye ca accmmdate. i.e. The smallest bject distace at which the eye ca maitai cus withut stai. Ote take as 5 cm. Optical istumets ae used t btai a lage etial image tha ca be btaied by viewig the bject diectly with the uaided eye. gula Magiicati etial agle subteded by image m istumet M = agle subteded by ea pit PHYSIC (WVE) OPTICS Iteeece ight et λ = wavelegth light d = slit sepaati θ = agle bsevati = scee distace y = distace alg scee m cete Maxima, bight iges, cstuctive iteeece ccu whe the dieece i distaces tavelled by light m the dieet slits = mλ (m = itege). i.e. The path legth dieece must be a itegal umbe wavelegths cstuctive iteeece t ccu. i The path legth dieece betwee csecutive slits spaced a distace d apat is d siθ. maxima ccu whe d siθ = mλ Whe θ is small, taθ, which equals y/, is appx. siθ F small agles, maxima ccu at scee psitis y such that d y = mλ mλ y = d MODERN PHYSICS Themal Radiati The ate at which a bject adiates themal eegy (e-m adiati) depeds its tempeatue (i Kelvi), its suace aea,, ad its emissivity, e, by the Stea-Bltzma aw: P = σ et 4 σ = Stea s cstat = W/m K 4 bject als absbs themal adiati m its suudigs at a ate give by: P abs = σ et su 4 Plack ppsed the quatum they i explaiig the bsevatis assciated with blackbdy adiati. Physical systems ca ly have cetai eegies, called quatum states. Plack s quatum they assumed that the atmic scillats emittig blackbdy adiati culd have ly the discete values eegy E =, h, h, 3h,..., h s applied t electmagetic waves, the quatum they becmes the pht (paticle) they light. d θ y The eegy a pht is give by: E = h = hc λ whee h = Plack s cstat = J s = ev s scee The pwe (ate eegy) emissi a mchmatic light suce ca be expessed as: P = eegy emitted = # phts emitted eegy pe pht

8 pe uit time pe uit time Phtelectic Eect Phts light stikig a mateial give thei eegy t elects with which they cllide. The maximum kietic eegy the ejected elects is give by the eegy the icidet pht mius the wk ucti the mateial (the miimum eegy equied t ee a elect). KE max = h W Cmpt Scatteig Phts scatteig m ee elects lse eegy i the cllisi. The lss eegy meas a icease i wavelegth. pplyig csevati eegy ad mmetum t the pht-elect cllisi (a paticle pcess): h λ= λ λ= θ mc ( cs ) whee λ is the shit i the wavelegth the phts, m is the mass the elect, ad θ is the scatteig agle. Bh Mdel the Hydge tm Bh applied the pht they t the atm ad als equied that the agula mmetum the elect i its bit aud the pt was quatized (i.e. culd ly have cetai values). The quatizati cditi quatized the elect bit adius ad the eegy the atm: h = mke = (. 59 m ) 4π E 8 ke 8. J 36. ev = = = The wavelegth the pht emitted absbed whe the eegy state a atm chages m i t is detemied m hc E = E E = i λ 36 λ =. ev = R hc i i R = Rydbeg cstat =.97 7 m X-ay Pducti The maximum eegy phts pduced whe elects KE = ev cllide with atms ad lse all thei eegy is: E max = KE = ev h = hc = ev λ NUCER PHYSICS The ucleus csists euts ad pts. The vlume the ucleus vaies diectly with the umbe ucles (euts ad pts, ). Sice a sphee vlume is R 3, the adius the ucleus vaies as /3. i.e. V s R 3 s R /3 The ucleus is bud tgethe by the stg uclea ce, which vecmes the electstatic epulsi betwee the pts. Sice the ucleus is a bud system, eegy must be added t a ucleus (i.e. wk must be de it) t sepaate it it its cmpet euts ad pts. This is called the BINDING ENERGY. Fm E = mc, i the eegy the bud ucleus is less tha that the sepaate ucles, the the mass the bud ucleus must be less tha the mass its cmpet ucles. This is called the MSS DEFECT MSS DEFICIT. Sice mass tables ae i tems the mass the eutal atm (icludig Z elects), athe tha the bae ucleus, the bidig eegy, BE, is calculated as llws: ( ) BE = ( Mass deect) c = Zm + Nm M c H Z

9 whee Z M is the atmic mass ad m H is the mass the hydge atm, (athe tha the pt mass), t iclude the Z elects. Systematics Stability dditial euts keep the pts apat m each the, but as me euts ae added they g it highe eegy states. I N is the umbe uclei that decay i a sht time iteval, the it is easable that N shuld be pptial t the iitial umbe ustable uclei, N, ad the size the time iteval: N N Realizig that N is egative (the umbe adiactive uclei emaiig deceases with time), ad deiig λ as the (+) cstat pptiality: N = λ N The sluti t this equati is: λ N = N e t Medium mass uclei have the highest bidig eegy pe ucle (they ae the mst tightly bud). Radiactivity Ustable uclei mve twad stability by emittig paticles ad/ eegy. Nucle umbe, electic chage, eegy, mmetum, ad agula mmetum ae all cseved i adiactive decay. lpha Decay emissi a α paticle (a α is a 4 He ucleus, pts ad euts) Beta Decay The tem beta decay is used t ee t tw dieet pcesses: emissi a elect (β ) ad emissi a psit (β + ). I bth cases, des t chage. F β decay, a eut chages it a pt ad a elect, the elect beig ejected m the ucleus. Z iceases by ad N deceases by. F β + decay, a pt chages it a eut ad a psit, the psit beig ejected m the ucleus. Z deceases by ad N iceases by. Gamma Decay elease e-m eegy (as a pht) by a excited ucleus Radiactive Decay Radiactive decay is a adm pcess. It is t pssible t pedict whe a paticula ustable ucleus will udeg decay. whee N is the umbe ustable uclei at t =, ad λ is the decay cstat. Hal-lie The hal-lie, T /, a adiactive mateial is deied as the time equied the umbe adiactive uclei t dp by a act. Hal-lie ad decay cstat ae elated by: 693. T / = λ ctivity The activity,, a adiactive sample is deied as the ate at which the adiactive uclei ae decayig. N λt λt = = λn = λn e = e whee = iitial activity = λn Nuclea Fissi Sice medium mass uclei have highe BE/ucle, i a heavy ucleus splits it medium uclei, the highe BE/ucle esults i a elease eegy. Nuclea Fusi Sice medium mass uclei have highe BE/ucle, i tw light uclei ca be made t ji, the highe BE/ucle esults i a elease eegy.