Homework 1

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Never enter units in your answers. Be sure to use the units indicated on the HW submission page!!!




Enter numbers without commas. 23,546 should be 23546 Always enter a * for multiplication. g(H+h) should be g*(H+h)


 

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Submitted Answers will be graded M 9/15, W 9/17, Th 9/18, F 9/19, and M 9/22 at 12:00 noon.




Problem 1.1
	What you should learn: Pressure is generally defined as force per unit area. As long as a force is exerted over an extended area, we can define an associated pressure.
	Problem: Mercury is placed in a vertical cylinder of radius r.  The height of the mercury in the cylinder is h. Find P, the pressure on the bottom of the cylinder.
	Constants and fixed variables: Density of mercury: rho = 13.53 g/ml Radius of cylinder: r = 2.00 cm Atmospheric pressure: P0 = 1.01 x 105 N/m2 Gravitational acceleration: g = 9.80 m/sec2 (Remember that you may use the names r, rho, P0, and g in your answer.)
	Variables that must be changed with each submission: Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
	Hints: What units do I use?
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Problem 1.2
	What you should learn: In fluid mechanics, pressure is defined as the force per unit area that results from the random motion of the fluid. We can think of it as the force on a surface flowing with the fluid. You will also need to know that the flow rate is the cress-sectional area multiplied by the velocity: Φ = Av
	Problem: Water passes through a pipe of radius r at a flow rate of Φ = 3.00 m3/s.  Water coming out of the pipe hits you in the back. The water is in the open air with a pressure of 1.00 atm. Find the force the water exerts on you as it strikes you. Plot this force as a function of r. To do this problem, you will need a model of how the water behaves as it strikes you. Let's assume that the water travels horizontally and that its velocity goes to zero upon impact . The water then falls to the ground. To find the force, we need to remember that force is dp/dt, the change of momentum per unit time. Enter the force of the water on you (F) for the value of r given below. Note that although the pressure is the same for all values of r, the force the water exerts on you does depend on r. This is because the pressure is related to the force that comes from the random motion of molecules in the water, not the force that comes from the bulk motion of the water. Making this distinction will help you understand pressure in the problems that follow. [Φ is Phi]
	Constants and fixed variables: Flow rate: Phi = 3.00 m3/s The density of water: rho=1000 kg/m3 Gravitational acceleration: g = 9.80 m/sec2
	Variables that must be changed with each submission: Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
	Hints: I still don't see how to calculate a force from these data.
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Problem 1.3
	What you should learn: The pressure at a depth h in static fluids of constant density is P=P0+ρgh [ρ is rho]
	Problem: Four unusually shaped vases are filled with water to the same height, h. Which vase has the largest pressure at the bottom? Using the drawings and value of h given below, estimate this largest pressure, P.
	Constants and fixed variables: Density of water rho = 1000 kg/m3 Gravitational acceleration: g = 9.8 m/s2
	Variables that must be changed with each submission: Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
	Hints: How does the shape of the container affect the force?
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Problem 1.4
	What you should learn: The buoyant force is the weight of the water displaced by an object. If an object floats, its buoyant force equals it's weight. While an object sinks it still experiences a buoyant force, but the buoyant force is less than its weight.
	Problem: A block of wood of height H floats in water with the top of the block a distance h above the water surface. Find h using the values of H and the density of the wood listed below.
	Constants and fixed variables: Density of water rho = 1000 kg/m3 Gravitational acceleration: g = 9.8 m/s2 Height of block: H = 12.4 cm.
	Variables that must be changed with each submission: Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
	Hints: What does "displaced water" mean?
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Problem 1.5
	What you should learn: For streamline flow, "total head" is conserved. This is a consequence of conservation of energy. Total head is defined as TH  =  P + ρ v2/2 + ρ gy. [ρ is rho]
	Problem: For the next few problems we will consider a water tank on top of a hill with a pipe extending down the hill, as shown to the right. The tank is open to the air. We choose to call y = 0 to be at the bottom of the tank. The height of the water level in the tank is H. The pipe extends down the hill a vertical distance h. We assume that the water is an ideal fluid and that no air enters into the pipe. Note that whenever we use the word "tank," we mean something that has large enough volume that we consider the velocity of the fluid everywhere in the tank (including right next to the pipe) to be zero. Write an algebraic expression for the total head TH at the top of the tank in terms of the variables below: H, the height of the water in the tank h, the vertical drop of the pipe g, the acceleration of gravity rho, the density of water P0, atmospheric pressure
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Problem 1.6
	What you should learn: For streamline flow, "total head" is conserved. That means that everywhere in the flow, total head has the same numerical value. Whenever water is in contact with air, the pressure is P0, atmospheric pressure, whether the water is moving or not.
