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Spring 2007

# Spring 2007 - Scott Pratt PHY231 Spring 2007 Introduct...

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Scott Pratt - PHY231, Spring 2007 - Introduct 1 Final Exam Scott Pratt Do not open exam until instructed to do so.

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Scott Pratt - PHY231, Spring 2007 - Introduct 3 Final Exam Quadratic Formula ax 2 + bx + c = 0, x = [ - b ± b 2 - 4 ac ] / (2 a ) Geometry Circle: circumference=2 π R , area= π R 2 Sphere: area=4 π R 2 , volume=4 π R 3 / 3 Trigonometry a A B b C sin α = A C , cos α = B C tan α = A B α γ β sin α A = sin β B = sin γ C A 2 + B 2 - 2 AB cos γ = C 2 Polar Coordinates x = r cos θ , y = r sin θ , r = x 2 + y 2 , tan θ = y/x SI Units and Constants quantity unit abbreviation Mass m kilograms kg Distance x meters m Time t seconds s Force F Newtons N=kg m/s 2 Energy E Joules J=N m Power P Watts W=J/s Temperature T C, K or F T F = 32 + (9 / 5) T C Pressure P Pascals Pa=N/m 2 1 cal=4.1868 J, 1 hp=745.7 W, 1 liter=10 - 3 m 3 g = 9 . 81 m/s 2 , G=6 . 67 × 10 - 11 Nm 2 /kg 2 1 atm=1 . 013 × 10 5 Pa, 0 C=273.15 K, N A = 6 . 023 × 10 23 R = 8 . 31 J/(mol K)=0.0821 L atm/(mol K), k B = R/N A = 1 . 38 × 10 - 23 J/K, σ = 5 . 67 × 10 - 8 W/(m 2 K 4 ) v sound = 331 T/ 273 m/s H 2 0: c ice , liq ., steam = { 0 . 5 , 1 . 0 , 0 . 48 } cal/g C L F,V = { 79 . 7 , 540 } cal/g, ρ = 1000 kg/m 3 . 1-d motion, constant a Δ x = (1 / 2)( v 0 + v f ) t v f = v 0 + at Δ x = v 0 t + (1 / 2) at 2 Δ x = v f t - (1 / 2) at 2 (1 / 2) v 2 f - (1 / 2) v 2 0 = a Δ x Range : R = ( v 2 0 /g ) sin 2 θ Forces, Work, Energy, Power, Momentum & Impulse F = ma , Gravity: F = mg, PE = mgh Friction: f = μN , Spring: F = - kx, PE = (1 / 2) kx 2 W = Fx cos θ , KE = (1 / 2) mv 2 , P = Δ E/ Δ t = Fv p = mv, I = F Δ t = Δ p X cm = ( m 1 x 1 + m 2 x 2 + · · · ) / ( m 1 + m 2 + · · · ) Elastic coll.s: v 1 - v 2 = - ( v 1 - v 2 ) Rotational Motion Δ θ = (1 / 2)( ω 0 + ω f ) t, ω f = ω 0 + α t Δ θ = ω 0 t + (1 / 2) α t 2 = ω f t - (1 / 2) α t 2 α Δ θ = (1 / 2) ω 2 f - (1 / 2) ω 2 0 ω = 2 π /T = 2 π f, f = 1 /T Rolling: a = α r, v = ω r a c = v 2 /r = ω v = ω 2 r τ = rF sin θ = I α , I point = mR 2 I cyl . shell = MR 2 , I sphere = (2 / 5) MR 2 I solid cyl . = (1 / 2) MR 2 , I sph . shell = (2 / 3) MR 2 L = I ω = mvr sin θ , ( θ = angle between v and r) KE = (1 / 2) I ω 2 = L 2 / (2 I ) , W = τ Δ θ Gravity and circular orbits PE = - G Mm r , Δ PE = mgh (small h ) F = G Mm r 2 , GM 4 π 2 = R 3 T 2 Gases, liquids and solids P = F/A, PV = nRT, Δ P = ρ gh (1 / 2) mv 2 = (3 / 2) k B T ideal monotonic gas: U = (3 / 2) nRT = (3 / 2) PV F bouyant = ρ displaced liq . V displaced liq . g Stress = F/A, Strain = Δ L/L, Y = Stress / Strain Δ L L = F/A Y , Δ V V = - Δ P B , Y = 3 B Continuity: ρ 1 A 1 v 1 = ρ 2 A 2 v 2 Bernoulli: P a + 1 2 ρ a v 2 a + ρ a gh a = P b + 1 2 ρ b v 2 b + ρ b gh b Thermal Δ L/L = α Δ T, Δ V/V = β Δ T, β = 3 α Q = mC v Δ T + mL (if phase trans . ) Conduction and Radiation P = kA ( T b - T a ) /L = A ( T b - T a ) /R , R L/k P = e σ AT 4 Thermodynamics Δ U = Q + W, W = - P Δ V , Q = T Δ S , Δ S > 0 Engines: W = | Q H | - | Q L | = W/Q H < ( T H - T L ) /T H < 1 Refrigerators and heat pumps: W = | Q H | - | Q L | = Q L /W < T L / ( T H - T L ) Simple Harmonic Motion and Waves f = 1 /T , ω = 2 π f x ( t ) = A cos( ω t - φ ) , v = - ω A sin( ω t - φ ) a = - ω 2 A cos( ω t - φ ) Spring: ω = k/m Pendulum: T = 2 π L/g Waves: y ( x, t ) = A sin[2 π ( ft - x/ λ ) + δ ], v = f λ I = const A 2 f 2 , I 2 /I 1 = R 2 1 /R 2 2 Standing waves: λ n = 2 L/n Strings: v = T/μ Solid/Liquid: v = B/ ρ Sound: I = Power /A = I 0 10 β / 10 , I 0 10 - 12 W/m 2 Decibels: β = 10 log 10 ( I/I 0 ) Beat freq.= | f 1 - f 2 | Doppler: f obs = f source ( V sound ± v obs ) / ( V sound ± v source ) Pipes: same at both ends: L = λ / 2 , λ , 3 λ / 2 Pipes: open at only one end: L = λ / 4 , 3 λ / 4 , 5 λ / 4 · · ·
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Spring 2007 - Scott Pratt PHY231 Spring 2007 Introduct...

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