For circular motion with tangential acceleration we get:

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Wesley Johnston
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1 FW Phys 13 E:\Exel files\h1-18 Fomulas eiew fo final4.do page 1 of 1 Last pinted 5/19/4 :4: PM Kinemati fomulas: x = a = onstant (1.1) = α = onstant Pojetile Motion: 1 The inemati equation eto t () = at + t+ has the following omponents: 1 y() t = yt gt + y;y = sin dy (1.) y() t = = y gt; x = os dt xt () = x t+ x We set the oigin equal to (,) and eliminate t fom equation (1.)a) by using x t = os We get y as a funtion of x: (poe it) (1.3) gx yx = xtan se Range: (1.4) R = 1 sin g Unetainty alulations: (1.5) df ln f ln f ln f = d ln f = dx + dy + f x y z dz Pola oodinates: d d du (1.6) = = u + = u + ω u dt dt dt d d = = u + ω = u + ω dt dt (1.7) (1.8) Fo iula motion with tangential aeleation we get: a = = = u + u = u u = u dt dt dt dt d d d dω ( ω ) ω α ω u

2 FW Phys 13 E:\Exel files\h1-18 Fomulas eiew fo final4.do page of 1 Last pinted 5/19/4 :4: PM (1.9) d d ω x y = + = + + z = u + ωu dt dt All onseatie foes deie fom a potential funtion U(x,y,z) (1.1) U U U F = gadu = U =,, x y z Veto opeatos: a) ( x, y, z) =,, x y z U U U b) U = gadu gadient of U =,, x y z (1.11) E E x y E z ) E = die = + + diegene of the etofield E x y z i j d) B = ulb = x y z B B B x y z Total enegy in gaitation; 1 mmg 1 d mmg (1.1) E = K + U = m = m + ω dt (1.13) The wo done by a foe oe a defined path is gien by: W = Fd = Pdx + Qdy + Rdz with F=( P, Q, R) defined path defined path The wo done on an objet with mass m by all exteio foes: (1.14) 1 1 W= K = mf mi (1.15) Gaitational foe on m by M, with the oodinate system enteed in (,,) the ente of mass of M: mmg F = u = mmg 3 ( x, y, z) ; U = mmg = mmg x + y + z x + y + z On the sufae of the eath: F=-mg dieted to the ente of the eath U=mgy with y= on the sufae of the eath

3 FW Phys 13 E:\Exel files\h1-18 Fomulas eiew fo final4.do page 3 of 1 Last pinted 5/19/4 :4: PM Keple s laws (ellipse, angula momentum onseation, gaity = entipetal foe with =a) Enegy onseation Linea momentum onseation Angula momentum onseation 1 (1.16) F = x; estoing foe of a sping; U= x (1.17) Conseation of mehanial enegy in the absene of fitional foes: E=K+U=onstant; K + U = ; K +U = K +U i i f f (1.18) Momentum and its onseation: m x p = m; elatiistially: px = If thee ae no exteio foes ating on a system, the momentum of the system is onseed. Fo all ollisions: m 1 1i+ m i= m 1 1f+ m f befoe afte In pefetly elasti ollisions the ineti enegy of the system is also onseed: (1.19) m1 1i+ mi= m1 1f+ mf befoe ollision Rotational fomulas: afte ollision (1.) The angula momentum L of a point mass m, moing with eloity is gien by the eto podut between its adius eto and the linea momentum eto p. d = = = ( + ω ) = dt ω L p m u m u u m ω = d dt

4 FW Phys 13 E:\Exel files\h1-18 Fomulas eiew fo final4.do page 4 of 1 Last pinted 5/19/4 :4: PM Note that the oss podut between two paallel etos is. (1.1) d L=m ω = m ; =u u dt Moment of inetia: (1.) N I = m dm fo the otation of objets with mass m o dm aound a fixed axis. i= 1 i i total mass i Kineti enegy of otation: (1.3) 1 K = IAω ; Paallel axis theoem: I = I + Md Angula momentum of a solid objet otating aound a fixed axis: (1.4) L= IAω A m F P = ; the pessue in a liquid of depth y, measued fom the sufae is: A N 5 lbs (1.5)Pessue: P = ρgy;[ P] = pasals = ;1atm = pa = 14.7 m in df = PdA Ahimedes piniple: The buoyant foe B equals the weight of the displaed liquid. ρ Vliquid (1.6) = ; m= ρv ρ V liquid Continuity equation; The uent density of the liquid is A j = ρ=density of the liquid times eloity of the uent. the flow pe unit time of a liquid though a oss setion A is gi dx dm dv ja = ρ A = (if is paallel to A)= ρa = = ρ dt dt dt The eto A has the alue of the oss setional aea and is pependiula to it.

