Reference Guide & Formula Sheet for Physics

Transcription

1 D. Hoselto & M. Pice Page of 8 #0 Heatig a Solid, Liquid o Gas #3 Compoets of a Vecto Q = m c T if V = 34 m/sec 48 the V i = 34 m/sec (cos 48 ); ad V J = 34 m/sec (si 48 ) #4 Weight = m g g = 9.8m/sec² ea the suface of the Eath = m/sec² i Fot Woth, TX Desity = mass / volume m 3 ρ = ( uit : kg / m ) V #7 Ave speed = distace / time = v = d/t Ave velocity = displacemet / time = v = d/t Ave acceleatio = chage i velocity / time #8 Fictio Foce F F = µ F N If the object is ot movig, you ae dealig with static fictio ad it ca have ay value fom zeo up to µ s F N If the object is slidig, the you ae dealig with kietic fictio ad it will be costat ad equal to µ K F N (o phase chages!) Q = the heat added c = specific heat. T = tempeatue chage, K # Liea Mometum mometum = p = m v = mass velocity mometum is coseved i collisios #3 Cete of Mass poit masses o a lie x cm = Σ(mx) / M total #5 Agula Speed vs. Liea Speed Liea speed = v = ω = agula speed #6 Pessue ude Wate P = ρ g h h = depth of wate ρ = desity of wate #8 Uivesal Gavitatio mm F = G G = 6.67 E- N m² / kg² #9 Toque τ = F L si θ Whee θ is the agle betwee F ad L; uit: Nm # Newto's Secod Law F et = ΣF Ext = m a # Wok = F D cos θ Whee D is the distace moved ad θ is the agle betwee F ad the diectio of motio, uit : J #6 Powe = ate of wok doe Wok Powe = time uit : watt Efficiecy = Wok out / Eegy i Mechaical Advatage = foce out / foce i M.A. = F out / F i #9 Costat-Acceleatio Liea Motio v = v ο + a t x (x-x ο ) = v ο t + ½ a t² v v ² = v ο ² + a (x - x ο ) t (x-x ο ) = ½ ( v ο + v) t a (x-x ο ) = v t - ½ a t² v ο #9 Mechaical Eegy PE Gav = P = m g h KE Liea = K = ½ m v² #30 Impulse = Chage i Mometum F t = (m v) #3 Sell's Law si θ = si θ Idex of Refactio = c / v c = speed of light = 3 E+8 m/s #3 Ideal Gas Law P V = R T = # of moles of gas R = gas law costat = 8.3 J / K mole. #34 Peiodic Waves v = f λ f = / T T = peiod of wave #35 Costat-Acceleatio Cicula Motio ω = ω ο + α t θ θ θ ο = ω ο t + ½ α t² ω ω = ω ο + α (θ θο ) t θ θ ο = ½ (ω ο + ω) t α θ θ ο = ω t - ½ α t² ω ο Vesio 5//005

