## CHM114: Exam #2 CHM 114, S2015 Exam #2, Version C 16 March 2015 Instructor: O. Graudejus Points: 100 Print Name Sign Name Student I.D. # 1. You are responsible for the information on this page. Please read it carefully. 2. Code your name and 10 digit affiliate identification number on the separate scantron answer sheet. Use only a #2 pencil 3. If you enter your ASU ID incorrectly on the scantron, a 3 point penalty will be assessed. 4. Do all calculations on the exam pages. Do not make any unnecessary marks on the answer sheet. 5. This exam consists of 25 multiple choice questions worth 4 points each and a periodic table. Make sure you have them all. 6. Choose the best answer to each of the questions and answer it on the computer-graded answer sheet. Read all responses before making a selection. 7. Read the directions carefully for each problem. 8. Avoid even casual glances at other students’ exams. 9. Stop writing and hand in your scantron answer sheet and your test promptly when instructed. LATE EXAMS MAY HAVE POINTS DEDUCTED. 10. You will have 50 minutes to complete the exam. 11. If you leave early, please do so quietly. 12. Work the easiest problems first. 13. A periodic table is attached as the last page to this exam. 14. Answers will be posted online this afternoon. Potentially useful information: K = ºC + 273.15 RH=2.18·10-18 J R=8.314 J·K-1·mol-1 1Å=10-10 m c=3·108 m/s Ephoton=h·n=h·c/l h=6.626·10-34 Js Avogadro’s Number = 6.022 × 1023 particles/mole DH°rxn = n DHf° (products) – n DHf° (reactants) ) 1 1 ( 2 2 f i H n n DE = R − \ -2- CHM114: Exam #2 1) Which one of the following is an incorrect orbital notation? A) 2s B) 2p C) 3f D) 3d E) 4s 2) The energy of a photon that has a frequency of 8.21 1015s 1 − × is __________ J. A) 8.08 10 50 − × B) 1.99 10 25 − × C) 5.44 10 18 − × D) 1.24×1049 E) 1.26 10 19 − × 3) The ground state electron configuration of Ga is __________. A) 1s22s23s23p64s23d104p1 B) 1s22s22p63s23p64s24d104p1 C) 1s22s22p63s23p64s23d104p1 D) 1s22s22p63s23p64s23d104d1 E) [Ar]4s23d11 4) Of the bonds N–N, N=N, and NN, the N-N bond is __________. A) strongest/shortest B) weakest/longest C) strongest/longest D) weakest/shortest E) intermediate in both strength and length 5) Of the atoms below, __________ is the most electronegative. A) Br B) O C) Cl D) N E) F 6) Of the following, __________ cannot accommodate more than an octet of electrons. A) P B) O C) S D) Cl E) I -3- CHM 114: Exam #2 7) Which electron configuration represents a violation of Hund’s Rule? A) B) C) D) E) 8) A tin atom has 50 electrons. Electrons in the _____ subshell experience the highest effective nuclear charge. A) 1s B) 3p C) 3d D) 5s E) 5p 9) In ionic compounds, the lattice energy_____ as the magnitude of the ion charges _____ and the radii _____. A) increases, decrease, increase B) increases, increase, increase C) decreases, increase, increase D) increases, increase, decrease E) increases, decrease, decrease 10) Which of the following ionic compounds has the highest lattice energy? A) LiF B) MgO C) CsF D) CsI E) LiI -4- CHM 114: Exam #2 11) For which one of the following reactions is the value of H°rxn equal to Hf° for the product? A) 2 C (s, graphite) + 2 H2 (g) C2H4 (g) B) N2 (g) + O2 (g) 2 NO (g) C) 2 H2 (g) + O2 (g) 2 H2O (l) D) 2 H2 (g) + O2 (g) 2 H2O (g) E) all of the above 12) Given the data in the table below, H rxn D ° for the reaction 3 2 3 PCl (g) + 3HCl(g)®3Cl (g) + PH (g) is __________ kJ. A) -570.37 B) -385.77 C) 570.37 D) 385.77 E) The f DH° of 2 Cl (g) is needed for the calculation. 13) Given the following reactions (1) 2 2 2NO® N +O H = -180 kJ (2) 2 2 2NO+O ®2NO H = -112 kJ the enthalpy of the reaction of nitrogen with oxygen to produce nitrogen dioxide 2 2 2 N + 2O ®2NO is __________ kJ. A) 68 B) -68 C) -292 D) 292 E) -146 14) Of the following transitions in the Bohr hydrogen atom, the __________ transition results in the absorption of the lowest-energy photon. A) n = 1 n = 6 B) n = 6 n = 1 C) n = 6 n = 5 D) n = 3 n = 6 E) n = 1 n = 4 -5- CHM 114: Exam #2 15) Which equation correctly represents the electron affinity of calcium? A) Ca (g) Ca+ (g) + e- B) Ca (g) Ca- (g) + e- C) Ca (g) + e- Ca- (g) D) Ca- (g) Ca (g) + e- E) Ca+ (g) + e- Ca (g) 16) Which of the following does not have eight valence electrons? A) Ca+ B) Rb+ C) Xe D) Br− E) All of the above have eight valence electrons. 