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### hw6_solutions

Course: APMAE 4200, Fall 2011
School: Columbia
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Columbia - APMAE - 4200
Columbia - APMAE - 4200
Columbia - APMAE - 4200
Columbia - APMAE - 4200
1/72/73/74/75/76/77/7
Columbia - APMAE - 4200
Columbia - APMAE - 4200
Columbia - APMAE - 4200
Columbia - APMAE - 4200
E4200 2011 MidtermProblem 1. Consider the wave equation for u(x, t): 2u 2u = c2 2 , t2 x u (0, t) = 0, x u(x, 0) = x u (x, 0) = t 0 &lt; x &lt; , u (, t) = 0, x 0&lt;x&lt; 1 -1 0&lt;x&lt; 2 t&gt;0 t &gt; 0, (1) &lt; x &lt; . 21.1. Using the method of separation of variables u(x, t)
Columbia - APMAE - 4200
Practice Final, APMA 4200, 2011.PROBLEM 1. Consider the function g(x) = x -x for for x (0, ), x (-, 0).1.1. Show that the function is even. 1.2. Calculate its Fourier coefficients. We recall that a 2 periodic function can be represented asf (x) = a0 +
Columbia - APMAE - 4200
E4200 Practice Midterm1. Let f (x) be the 2-periodic function defined on (-1, 1) by f (x) = 1.1. Sketch the graph of f (x). 1.2. Show that f (x) is even. 1.3. Show that11+x 1-xon on(-1, 0) (0, 1).(1 - x) cos(nx)dx =0 0 2 (n)2if ifn 2 and n is e
Columbia - APMAE - 4200
ME 6700 Carbon Nanotube Science and TechnologyProblem Set 1: Basics of quantum mechanics Due Date: September 21, 2011 1. For an 1D particle in a box, calculate:for the two lowest-energy eigenstates. Then calculate and compare to the minimum set by the H
Columbia - APMAE - 4200
MECE 6700Carbon Nanotube Sci &amp; TechJames Hone, Fall 2011Problem Set 1 Due on Sept. 21th1.For an 1D particle in a box, calculate: x2 = p2 = -2 x2 dx 2 dx x2for the two lowest-energy eigenstates. Then calculate xp = x2 - x2p2 - p2and compare to
Columbia - APMAE - 4200
MECE 6700 Problem Set 2 Due on Oct. 12thCarbon Nanotube Sci &amp; TechJames Hone, Fall 2011Determine the real-space unit vectors Ch and T, and the reciprocal lattice vectors K1 and K2 for the following tubes: (a) amchair (5,5) (b) zigzag (6,0) (c) zigzag (
Columbia - APMAE - 4200
Homework Solution #2 MECE 4212 Problem #1 Bulk When the etching process cuts deeply into the substrate. Surface Process that removes sacrificial layers from beneath thin film structures leaving free standing mechanical structures. Examples: Bulk Pressure
Columbia - APMAE - 4200
MECE 6700 Problem Set 3 Due on Nov. 5thCarbon Nanotube Sci &amp; TechJames Hone, Fall 20111.Plot the lines of allowed k-vectors for the following tubes:(a) amchair (5,5) (b) zigzag (6,0) (c) zigzag (7,0) (d) chiral (6,2) (e) chiral (5,2) SolutionPage 1
Columbia - APMAE - 4200
Homework Solution #3 MEMS MECE E4212 Problem #1 (a) KOH wet etching: Anisotropic wet etch. Highly selective to cfw_111 planes. Geometry constrained by crystallographic planes. Convex corners can be etched to undercut masking layers leaving a free standing
Columbia - APMAE - 4200
MECE 6700 Problem Set 4Carbon Nanotube Sci &amp; TechJames Hone, Fall 20111. Using the force constant parameters for 2D graphite in table 9.1, numerically calculate thephonon frequencies for 10 evenly spaced k-points along the -M direction, using only fir
Columbia - APMAE - 4200
Homework Solution #4 MEMS MECE E4212Problem #1 F (a) = kk=3EI L3=3 * 160 E 9 *1 (50 E - 6)(10 E - 6) 3 12 = 2N / m (1E - 3) 3 =1E - 6 N = 0.5m 2N / m(b) Neglecting Point Load:(2n - 1) E2L E ~ 160E9Pan =10 -3 100cm Kg = 2.3 g cm * = 2300 m3 1g
Columbia - APMAE - 4200
Homework Solution #5 MEMS MECE E4212 Problem #1 V PI = 8kg 0 3 27A (3) g 0 : the gap in the absence of any voltage or spring constant k : the spring constant A: the surface area 3EI Ebh 3 k= 3 = L 4 L3 (160 E 9)(50 E - 6)(10 E - 6)3 k= 3 4(1000 E - 6) k =
