Tacoma Narrows Bridge Collapse "Gallopin' Gertie" https://video.search.yahoo.com/yhs/search?fr=yhs-adk-adk_sbnt&hsimp=yhsadk_sbnt&hspart=adk&p=youtube+tacoma+bridge+collapse#id=4&vid=ffea498afb4251333b6f5b57baed 644b&action=click AC Circuits (You Do due 10.31.17) 1 2 3 4 5 6 An LC circuit containing a 0.355 H inductor has a maximum current of 7.88 A. When the energy stored in the inductor is at its maximum, what is (a) the energy stored in the inductor? (b) in the capacitor? (c) the total energy? ((a) 11.0 J; (b) 0 J; (c) 11.0 J). The maximum charge across the 2.70×10−5 F capacitor in an LC circuit is 4.83×10−4 C at t = 0 when the switch is closed. The circuit contains a 3.22×10−1 H inductor. At time t =22.3ms, (a) what is the energy stored in the capacitor? (b) in the inductor? ((a) 3.56e−4 J; (b) 3.96e−3 J) An AC generator generates a maximum emf of 35 V and runs at a frequency of 12 Hz. At a time t, the emf is at a maximum. How many seconds after t does it take for the emf to fall to zero? (0.021 s) At time t = 0 s, the potential difference across the capacitor in an AC capacitor circuit is 0V, and it is increasing. The angular frequency of the AC generator is 18.5 rad/s. What is the first time when the current in the circuit reaches 0 A? (8.49e−2 s) A capacitor of capacitance 3.4×10−6 F is part of an AC capacitor circuit with angular frequency of 8.8×102 rad/s. At time t = 0 s, the current is at its maximum value of 0.21A. What is the potential difference across the capacitor at time t = 2.6 ms? (53 V) The diagram on the right is an RLC circuit with two capacitors. If R = 387 Ω, L = 3.65×10−1 H, C1 = 1.12×10−6 F and C2 = 3.27×10−6 F, what is the resonance frequency of this circuit? (790 rad/s) 7. An AC circuit has a 1.00×10−11 F capacitor in parallel with a 3.00×10−11 F capacitor and a 6.00×10−2 H inductor in parallel with a 2.00×10−2 H inductor as shown below. What is the resonant frequency of this circuit? Hint: Combine the inductors in the same manner as you would resistors. (1.29e6 rad/s) 8. What is the instantaneous power, at a time of 0.00350 seconds, dissipated by a 200 ohm resistor in a resonant AC circuit powered by a generator with a 21.0 V maximum potential difference operating at 5000 rad/s? (2.10 W) 9. The emf of an AC circuit has an rms value of 120 V. (a) What is the maximum positive emf? (b) What is the most negative emf? ((a) 170 V; (b) -170 V) 10. A series RLC circuit contains an AC generator (120 volts rms, frequency = 60 Hz), a −5 35.0 ohm resistor, a 1.49×10 F capacitor, and a 0.0300 H inductor. (a) What is the resonant frequency of the circuit? Express your answer in Hz. (b) Determine the capacitive reactance. (c) Determine the inductive reactance. (d) Determine the maximum current in the circuit. (e) Determine the phase constant. (f) Determine the average power. (g) What driving frequency minimizes the circuit impedance? (h) What driving frequency maximizes the average power consumed? ((a) 238 Hz; (b) 178 Ω; c) 11.3 Ω; (d) 0.996 A; (e) −1.36 rad; (f) 17.4 W; (g) 238 Hz; (h) 238 Hz) Name: ___________________________ Period:_____ Mag Lab Go to the following website http://phet.colorado.edu/simulations/sims.php?sim=Magnets_and_Electromagnets ^ ^ (those are underscores) Click Move the compass around the bar magnet. 1. Which pole of the magnet does the red compass needle point towards? Click “Flip Polarity” in the right menu. 2. Now which pole of the magnet does the red needle point towards? Does it still point toward the same pole? 3. This means that the red part of the needle is a _________ pole. Click in the right menu. Check the box “Show Field Meter” in the right menu. A blue box should appear. This measures the Magnetic Field around the magnet (which is known as ‘B’). The Magnetic field is measured in Gauss (G). Move the field meter around the magnet. 4. Does the field increase or decrease as you move the meter closer to the magnet? Move your meter so that it is about one inch (on your computer screen) away from the North end of your magnet. 5. What is the magnitude of field strength (B) in Gauss? Now move the meter the same distance away from the South end of your magnet. 6. Is the amount of magnetic field the same for both North and South ends of a magnet? At the top left of the simulation window, click the “Electromagnet” tab. You should see a battery connected to a wire with loops that has current running through it. Move your compass around the electromagnet. 7. Is the left side of the electromagnet the North end or the South end? How do you know? 1 Mag Lab Click “Show Field Meter” in the right menu. 8. Move the meter around the electromagnet. Does the field strength increase or decrease as you move the meter closer to the electromagnet? Place the meter about one inch (on your screen) from the left side of the electromagnet. In the right menu you can adjust the number of loops in your electromagnet. For each number of loops (1-4) write down the field strength in the table below. # Loops Field Strength (in G) 1 2 3 4 9. Based on your data above, does the amount of field strength increase or decrease as you increase the number of loops in an electromagnet? Set the number of loops for your electromagnet back to 4 and make sure your field meter is still one inch from the left side of the coils. Your battery has a sliding bar on it that lets you adjust the voltage in your electromagnet. Complete the table below by adjusting the voltage on the battery and writing down the field strength at each voltage. Voltage (in V) Field Strength (in G) 0 2 4 6 8 10 10. Based on your data above, what is the relationship between voltage and field strength in an electromagnet? 2 Morgan Extra Pages Graphing with Excel to be carried out in a computer lab, 3rd floor Calloway Hall or elsewhere Name Box Figure 1. Parts of an Excel spreadsheet. 