5 FW Phys 13 E:\Exel files\h1-18 Fomulas eiew fo final4.do page 5 of 1 Last pinted 5/19/4 :4: PM It is theefoe only the pependiula omponents of the steamlines of, whih ontibute to the flow aoss A. The ontinuity equation says that the amount pe unit time of liquid enteing a pipe on one end is equal to the amount of liquid leaing it at anothe oss setion. If no liquid is lost o geneated, so alled steady state flow, we an expess this fat mathematially as follows: (1.7) A11= A Benoulli equation: Wo enegy theoem applied to a potion of flowing liquid. F (1.8) PdV = Adx = Fdx = dw A 1 1 P1+ ρ1 + ρgy1 = P + ρ + ρgy (1.9) Osillations: af ) = b( t) bx damping The sum of the exteio foes on the sping is then: bf ) = x bx = mx o b b t m ) x+ x + x = ; ω = ; x( t) = Ae os( ω1t+ ϕ) m m m d)diing foe: Fosωt b F iωt x+ x + x = e m m m F / m A = b ( ω ω ) + m g m (1.3) simple pendulum: ω = physial pendulum ω = g l I b t b m (1.31) xt () = Ae os ω1t; ω1 = ω 1 ; 4m Complex Numbes: (1.3) i zˆ = Ae = A + i = a+ ib + b = zz ˆˆ = i zˆ = a ib= Ae = A(os isin ) (os sin ) ; with A= a atan A b a

6 FW Phys 13 E:\Exel files\h1-18 Fomulas eiew fo final4.do page 6 of 1 Last pinted 5/19/4 :4: PM Wae equation: (1.33) y 1 y = ; y = y( xt, ) x t (1.34) yxt (, ) = ysin( x ωt+ ϕ) o i( x ωt+ ϕ) λ ω yxt ˆ(, ) = ye ; = λ f= = ; the tem in (..) is alled the phase; T The elationship between waelength, fequeny, and speed is oet fo all suh waes, whethe they ae mehanial, aousti, o eletomagneti, only if thee is no dispesion. Dispesion indiates the fat that the eloity depends on λ. Then one must ω distinguish between phase eloity = and goup eloity = d ω d T a) = ; waes on a sting; = tension diided by linea density µ B b) = ; waes in a gas with bulmodulus B and density ρ ρ (1.35) 331m TC ) = 1 + sound speed in ai, also dependent on tempeatue T s 73 Y d) = ; soundwae in a solid mateial with Young's modulus Y ρ Pessue waes: P ρ P 1 P = = ; x B t t (1.36) B ω Pxt (, ) = P sin( x ωt+ ϕ);= = ρ π asxt ) (, ) = smax os( x ωt); Pand s ae out of phase by (1.37) bpxt ) (, ) = P sin( x ωt) max P = ρ ωs, whee we put = ω/ (1.38) max max Powe 1 I ntensity= = I = ρ ωs oss setional aea A (1.39) P (1.4) I = ρ Sound leel β: max

7 FW Phys 13 E:\Exel files\h1-18 Fomulas eiew fo final4.do page 7 of 1 Last pinted 5/19/4 :4: PM I 1 W β = 1log deibels, db, with I = 1 I m (1.41) In this way the theshold of heaing lies at deibels: 1 1 db 1 = = β =1log 1 log1 db 1 Dopple effet: soue moes, eeie is stationay; Dopple effet fo sound, soue moes, obsee is stationay: s a) λ = λ 1 soue appoahing with speed s s (1.4) b ) λ = λ 1 + soue eeding with speed s To get fom waelengths λ to fequenies f, we just emembe the geneal elationship λf = and λ f = ; whee is the speed of the wae. Relatiisti Dopple effet fo light: (1.43) Dopple effet fo light (elatiisti): a) λ = λ ; f = f b) λ = λ ; f = f 1+ a)fo a sta appoahing with the speed ; blue shift b)fo a sta eeding with the speed ; ed shift 1+ Hubble onstant: One we now the speed of a galaxy we an appoximate the distane R of the galaxy to ou own with Hubble s fomula: R = H.17m 15 (1.44) H = ± 5%;1lightyea = m s lightyeas 8 8 m s lightyea R = = lty = 3.lty ± 1.6lty s.17m.17