2 D. Hoselto & M. Pice Page of 8 #53 Resisto Combiatios #36 Buoyat Foce - Buoyacy SERIES F B = ρ V g = m Displaced fluid g = weight Displaced fluid R eq = R + R + R ρ = desity of the fluid PARALLEL V = volume of fluid displaced = + + K + = #37 Ohm's Law V = I R V = voltage applied I = cuet R = esistace Resistace of a Wie R = ρ L / A x ρ = esistivity of wie mateial L = legth of the wie A x = coss-sectioal aea of the wie #39 Heat of a Phase Chage Q = m L L = Latet Heat of phase chage #4 Hooke's Law F = k x Potetial Eegy of a spig W = ½ k x² = Wok doe o spig #4 Electic Powe P = I² R = V ² / R = I V #44 Speed of a Wave o a Stig mv T = L T = tesio i stig m = mass of stig L = legth of stig #45 Pojectile Motio Hoizotal: x-x ο = v ο t + 0 Vetical: y-y ο = v ο t + ½ a t² R eq R R R i = #54 Newto's Secod Law ad Rotatioal Ietia τ = toque = I α I = momet of ietia = m ² (fo a poit mass) (See table i Lesso 58 fo I of 3D shapes.) #55 Cicula Ubaked Tacks mv = µ mg #56 Cotiuity of Fluid Flow A i v i = A out v out #58 Momet of Ietia - I cylidical hoop m solid cylide o disk ½ m solid sphee / 5 m hollow sphee ⅔ m thi od (cete) / m L thi od (ed) ⅓ m L R i A= Aea v = velocity #59 Capacitos Q = C V Q = chage o the capacito C = capacitace of the capacito V = voltage applied to the capacito RC Cicuits (Dischagig) V c = V o e t/rc V c I R = 0 #60 Themal Expasio Liea: L = L o α T Volume: V = V o β T #46 Cetipetal Foce mv F = = mω #47 Kichhoff s Laws Loop Rule: Σ Aoud ay loop V i = 0 Node Rule: Σ at ay ode I i = 0 #5 Miimum Speed at the top of a Vetical Cicula Loop v = g #6 Beoulli's Equatio P + ρ g h + ½ ρ v ² = costat Q Volume Flow Rate = A v = A v = costat #6 Rotatioal Kietic Eegy (See LEM, pg 8) KE otatioal = ½ I ω = ½ I (v / ) KE ollig w/o slippig = ½ m v + ½ I ω Agula Mometum = L = I ω = m v si θ Agula Impulse equals CHANGE IN Agula Mometum L = τ oque t = (I ω) Vesio 5//005

3 D. Hoselto & M. Pice Page 3 of 8 #75 Thi Les Equatio #63 Peiod of Simple Hamoic Motio T = π m k whee k = spig costat f = / T = / peiod #64 Baked Cicula Tacks v = g ta θ = f D o + D i = + o i Magificatio M = D i / D o = i / o = H i / H o f = focal legth i = image distace o = object distace #66 Fist Law of Themodyamics U = Q Net + W Net Chage i Iteal Eegy of a system = +Net Heat added to the system +Net Wok doe o the system Flow of Heat though a Solid Q / t = k A T / L k = themal coductivity A = aea of solid L = thickess of solid #68 Potetial Eegy stoed i a Capacito P = ½ C V² RC Cicuit fomula (Chagig) V c = V cell ( e t / RC ) R C = τ = time costat V cell - V capacito I R = 0 #7 Simple Pedulum L T = π ad f = / T g #7 Siusoidal motio x = A cos(ω t) = A cos( π f t) ω = agula fequecy f = fequecy #73 Dopple Effect Towad 343 ± Away v o f = f Towad 343 m Away v s v o = velocity of obseve: v s = velocity of souce #74 d Law of Themodyamics The chage i iteal eegy of a system is U = Q Added + W Doe O Q lost W Doe By Maximum Efficiecy of a Heat Egie (Caot Cycle) (Tempeatues i Kelvi) Tc % Eff = ( ) 00% T h Helpful emides fo mios ad leses Focal Legth of: positive egative mio cocave covex les covegig divegig Object distace = o all objects Object height = H o all objects Image distace = i eal vitual Image height = H i vitual, upight eal, iveted Magificatio vitual, upight eal, iveted #76 Coulomb's Law qq F = k N m k = = 9E9 4 πε o C #77 Capacito Combiatios PARALLEL C eq = C + C + C 3 + SERIES = + + K + C eq C C C i = #78 Wok doe o a gas o by a gas W = P V = C #80 Electic Field aoud a poit chage q E = k N m k = = 9E 9 4 πε o C #8 Magetic Field aoud a wie µ o I B = π Magetic Flux Φ = B A cos θ Foce caused by a magetic field o a movig chage F = q v B si θ #83 Etopy chage at costat T S = Q / T (Phase chages oly: meltig, boilig, feezig, etc) i Vesio 5//005