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) Figure 2. Tabs. ___________________________________________________________________ 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. 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. Figure 4. A formula. 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. 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”). Figure 5. Plus-dragging formulas. Figure 6. Creating a scatter graph. 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 7. A graph with a fitted curve. 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 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). ing 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 Figure 9. A spring with a weight stretching it 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 giv- 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 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) y = mx+ b (Eq. 1′) 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 y o. 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. Laboratory Simulation: Reflection and Refraction (due 11.14.17) Learning goals     Familiarize with simulations of physical processes. Log raw data and plot graphs. Partially familiarize with the scientific method (phenomenon, prediction, experiment conclusion). Derive the dependence of the angle of refraction on the angle of incidence and the index of refraction. Simulation used “Refraction of light” (“bending-light_el.jar”) Theory / Definitions 1. Optical (or transparent) medium: ________________________________________________________________________ ___________________________________________________________________________________________ ___________________________________________________________________________________________ ___________________________________________________________________________________________ 2. Index of refraction: ____________________________________________________________________ ___________________________________________________________________________________________ __________________________________________ __________________________________________ 4. Denote the angles of incidence and refraction . __________________________________________ 3. Refraction: _________________________________________ _________________________________________ __________________________________________ _________________________________________ ___________________________________________________________________________________________ Experiment 1: Dependence of angle of refraction on the angle of incidence Laser light falls from air to a transparent medium. Prediction: What do we expect to happen to the angle of refraction as the index of refraction of the transparent medium increases? Explain. ___________________________________________________________________________________________ ___________________________________________________________________________________________ 1 Set the angle of incidence at and leave it Don’t change unchanged 2 Measure the angle of refraction 1 Vary from 1.00 to 1.60 The laser is placed so that the angle of incidence equals 50 degrees. Table 1: Angle of reflaction vs the index of refraction Measurement Index of refraction 1 2 3 4 5 6 7 2 Angle of refraction (degrees) Graph 1: Angle of reflaction vs the index of refraction Conclusion compared to our prediction: ___________________________________________________________________________________________ ___________________________________________________________________________________________ 3 Experiment 2: Dependence of the angle of refraction on the angle of incidence Predictions/Hypothesis: What do we expect to happen to the angle of refraction as the angle of incidence increases? Explain. ___________________________________________________________________________________________ ___________________________________________________________________________________________ Vary the angle of incidence ( 1 Keep unchanged ) 2 Measure the angle of refraction Table 2: Angle of refraction vs angle of incidence Measurement Angle of incidence (degrees) 1 2 3 4 5 6 7 8 4 Angle of refraction (degrees) Graph 2: Angle of refraction vs angle of incidence Conclusion compared to the prediction: ___________________________________________________________________________________________ ___________________________________________________________________________________________ ___________________________________________________________________________________________ 5 Conclusion: Theory predictions versus the results of the 2 experiments. A number of scientists between the 10th and the 17th centuries (Sahl, Snellius, Descartes) concluded that the following formula should relate the angle of incidence with the angle of refraction : Explain if the results of the 2 simulated experiments above are compatible with the formula. Experiment 1: __________________________________________________________________________________________ __________________________________________________________________________________________ _________________________________________________________________________________________ ___________________________________________________________________________________________ ___________________________________________________________________________________________ Experiment 2: ___________________________________________________________________________________________ _________________________________________________________________________________________ ___________________________________________________________________________________________ ___________________________________________________________________________________________ ___________________________________________________________________________________________ ___________________________________________________________________________________________ 6