Columbia - APMAE - 4200
Homework Solution #6 MEMS MECE E4212 Problem #1zha RPiP0The force balance on the beam must take into account the surface tension, , over the circumference of the drop, and the pressure difference between the drop and the ambient air. F = - 2R sin +
Columbia - APMAE - 4200
Quiz 11. Consider a concentration of a chemical c(t, x). We denote by (t, x) the flux of the chemical (amount of chemical per unit surface flowing to the right per unit time). 1.1. Write the conservation of chemical in a rod (a, b) of cross-sectional are
Columbia - APMAE - 4200
Quiz 2Consider the heat equation u 2u = k 2 , 0 &lt; x &lt; , t &gt; 0 t x u u (0, t) = 0, (, t) = 0, t &gt; 0, x x u(x, 0) = 1 for 0 &lt; x &lt; , and u(x, 0) = -1 2(1) for &lt; x &lt; . 21. Using the method of separation of variables u(x, t) = G(t)(x), write the ordinary di
Columbia - APMAE - 4200
Quiz 3We consider the problem for u(x, t): u 2u + Q(t, x), = t x2 u(0, t) = 0, u(, t) = 0 u(x, 0) = 0 0 &lt; x &lt; , t&gt;0 0 &lt; x &lt; . t&gt;0 (1)u (x, t) is an even function in x and x 2 2 u u 2 2u that (x, t) is again odd in x, so that (x, t) = (y, t) sin nydy. n
Columbia - APMAE - 4200
Quiz 4.Consider the problem u u k(x) - (x)u, = t x x u(0, t) = 0, u(L, t) = 0, u(x, 0) = f (x), 0 &lt; x &lt; L. 0 &lt; x &lt; L, t&gt;0 t&gt;0 (1)We assume that k(x) &gt; 0. 1. Using the method of separation of variables u(x, t) = G(t)(x), find the equations that G and mus
Columbia - APMAE - 4200
Quiz 5We consider the equation for all 0 &lt; y &lt; L: - d2 G (x, y) = (x - y), 0 &lt; x &lt; L, dx2 dG (0, y) = 0, G(L, y) = 0. dx (1) (2)1. For fixed y, find the general solution of (1) on x (0, y) and x (y, L). 2. Use the boundary conditions (2) on G to simplif
Columbia - APMAE - 4200
Quiz 6.We recall the Fourier transform and inverse Fourier transform formulas: Fcfw_f () = F () = 1. 2. 3. 4. 1 2 eix f (x)dx,-F -1 cfw_F (x) = f (x) =-e-ix F ()d.For F () = e-| with &gt; 0, calculate the inverse Fourier transform f (x). Show that th
Columbia - APMAE - 4200
MECE 4212 - MEMS Midterm Fall 2010 November 3rd 2010, 2:30 4 pm in 227 Mudd Section I-Short paragraph answers [100 total points: 30-45 minutes] Please provide short paragraph answers for the questions in this section. Also, use math and sketches where nee
Arkansas Little Rock - SCIENCE - sci 241
Three Day Diet analysis1SCI/241 Janurary 27,2012I just recently decided that I was going to start eating healthier. I have made quite a bit of progress with changing how I eat. The recommendation that I got were not that far of what I'm trying to accom
Arkansas Little Rock - SCIENCE - sci 241
hcr240r5hcr240r5Welcome to WritePoint, the automated review system that recognizes errors most commonly made by university students in academic essays. The system embeds comments into your paper and suggests possible changes in grammar and style. Please
Arkansas Little Rock - SCIENCE - sci 241
Associate Level MaterialAppendix B Health Care TerminologyDirections: Complete Parts I and II and post to your Individual forum as an attachment.Part I: Defining Health Care TerminologyPart I. Refer to pages 23-24 in your text to complete the followin
Arkansas Little Rock - SCIENCE - sci 241
Associate Level MaterialAppendix B Health Care TerminologyDirections: Complete Parts I and II and post to your Individual forum as an attachment.Part I: Defining Health Care TerminologyPart I. Refer to pages 23-24 in your text to complete the followin
Arkansas Little Rock - SCIENCE - sci 241
HIPAA and Information Technology Checkpoint1HIPAA and Information Technology Chec HCR/24HIPAA and Information Technology Checkpoint2There are many advantages of having a standardized database for health records. Some of those advantages are a common
