PHYSICS FORMULA LIST

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1 0.: Phsc onsns Spee of gh c m/s Pnck consn h J s hc 4 ev-nm Gon consn G m 3 kg s ozmnn consn k J/K Mo gs consn 8.34 J/(mo K) ogo s numbe mo hge of eecon e Pemeb of cuum 0 4π 0 7 / Pem of cuum ɛ F/m ouomb consn 4πɛ m / F consn F /mo Mss of eecon m e kg Mss of poon m p kg Mss of neuon m n kg omc mss un u kg omc mss un u MeV/c Sefn-ozmnn σ W/(m K 4 ) consn beg consn m oh mgneon J/ oh us m Sn mosphee m P Wen spcemen b m K consn MEHS.: Vecos oon: = î + ĵ + z ˆk Mgnue: = = + + z o pouc: b = b + b + z b z = b cos PHYSS FOMU S Pojece Moon: u sn O u cos = u cos, = u sn g g = n u cos u sn =, = u sn, H = u sn g g g.3: ewon s ws n Fcon ne momenum: p = m ewon s fs w: ne fme. ewon s secon w: F = p, ewon s h w: F = F Fcon foce: f sc, m = s, F = m nkng nge: g = n, g = +n n enpe foce: F c = m, c = Pseuo foce: Fpseuo = m 0, u H f knec = k F cenfug = m Mnmum spee o compee ec cce: mn, boom = 5g, cos onc penuum: = π g mn, op = g oss pouc: î b b ˆk ĵ b = ( b z z b )î+( z b b z )ĵ +( b b )ˆk b = b sn.: Knemcs ege n nsnneous Ve. n cce.:.4: Wok, Powe n Eneg Wok: W = F S = F S cos, Knec eneg: K = m = p m W = F S Poen eneg: F = U/ fo consee foces. U gon = mgh, mg U spng = k = /, = / ns = / ns = / Wok one b consee foces s ph nepenen n epens on on n n fn pons: Fconsee = 0. Moon n sgh ne wh consn : = u +, s = u +, u = s Wok-eneg heoem: W = K Mechnc eneg: E = U + K. onsee f foces e consee n nue. ee Veoc: / = Powe P = W, P ns = F

2 .5: ene of Mss n oson m m 3 m 5 m m m m m( +b ) ene of mss: cm = m m m, cm = m b M of few usefu confguons: ng sk she sphee o hoow so ecnge. m, m sepe b : m m m m +m m m +m heoem of Pe es: = cm + m c cm. nge (M eno) c = h 3 h 3 h heoem of Pep. es: z = + z 3. Semccu ng: c = π 4. Semccu sc: c = 4 5. Hemsphec she: c = 6. So Hemsphee: c = 3 8 π 3π 4 3π 7. one: he hegh of M fom he bse s h/4 fo he so cone n h/3 fo he hoow cone. Moon of he M: M = m cm = m M, p F cm = M cm, cm = e M 3 8 us of Gon: k = /m ngu Momenum: = p, = ω oque: τ = F, τ =, τ = α P F O onseon of : τ e = 0 = = cons. Equbum conon: F = 0, τ = 0 Knec Eneg: K o = ω nmcs: τ cm = cm α, Fe = m cm, p cm = m cm K = m cm + cmω, = cm ω + cm m cm mpuse: J = F = p efoe coson oson: m m fe coson m m.7: Gon Gon foce: F = G mm m F F m Momenum conseon: m +m = m +m Esc oson: m + m = m + m oeffcen of esuon: e = ( ) {, compee esc = 0, compee n-esc f = 0 n m m hen =. f = 0 n m m hen =. Esc coson wh m = m : = n =..6: g o nmcs Poen eneg: U = GMm Gon cceeon: g = GM Von of g wh eph: g nse g ( h Von of g wh hegh: g ouse g ( h Effec of non-sphec eh shpe on g: g poe > g equo ( e p km) Effec of eh oon on ppen wegh: ω ) ) ngu eoc: ω =, ngu cce.: α = ω, ω =, α = ω, oon bou n s wh consn α: = ω = α ω = ω 0 + α, = ω + α, ω ω 0 = α Momen of ne: = m, = m mg =mg mω cos Ob eoc of see: o = Escpe eoc: e = GM mg GM mω cos