8 FW Phys 13 E:\Exel files\h1-18 Fomulas eiew fo final4.do page 8 of 1 Last pinted 5/19/4 :4: PM (1.45) Supeposition (addition) of tigonometi funtions: aa ) 1os1+ A1os = A1os os ba ) 1sin1+ A1sin = A1os sin zˆ1+ zˆ = Ae + Ae = Aos e + 1 i i 1 i 1 1 i + nπ zˆ = e = a+ ib= os + nπ + isin + nπ has solutions: (1.46) 1 ( + nπ) i ˆ z = e fo n=, Standing Waes: When waes ae efleted they supeimpose and unde etain onditions fom standing waes. Standing waes on a sting unde tension, fo example ou on sting instuments guitas, iolins, ellos, et. Two waes taeling in opposite dietions intefee aoding to: Asin( x ωt) + Asin( x+ ωt) = Aos( ωt) sin x (1.47) I f the sting has length L, we must hae a alue fo x=l L sin L = L = nπ λn = ; fn = n; n een and odd integes. n L Double slit expeiment intefeene: (1.48) dsin = nλ yields a maximum fo n=, ± 1, ±...et Beats and goup eloity: (1.49) Supeposition of two waes with simila and ω ω y1+ y = Aos x t sin x ωt phase goup ω dω ω g = ;p = d

9 FW Phys 13 E:\Exel files\h1-18 Fomulas eiew fo final4.do page 9 of 1 Last pinted 5/19/4 :4: PM Wae patile elationships:(1.5) hf m 4 a) E = ω = hf = γhf = =(b)= =() p + m Eey mateial patile with mass m has a fequeny f, a waelength λ, m h and a momentum () p= ( d) p= = λ dω ω (1.51) Goup eloity: g = =patile eloity Phase eloity: p = d 4 ω ω E p + m m (1.5) p = = = = = 1+ > P p Heisenbeg s unetainty elationships: (1.53) x p and t E ; Equilibium onditions: Sum of the exteio toques = ; Sum of the exteio foes = F l V Stess=Modulus Stain = Y ; P = B (1.54) A l V N 1 i( x ωt) iϕ iϕ i3 ϕ i( N 1) ϕ i( x ωt) ilϕ (1.55) R ( 1... ) y = e + e + e + e + + e = e e We hae to eall the ules fo a geometi seies: N 1 N l x l a (1.56) sn = ax = ; x < 1 s = ax = x x MLauin seies: (1.57) Taylo seies: (1.58) l= l= l= ( ) x ( ) d f ( x) = f( x) = f ; with f x! dx = = x a ( ; with ) d f x f x = f a f ( a) x= a! dx

10 FW Phys 13 E:\Exel files\h1-18 Fomulas eiew fo final4.do page 1 of 1 Last pinted 5/19/4 :4: PM Poofs: Gaitation: Effetie potential enegy, total enegy Supeposition of two waes: omplex numbes appoah Simple pendulum d.e. ange fomula, inemati equations goup eloity fo elatiisti matte waes amplitude of foed osillations gien d.e. and i t Ae ω ; ω is the fequeny of the diing foe Fsinωt

Moment of inertia: (1.3) Kinetic energy of rotation: Angular momentum of a solid object rotating around a fixed axis: Wave particle relationships: ω =

Moment of inertia: (1.3) Kinetic energy of rotation: Angular momentum of a solid object rotating around a fixed axis: Wave particle relationships: ω = FW Phys 13 E:\Exel files\h 18 Reiew of FormulasM3.do page 1 of 6 Rotational formulas: (1.1) The angular momentum L of a point mass m, moing with eloity is gien by the etor produt between its radius etor