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Health Pyramid1Welcome to WritePoint, the automated review system that recognizes errors most commonly made by university students in academic essays. The system embeds comments into your paper and suggests possible changes in grammar and style. Please
Arkansas Little Rock - SCIENCE - sci 241
SCI/241 January 26, 201I just recently decided that I was going to start eating healthier. I am not on a diet! That is just too much stress. However, I have made quite a bit of progress with changing how I eat. The recommendation that I got were not that
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Individual Healthy Eating Plan1Welcome to WritePoint, the automated review system that recognizes errors most commonly made by university students in academic essays. The system embeds comments into your paper and suggests possible changes in grammar an
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HIPAA and Information Technology Checkpoint1HIPAA and Information Technology Checkpoint Priscilla Sanchez HCR/240 February 2, 2012HIPAA and Information Technology Checkpoint2There are many advantages of having a standardized database for health recor
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HIPAA and Information Technology Checkpoint1Welcome to WritePoint, the automated review system that recognizes errors most commonly made by university students in academic essays. The system embeds comments into your paper and suggests possible changes
Arkansas Little Rock - SCIENCE - sci 241
Individual Healthy Eating Plan Healthy Eat Content / Development 70 Points Points Earned 70/70 Additional Comments: Word count: 19441All key elements of the assignment are covered in a substantive way. Identifies current eating habits (as documented by
Arkansas Little Rock - SCIENCE - sci 241
( Gentl &amp; Hyers/Photonica)CHAPTER 4 CONCEPTSI I I I I I I IEating more unrefined grains, fruits, and vegetables and less added sugars increases the nutrient density of your diet. Sugars are the simplest type of carbohydrate. Complex carbohydrates are m
Arkansas Little Rock - SCIENCE - sci 241
Three Day Diet analysisDocument what you eat during a 3-day period by using the free food intake assessment tool located on http:/www.mypyramidtracker.gov. Set up an account with the MyPyramid Tracker website by using the registration instructions in App
Arkansas Little Rock - SCIENCE - sci 241
Assignment: Electronic Medical Record Speech1Welcome to WritePoint, the automated review system that recognizes errors most commonly made by university students in academic essays. The system embeds comments into your paper and suggests possible changes
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HIPAA and Information Technology Checkpoint1Human DigestionSCI/241 February 3, 2012CheckPoint: Human Digestion2The digestive system is a chain of organs and glands that breaks down the food we eat in order for our bodies to receive the nutrients it
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HIPAA and Information Technology CheckpointHuman Digestio1SCI/241 February 3, 2012Describe, in 200 to 300 words, the path food follows through the digestive system and how digestion occurs in each of the following parts of the digestive system. Includ
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Washington - AA - 311
AA 311 Homework 1. Due October 3, 20111. Write a brief essay about why you chose aeronautics and astronautics as your academic career path. (There is no right or wrong answer here.) What is your goal after graduation? 2. Pick three fixed-wing aircraft of