) cceeon: = me peo: = π ω = k m = ω = π m k spcemen: = sn(ω + φ) Veoc: = ω cos(ω + φ) = ±ω Sufce enson: S = F/ Sufce eneg: U = S Ecess pessue n bubbe: p se: h = p = S/, S cos ρg p sop = 4S/ Poen

3 Kepe s ws: Fs: Epc ob wh sun one of he focus. Secon: e eoc s consn. ( / = 0). h: 3. n ccu ob = 4π GM 3..8: Smpe Hmonc Moon o.9: Popees of Me Mouus of g: Y = F/ P /, = V V, η = F ompessb: K = = V Posson s o: σ = V P e sn ongun sn = / / Esc eneg: U = sess sn oume Hooke s w: F = k (fo sm eongon.) cceeon: = me peo: = π ω = k m = ω = π m k spcemen: = sn(ω + φ) Veoc: = ω cos(ω + φ) = ±ω Sufce enson: S = F/ Sufce eneg: U = S Ecess pessue n bubbe: p se: h = p = S/, S cos ρg p sop = 4S/ Poen eneg: U = k U 0 Knec eneg K = m K 0 o eneg: E = U + K = mω Hosc pessue: p = ρgh uon foce: F = ρv g = Wegh of spce qu Equon of connu: = enou s equon: p + ρ + ρgh = consn Smpe penuum: = π g oce s heoem: effu = gh Vscous foce: F = η Phsc Penuum: = π mg Soke s w: F = 6πη F oson Penuum = π k Poseu s equon: Voume fow me = πp4 8η emn eoc: = (ρ σ)g 9η Spngs n sees: k eq = k + k k k Spngs n pe: k eq = k + k k k Supeposon of wo SHM s: ɛ δ = sn ω, = sn(ω + δ) = + = sn(ω + ɛ) = + + cos δ n ɛ = sn δ + cos δ

4 .: Wes Moon Wes Gene equon of we: =. oon: mpue, Fequenc ν, Weengh, Peo, ngu Fequenc ω, We umbe k, 4. s oeone/ n hmoncs: ν = 5. n oeone/3 hmoncs: ν = 3 6. hmoncs e pesen. = ν = π ω, = ν, k = π Sng fe one en: Pogesse we eng wh spee : =f( /), +; Pogesse sne we: =f( + /), = sn(k ω) = sn(π (/ / )).: Wes on Sng Spee of wes on sng wh mss pe un engh n enson : = / nsme powe: P = π ν nefeence: = sn(k ω), = sn(k ω + δ) = + = sn(k ω + ɛ) = + + cos δ sn δ n ɛ = + cos δ { nπ, consuce; δ = (n + )π, esuce. Snng Wes: = sn(k ω), cos k /4 = sn(k + ω) = + = ( cos k) sn ω { ( n + = ), noes; n = 0,,,..., nnoes. n = 0,,,... n Sng fe boh ens: /. oun conons: = 0 = 0 n =. owe Feq.: = n, ν = n, n =,, 3, Funmen/ s hmoncs: ν 0 =. oun conons: = 0 = 0 /. owe Feq.: = (n + ) 4, ν = n+ 4 0,,, Funmen/ s hmoncs: ν 0 = 4 4. s oeone/3 hmoncs: ν = n oeone/5 h hmoncs: ν = On o hmoncs e pesen. Sonomee: ν, ν, ν. ν = n.3: Soun Wes spcemen we: s = s 0 sn ω( /), n = Pessue we: p = p 0 cos ω( /), p 0 = (ω/)s 0 Spee of soun wes: qu = nens: = π ρ, so = s 0 ν = p0 Snng ongun wes: p = p 0 sn ω( /), Y ρ, γp gs = ρ = p0 ρ p = p + p = p 0 cos k sn ω ose ogn ppe:. oun conon: = 0 = 0 p = p 0 sn ω( + /). owe feq.: = (n + ) 4, ν = (n + ) 4, n = 0,,, Funmen/ s hmoncs: ν 0 = 4 4. s oeone/3 hmoncs: ν = 3ν 0 = 3 4