17) The specific heat of liquid bromine is 0.226 J/g · K. The molar heat capacity (in J/mol-K) of liquid bromine is __________. A) 707 B) 36.1 C) 18.1 D) 9.05 E) 0.226 18) Given the electronegativities below, which covalent single bond is least polar? Element: H C N O F Electronegativity: 2.1 2.5 3.0 3.5 4.0 A) C-H B) C-F C) O-H D) O-C E) F-H 19) The bond length in an HCl molecule is 1.27 Å and the measured dipole moment is 1.08 D. What is the magnitude (in units of e) of the negative charge on Cl in HCl? (1 debye = 3.34 10 30 coulomb-meters − × ; e=1.6 10 19 coulombs − × ) A) 1.6 10 19 − × B) 0.057 C) 0.18 D) 1 E) 0.22 -6- CHM 114: Exam #2 20) The F-B-F bond angle in the BF3 molecule is approximately __________. A) 90° B) 109.5° C) 120° D) 180° E) 60° 21) Which isoelectronic series is correctly arranged in order of increasing radius? A) K+ < Ca2+ < Ar < Cl- B) Cl- < Ar < K+ < Ca2+ C) Ca2+ < Ar < K+ < Cl- D) Ca2+ < K+ < Ar < Cl- E) Ca2+ < K+ < Cl- < Ar 22) What is the electron configuration for the Fe2+ ion? A) [Ar]4s03d6 B) [Ar]4s23d4 C) [Ar]4s03d8 D) [Ar]4s23d8 E) [Ar]4s63d2 23) The formal charge on carbon in the Lewis structure of the NCS - ion is __________: A) -1 B) +1 C) +2 D) 0 E) +3 -7- CHM 114: Exam #2 24) Using the table of bond dissociation energies, the H for the following gas-phase reaction is __________ kJ. A) 291 B) 2017 C) -57 D) -356 E) -291 25) According to VSEPR theory, if there are six electron domains in the valence shell of an atom, they will be arranged in a(n) __________ geometry. A) octahedral B) linear C) tetrahedral D) trigonal planar E) trigonal bipyramidal -8- CHM 114: Exam #2

## SUMMARY ASSIGNMENT SHEET Upload to Turnitin.com by due date. Create an account using Class ID: 10423941 and Password: English. Due Dates: • Peer Response Workshop with Rough Draft: Tuesday, September 8th • First Draft Due (submit for feedback): Thursday, September 10th • Final Draft Due (highlighting, labeling & reflection done in class): Thursday, September 17th What is Summary? Summary is a comprehensive and objective restatement of the main ideas of a text (an article, book, movie, event, etc.) and while the act of summarizing might seem like an easy and obvious undertaking there is a noticeable difference between summarizing and summarizing well. For example, many students, without even realizing it, leave out key information when they summarize because they forget to consider how much their reader already knows about the topic or reading. Be careful to explain fully so readers do not have to guess what you mean. In his textbook, A Brief Guide to Writing from Readings, Stephen Wilhoit suggests that to avoid the pitfalls of unclear or disjointed summaries, they should keep in mind the qualities of a good summary. These qualities include: Neutrality – The writer avoids inserting his or her opinion or interpreting the original text’s content in any way. This requires the writer avoid language that is evaluative, such as: good, bad, effective, ineffective, interesting, boring, etc. Also, keep 1st person (I, we, our, us) out of the summary; instead, summary should be written in grammatical 3rd person (For example, he she, the author, they, etc). Brevity – The summary should not be longer than the original text, but rather highlight the most important information from that text while leaving out unnecessary details and still maintaining accuracy. Independence – The summary should make sense to someone who has not read the text. There should be no confusion about the main content and organization of the original source. This also requires that the summary be accurate—that it not misinterpret any part of the original text. Mastering the craft of summarizing puts students in the position to do well on many assignments in college, not just English essays. In most fields, from the humanities to the sciences, summary is a required task. Being able to summarize lab results accurately and briefly, for example, is critical in a chemistry or engineering class. Summarizing the various theories of sociology or education helps a person apply them to his or her fieldwork. In business, summarizing ideas effectively can be useful in numerous scenarios. The assignment: • Your summary will follow guidelines that integrate both informative and explanatory summary components: Informative summary “simply conveys the author’s main ideas, data, [and] arguments” (Wilhoit 62). Explanatory summary maintains the order of the author’s main points, and also includes the author, title and publication information. • The text: “Who are You and What are You Doing Here?” by Mark Edmundson, pgs. 115-27 The