4 D. Hoselto & M. Pice Page 4 of 8 #95 Relativistic Time Dilatio t = t o / β #84 Capacitace of a Capacito C = κ ε o A / d κ = dielectic costat A = aea of plates d = distace betwee plates ε o = 8.85 E(-) F/m #85 Iduced Voltage N = # of loops Φ Emf = N t Lez s Law iduced cuet flows to ceate a B-field opposig the chage i magetic flux. #96 Relativistic Legth Cotactio x = β x o Relativistic Mass Icease m = m o / β #97 Eegy of a Photo o a Paticle E = h f = m c h = Plack's costat = 6.63 E(-34) J sec f = fequecy of the photo #86 Iductos duig a icease i cuet t / (L / R) V L = V cell e I = (V cell /R) [ - e t / (L / R) ] L / R = τ = time costat #88 Tasfomes N / N = V / V I V = I V #89 Decibel Scale B (Decibel level of soud) = 0 log ( I / I o ) I = itesity of soud I o = itesity of softest audible soud #9 Poiseuille's Law P = 8 η L Q/(π 4 ) η = coefficiet of viscosity L = legth of pipe = adius of pipe Q = flow ate of fluid Stess ad Stai Y o S o B = stess / stai stess = F/A Thee kids of stai: uit-less atios I. Liea: stai = L / L II. Shea: stai = x / L III. Volume: stai = V / V #93 Postulates of Special Relativity. Absolute, uifom motio caot be detected.. No eegy o mass tasfe ca occu at speeds faste tha the speed of light. #94 Loetz Tasfomatio Facto β = c v #98 Radioactive Decay Rate Law A = A o e k t = (/ ) A 0 (afte half-lives) Whee k = (l ) / half-life #99 Blackbody Radiatio ad the Photoelectic Effect E= h f whee h = Plack's costat #00 Ealy Quatum Physics Ruthefod-Boh Hydoge-like Atoms f = R metes s o c = = cr Hz λ s R = Rydbeg's Costat = E7 m - s = seies itege ( = Balme) = a itege > s λ Mass-Eegy Equivalece m v = m o / β Total Eegy = KE + m o c = m o c / β Usually witte simply as E = m c de Boglie Matte Waves Fo light: E p = h f = h c / λ = p c Theefoe, mometum: p = h / λ Similaly fo paticles, p = m v = h / λ, so the matte wave's wavelegth must be λ = h / m v Eegy Released by Nuclea Fissio o Fusio Reactio E = m o c Vesio 5//005

5 D. Hoselto & M. Pice Page 5 of 8 MISCELLANEOUS FORMULAS Quadatic Fomula if a x²+ b x + c = 0 b ± x = the b 4ac a Tigoometic Defiitios si θ = opposite / hypoteuse cos θ = adjacet / hypoteuse ta θ = opposite / adjacet sec θ = / cos θ = hyp / adj csc θ = / si θ = hyp / opp cot θ = / ta θ = adj / opp Ivese Tigoometic Defiitios θ = si - (opp / hyp) θ = cos - (adj / hyp) θ = ta - (opp / adj) Law of Sies a / si A = b / si B = c / si C o si A / a = si B / b = si C / c Law of Cosies a = b + c - b c cos A b = c + a - c a cos B c² = a² + b² - a b cos C T-Pots Fo the fuctioal fom = + A B C You may use "The Poduct ove the Sum" ule. B C A = B + C Fo the Alteate Fuctioal fom = A B C You may substitute T-Pot-d B C B C A = = C B B C Fudametal SI Uits Uit Base Uit Symbol. Legth mete m Mass kilogam kg Time secod s Electic Cuet ampee A Themodyamic Tempeatue kelvi K Lumious Itesity cadela cd Quatity of Substace moles mol Plae Agle adia ad Solid Agle steadia s o st Some Deived SI Uits Symbol/Uit Quatity Base Uits. C coulomb Electic Chage A s F faad Capacitace A s4/(kg m ) H hey Iductace kg m /(A s ) Hz hetz Fequecy s - J joule Eegy & Wok kg m /s = N m N ewto Foce kg m/s Ω ohm Elec Resistace kg m /(A s ) Pa pascal Pessue kg/(m s ) T tesla Magetic Field kg/(a s ) V volt Elec Potetial kg m /(A s 3 ) W watt Powe kg m /s 3 No-SI Uits o C degees Celsius ev electo-volt Tempeatue Eegy, Wok Vesio 5//005

