FW Phys 130 G:\130 lecture\130 tests\formulas final03.docx page 1 of 7

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1 FW Phys 13 G:\13 leture\13 tests\forulas final3.dox page 1 of 7 dr dr r x y z ur ru (1.1) dt dt All onseratie fores derie fro a potential funtion U(x,y,z) (1.) U U U F gradu U,, x y z 1 MG 1 dr MG E K U r r dt r (1.3) The wor done by a fore oer a defined path is gien by: W Fdr Pdx Qdy Rdz with F=( P, Q, R) defined path defined path The wor done on an objet with ass by all exterior fores: (1.4) 1 1 W= K f i (1.5) Graitational fore on by M, with the oordinate syste entered in (,,) the enter of ass of M: F MG u MG x, y, z; U MG MG r r x y z x y z r 3 On the surfae of the earth: F=-g direted to the enter of the earth U=gy with y= on the surfae of the earth Kepler s laws Energy onseration Linear oentu onseration Angular oentu onseration 1 (1.6) F x; restoring fore of a spring; U= x (1.7) Conseration of ehanial energy in the absene of fritional fores: E=K+U=onstant; K U ; K +U K +U i i f f

2 FW Phys 13 G:\13 leture\13 tests\forulas final3.dox page of 7 (1.8) Moentu and its onseration: x p ; relatiistially: px 1 If there are no exterior fores ating on a syste, the oentu of the syste is onsered. For all ollisions: 1 1i i 11f f before after In perfetly elasti ollisions the ineti energy of the syste is also onsered: (1.9) i i 1 1f f before ollision Rotational forulas: after ollision (1.1) The angular oentu L of a point ass, oing with eloity is gien by the etor produt between its radius etor r and the linear oentu etor p. L r d dt p r ru r dr ( u dt r r u Note that the ross produt between two parallel etors is. (1.11) d L=r r ; =ur u dt ) r ) Moent of inertia: (1.1) N I r r d for the rotation of objets with ass or d around a fixed axis. i1 i i total ass i Kineti energy of rotation: (1.13) 1 K I ; I I Md A A Angular oentu of a solid objet rotating around a fixed axis:

3 FW Phys 13 G:\13 leture\13 tests\forulas final3.dox page 3 of 7 (1.14) L IA Bernoulli Equation: 1 1 (1.15) P gy P gy Continuity equation: (1.16) 1A1 A ; inopressible fluids Wae partile relationships:(1.17) hf 4 a) E hf hf =(b)= =() p 1 1 Eery aterial partile with ass has a frequeny f, a waelength, h and a oentu () p= ( d) p 1 d (1.18) Group eloity: g =partile eloity Phase eloity: p d 4 E p (1.19) p 1 P p a) F b( t) bx daping The su of the exterior fores on the spring is then: b) F x bx x or b b t ) x x x ; ; x( t) Ae os1t d)driing fore: Fost b F x x x e F/ A b it (1.) Osillations:

4 FW Phys 13 G:\13 leture\13 tests\forulas final3.dox page 4 of 7 (1.1) siple pendulu: g gr physial pendulu l I b t b (1.) x( t) Ae os 1t; 1 1 ; 4 Coplex Nubers: (1.3) i zˆ Ae A i a ib b zz ˆˆ i zˆ a ib Ae A(os isin ) Wae equation: (1.4) (os sin ) ; with A= a artan y 1 y ; (, ) y y x t x t (1.5) y( x, t) y sin( x t ) or i( xt) yˆ( x, t) ye ; = f ; the ter in (..) is alled the phase; T The relationship between waelength, frequeny, and speed is orret for all suh waes, whether they are ehanial, aousti, or eletroagneti, only if there is no dispersion. Dispersion indiates the fat that the eloity depends on. Then one ust distinguish between phase eloity = and group eloity = d d T a) ; waes on a string; tension diided by linear density B b) ; waes in a gas with bulodulus B and density (1.6) 331 TC ) 1 sound speed in air, also dependent on teperature T s 73 Y d) ; soundwae in a solid aterial with Young's odulus Y Pressure waes: P P 1 P ; x B t t (1.7) B P( x, t) P sin( x t ); A b a

5 FW Phys 13 G:\13 leture\13 tests\forulas final3.dox page 5 of 7 a) s( x, t) sax os( x t); P and s are out of phase by (1.8) b) P( x, t) P sin( x t) ax P s, where we put = / (1.9) ax ax Power 1 Intensity= I s ross setional area A (1.3) P ax (1.31) I Sound leel β: I 1 W 1log deibels, db, with I 1 I In this way the threshold of hearing lies at deibels: (1.3) 1 1 db 1 =1log 1 log1 db 1 Doppler effet: soure oes, reeier is stationary; Doppler effet for sound, soure oes, obserer is stationary: s a) 1 soure approahing with speed s s (1.33) b) 1 soure reeding with speed s To get fro waelengths to frequenies f, we just reeber the general (1.34) relationship f and f; where is the speed of the wae. +O f ' f - s Relatiisti Doppler effet for light: (1.35) Doppler effet for light (relatiisti): a) ; f f b) ; f f a)for a star approahing with the speed ; blue shift b)for a star reeding with the speed ; red shift

6 FW Phys 13 G:\13 leture\13 tests\forulas final3.dox page 6 of 7 Hubble onstant: One we now the speed of a galaxy we an approxiate the distane R of the galaxy to our own with Hubble s forula: R H (1.36) H 5%;1lightyear s lightyears 8 8 s lightyear R.1831 ltyr 3.ltyr 1.6ltyr s (1.37) Superposition (addition) of trigonoetri funtions: 1 1 a) A1 os1 A1 os A1 os os 1 1 b) A1 sin1 A1 sin A1 os sin 1 i i1 i 1 1 zˆ1 zˆ Ae Ae Aos e i n zˆ re a ib os n i sin n has solutions: (1.38) 1 n i ˆ z r e for n=, n Standing Waes: When waes are refleted they superipose and under ertain onditions for standing waes. Standing waes on a string under tension, for exaple our on string instruents guitars, iolins, ellos, et. Two waes traeling in opposite diretions interfere aording to: Asin( x t) Asin( x t) Aos tsin x (1.39) If the string has length L, we ust hae a alue for x=l L sin L L n n ; fn n; n een and odd integers. n L (1.4)For standing sound waes in a tube, losed at one end and open at the other we need a axiu at the open end. (Flutes, organ pipes et.) Asin( x t) Asin( x t) Aos t sin x If the tube has length L, we ust hae a L= 1 alue at x=l. n1 4L sin L 1 L n 1 ; fn 1 n 1 ; for n=.1... this forula results in n 1 4L odd ultiples of the fundaental frequeny.

7 FW Phys 13 G:\13 leture\13 tests\forulas final3.dox page 7 of 7 Double slit experient interferene: (1.41) dsin n yields a axiu for n=, 1,...et Beats and group eloity: (1.4) Superposition of two waes with siilar and y1 y Aos x t sin x t phase group d g ; p d The beat frequeny is the group freqeny, and is equal to the differene between the two wae frequenies. Heisenberg s unertainty relationships: (1.43) xp and te ; Equilibriu onditions: Su of the exterior torques = ; Su of the exterior fores = F l V Stress=Modulus Strain Y ; P B (1.44) A l V N1 i( xt ) i i i3 i( N1) i( xt ) il (1.45) R 1... y e e e e e e e We hae to reall the rules for a geoetri series: N1 N l 1 x l a (1.46) sn ax ; x 1 s ax 1x 1x MLaurin series: (1.47) l l x x ( ) ; with d f f x f f x! dx l