Physics Final Exam Formulas

Transcription

1 INSTRUCTIONS: Write your NAME on the front of the blue exam booklet. The exam is closed book, and you may have only pens/pencils and a calculator. Show all of your work in the blue book. For problems II-VI, an answer alone is worth very little credit, even if it is correct so show how you get it. Small print: Do not assume ANYTHING is obvious if it s not clear, please ask. It s hard to write these questions, and we often forget things. Suggestions: Draw a diagram when possible, circle or box your final answers, and cross out parts which you do not want us to consider. Physics Final Exam Formulas Kinematics : Constant a x : x = x 0 + v 0,x t a xt 2 v x = v 0,x + a x t v 2 x = v2 0,x + 2a x(x x 0 ) 3D : v = d r = v xî + v yĵ + v zˆk = dx î + dy ĵ + dz ˆk at 2 + bt + c = 0 t = b ± b 2 4ac ; Newton : 2a a = d v = a xî +... = dv x î +... = d2 x 2 î +... F = Ftot = m a Work : W on object = Fon object d r U = W K = W U gravity = mg h Power = F v = τ ω = W/ t = dw/ F x, spring = kx U spring = 1 2 kx2 F centrip = mv2 r Conservation of Energy : K f + U f = K i + U i + W into system K lin = 1 2 mv2 F kinetic friction = µ k F normal F static friction µ s F normal p = m v J = p = pf p i Systems : M tot x c.m. = i m i x i p tot = M tot v c.m. v c.m. = d r c.m. U grav = M tot gy c.m. Rotation : α = const. ω = ω 0 + αt ω 2 = ω α(θ θ 0 ) θ = θ 0 + ω 0 t αt2 1 rev. = 2π rad τ = Iα vt = rω a t = rα a r = rω 2 I = i m i r 2 i K rot = 1 2 Iω2 L = r p = I ω τ = r F = r F sin φ = r F I parallel = I c.m. + Mh 2 F net = d P c.m. τ net = d L c.m. Elastic : v i + v f = V i + V f f = ω/2π f = 1/T F grav = GM 1M 2 r 2 U grav = GM 1M 2 r (U = 0 at r = ) G = Nm 2 /kg 2 Fluid Statics : p 2 = p 1 + ρg h Pascal : p 1 = p 2 F 1 A 1 = F 2 A 2 Archimedes : F b = ρ W gv displ Fluid Dynamics : Continuity : A 1 v 1 = A 2 v 2 Bernoulli : p 1 + ρgy ρv2 1 = p 2 + ρgy ρv2 2 d SHM : 2 x = ( ) k 2 m x ω = k/m = 2πf = 2π/T x = A cos(ωt + φ) Waves : y = A sin(kx ωt + φ) k = 2π/λ fλ = v = ω/k v = F/µ φ/2π = x/λ P ave E 2 A 2 Intensity : I = P ave /Area A sin α + A sin β = 2A cos [ 1 (α 2 β)] sin [ 1(α + β)] 2 Decibels : SL = 10 log I/I 0 I 0 = W/m 2 Doppler : f = f 0 v v±v s Shock : sin θ = v/v s Expansion : L = αl T V = 3αV T pv = Nk B T = nrt M = mn A P ave = 1 2 µvω2 A 2 K ave = 3k 2 BT K tot = 3nRT E 2 i = 1k 2 BT v rms = 3p/ρ = 3RT/M HeatConduction : dq = ka dt dx First Law : dq + dw = de int Q = mc T Q f = ml f Page 1 of 6

2 W = p dv pv γ = const W = 1 γ 1 (p fv f p i V i ) C V + R = C P γ = C P C V S = dq T S = nr ln V f V i + nc V ln T f T i ε = W Q in = Q H Q C Q H = 1 Q C Q H = 1 T C TH (engine) multiplicity : w = N! (N r)!r! S = k B ln w ln N! N ln N N γ = 1 1 v 2 /c 2 Length : L = L 0 /γ Time : t = γt 0 E tot = γmc 2 E rest = mc 2 p = mvγ Lorentz Transformations :( v = vˆx) t = γ(t vx/c 2 ) x = γ(x vt) y = y z = z Some Constants: Velocity Addition : v = v u x 1 vu x /c 2 E 2 = p 2 c 2 + m 2 c 4 Mass of the Earth kg Radius of Earth m Mass of the Sun kg Radius of Earth s Orbit m Density of Water 1000 kg/m 3 Boltzmann k B = J/K N A R 8.31 J/mol K L f ice 333 kj/kg L f steam 2256 kj/kg c water 4190 J/kg K 1 atm Pa Page 2 of 6

3 I. Multiple Choice Questions (4 points each) Please write the letter corresponding to your answer for each question in the grid you made on the first inside page of your blue book. No partial credit is given for these questions. 1. For the Carnot cycle shown in the S-T diagram below, the total heat that enters the system during one entire cycle is T (K) S (J/K) (a) 75J (b) 200J (c) 125J (d) 125J (e) 75J 2. Oxygen has a molar mass of 32 g/mol and nitrogen has a molar mass of 28 g/mol. The oxygen and nitrogen molecules in a room of uniform temperature have (a) equal average kinetic energies, but the oxygen molecules are faster. (b) equal average kinetic energies, but the nitrogen molecules are faster. (c) equal average kinetic energies and speeds. (d) equal average speeds, but the oxygen molecules have a higher average kinetic energy. (e) equal average speeds, but the nitrogen molecules have a higher average kinetic energy. 3. Water flows out of a huge tank through the plumbing shown in the diagram. At which point is the pressure the least? (a) A (b) B (c) C (d) D (e) All of the pressures are the same. A B C D Page 3 of 6