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AA311 Homework 2. Due October 10, 2011Give all your answers in SI units. 1. (Problem 2.1 in [1]) Consider the low-speed flight of the Space Shuttle as it is nearing a landing. If the air pressure and temperature at the nose are p = 1.2 atm and T = 300 K,
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AA311 Homework 2. Due October 10, 2011Give all your answers in SI units. 1. (Problem 2.1 in [1]) Consider the low-speed flight of the Space Shuttle as it is nearing a landing. If the air pressure and temperature at the nose are p = 1.2 atm and T = 300 K,
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AA311 Homework 3. Due October 17, 2011Give all your answers in SI units. Problem 1. Recall that, when developing the equations for the ISA and the notion of &quot;geopotential height,&quot; we assumed that g(hG ) g0 . The geopotential height h was then defined suc
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AA311 Homework 3. Due October 17, 2011. Solution.Give all your answers in SI units. Problem 1. Recall that, when developing the equations for the ISA and the notion of &quot;geopotential height,&quot; we assumed that g(hG ) g0 . The geopotential height h was then
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AA311 Homework 4. Due November 7, 2011Give all your answers in SI units. Problem 1. An aircraft is flying at a true airspeed of 400 ft/s in standard sea-level conditions. Estimate the pitot pressure (P0 ) based on the incompressible flow form of the Bern
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AA311 Homework 5. Due November 14, 2011Give all your answers in SI units. Problem 1. An aircraft weighs 12, 500 N and has a 20 m2 wing area, an aspect ratio of 6 and efficiency factor e = 0.9. If CD0 = 0.02, (a) Calculate the values of the minimum drag f
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AA311 - Atmospheric Flight Mechanics Autumn 2010 University of Washington Homework 2 SolutionsProblem 1Part a.From the problem statement we know that g0 = 24.9 m s2 R = 4157J kg KT = 150 Kg0 = 24.9; R = 4157; T = 150;Since the atmosphere is assumed
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AA311 - Atmospheric Flight Mechanics Autumn 2010 University of Washington Homework 3 SolutionsProblem 1Part a.From Appendix B, at h = 50 000 ft, the temperature is T = 389.99 RTinfty = 389.99;The free stream speed of sound is a= gRTwhere g = 1.4 R =
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AA311 - Atmospheric Flight Mechanics Autumn 2010 University of Washington Homework 4 SolutionsProblem 1We can find pressure at free stream as using Appendix A with hG = 3000 m p = 7.0121 104 N m2pinfty = 7.0121 104 ;Because the flow is assumed isentro
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AA311 - Atmospheric Flight Mechanics Autumn 2010 University of Washington Homework 5 SolutionsProblem 1Part a.For standard sea level conditions T = 288.16 K r = 1.225 kg m3p = 1.01325 105 N m2Tinfty = 288.16; infty = 1.225; pinfty = 1.01325 105 ;We
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AA311 - Atmospheric Flight Mechanics Autumn 2010 University of Washington Homework 6 SolutionsProblem 1Part a.We can first convert to consistent, metric units F-14 S = 54.5 m2 W = 52 000 lbf .4.44822162 N 1 lbf. = 231 308 N . = 123 661 N i = 1, 2 (th
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AA311 - Atmospheric Flight Mechanics Autumn 2010 University of Washington Homework 7 SolutionsProblem 1We can plot the data. From this we see that the drag polar should be described by CD = CD,0 +CL 2 p e ARNote that we can write this as y = a1 x2 + a