6 D. Hoselto & M. Pice Page 6 of 8 Αα Alpha agula acceleatio, coefficiet of liea expasio, Ββ Beta coefficiet of volume expasio, Loetz tasfomatio facto, Χχ Chi Aa acceleatio, Aea, A x =Coss-sectioal Aea, Ampees, Amplitude of a Wave, Agle, Bb Magetic Field, Decibel Level of Soud, Agle, Cc specific heat, speed of light, Capacitace, Agle, Coulombs, o Celsius, Celsius Degees, cadela, Dd displacemet, diffeetial chage i a vaiable, Distace, Distace Moved, distace, Ee base of the atual logaithms, chage o the electo, Eegy, Ff Foce, fequecy of a wave o peiodic motio, Faads, Gg Uivesal Gavitatioal Costat, acceleatio due to gavity, Gauss, gams, Giga-, Hh depth of a fluid, height, vetical distace, Heys, Hz=Hetz, Ii Cuet, Momet of Ietia, image distace, Itesity of Soud, Jj Joules, Kk K o KE = Kietic Eegy, foce costat of a spig, themal coductivity, coulomb's law costat, kg=kilogams, Kelvis, kilo-, ate costat fo Radioactive decay =/τ=l / half-life, Ll Legth, Legth of a wie, Latet Heat of Fusio o Vapoizatio, Agula Mometum, Thickess, Iductace, Mm mass, Total Mass, metes, milli-, Mega-, m o =est mass, mol=moles, N idex of efactio, moles of a gas, Newtos, Numbe of Loops, ao-, Oo Pp Powe, Pessue of a Gas o Fluid, Potetial Eegy, mometum, Powe, Pa=Pascal, Qq Heat gaied o lost, Maximum Chage o a Capacito, object distace, Flow Rate, R adius, Ideal Gas Law Costat, Resistace, magitude o legth of a vecto, ad=adias Ss speed, secods, Etopy, legth alog a ac, Tt time, Tempeatue, Peiod of a Wave, Tesio, Teslas, t / =half-life, Uu Potetial Eegy, Iteal Eegy, Vv velocity, Velocity, Volume of a Gas, velocity of wave, Volume of Fluid Displaced, Voltage, Volts, Ww weight, Wok, Watts, Wb=Webe, Xx distace, hoizotal distace, x-coodiate east-ad-west coodiate, Yy vetical distace, y-coodiate, oth-ad-south coodiate, Zz z-coodiate, up-ad-dow coodiate, δ Delta =chage i a vaiable, Εε Epsilo ε ο = pemittivity of fee space, Φφ Phi Magetic Flux, agle, Γγ Gamma suface tesio = F / L, / γ = Loetz tasfomatio facto, Ηη Eta Ιι Iota ϑϕ Theta ad Phi lowe case alteates. Κκ Kappa dielectic costat, Λλ Lambda wavelegth of a wave, ate costat fo Radioactive decay =/τ=l/half-life, Μµ Mu fictio, µ o = pemeability of fee space, mico-, Νν Nu alteate symbol fo fequecy, Οο Omico Ππ Pi , Θθ Theta agle betwee two vectos, Ρρ Rho desity of a solid o liquid, esistivity, Σσ Sigma Summatio, stadad deviatio, Ττ Tau toque, time costat fo a expoetial pocesses; eg τ=rc o τ=l/r o τ=/k=/λ, Υυ Upsilo ςϖ Zeta ad Omega lowe case alteates Ωω Omega agula speed o agula velocity, Ohms Ξξ Xi Ψψ Psi Ζζ Zeta Vesio 5//005