5 5. n oeone/5 h hmoncs: ν = 5ν 0 = On o hmoncs e pesen. Ph ffeence: = S S P Phse ffeence: δ = π Open ogn ppe: nefeence onons: fo nege n, { nπ, consuce; δ = (n + )π, esuce,. oun conon: = 0 = 0 owe feq.: = n, ν = n 4, n =,,.... Funmen/ s hmoncs: ν 0 = 3. s oeone/ n hmoncs: ν = ν 0 = 4. n oeone/3 hmoncs: ν = 3ν 0 = 3 5. hmoncs e pesen. nens: = { n, consuce; ( n + ), esuce = + + cos δ, ( m = + ) (, mn = ) = : = 4 0 cos δ, m = 4 0, mn = 0 esonnce coumn: + + Fnge wh: w = Opc ph: = nefeence of wes nsme hough hn fm: + =, + = 3 4, = ( )ν es: wo wes of mos equ fequences ω ω oppe Effec: p = p 0 sn ω ( /), p = p 0 sn ω ( /) p = p + p = p 0 cos ω( /) sn ω( /) ω = (ω + ω )/, ω = ω ω (bes feq.) ν = + u o u s ν 0 whee, s he spee of soun n he meum, u 0 s he spee of he obsee w... he meum, consee pose when moes ows he souce n nege when moes w fom he souce, n u s s he spee of he souce w... he meum, consee pose when moes ows he obsee n nege when moes w fom he obsee. = = ffcon fom snge s: { n, consuce; ( n + ), esuce. Fo Mnm: n = b sn b(/) esouon: sn =. b w of Mus: = 0 cos 0 b.4: gh Wes Pne We: E = E 0 sn ω( ), = 0 Sphec We: E = E0 sn ω( 0 ), = Young s oube s epemen

6 Opcs 3.: efecon of gh ws of efecon: ncen nom efece ncen, efece, n nom e n he sme pne () = Pne mo: () he mge n he objec e equsn fom mo () u mge of e objec () [ ] ens mke s fomu: f = ( ) ens fomu: u = f, m = u Powe of he ens: P = f, P n ope f f n mee. wo hn enses sepe b snce : F = f + f f f f f u f Sphec Mo: O u f 3.3: Opc nsumens Smpe mcoscope: m = /f n nom jusmen.. Foc engh f = /. Mo equon: + u = f 3. Mgnfcon: m = u ompoun mcoscope: Objece O Eepece u f e 3.: efcon of gh efce ne: = spee of gh n cuum spee of gh n meum = c ncen Sne s w: sn sn = ppen eph: = e eph ppen eph = efece efce O. Mgnfcon n nom jusmen: m = u. esong powe: = sn = sonomc eescope: f o f e f e c nge: c = sn c. n nom jusmen: m = fo f e, = f o + f e. esong powe: = =. eon b psm: δ 3.4: speson uch s equon: = 0 +, > 0 speson b psm wh sm n : δ = +, = sn +δm sn δ m = ( ), gene esu, = fo mnmum eon fo sm efcon sphec sufce: δ δ m P O Q u. Men eon: δ = ( ). ngu speson: = ( ) spese powe: ω = δ speson whou eon: ( ) + ( ) = 0 eon whou speson: ( ) = ( ) (f n sm) u =, m = u

7 He n hemonmcs 4.: He n empeue emp. sces: F = , K = e gs equon: pv = n, n : numbe of moes n e Ws equon: ( p + V ) (V b) = n hem epnson: = 0 ( + α ), = 0 ( + β ), V = V 0 ( + γ ), γ = β = 3α hem sess of me: F = Y 4.: Knec heo of Gses 4.4: heomonmc Pocesses Fs w of hemonmcs: Q = U + W Wok one b he gs: W = p V, W = W sohem = n n W sobc = p(v V ) V V ( V V W bc = p V p V γ W sochoc = 0 pv ) Gene: M = m, k = / n Effcenc of he he engne: Q W Mwe sbuon of spee: Q MS spee: ms = ege spee: = 3k m 8k = 3 M πm = Mos pobbe spee: p = Pessue: p = 3 ρ ms 8 πm k m p ms Equpon of eneg: K = k fo ech egee of feeom. hus, K = f k fo moecue hng f egees of feeoms. nen eneg of n moes of n e gs s U = f n. η = wok one b he engne he suppe o η cno = Q Q = oeff. of pefomnce of efgeo: OP = Q W = Q Q Q Enop: S = Q, S f S = f Q ons. : S = Q, = Q Q Q Q W Q Vng : S = ms n f bc pocess: Q = 0, pv γ = consn 4.3: Specfc He Specfc he: s = Q m en he: = Q/m Specfc he consn oume: = Q n V Specfc he consn pessue: p = Q n p 4.5: He nsfe onucon: Q = K hem essnce: = sees = + = K ( ) K + K K K eon beween p n : p = o of specfc hes: γ = p / eon beween U n : U = n pe = + = (K K + K ) K Kchhoff s w: emsse powe bsope powe = E bo bo = E bckbo Specfc he of gs mue: = n + n n + n, γ = n p + n p n + n Wen s spcemen w: m = b E m Mo nen eneg of n e gs: U = f, f = 3 fo monomc n f = 5 fo omc gs. Sefn-ozmnn w: Q = σe 4 ewon s w of coong: = b( 0)