basic structure is as follows: • An Introductory Paragraph including: o The title of the essay, the author’s name, and a brief bio o The topic of the essay—what the text is about o The author’s main idea (essentially your thesis statement) • Body Paragraph(s) including: o Topic Sentences with transitions for each main idea the author addresses o Supporting points following the same order as the article o A concluding sentence that expounds upon, or echoes, the main idea • A Work Cited page that o Includes only the source text Evaluation: A successful summary will include all of the following: • Briefly summarizes the main ideas of the text • Comprehensive, accurate and independent summarization of the text • Objective and neutral restatement of the main ideas • Contain no directly quoted sentences • Clearly introduces the author and title • Body paragraphs that reflect the text’s main ideas • Clear and effective transitions • Mostly free of mechanical and grammatical errors • A correct Work Cited page • Formatted with correct MLA standards • Follows the basic structure of a summary Possible Points: • Peer Response Workshop w/ Rough Draft: 10 points • Final Draft: 80 points • Highlighted revisions and labeling: 5 points • Reflection: 5 points _______________________________________________________________ • Total possible points for the Summary Assignment: 100 Points

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## Project 1: Particle Trajectory in Pleated Filters Due: 12:30 pm, Dec. 1, 2015, submission through blackboard Course: Numerical Methods Instructor: Dr. Hooman V. Tafreshi Most aerosol filters are made of pleated fibrous media. This is to accommodate as much filtration media as possible in a limited space available to an air filtration unit (e.g., the engine of a car). A variety of parameters contribute to the performance of a pleated filter. These parameters include, but are not limited to, geometry of the pleat (e.g., pleat height, width, and count), microscale properties of the fibrous media (e.g., fiber diameters, fiber orientation, and solid volume fraction), aerodynamic and thermal conditions of the flow (e.g., flow velocity, temperature, and operating pressure), and particle properties (e.g., diameter, density, and shape). Figure 1: Examples of pleated air filters [1‐2]. In this project you are asked to calculate the trajectory of aerosol particles as they travel inside a rectangular pleat channel. Due to the symmetry of the pleat geometry, you only need to simulate one half of the channel (see Figure 2). Figure 2: The simulation domain and boundary conditions (the figure’s aspect ratio is altered for illustration purposes). Trajectory of the aerosol particles can be calculated in a 2‐D domain by solving the Newton’s 2nd law written for the particles in the x‐ and y‐directions, v(h) inlet velocity fibrous media v(y) y tm l h x Ui u(l) u(x) 2 2 p 1 p 1 ( , ) d x dx u x y dt dt 2 2 p 1 p 1 ( , ) d y dy v x y dt dt where 2 1/18 p p d is the particle relaxation time, 10 μm p d is the particle diameter, 1000 kg/m3 p is the particle density, and 1.85105 Pa.s is the air viscosity. Also, u(x, y) and v(x, y) represent the components of the air velocity in the x and y directions inside the pleat channel, respectively. The x and y positions of the particles are denoted by xp and yp, respectively. You may use the following expressions for u(x, y) and v(x, y) . 2 3 1 2 u x, y u x y h sin 2 v x,y v h π y h where i 1 u x U x l h is the average air velocity inside the pleat channel in the x‐direction and Ui is the velocity at the pleat entrance (assume 1 m/s for this project). l = 0.0275 m and h =0.0011 m are the pleat length and height, respectively. Writing the conservation of mass for the air flowing into the channel, you can also obtain that i v h U h l h . These 2nd order ODEs can easily be solved using a 4th order Rung‐Kutta method. In order to obtain realistic particle trajectories, you also need to consider proper initial conditions for the velocity of the particles: x(t 0) 0 , ( 0) i p p y t y , p ( 0) cos i i dx t U dt , p ( 0) sin i i dy t U dt . where i is the angle with respect to the axial direction by which a particle enters the pleat channel (see Figure 3). The inlet angle can be obtained from the following equation: 2 75 0.78 +0.16 1.61St i i p p i y y e h h where 2 St 18 2 ρPdPUi μ h is the particles Stokes number. Figure 3: An illustration of the required particle trajectory calculation inside a rectangular pleated filter. You are asked to calculate and plot the trajectories of particles released from the vertical positions of ?? ? ? 