7 D. Hoselto & M. Pice Page 7 of 8 Values of Tigoometic Fuctios fo st Quadat Agles Pefixes (simple mostly-atioal appoximatios) θ si θ cos θ ta θ Facto Pefix Symbol Example 0 o o /6 65/66 / exa- E 38 Es (Age of 5 o /4 8/9 9/08 the Uivese 0 o i Secods) /3 6/7 7/47 9 o 5 / /8 7/8 5 / 0 5 peta- P /7 30 o / 3 / / /3 / 37 o 0 tea- T 0.3 TW (Peak 3/5 4/5 3/4 4 o powe of a /3 3/4 8/9 ps pulse 45 o / / / / fom a typical 49 o 3/4 /3 9/8 Nd-glass lase) 53 o 4/5 3/5 4/ / / / 3 / 0 9 giga- G G$ (Size of 6 o 7/8 5 / /8 7/5 / Bill & Melissa 70 o 6/7 /3 47/7 Gates Tust) 75 o 8/9 /4 08/9 80 o 65/66 /6 65/ 0 6 mega- M 6.37 Mm (The 90 o 0 adius of the Eath) (Memoize the Bold ows fo futue efeece.) 0 3 kilo- k kg (SI uit Deivatives of Polyomials of mass) Fo polyomials, with idividual tems of the fom Ax, we defie the deivative of each tem as ( ) d Ax = Ax dx To fid the deivative of the polyomial, simply add the deivatives fo the idividual tems: ( ) 6 d 3x + 6x 3 = 6x + dx Itegals of Polyomials Fo polyomials, with idividual tems of the fom Ax, we defie the idefiite itegal of each tem as + ( ) Ax dx = Ax + To fid the idefiite itegal of the polyomial, simply add the itegals fo the idividual tems ad the costat of itegatio, C. ( 6x + 6) dx = [ 3x + 6x C] deci- d 0 cm 0 - ceti- c.54 cm (= i) 0-3 milli- m mm (The smallest divisio o a mete stick) 0-6 mico- µ 0-9 ao- 50 m (Wavelegth of gee light) 0 - pico- p pg (Typical mass of a DNA sample used i geome studies) 0-5 femto- f 0-8 atto- a 600 as (Time duatio of the shotest lase pulses) Vesio 5//005

8 D. Hoselto & M. Pice Page 8 of 8 Liea Equivalet Mass Rotatig systems ca be hadled usig the liea foms of the equatios of motio. To do so, howeve, you must use a mass equivalet to the mass of a o-otatig object. We call this the Liea Equivalet Mass (LEM). (See Example I) Fo objects that ae both otatig ad movig liealy, you must iclude them twice; oce as a liealy movig object (usig m) ad oce moe as a otatig object (usig LEM). (See Example II) The LEM of a otatig mass is easily defied i tems of its momet of ietia, I. LEM = I/ Fo example, usig a stadad table of Momets of Ietia, we ca calculate the LEM of simple objects otatig o axes though thei cetes of mass: I LEM Cylidical hoop m m Solid disk ½m ½m Hollow sphee 5 m 5 m Solid sphee ⅔m ⅔m Example I A flywheel, M = 4.80 kg ad = 0.44 m, is wapped with a stig. A hagig mass, m, is attached to the ed of the stig. Whe the hagig mass is eleased, it acceleates dowwad at.00 m/s. Fid the hagig mass. To hadle this poblem usig the liea fom of Newto s Secod Law of Motio, all we have to do is use the LEM of the flywheel. We will assume, hee, that it ca be teated as a uifom solid disk. The oly exteal foce o this system is the weight of the hagig mass. The mass of the system cosists of the hagig mass plus the liea equivalet mass of the fly-wheel. Fom Newto s d Law we have F = ma, theefoe, If a = g/ = m/s, If a = ¾g = m/s, mg = [m + (LEM=½M)]a mg = [m + ½M] a (mg ma) = ½M a m(g a) = ½Ma m = ½ M a / (g a) m = ½ / (9.8 ) m = 0.7 kg m =.4 kg m = 7. kg Note, too, that we do ot eed to kow the adius uless the agula acceleatio of the fly-wheel is equested. If you eed α, ad you have, the α = a/. Example II Fid the kietic eegy of a disk, m = 6.7 kg, that is movig at 3. m/s while ollig without slippig alog a flat, hoizotal suface. (I DISK = ½m ; LEM = ½m) The total kietic eegy cosists of the liea kietic eegy, K L = ½mv, plus the otatioal kietic eegy, K R = ½(I)(ω) = ½(I)(v/) = ½(I/ )v = ½(LEM)v. Fial Note: KE = ½mv + ½ (LEM=½m) v KE = ½ ½ (½ 6.7) 3. KE = = 5 J This method of icopoatig otatig objects ito the liea equatios of motio woks i evey situatio I ve tied; eve vey complex poblems. Wok you poblem the classic way ad this way to compae the two. Oce you ve veified that the LEM method woks fo a paticula type of poblem, you ca cofidetly use it fo solvig ay othe poblem of the same type. Vesio 5//005