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Physics 101 DIY : Make and Use a Barometer to Measure Air Pressure Overview Air pressure is the result of the weight of tiny particles of air (air molecules) pushing down on an area. While invisible to the naked eye (i.e. microscopic), they nevertheless take up space and have weight. For example, take a deep breath while holding your hand on your ribs and observe what happens. Did you feel your chest expand? Why did it expand? Air pressure expands because the air molecules take up space in your lungs, causing your chest to expand. Furthermore, air can be compressed to fit in a smaller volume since there's a lot of empty space between the air molecules. When compressed, air is placed under high pressure. Meteorologists measure these changes in the air to forecast weather, and the tool they use is a barometer. The common units of measurement that barometers use are millibars (mb) or inches of mercury. DIY Make a Barometer A. Materials o B. Theory How does this measure air pressure? C. Procedure 1. Place the completed barometer and scale in a shaded location free from temperature changes (i.e. not near a window as sunlight will adversely affect the barometer's results). 2. In your notebook or the table below, record the current date, time, the weather conditions, and air pressure (i.e. the level where the end of the straw measures on the scale). 3. Continue checking the barometer twice a day (if possible) each day over a four days period. Date Data Table Weather Air Time Conditions Pressure Date June 4, 2003 June 4, 2003 June 5, 2003 Sample Data Table Weather Air Time Conditions Pressure Clear and 9:30 am 4 Sunny 2:30 pm Cloudy 3 9:30 am Rainy 1 Barometer Analysis Answer the following questions in complete sentences. 1. What problem were you trying to solve with your barometer? ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 2. Were there any changes in the weather during the week? ___________________________________________________________________________ ___________________________________________________________________________ 3. Did the barometer measurement change when the weather changed? How much? ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 4. Did the barometer measurement change without a change in weather? Why do you think that happened? ___________________________________________________________________________ ___________________________________________________________________________ 5. How well did your barometer work? ___________________________________________________________________________ ___________________________________________________________________________ 6. What would you change if you could design the barometer again? ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 7. How do the barometer measurements help us understand the system of weather around us? ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ The Acceleration Due to Gravity Prof. M.L.C. Murdock version: Sept2015 Introduction To sensibly describe the motion of objects in our universe, we need to understand displacement, velocity and acceleration. This lab will emphasize the latter. When we use the word “acceleration” we mean the rate at which the velocity of a moving object changes with time. Accelerations are always caused by “forces.” (For those who missed lecture, think of a force as a push or a pull.) This is the essence of Newton’s first law. In today’s lab we will measure the acceleration due to the gravitational force exerted by the earth on two different types of objects, a tennis ball and a ping-pong ball. Theory Newton's second law of force relates the amount of force on an object to its mass and acceleration. F = ma (1) The greater the force on an object, the larger is its acceleration. Beyond that, this equation doesn’t say all that much until we know what to write on the left-hand side! Fortunately, that is the case for today. Probably the most apparent (and yet the weakest!) of the known forces in nature is the gravitational force. Newton's Universal Law of Gravitation describes the mutual attractive gravitational force (Fg) exerted on each other by two objects with masses m and m’ that are separated by a distance r. The magnitude of this force is simply Fg = G m(1) m(2) d^2 (2) The constant G is called the gravitational constant and, as its name implies, does not depend on variables like mass or distance. This course will not explain where this formula comes from or even the experimental tests done to verify it. Instead, we will just use it. To compute the magnitude of the gravitational force between the earth and an object, we substitute the mass of the earth (ME) and the distance from the object to the center of the earth (r). When the objects are on or near the earth's surface, this distance can be approximated by the value for the radius of the earth (RE), so that Equation (2) becomes Fg = (3) We see that the force on the object depends only on the mass of the object, because G, ME, and RE are all constants and remain the same if we change our object. This force (measured at the earth's surface) is called the weight of the object. Looking at Equation (1) and equating F to the gravitational force (Fg), we see that: ma = = mg , (4) where a= . This quantity g is a special acceleration and is itself a constant because it depends on quantities that do not change with time. We call this special acceleration the gravitational acceleration and is, for example, the same acceleration a book experiences when you drop it. Allegedly, Galileo first demonstrated this result when he dropped cannonballs of different masses (weights) from the Leaning Tower of Pisa to show that although they had different masses, when dropped together, they landed together. This happened in this manner because they both experienced the same acceleration. A similar experiment can be done by dropping a coin and a feather. When dropped in air, the coin always lands first, but when they are dropped in a vacuum, an environment where there is no air, they land together! In the coin and feather case, the different velocities are due to the presence of a force, the frictional force on the coin and the feather due to the presence of air. Our Equation (4) equates the total force to the gravitational force and therefore neglects the effects of air friction. Let’s now