	Problem: Consider the tank of problem 1.5. Write an algebraic expression for the velocity of the water coming out the pipe in terms of the same variables as in Problem 1.5. How does the velocity compare with the velocity of a ball dropped from rest through a distance H+h?
	Hints: How do I apply Bernoulli's equation here?
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Problem 1.7
	What you should learn: Once again total head is conserved. Bernoulli's equation applies to static bodies of water as well as to streamline flow. Gauge pressure is true pressure minus atmospheric pressure.
	Problem: Consider the tank of problem 1.5. Find the numerical value for the gauge pressure of the water at the bottom of the tank near where the water enters the pipe. It may easiest to solve the problem algebraically first rather than find the numerical value of total head first; however, either method works.
	Constants and fixed variables: Density of water: rho = 1000 kg/m3 Gravitational acceleration: g = 9.8 m/s2 Atmospheric pressure: P0 = 1.01 x 105 N/m2 Vertical drop of the pipe: h=5.40 m
	Variables that must be changed with each submission: Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
	Hints: How do I apply Bernoulli's equation here?
	Results of previous submissions: Submission Score Your Answer Correct Answer Comments Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
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Problem 1.8
	What you should learn: This is one more application of Bernoulli's equation. Gauge pressure is the true pressure minus atmospheric pressure. Here we must also apply the continuity equation. Since the flow rate (in m3/s) must be the same at every point through a pipe, we know A1v1 = A2 v2.
	Problem: Consider the tank of problem 1.5. Find a numerical value for the gauge pressure just inside the pipe as it leaves the tank. Note that there is a sudden drop in pressure as the water enters the pipe. This pressure difference is what accelerates the water into the pipe. Another way of looking at it is that the kinetic energy of the water comes at the cost of losing pressure.
	Constants and fixed variables: Density of water: rho = 1000 kg/m3 Gravitational acceleration: g = 9.8 m/s2 Atmospheric pressure: P0 = 1.01 x 105 N/m2 Vertical drop of the pipe: h=5.40 m
	Variables that must be changed with each submission: H, as given above.
	Hints: What is the velocity of the water at the top of the pipe? How can the pressure be negative?
	Results of previous submissions: Submission Score Your Answer Correct Answer Comments Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
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Problem 1.9
	What you should learn: Ideal fluids are non-viscous. Real water, however, has significant viscosity effects.
	Problem: You can make water spray faster out of a hose by partially covering the hose with your thumb. If water had no viscosity, what would happen to the pressure of the water coming out of the hose and the pressure of water just inside the hose when you partially cover the end with your thumb.  If water were ideal, after you out your thumb over the hose, A. the water outside the hose would go faster and the water inside the hose would go the same speed. B. the water outside the hose would go faster and the water inside the hose would go slower. C. the water outside the hose would go the same speed and the water inside the hose would go slower. D. the water outside the hose and the water inside the hose would go the same speed. For multiple choice questions, enter the letter of the answer. The form will only accept upper case letters.
	Hints: I can't figure out how this relates to anything we discussed in class.
	Results of previous submissions: Submission Score Your Answer Correct Answer Comments Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
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Problem 1.10
	What you should learn: Flow rate is given by Φ  = Av. Flow rate of an incompressible fluid through a pipe is the same at every point through a pipe. With all else equal, pressure decreases when velocity increases. [Φ is Phi]
	Problem: An artery of diameter D has blockage at one point that reduces the diameter of the opening to d. In the unblocked region, the gauge pressure is 140 mmHg (atmospheric pressure is 760 mmHg) and the velocity of the blood is 0.500 m/s. What is the pressure in the region of constriction? Enter for P the gauge pressure in mmHg. Note that the pressure of the body cavity can cause the artery to collapse if the pressure in the constricted region drops too low.
	Constants and fixed variables: Unblocked artery diameter: D is 6.00 mm Atmospheric pressure: P0 = 1.01 x 105 N/m2 Density of blood: rho = 1060 kg/m3 Gravitational acceleration: g = 9.8 m/s2 Blood velocity: v = 0.500 m/s
	Variables that must be changed with each submission: Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
	Hints: Bernoulli's equation doesn't give me enough information to solve the problem. What should I do?
	Results of previous submissions: Submission Score Your Answer Correct Answer Comments Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
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Problem 1.11
	What you should learn: The trajectory of a stream of water after it leaves a hose is the same as the trajectory of a solid mass.
	Problem: A pipe has its end turned upward so water leaves at an angle of 60° with respect to the ground. The speed of the water as it leaves the pipe is v0. To what vertical distance above the end of the pipe does the stream of water rise? (Ignore air resistance and assume the water is an ideal fluid.)