0.05?, ?? ? ? 0.25?, ?? ? ? 0.5?, ?? ? ? 0.75? , and ?? ? ? 0.95? in one single figure. To do so, you need to track the trajectories until they reach one of the channel walls (i.e., stop when xp l or p y h ). Use a time step of 0.00001 sec. For more information see Ref. [3]. For additional background information see Ref. [4] and references there. In submitting your project please stick to following guidelines: 1‐ In blackboard, submit all the Matlab files and report in one single zip file. For naming your zip file, adhere to the format as: Lastname_firstname_project1.zip For instance: Einstein_albert_project1.zip 2‐ The report should be in pdf format only with the name as Project1.pdf (NO word documents .docx or .doc will be graded). 3‐ Your zip file can contain as many Matlab files as you want to submit. Also please submit the main code which TA’s should run with the name as: Project1.m (You can name the function files as you desire). Summary of what you should submit: 1‐ Runge–Kutta 4th order implementation in MATLAB. 2‐ Plot 5 particle trajectories in one graph. 3‐ Report your output (the x‐y positions of the five particles at each time step) in the form of a table with 11 columns (one for time and two for the x and y of each particle). Make sure the units are second for time and meter for the x and y. 4‐ Write a short, but yet clean and professional report describing your work. Up to 25% of your grade will be based solely on the style and formatting of your report. Use proper heading for each section of your report. Be consistent in your font size. Use Times New Roman only. Make sure that figures have proper self‐explanatory captions and are cited in the body of the report. Make sure that your figures have legends as well as x and y labels with proper and consistent fonts. Don’t forget that any number presented in the report or on the figures has to have a proper unit. Equations and pages in your report should be numbered. Embed your figures in the text. Make sure they do not have unnecessary frames around them or are not plotted on a grey background (default setting of some software programs!). inlet angle Particle trajectory i p y i 0 p x Important Note: It is possible to solve the above ODEs using built‐in solvers such as ode45 in MATLAB, and you are encouraged to consider that for validating your MATLAB program. However, the results that you submit for this project MUST be obtained from your own implementation of the 4th order Runge‐Kutta method. You will not receive full credit if your MATALB program does not work, even if your results are absolutely correct! References: 1. http://www.airexco.net/custom‐manufacturedbr12‐inch‐pleated‐filter‐c‐108_113_114/custommadebr12‐ inch‐pleated‐filter‐p‐786.html 2. http://www.ebay.com/itm/Air‐Compressor‐Air‐Filter‐Element‐CFE‐275‐Round‐Pleated‐Filter‐ /251081172328 3. A.M. Saleh and H.V. Tafreshi, A Simple Semi‐Analytical Model for Designing Pleated Air Filters under Loading, Separation and Purification Technology 137, 94 (2014) 4. A.M. Saleh, S. Fotovati, H.V. Tafreshi, and B. Pourdeyhimi, Modeling Service Life of Pleated Filters Exposed to Poly‐Dispersed Aerosols, Powder Technology 266, 79 (2014)

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## Morgan Extra Pages Graphing with Excel to be carried out in a computer lab, 3rd floor Calloway Hall or elsewhere The Excel spreadsheet consists of vertical columns and horizontal rows; a column and row intersect at a cell. A cell can contain data for use in calculations of all sorts. The Name Box shows the currently selected cell (Fig. 1). In the Excel 2007 and 2010 versions the drop-down menus familiar in most software screens have been replaced by tabs with horizontally-arranged command buttons of various categories (Fig. 2) ___________________________________________________________________ Open Excel, click on the Microsoft circle, upper left, and Save As your surname. xlsx on the desktop. Before leaving the lab e-mail the file to yourself and/or save to a flash drive. Also e-mail it to your instructor. Figure 1. Parts of an Excel spreadsheet. Name Box Figure 2. Tabs. Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com EXERCISE 1: BASIC OPERATIONS Click Save often as you work. 1. Type the heading “Edge Length” in Cell A1 and double click the crack between the A and B column heading for automatic widening of column A. Similarly, write headings for columns B and C and enter numbers in Cells A2 and A3 as in Fig. 3. Highlight Cells A2 and A3 by dragging the cursor (chunky plus-shape) over the two of them and letting go. 2. Note that there are three types of cursor crosses: chunky for selecting, barbed for moving entries or blocks of entries from cell to cell, and tiny (appearing only at the little square in the lower-right corner of a cell). Obtain a tiny arrow for Cell A3 and perform a plus-drag down Column A until the cells are filled up to 40 (in Cell A8). Note that the two highlighted cells set both the starting value of the fill and the intervals. 3. Click on Cell B2 and enter a formula for face area of a cube as follows: type =, click on Cell A2, type ^2, and press Enter (note the formula bar in Fig. 4). 4. Enter the formula for cube volume in Cell C2 (same procedure, but “=, click on A2, ^3, Enter”). 5. Highlight Cells B2 and C2; plus-drag down to Row 8 (Fig. 5). Do the numbers look correct? Click on some cells in the newly filled area and notice how Excel steps the row designations as it moves down the column (it can do it for horizontal plusdrags along rows also). This is the major programming development that has led to the popularity of spreadsheets. Figure 3. Entries. Figure 4. A formula. Figure 5. Plus-dragging formulas. Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com 6. Now let’s graph the Face Area versus Edge Length: select Cells A1 through B8, choose the Insert tab, and click the Scatter drop-down menu and select “Scatter with only Markers” (Fig. 6). 7. Move the graph (Excel calls it a “chart”) that appears up alongside your number table and dress it up as follows: a. Note that some Chart Layouts have appeared above. Click Layout 1 and alter each title to read Face Area for the vertical axis, Edge Length for the horizontal and Face Area vs. Edge Length for the Graph Title. b. Activate the Excel Least squares routine, called “fitting a trendline” in the program: right click any of the data markers and click Add Trendline. Choose Power and also check “Display equation on chart” and “Display R-squared value on chart.” Fig. 7 shows what the graph will look like at this point. c. The titles are explicit, so the legend is unnecessary. Click on it and press the delete button to remove it. Figure 6. Creating a scatter graph. Figure 7. A graph with a fitted curve. Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com 8. Now let’s overlay the Volume vs. Edge Length curve onto the same graph (optional for 203L/205L): Make a copy of your graph by clicking on the outer white area, clicking ctrl-c (or right click, copy), and pasting the copy somewhere else (ctrl-v). If you wish, delete the trendline as in Fig. 8. a. Right click on the outer white space, choose Select Data and click the Add button. b. You can type in the cell ranges by hand in the dialog box that comes up, but it is easier to click the red, white, and blue button on the right of each space and highlight what you want to go in. Click the red, white, and blue of the bar that has appeared, and you will bounce back to the Add dialog box. Use the Edge Length column for the x’s and Volume for the y’s. c. Right-click on any volume data point and choose Format Data Series. Clicking Secondary Axis will place its scale on the right of the graph as in Fig. 8. d. Dress up your graph with two axis titles (Layout-Labels-Axis Titles), etc. Figure 8. Adding a second curve and y-axis to the graph Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com EXERCISE 2: INTERPRETING A LINEAR GRAPH Introduction: Many experiments are repeated a number of times with one of the parameters involved varied from run to run. Often the goal is to measure the rate of change of a dependent variable, rather than a particular value. If the dependent variable can be expressed as a linear function of the independent parameter, then the slope and yintercept of an appropriate graph will give the rate of change and a particular value, respectively. An example of such an experiment in PHYS.203L/205L is the first part of Lab 20, in which weights are added to the bottom of a suspended spring (Figure 9). This experiment shows that a spring exerts a force Fs proportional to the distance stretched y = (y-yo), a relationship known as Hooke’s Law: Fs = – k(y – yo) (Eq. 1) where k is called the Hooke’s Law constant. The minus sign shows that the spring opposes any push or pull on it. In Lab 20 Fs is equal to (- Mg) and