Physics Exam 2 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 (no stored equations or programs and no graphing). Show all of your work in the blue book. For problems II-V, an answer alone is worth very little credit, even if it is correct so show how you get it. 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 Exam 2 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 = v 2 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 = ω 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 = r/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 Sound : speed : v = B ρ Intensity : I = 1 2 ρω2 vs 2 max v = ωacos(kx ωt + φ) Some Constants: Boltzmann k B = J/K, N A = , R = 8.31 J/mol K.

2 I. Multiple Choice Questions (4 points each) Please write the letter corresponding to your answer for each question in the grid stamped on the first inside page of your blue book. No partial credit is given for these questions. 1. In class, we explored the physics of simple harmonic motion if a hole were to be drilled through the earth and you jumped in. We found that the period was about 5000 seconds for this trip. The radius of the earth is m. You carry a laser that emits light at a wavelength of nm that you shine at us as you oscillate. What is the maximum wavelength we see from your laser throughout the trip? (non-relativistic formulæ are ok.) (a) nm (b) nm (c) nm (d) nm (e) nm 2. Two strings of different mass densities are strung between two walls, as shown. The mass densities are such that 1 = 4 2, (i.e., string 1 is has four times the mass per unit length of string 2.) and the total length is 3 meters. You wish to have the between the two strings be stationary for a standing wave at a given frequency. Which of the following diagrams indicates a possible situation where this is true? (Ignore the possible reflections off of the.) (a) (b) L 1 = 2m (c) (d) L 1 = 1m (e) Page 2 of 5

3 3. An organ pipe is closed at one end, open on the other. Take the speed of sound to be 343 m/s. Two successive audible harmonics have frequencies 1029 Hz and 1715 Hz. How long is the pipe (not the wavelength)? (a) 0.25 m (b) 0.5 m (c) 1m (d) 2.0 m (e) 4.0 m 4. A cantaloupe of radius 2 cm is placed inside of a pipe of radius 3 cm. Far from the cantaloupe, the fluid flows at a velocity of 4 m/s. If the fluid has a density of 1200 kg/m 3, what is the maximum pressure change as the fluid flows past the cantaloupe? (a) Pa (b) Pa (c) Pa (d) Pa (e) Pa 5. The magnet in a speaker moves with an amplitude of 2mm at a frequency of 30 Hz. Assuming the density of air is 1.2 kg/m 3, and the velocity of sound is 343 m/s, what is the sound level of the speaker? (a) 29.2 db (b) 127 db (c) 135 db (d) 13.5 db (e) 119 db Problems (20 points each) II. A hydroelectric plant consists of a two-level system, as shown in the figure. On the upper level, a dam holds back water of density. The intake for the power system is a long pipe of cross-sectional area A 1 that leads down to the lower level. The mouth of the pipe is a distance h below the surface of the water. The turbine station is located a distance D below the intake pipe. Here, the pipe narrows to an area A 2 before it enters the turbine station. A 1 1 h D turbine A 2 (a) Find the pressure at the bottom of the dam at the input to the intake pipe (labeled 1 ). (b) The total flow through the turbine is T m 3 /s. Use this to find v 2, the velocity of the water at the turbine in terms of the given quantities. (c) Find v 1, the velocity of the water just inside the intake pipe in terms of the given quantities. (d) Find the pressure inside the intake pipe just before the water reaches the turbine. Page 3 of 5

4 III. A wooden cube of side A floats partially submerged in a large reservoir of water. The bottom of the cube is a distance L below the water. The density of the water is. Attached to the bottom of the cube is a spring of force constant k. The spring is initially neither stretched nor compressed. (a) Find the buoyancy force on the block. What is the net force on the block? Now, the cube is pushed down a small distance h from this equilibrium position. (b) Find the net force (magnitude and direction) on the cube. Take downward to be the positive direction. (c) Write Newton s 2 nd Law (F = ma) for this system demonstrating that it has an equation of motion characteristic of simple harmonic motion. (d) The block is released. Find the angular frequency of its oscillation. A L IV. A transverse harmonic wave traveling down a string is shown in the figure below. y 4.0 cm 3.0 cm x (a) The tension in the rope is 10 N, and the mass density is 0.1 kg/m. Find the velocity of the wave. (b) The figure shows the wave at time t = 0. At this time, the wave amplitude at x = 0 is half of the maximum value of 4.0 cm (i.e., y = 2.0 cm at x = t = 0.). If the wave travels to the right, what is the phase angle that must be included? Give the phase in radians, not degrees. (c) Find the wavelength of the wave if the amplitude is zero at x = 3.0 cm at time t = 0. (d) Write an equation describing this wave mathematically. continued Page 4 of 5

5 V. In optics, non-reflective coatings are used to cancel spurious reflections from surfaces. This problem explores an analogous system using sound waves and water. First, consider a sound wave incident on a wall, like the end wall of the pipe shown above. (Ignore the gray area for now.) Given your knowledge of how a sound wave propagates and our discussion of phase flips at the ends of strings, etc., (a) Does the sound wave get reflected as is or with a 180 phase shift? Explain your reasoning by drawing sketches of the displacement wave as the wave reaches then reflects off of the wall. (Hint: what must be the magnitude of the displacement wave at the wall, always?) Now, we want to include a thickness t of water into the problem. Since the gas and the water have different compressibilities, the speed of the sound wave in the two media will be different. (b) If B G and G are the bulk compressibility and density of the gas, and B W and W are the bulk compressibility and density of water, find the wavelength of a sound wave of frequency f in each of the media. (c) The sound wave in the water travels a distance t, is reflected off of the wall, and travels back a distance t to the air-water interface. What is the total phase change of the sound wave during this process? (d) The portion of the original wave that does not travel into the water is reflected off of the airwater interface back into the tube. We want to adjust the thickness t so that this reflected wave and the returning water wave are 180 out of phase, which would cancel all of the reflection. Find the minimum thickness t so that this is the case. (Note: water is a denser medium than air do you also expect a phase shift for the wave reflected off of the air-water interface?) (e) Bonus: B W = N/m 2, B G = N/m 2, W = 1000 kg/m 3, G = 1.21 kg/m 3, f = 1 khz; find a numeric value for t. t Page 5 of 5