8 Eecc n Mgnesm 5.3: pcos 5.: Eecoscs pcnce: = q/v ouomb s w: F = q q 4πɛ 0 ˆ q q Eecc fe: E( ) = q 4πɛ 0 ˆ Eecosc eneg: U = q q 4πɛ 0 Eecosc poen: V = 4πɛ 0 q q E Pe pe cpco: = ɛ 0 / Sphec cpco: = 4πɛ0 q q +q +q V = E, Eecc poe momen: p = q V ( ) = E Poen of poe: V = p cos 4πɛ 0 p q +q p V () nc cpco: = πɛ0 n( / ) pcos n pe: eq = + pcos n sees: eq = + Fe of poe: E = p cos 4πɛ 0, E 3 = p sn 4πɛ 0 3 oque on poe pce n E: τ = p E p Po. eneg of poe pce n E: U = p E E E Foce beween pes of pe pe cpco: F = Q ɛ 0 Eneg soe n cpco: U = V = Q = QV Eneg ens n eecc fe E: U/V = ɛ 0E pco wh eecc: = ɛ0k 5.: Guss s w n s ppcons Eecc fu: φ = E S 5.4: uen eecc uen ens: j = / = σe Guss s w: E S = qn /ɛ 0 Fe of unfom chge ng on s s: E P = q 4πɛ 0 ( + ) 3/ q E n V { of unfom chge sphee: Q 4πɛ E = 0, fo < 3 E Q 4πɛ 0, fo O V = { Q 4πɛ 0, fo < 3 V Q 4πɛ 0, fo O P E E n V of unfom chge sphec she: { 0, fo < E = E Q 4πɛ 0, fo O V = { Q 4πɛ 0, fo < V Q 4πɛ 0, fo O Fe of ne chge: E = πɛ 0 Fe of n nfne shee: E = σ ɛ 0 Fe n he cn of conucng sufce: E = σ ɛ 0 f spee: = ee m τ = ne essnce of we: = ρ/, whee ρ = /σ emp. epenence of essnce: = 0 ( + α ) Ohm s w: V = Kchhoff s ws: () he Juncon w: he gebc sum of he cuens ece ows noe s zeo.e., Σ noe = 0. ()he oop w: he gebc sum of he poen ffeences ong cose oop n ccu s zeo.e., Σ oop V = 0. essos n pe: eq = + essos n sees: eq = + Whesone bge: nce f / = 3 / 4. Eecc Powe: P = V / = = V 3 V G 4

9 Gnomee s n mmee: g G = ( g )S Gnomee s Vomee: V = g ( + G) g G g S G g Eneg of mgnec poe pce n : U = H effec: V w = ne w 5.6: Mgnec Fe ue o uen z hgng of cpcos: q() = V [ ] e V o-s w: = 0 4π 3 schgng of cpcos: q() = q 0 e q() Fe ue o sgh conuco: me consn n ccu: τ = = 0 4π (cos cos ) Fe ue o n nfne sgh we: = 0 π Pee effec: emf e = H Q = Seebck effec:. hemo-emf: e = + b Pee he chge nsfee. e 0 n. hemoeecc powe: e/ = + b. 3. eu emp.: n = /b. 4. neson emp.: = /b. homson effec: emf e = H Q = homson he chge nsfee = σ. F s w of eecoss: he mss epose s m = Z = F E whee s cuen, s me, Z s eecochemc equen, E s chemc equen, n F = /g s F consn. Foce beween pe wes: F Fe on he s of ng: P = 0 ( + ) 3/ = 0 π Fe he cene of n c: = 0 4π Fe he cene of ng: = 0 mpee s w: = 0 n Fe nse soeno: = 0 n, n = P 5.5: Mgnesm oenz foce on mong chge: F = q + q E Fe nse oo: = 0 π hge pce n unfom mgnec fe: q = m q, = πm q Fe of b mgne: = 0 M 4π, 3 = 0 M 4π 3 S Foce on cuen cng we: F = Mgnec momen of cuen oop (poe): = F oque on mgnec poe pce n : τ = nge of p: h = cos δ Hozon ngen gnomee: h n = 0n, = K n Mong co gnomee: n = k, = k me peo of mgneomee: = π Pemeb: = H M h δ n h