y is given by the reading on a meter stick. Masses were added to the bottom of the spring in 50-g increments giving weights in newtons of 0.49, 0.98, etc. The weight pan was used as the pointer for reading y and had a mass of 50 g, so yo could not be directly measured. For convenient graphing Equation 1 can be rewritten: -(Mg) = – ky + kyo Or (Mg) = ky – kyo (Eq. 1′) Procedure 1. On your spreadsheet note the tabs at the bottom left and double-click Sheet1. Type in “Basics,” and then click the Sheet2 tab to bring up a fresh worksheet. Change the sheet name to “Linear Fit” and fill in data as in this table. Hooke’s Law Experiment y (m) -Fs = Mg (N) 0.337 0.49 0.388 0.98 0.446 1.47 0.498 1.96 0.550 2.45 2. Highlight the cells with the numbers, and graph (Mg) versus y as in Steps 6 and 7 of the Basics section. Your Trendline this time will be Linear of course. If you are having trouble remembering what’s versus what, “y” looks like “v”, so what comes before the “v” of “versus” goes on the y (vertical) axis. Yes, this graph is confusing: the horizontal (“x”) axis is distance y, and the “y” axis is something else. 3. Click on the Equation/R2 box on the graph and highlight just the slope, that is, only the number that comes before the “x.” Copy it (control-c is a fast way to Figure 9. A spring with a weight stretching it Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com do it) and paste it (control-v) into an empty cell. Do likewise for the intercept (including the minus sign). SAVE YOUR FILE! 5. The next steps use the standard procedure for obtaining information from linear data. Write the general equation for a straight line immediately below a hand-written copy of Equation 1′ then circle matching items: (Mg) = k y + (- k yo) (Eq. 1′) y = m x + b Note the parentheses around the intercept term of Equation 1′ to emphasize that the minus sign is part of it. Equating above and below, you can create two useful new equations: slope m = k (Eq. 2) y-intercept b = -kyo (Eq. 3) 6. Solve Equation 2 for k, that is, rewrite left to right. Then substitute the value for slope m from your graph, and you have an experimental value for the Hooke’s Law constant k. Next solve Equation 3 for yo, substitute the value for intercept b from your graph and the value of k that you just found, and calculate yo. 7. Examine your linear graph for clues to finding the units of the slope and the yintercept. Use these units to find the units of k and yo. 8. Present your values of k and yo with their units neatly at the bottom of your spreadsheet. 9. R2 in Excel, like r in our lab manual and Corr. in the LoggerPro software, is a measure of how well the calculated line matches the data points. 1.00 would indicate a perfect match. State how good a match you think was made in this case? 10. Do the Homework, Further Exercises on Interpreting Linear Graphs, on the following pages. Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com Eq.1 M m f M a g , (Eq.2) M slope m g (Eq.3) M b f Morgan Extra Pages Homework: Graph Interpretation Exercises EXAMPLE WITH COMPLETE SOLUTION In PHYS.203L and 205L we do Lab 9 Newton’s Second Law on Atwood’s Machine using a photogate sensor (Fig. 1). The Atwood’s apparatus can slow the rate of fall enough to be measured even with primitive timing devices. In our experiment LoggerPro software automatically collects and analyzes the data giving reliable measurements of g, the acceleration of gravity. The equation governing motion for Atwood’s Machine can be written: where a is the acceleration of the masses and string, g is the acceleration of gravity, M is the total mass at both ends of the string, m is the difference between the masses, and f is the frictional force at the hub of the pulley wheel. In this exercise you are given a graph of a vs. m obtained in this experiment with the values of M and the slope and intercept (Fig. 2). The goal is to extract values for acceleration of gravity g and frictional force f from this information. To analyze the graph we write y = mx + b, the general equation for a straight line, directly under Equation 1 and match up the various parameters: Equating above and below, you can create two new equations: and y m x b M m f M a g Figure 1. The Atwood’s Machine setup (from the LoggerPro handout). Figure 2. Graph of acceleration versus mass difference; data from a Physics I experiment. Atwood’s Machine M = 0.400 kg a = 24.4 m – 0.018 R2 = 0.998 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 0.000 