	Constants and fixed variables: Density of water: rho = 1000 kg/m3 Gravitational acceleration: g = 9.8 m/s2 Atmospheric pressure: P0 = 1.01 x 105 N/m2
	Variables that must be changed with each submission: Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
	Hints: Some general help
	Results of previous submissions: Submission Score Your Answer Correct Answer Comments Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
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Problem 1.12
	What you should learn: Any well-behaved function of x-vt represents a wave traveling to the right with a velocity v. Similarly, any function of x+vt represents a wave traveling to the left with a velocity v. Waves do not have to be sine functions.
	Problem: A wave at time t = 0.300 s has the mathematical form y(x,t=0.300 s) = A exp(kx-c). The wave moves to the right at a speed v.  What is the value of y for x = 0 at the time t2 given below?
	Constants and fixed variables: A =  0.00347 m k = 4.32 m-1  c = 5.67 v = 6.00 m/s
	Variables that must be changed with each submission: Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
	Hints: How do I put t2 into the equation since there's no t there?
	Results of previous submissions: Submission Score Your Answer Correct Answer Comments Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
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Problem 1.13
	What you should learn: The basic equation for of a sine wave is y(x,t)=A sin(kx-ωt+φ) where A is the amplitude, k is the wave number, ω is the angular frequency, φ is the phase angle (we will let φ = 0 this assignment), k = 2π/λ and ω=2π/f where λ is wavelength and f frequency. You should also know that the period T = 1/f and the velocity v = λf = ω/k. Angular frequency is radians per second. Wave number is radians per meter. It is in space what angular frequency is in time. Period is seconds per cycle. Frequency is meters per cycle. It is in space what period is in time. Be sure you memorize all of these concepts and equations and are very familiar with them!   [ω is omega, λ is lambda, and π is pi, φ is phi].
	Problem: A wave on a string has an amplitude (maximum displacement) y0 and a speed v. Write the equation of the wave in the form y = A sin(kx-ωt) with all the constants replaced by numbers in SI units.
	Constants and fixed variables: Phase angle: φ = 0 Amplitude: y0 =  3.00 mm Wave speed: v = 452 m/s
	Variables that must be changed with each submission: Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
	Hints: Be sure all your numbers are in SI units.
	Results of previous submissions: Submission Score Your Answer Correct Answer Comments Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
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Problem 1.14
	What you should learn: See Problem 1.13.
	Problem: A wave on a string has an amplitude y0 and a wave number k. Write the equation of the wave in the form y = A sin(kx-ωt) with all the constants replaced by numbers in SI units.
	Constants and fixed variables: Amplitude: y0 =  3.00 mm Wave number: k = 2.58 m-1
	Variables that must be changed with each submission: Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
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Problem 1.15
	What you should learn: Waves in matter travel with a speed that is always given by a general for v2 = {elastic or restoring property}/{inertial property}. On a string v2 = T/μ  where T is the tension in string and μ is the linear density (mass/meter) of the string. Be careful not to confuse period and tension in equations, since we use the same symbol for both of these quantities. [μ is mu]
	Problem: A vertical rope has a linear mass density of mu. The length of the rope is L. Find the wave speed v as a function of x where x is the distance from the top of the rope. Your answer should be given in terms of mu, L, x, and  g, the gravitational acceleration. (Note that when I give you a list of the variables to use in a problem, you may not have to use all these variables in your answer.)
	Hints: How can I find the tension in the rope?
	Results of previous submissions: Submission Score Your Answer Correct Answer Comments Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
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Problem 1.16
	What you should learn: The wave equation (for a one-dimensional wave with no energy loss) is (d2y)/(dx2) = 1/v2 * (d2y)/(dt2). Let's interpret this for a string: If the displacement is small and restricted to the y direction, then y' is close to zero and y'' is approximately equal to the curvature of the function. That is, the radius of the osculating circle at x is related to y'' by the relation y'' = 1/|R|. the wave equation says that the restoring force and hence the acceleration of a point on the string is proportional to the curvature of the string.
	Problem: The curvature of a string at a point is given along with the tension and the linear mass density of the string. What is the acceleration a of the string at that point. Assume the string displacement is entirely transverse. Be careful of signs.
	Constants and fixed variables: Curvature (d2y)/(dx2):  ypp =  0.65 Tension: T = 3.28 N
	Variables that must be changed with each submission: Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
	Hints: This is just an exercise to help you understand the terms in the wave equation.
