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

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1 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 r and the linear momentum etor p. L r dθ ω dt p r m ru r dr m( u dt r + rωu θ ) Note that the ross produt between two parallel etors is. (1.) dθ Lmr ω mr ; u r uθ dt mr ω ) Moment of inertia: (1.3) N I mr r dm for the rotation of objets with mass m or dm around a fixed axis. i 1 i i total mass i Kineti energy of rotation: (1.4) 1 K I ω ; I I + Md A A m Angular momentum of a solid objet rotating around a fixed axis: (1.5) L IAω Wae partile relationships: hf m 4 a) E ω hf γhf (b) () p + m (1.6) Eery material partile with mass m has a frequeny f, a waelength λ, m h h and a momentum () p ( d) p λ λ m dω ω (1.7) Group eloity: g partile eloity Phase eloity: p d

2 FW Phys 13 E:\Exel files\h 18 Reiew of FormulasM3.do page of 6 4 ω ω E p + m m (1.8) p 1+ > P p (1.9) Osillations: af ) b( t) bx dampening The sum of the exterior fores on the spring is then: bf ) x bx mx or b b t m x ) + x + x ; ω ; xt ( ) Ae os( ω1t+ ϕ) m m m ( ) g m (1.1) simple pendulum: ω physial pendulum ω gr l I b t b m (1.11) xt () Ae os ω1t; ω1 ω 1 ; 4m Complex Numbers: (1.1) iθ zˆ Ae A θ + i θ a+ ib + b zz ˆˆ θ iθ zˆ a ib Ae A(osθ isin θ ) Wae equation: (1.13) (os sin ) ; with A a artan y 1 y ; y y( xt, ) x t (1.14) yxt (, ) ysin( x ωt+ ϕ) or i( x ωt+ ϕ) λ ω yxt ˆ(, ) ye ; λ f ; the term in (..) is alled the phase; T The relationship between waelength, frequeny, and speed is orret for all suh waes, whether they are mehanial, aousti, or eletromagneti, only if there is no dispersion. Dispersion indiates the fat that the eloity depends on λ. Then one must ω distinguish between phase eloity and group eloity d ω d A b a

3 FW Phys 13 E:\Exel files\h 18 Reiew of FormulasM3.do page 3 of 6 a T µ ) ; waes on a string; tension diided by linear density B b) ; waes in a gas with bulmodulus B and density ρ ρ (1.15) 331m TC ) 1 + sound speed in air, also dependent on temperature T s 73 Y d) ; soundwae in a solid material with Young's modulus Y ρ Pressure waes: P ρ P 1 P ; x B t t (1.16) B ω Pxt (, ) P sin( x ωt+ ϕ); ρ π asxt ) (, ) smax os( x ωt); Pand s are out of phase by (1.17) bpxt ) (, ) P sin( x ωt) max P ρ ωs, where we put ω (1.18) max max Power 1 I ntensity I ρ ωs ross setional area A (1.19) ( ) (1.) I P ρ max Sound intensity: I 1 W β 1log deibels, db, with I 1 I m (1.1) In this way the threshold of hearing lies at deibels: 1 1 db 1 β 1log 1 log1 db 1

4 FW Phys 13 E:\Exel files\h 18 Reiew of FormulasM3.do page 4 of 6 Doppler effet: soure moes, reeier is stationary; Doppler effet for sound, soure moes, obserer is stationary: s a) λ λ 1 soure approahing with speed s s (1.) b ) λ λ 1 + soure reeding with speed s To get from waelengths λ to frequenies f, we just remember the general relationship λf and λ f ; where is the speed of the wae. Doppler effet: reeier moes, soure is stationary The relationship between speed of sound and waelength is in the referene frame of the soure: λ f or a) f λ In the referene frame of the moing reeier, it is ' f ' ' b)' λ f ' or a) f' λ f As we hae ' ± V, we get: R ± VR VR ) f ' f f 1± The frequeny inreases when the reeier moes towards the stationary soure. Relatiisti Doppler effet for light: (1.3) Doppler effet for light (relatiisti): a) λ λ ; f f b) λ λ ; f f 1+ a)for a star approahing with the speed ; blue shift b)for a star reeding with the speed ; red shift 1+ Hubble onstant: One we now the speed of a galaxy we an approximate the distane R of the galaxy to our own with Hubble s formula:

5 FW Phys 13 E:\Exel files\h 18 Reiew of FormulasM3.do page 5 of 6 R H.17m 15 (1.4) H ± 5%;1lightyear m s lightyears 8 8 m s lightyear R ltyr 3.ltyr ± 1.6ltyr s.17m.17 (1.5) Superposition (addition) of trigonometri funtions: θ θ1 θ + θ1 aa ) 1osθ1+ A1osθ A1os os θ θ1 θ + θ1 ba ) 1sinθ1+ A1sinθ A1os sin iθ θ θ θ zˆ1+ zˆ Ae + Ae Aos e θ + θ 1 i 1 i 1 1 ( ) ( ) ( ) i θ+ nπ zˆ re a+ ib os θ + nπ + isin θ + nπ has solutions: (1.6) 1 ( θ+ nπ) i ˆ z r e for n, Standing Waes: When waes are refleted they superimpose and under ertain onditions form standing waes. Standing waes on a string under tension, for example our on string instruments guitars, iolins, ellos, et. Two waes traeling in opposite diretions interfere aording to: Asin( x ωt) + Asin( x+ ωt) Aos( ωt) sin x (1.7) I f the string has length L, we must hae a alue for xl L sin L L nπ λn ; fn n; n een and odd integers. n L If we are dealing with a situation lie a solid bar, or a tube whih is open at (1.8) both ends we must hae a osine funtion whih has ± 1 alues for xl L os L ± 1 L nπ; λn ; fn n; n een and odd integers n L

6 FW Phys 13 E:\Exel files\h 18 Reiew of FormulasM3.do page 6 of 6 If we hae a system whih is losed at one end and open at the other, we must hae a sin L ± 1 at the open end, whih means that n+1 π n+1 4L L π L π λ ; f ( n+1 ), n,1,... n+1 4L n+1 n+1 n+1 λn+1 We get only odd numbers: 3 5 f1 ; f3 ; f5 ; et 4L 4L 4L Double slit experiment interferene: (1.9) dsin θ nλ yields a maximum for n, ± 1, ±...et Beats and group eloity: (1.3) Superposition of two waes with similar and ω ω y1+ y Aos x t sin x ωt phase group ω dω ω g ;p d Heisenberg s unertainty relationships: (1.31) x p and t E ; Equilibrium onditions: Sum of the exterior torques ; Sum of the exterior fores F l V StressModulus Strain Y ; P B (1.3) A l V N 1 i( x ωt) iϕ iϕ i3 ϕ i( N 1) ϕ i( x ωt) ilϕ (1.33) R ( 1... ) y e + e + e + e + + e e e We hae to reall the rules for a geometri series: N 1 N l x l a (1.34) sn ax ; x < 1 s ax x x MLaurin series: (1.35) l l l ( ) x ( ( ) ) d f ( x) ( ) f( x) f ; with f x! dx

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

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