0.010 0.020 0.030 0.040 0.050 0.060 m (kg) a (m/s2) Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com 2 2 9.76 / 0.400 24.4 /( ) m s kg m kg s g Mm To handle Equation 2 it pays to consider what the units of the slope are. A slope is “the rise over the run,“ so its units must be the units of the vertical axis divided by those of the horizontal axis. In this case: Now let’s solve Equation 2 for g and substitute the values of total mass M and of the slope m from the graph: Using 9.80 m/s2 as the Baltimore accepted value for g, we can calculate the percent error: A similar process with Equation 3 leads to a value for f, the frictional force at the hub of the pulley wheel. Note that the units of intercept b are simply whatever the vertical axis units are, m/s2 in this case. Solving Equation 3 for f: EXERCISE 1 The Picket Fence experiment makes use of LoggerPro software to calculate velocities at regular time intervals as the striped plate passes through the photogate (Fig. 3). The theoretical equation is v = vi + at (Eq. 4) where vi = 0 (the fence is dropped from rest) and a = g. a. Write Equation 4 with y = mx + b under it and circle matching factors as in the Example. b. What is the experimental value of the acceleration of gravity? What is its percent error from the accepted value for Baltimore, 9.80 m/s2? c. Does the value of the y-intercept make sense? d. How well did the straight Trendline match the data? 2 / 2 kg s m kg m s 0.4% 100 9.80 9.76 9.80 100 . . . % Acc Exp Acc Error kg m s mN kg m s f Mb 7.2 10 / 7.2 0.400 ( 0.018 / ) 3 2 2 Figure 3. Graph of speed versus time as calculated by LoggerPro as a picket fence falls freely through a photogate. Picket Fence Drop y = 9.8224x + 0.0007 R2 = 0.9997 0 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 1.2 t (s) v (m/s) Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com EXERCISE 2 This is an electrical example from PHYS.204L/206L, potential difference, V, versus current, I (Fig. 4). The theoretical equation is V = IR (Eq. 5) and is known as “Ohm’s Law.” The unit symbols stand for volts, V, and Amperes, A. The factor R stands for resistance and is measured in units of ohms, symbol (capital omega). The definition of the ohm is: V (Eq. 6) By coincidence the letter symbols for potential (a quantity ) and volts (its unit) are identical. Thus “voltage” has become the laboratory slang name for potential. a. Rearrange the Ohm’s Law equation to match y = mx + b.. b. What is the experimental resistance? c. Comment on the experimental intercept: is its value reasonable? EXERCISE 3 This graph (Fig. 5) also follows Ohm’s Law, but solved for current I. For this graph the experimenter held potential difference V constant at 15.0V and measured the current for resistances of 100, 50, 40, and 30 Solve Ohm’s Law for I and you will see that 1/R is the logical variable to use on the x axis. For units, someone once jokingly referred to a “reciprocal ohm” as a “mho,” and the name stuck. a. Rearrange Equation 5 solved for I to match y = mx + b. b. What is the experimental potential difference? c. Calculate the percent difference from the 15.0 V that the experimenter set on the power supply (the instrument used for such experiments). d. Comment on the experimental intercept: is its value reasonable? Figure 4. Graph of potential difference versus current; data from a Physics II experiment. The theoretical equation, V = IR, is known as “Ohm’s Law.” Ohm’s Law y = 0.628x – 0.0275 R2 = 0.9933 0 0.1 0.2 0.3 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 Current, I (A) Potential difference, V (V) Figure 5. Another application of Ohm’s Law: a graph of current versus the inverse of resistance, from a different electric circuit experiment. Current versus (1/Resistance) y = 14.727x – 0.2214 R2 = 0.9938 0 100 200 300 400 500 600 5 10 15 20 25 30 35 R-1 (millimhos) I (milliamperes) Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com EXERCISE 4 The Atwood’s Machine experiment (see the solved example above) can be done in another way: keep mass difference m the same and vary the total mass M (Fig. 6). a. Rewrite Equation 1 and factor out (1/M). b. Equate the coefficient of (1/M) with the experimental slope and solve for acceleration of gravity g. c. Substitute the values for slope, mass difference, and frictional force and calculate the experimental of g. d. Derive the units of the slope and show that the units of g come out as they should. e. Is the value of the experimental intercept reasonable? EXERCISE 