10 5.7: Eecomgnec nucon Mgnec fu: φ = S F s w: e = φ ccu: e 0 sn ω Z = + (/ω), n φ = ω ω Z φ enz s w: nuce cuen cee -fe h opposes he chnge n mgnec fu. Moon emf: e = + ccu: Z = + ω, n φ = ω e 0 sn ω ω φ Z Sef nucnce: φ =, e = Sef nucnce of soeno: = 0 n (π ) ] Gowh of cuen n ccu: = [ e / e S 0.63 e ec of cuen n ccu: = 0 e 0 e / cu: e 0 sn ω ω ω Z = + ( ω ω), n φ = ν esonnce = π Powe fco: P = e ms ms cos φ ω ω Z ω ω φ nsfome: = e e, e = e Spee of he EM wes n cuum: c = / 0 ɛ 0 e e S me consn of ccu: τ = / Eneg soe n n nuco: U = Eneg ens of fe: u = U V = 0 Muu nucnce: φ = M, e = M EMF nuce n ong co: e = ω sn ω enng cuen: = 0 sn(ω + φ), = π/ω ege cuen n : ī = 0 = 0 MS cuen: ms = [ 0 ] / = 0 Eneg: E = ms pce ecnce: X c = ω nuce ecnce: X = ω mepence: Z = e 0 / 0

11 Moen Phscs 6.: Phoo-eecc effec Popuon me : = 0 e 0 0 Phoon s eneg: E = hν = hc/ O / Phoon s momenum: p = h/ = E/c M. KE of ejece phoo-eecon: K m = hν φ hesho feq. n phoo-eecc effec: ν 0 = φ/h Hf fe: / = 0.693/ ege fe: = / Popuon fe n hf es: = 0 / n. Soppng poen: V o = hc e e oge weengh: = h/p ( ) φ e V 0 φ e φ hc hc e Mss efec: m = [Zm p + ( Z)m n ] M nng eneg: = [Zm p + ( Z)m n M] c Q-ue: Q = U U f Eneg eese n nuce econ: E = mc whee m = m ecns m poucs. 6.: he om Eneg n nh oh s ob: 6.4: Vcuum ubes n Semconucos E n = mz e 4 8ɛ 0 h n, E n = 3.6Z n ev us of he nh oh s ob: Hf We ecfe: Oupu n = ɛ 0h n πmze, n = n 0 Z, 0 = 0.59 Å Fu We ecfe: Oupu Qunzon of he ngu momenum: = nh π Phoon eneg n se nson: E E = hν oe Ve: hoe Fmen G Pe E E hν E Emsson hν E bsopon Pe essnce of oe: p = Vp p Vg=0 Weengh of eme on: fo nson fom nh o mh se: [ = Z n ] m nsconucnce of oe: g m = p V g Vp=0 mpfcon b oe: = Vp V g p=0 eon beween p,, n g m : = p g m X- specum: mn = hc ev Mosee s w: ν = (Z b) X- ffcon: sn = n Hesenbeg uncen pncpe: p h/(π), E h/(π) 6.3: he uceus uce us: = 0 /3, ec e: = mn K α K β α m uen n nsso: e = b + c α n β pmees of nsso: α = c e, β = c b, β = α α nsconucnce: g m = c V be ogc Ges: O O XO Ā e b c

Chapter 6 Plane Motion of Rigid Bodies

Chapter 6 Plane Motion of Rigid Bodies Chpe 6 Pne oon of Rd ode 6. Equon of oon fo Rd bod. 6., 6., 6.3 Conde d bod ced upon b ee een foce,, 3,. We cn ume h he bod mde of e numbe n of pce of m Δm (,,, n). Conden f he moon of he m cene of he