	Results of previous submissions: Submission Score Your Answer Correct Answer Comments Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
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Problem 1.17
	What you should learn: When a pulse passes from one medium to another: 1) there is a reflected pulse and and a transmitted pulse, and 2) the time over which the pulse occurs remains the same for incident pulse, the transmitted pulse, and the reflected pulse. Energy is related to the square of the amplitude of the pulses and energy must be conserved in this process. If the pulse reaches the end of a string where the string is fixed or if the pulse reaches a heavier string, it inverts upon reflection. Similarly, when a sine wave passes from one medium to another, 1) part of the wave is reflected and part of the wave is transmitted, and 2) the frequency of the incident wave is the same as the frequencies of the reflected and transmitted waves. This will be important when we learn about light.
	Problem: A wave is moving toward the right along a light (smaller linear density) string and passes on to a heavier string as indicated in the top figure. Which of the bottom figures is qualitatively accurate? Enter the letter corresponding to the correct answer.
	Hints: What can I say about the heights and widths of the reflected and transmitted waves?
	Results of previous submissions: Submission Score Your Answer Correct Answer Comments Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
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Problem 1.18
	What you should learn: The speed of a wave in a string is v2 = T/μ . Similarly, the speed of sound in matter is v2 = B/ρ where B is the bulk modulus and  ρ is the density. You might expect the speed of sound to be slower in iron than in air because the density is larger. However, the bulk modulus is a quantity that is inversely proportional to compressibility. Since iron isn't very compressible, it's bulk modulus is very large, and the speed of sound correspondingly high. This problem is designed to give you a feeling for the speed of sound in air and iron. It's really quite easy. [μ is mu, ρ is rho]
	Problem: You are on a straight section of railroad track. A friend is quite a distance down the tracks where you can't see him very clearly, but you can hear the sound of him hitting the track with a hammer. You wish to determine the distance to your friend. You decide to put one ear on the tracks and then carefully measure the time between the sounds of the hammer in each of your ears. The temperature is 20° C. Find the distance to your friend, d, given the difference in time between the two sound pulses, Deltat.
	Constants and fixed variables: Speed of sound in air at 20° C: v = 343 m/s. Speed of sound in iron: vFe = 5950 m/s.
	Variables that must be changed with each submission: Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
	Hints: Set up two equations in two unknowns.
	Results of previous submissions: Submission Score Your Answer Correct Answer Comments Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
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Problem 1.19
	What you should learn: Sound intensity is power/unit area or energy/(unit area x unit time) From a point source, sound intensity falls off as 1/r2. Sound levels are measured in units of decibels (dB) using the equation β = 10 log10 (I/I0) where I0 is the threshold of hearing, 1.00 x 10-12 W/m2. This problem will also give a little review in using logarithms.
	Problem: Fireworks exploding at a height of h are measured to have a sound level of 108 dB directly below them. How far would you have to move so the sound level would be at 90 dB? Enter x , the distance you move along the ground (not the distance from the explosions to your ear).
	Variables that must be changed with each submission: Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
	Hints: I can't remember rules for logarithms.
	Results of previous submissions: Submission Score Your Answer Correct Answer Comments Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
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Problem 1.20
	What you should learn: Learn to use the Doppler shift formula: f ' = f (v±vo)/(v±vs). To get the signs right, ask yourself if the motion of the observer of the source tends to increase or to decrease the frequency. I purposely used different symbols for the velocities in this problem so you will have to think about what velocities to use and what signs to choose. Relate the Doppler formula to the times it takes signals to arrive.
	Problem: A rumble strip consists of 101 grooves cut into the pavement on the edge of a highway. The grooves are spaced a distance d apart. A car travels at a speed v over the rumble strip. A listener is located at a distance L directly in front of the car at a distance L from the first groove. Let t = 0 be the time the car passes over the first groove. Find the following: The time when the car passes over the last groove. The times when the first and last pulses reach the listener on the road. The frequency of the sound pulses as heard in the car. The frequency of the sound pulses as heard by the listener on the road. Calculate the frequency heard by the listener using the Doppler formula. Enter f the frequency of the pulses as heard by the listener on the road.
	Constants and fixed variables: Distance between grooves: d = 30.0 cm Distance from the first groove to the listener: L = 200 m Speed of sound: v0 = 343 m/s
	Variables that must be changed with each sub Database Results Wizard Error The operation failed. If this continues, please contact your server administrator. mission:
	Hints: How can I calculate the frequencies from this information?
	Results of previous submissions: Submission Score Your Answer Correct Answer Comments Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
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Problem 1.21
	What you should learn: See Problem 1.20.
	Problem: Repeat Problem 1.20 with the listener on the road located a distance L behind the first groove.
	Results of previous submissions: Problem Submission Score Your Answer Correct Answer Comments Database Results Wizard Error The operation failed. If this continues, please contact your server administrator.
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