5 In the previous two exercises the reciprocal of a variable was used to make the graph come out linear. In this one the trick will be to use the square root of a variable (Fig. 7). In PHYS.203L and 205L Lab 19 The Pendulum the theoretical equation is where the period T is the time per cycle, L is the length of the string, and g is the acceleration of gravity. a. Rewrite Equation 7 with the square root of L factored out and placed at the end. b. Equate the coefficient of √L with the experimental slope and solve for acceleration of gravity g. c. Substitute the value for slope and calculate the experimental of g. d. Derive the units of the slope and show that the units of g come out as they should. e. Is the value of the experimental intercept reasonable? 2 (Eq . 7) g T L Figure 6. Graph of acceleration versus the reciprocal of total mass; data from a another Physics I experiment. Atwood’s Machine m = 0.020 kg f = 7.2 mN y = 0.1964x – 0.0735 R2 = 0.995 0.400 0.600 0.800 1.000 2.000 2.500 3.000 3.500 4.000 4.500 5.000 1/M (1/kg) a (m/s2) Effect of Pendulum Length on Period y = 2.0523x – 0.0331 R2 = 0.999 0.400 0.800 1.200 1.600 2.000 2.400 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 L1/2 (m1/2) T (s) Figure 7. Graph of period T versus the square root of pendulum length; data from a Physics I experiment. Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com EXERCISE 6 In Exercise 5 another approach would have been to square both sides of Equation 7 and plot T2 versus L. Lab 20 directs us to use that alternative. It involves another case of periodic or harmonic motion with a similar, but more complicated, equation for the period: where T is the period of the bobbing (Fig. 8), M is the suspended mass, ms is the mass of the spring, k is a measure of stiffness called the spring constant, and C is a dimensionless factor showing how much of the spring mass is effectively bobbing. a. Square both sides of Equation 8 and rearrange it to match y = mx + b. b. Write y = mx + b under your rearranged equation and circle matching factors as in the Example. c. Write two new equations analogous to Equations 2 and 3 in the Example. Use the first of the two for calculating k and the second for finding C from the data of Fig. 9. d. A theoretical analysis has shown that for most springs C = 1/3. Find the percent error from that value. e. Derive the units of the slope and intercept; show that the units of k come out as N/m and that C is dimensionless. 2 (Eq . 8) k T M Cm s Figure 8. In Lab 20 mass M is suspended from a spring which is set to bobbing up and down, a good approximation to simple harmonic motion (SHM), described by Equation 8. Lab 20: SHM of a Spring Mass of the spring, ms = 25.1 g y = 3.0185x + 0.0197 R2 = 0.9965 0.0000 0.2000 0.4000 0.6000 0.8000 1.0000 0 0.05 0.1 0.15 0.2 0.25 0.3 M (kg) T 2 2 Figure 9. Graph of the square of the period T2 versus suspended mass M data from a Physics I experiment. Click to buy NOW! PDF-XChange Viewer www.docu-track.com Click to buy NOW! PDF-XChange Viewer www.docu-track.com EXERCISE 7 This last exercise deals with an exponential equation, and the trick is to take the logarithm of both sides. In PHYS.204L/206L we do Lab 33 The RC Time Constant with theoretical equation: where V is the potential difference at time t across a circuit element called a capacitor (the is dropped for simplicity), Vo is V at t = 0 (try it), and (tau) is a characteristic of the circuit called the time constant. a. Take the natural log of both sides and apply the addition rule for logarithms of a product on the right-hand side. b. Noting that the graph (Fig. 10) plots lnV versus t, arrange your equation in y = mx + b order, write y = mx + b under it, and circle the parts as in the Example. c. Write two new equations analogous to Equations 2 and 3 in the Example. Use the first of the two for calculating and the second for finding lnVo and then Vo. d. Note that the units of lnV are the natural log of volts, lnV. As usual derive the units of the slope and interecept and use them to obtain the units of your experimental V and t. V V e (Eq. 9) t o Figure 10. Graph of a logarithm versus time; data from Lab 33, a Physics II experiment. Discharge of a Capacitor y = -9.17E-03x + 2.00E+00 R2 = 9.98E-01 0.00 0.50 1.00 1.50 2.00 2.50

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## . What behaviors indicate psychological distress? Name 5 and explain.

The term ‘distress’ is commonly used in nursing literature to … Read More...