4 4. Five satellites orbit the earth in circular orbits, each with different orbit radius and mass, as shown. Which of these would require the most energy to escape from earth orbit? ½ m (a) R 2R m (b) 3R (e) 5R (c) 3m 4R (d) 2m ½ m 5. Astronauts heading towards Jupiter at a speed of 0.6c relative to the earth see a cosmic ray shower that contains protons moving toward them. From earth, the speed of the protons is measured to be 0.2c. The astronauts measure the speed of the protons to be (a) 0.8c (b) 0.71c (c) 0.95c (d) 0.69c (e) 0.45c 6. The transverse displacement for a standing wave on a string is given by: y(x,t) = (4.2m) sin[(0.20 m 1 )x] cos[(3.0 s 1 )t] Which of the following is NOT true? (a) When t = π/2 seconds, the string is flat. (b) The wavelength is 10π meters. (c) The maximum transverse velocity v y of a point on the string is 2.1 m/s. (d) The point at x = 5π meters is a node. (e) The period of the oscillation is 2π/3 seconds. Page 4 of 6

5 Problems (15 Points for each problem) II. An ideal fluid sits inside of a cylindrical container which is open to the atmosphere at the top. The fluid has volume V, mass M, and temperature T 0. Its specific heat is c F K/kg, and its coefficient of thermal expansion is α. Q Joules of heat are applied in a reversible manner to the fluid, which expands at constant (atmospheric) pressure. (a) Find the final temperature of the fluid. (b) How much work does the fluid do on the atmosphere during its expansion? (c) Find the change in entropy of the fluid. III. In a binary star system, two stars with masses M and 3M orbit each other in circular orbits about their center-of-mass. The distance between the centers of the two stars is D. Newton s gravitational constant is G. D CofM (a) Find the tangential velocity of the lighter star. (Hint: gravity provides the centripetal force.) (b) Find the period of the orbital motion. (c) A system like this emits gravity waves, which travel outward at the speed of light. This causes the orbit to shrink and the period to change with time, providing us with the best evidence yet that gravity waves exist, since we see this happening in nature. The rate of energy loss due to gravitational radiation is: de/ = (128G/5c 5 )ω n M m D p. Find the values of the exponents n, m, and p such that de/ has the correct units. Here, ω is the orbital frequency, M is the mass of a star, and D is the distance between the two stars. IV. In a nuclear reactor, 235 U (Uranium 235) splits (at rest) into two smaller identical nuclei, 116 Ag. The molar masses of the three elements involved are 235g, 115.9g, respectively. (a) If one mole of 235 U undergoes fission, how much energy is released? (Note: Momentum and energy must be conserved! What are the final kinetic energies of the decay products?) Reactors are cooled by water. The kinetic energy of the reactor products are absorbed by the cooling water, heating it up. Assume that the cooling water is at 100 C. (If you didn t get part (a) assume a value of 3 J.) (b) How many moles of steam can be generated from the fission energy? (c) If the space into which the steam can expand is 1 liter/mole, what is the steam pressure? Assume a constant temperature of 100 C. (d) At some point later in the cycle, the steam is condensed to liquid at 100 C by passing it through a heat exchanger that contains coils of cooler water. If the cooling water is pulled from a reservoir at a temperature of 22 C, how many moles of cooling water are required to condense one mole of steam if the exit temperature of the cooling water is 40 C? (i.e., each mole of cooling water is only heated from 22 C to 40 C.) Page 5 of 6

6 V. Following up on the reactor problem, let s consider what happens when the steam is used to drive a turbine. The steam flows along a pipe of cross sectional area 300 cm 2 with velocity 100m/s. The pressure in the steam is 100atm, and it is at a temperature of 200 C. (a) Find the density of the steam in kg/m 3. The molar mass of water is 18g. (b) Find the rate of mass flow in kg/sec past an arbitrary point in the pipe. The steam then expands across a turbine, making it spin to generate electricity. The cross-sectional area of the new pipe after the expansion is 600 cm 2. (c) Assume an adiabatic expansion; γ = 4/3 for steam. What is the final density of the steam? (d) What is the final velocity of the steam? (Hint: conserve mass in the flow.) VI. Astérix and Obélix, heroes of Gaul, have laid siege to the castle of their sworn enemy, Nastius Maximus. In order to get a better view of the castle, they devise a simple device to throw Obélix into the air. It is based on a massless vertical spring, spring constant k=2000 N/m, with a small massless platform on top. (a) Obélix, whose mass is M 0 = 300 kg, sits on top of the spring. How far is the spring compressed? (b) To fire the catapult, Astérix pushes Obélix down another 0.2 meters and releases him from rest. Obélix, scared out of his mind, holds on for dear life. What is the angular frequency of his resulting oscillation? (c) At what elapsed time after his release does he reach the equilibrium position of his oscillation? Page 6 of 6