try to discover a quantitative method for determining the gravitational acceleration, g. We first look at the equation for displacement x in a one dimensional universe: x(t) = vo t + (1/2) at 2 (5) Here, the quantities vo and a are, respectively, the initial velocity and the acceleration. In this experiment, like Galileo, we will be dropping an object from rest, so that vo will be zero. We will assume that the only acceleration present is due to the force of gravity so that we can set a = g. We then obtain the relation that describes the distance the object falls as a function of time x(t) = (1/2) gt2 . (6) Equation (6) provides a means to measure g. All we need to do is to drop an object through a known distance x and then measure the time t it takes to fall. If we know both x and t, we can solve equation (6) for g, g = (7) Hence, by measuring the travel distance and the travel time for an object falling from rest, we can measure the gravitational acceleration. Procedure Data Use the given lab form to record all data and extended data (t 2 ). You should also record m, the mass of the ball. Remember to report all data with units. For the last column of all four tables, the value of g to be used in computing |gave – g| is your value of g found from that particular drop. Calculations 1. Using the measured values for x and the calculated values for t 2, insert these values into Equation (7) to get the measured values for g. Show all calculations and remember to label the appropriate quantities with their respective units. 2. Find the average of the five measurements of g and record this in the table below. 3. As a measure of precision, we will be using the average deviation from the mean for the measured value of the gravitational acceleration. Reread the lab write-up for Measurement and Measurement Error to remind yourself on how to do this type of error analysis. Your final result for the gravitational acceleration should be reported as: g = gave ±  As a measure of accuracy, use the given accepted value for g (9.8 m/s2) and your measured average value of g to compute the percentage error. Again, reread the Measurement and Measurement Error write-up to refresh your memory on how to do this. Conclusion 1. How did the gravitational acceleration vary with the different heights? Did it become smaller, larger or remain roughly constant? Why do you think this is? 2. What did you discover in this experiment? Error Analysis 1. How well do you think you can measure distance in this lab? Explain. 2. How well do you think you can measure drop time in this lab? Is this a matter of accuracy, precision or both? Explain. 3. Examine equation (7). Do you think it is more important to measure accurately the drop time or the fall distance of the ball to determine g with the greatest possible accuracy? Explain. 4. Compare your value of g with the accepted value of 9.8 m/s2. Is your value lower or higher than this value? Why do you think this is? Try to identify the experimental conditions or errors that may cause this discrepancy. Ball type: ______________________ trial 1 X t t2 g |gave-g | 2 3 4 Ball mass Average measured value for g= _______________ Accepted value for g= _______ % error: _______ Calculations Conclusions 1. 2. 3. Error Analysis 1. 2. 3.  Morgan Extra Pages Graphing with Excel (revised 6/10/13) 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. 1 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 vertical plus-drags 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. 2 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. 3 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 4 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 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. 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- 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 5 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′) 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. Note the parentheses around the intercept term of Equation 1′ to emphasize that the minus sign is part of it. 8. Present your values of k and yo with their units neatly at the bottom of your spreadsheet. Equating above and below, you can create two useful new equations: 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? slope m = k (Eq. 2) y-intercept b = -kyo (Eq. 3) 10. Do the Homework, Further Exercises on Interpreting Linear Graphs. 6 Name . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section . . . . . . . . . . . . . . . . Date . . . . . . . . . . . . . . . . Lab Partners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 L A B O R A T O R Y 1 Measurement of Length LABORATORY REPORT Data and Calculations Table 1 (nearest 0.0001 m, which is 0.1 mm) Trial X1 (m) X2 (m) Li  L (m) Li = X2 – X1 (m) n X ðLi  LÞ2 (m2 ) ðLi  LÞ2 ¼ COPYRIGHT ª 2008 Thomson Brooks/Cole 1 L¼ sLn1 ¼ L  sLn1 ¼ L þ sLn1 ¼ aL ¼ 19 20 Physics Laboratory Manual n Loyd Data and Calculations Table 2 (nearest 0.0001 m, which is 0.1 mm) Trial Y1 (m) Y2 (m) Wi  W (m) Wi = Y2 – Y1 (m) n X ðWi  WÞ2 (m2 ) ðWi  WÞ2 ¼ 1 W¼ sW n1 ¼ W  sW n1 ¼ A¼LW ¼ SAMPLE CALCULATIONS 1. L1 ¼ X21  X11 ¼ 2. W1 ¼ Y12  Y11 ¼ 10 1 X Li ¼ 3. L ¼ 10 1 10 1 X 4. W ¼ Wi ¼ 10 1 5. L1  L ¼ 6. ðL1  LÞ2 ¼ 7. W1  W ¼ 8. ðW1  WÞ2 ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1 X ðLi  LÞ2 9. sLn1 ¼ n1 1 10. L  sLn1 ¼ 11. L þ sLn1 ¼ W þ sW n1 ¼ sA ¼ aW ¼ Laboratory 1 n Measurement of Length 12. sW n1 21 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 1 X ¼ ðWi  WÞ2 n1 1 13. W  sW n1 ¼ 14. W þ sW n1 ¼ 15. A ¼ L  W ¼ 16. sA = QUESTIONS 1. According to statistical theory, 68% of your measurements of the length of the table should fall in the range from L  sLn1 to L þ sLn1 . About 7 of your 10 measurements should fall in this range. What is the range of these values for your data? From __________m to __________m. How many of your 10 measurements of the length of the table fall in this range? __________? State clearly the extent to which your data for the length agree with the theory. What is your evidence for your statement? W 2. Answer the same question for the width. Range of W  sW n1 to W þ sn1 is from __________m to __________m. The number of measurements that fall in that range is __________. Do your data for the width of the table agree with the theory reasonably well? State your evidence for your opinion. COPYRIGHT ª 2008 Thomson Brooks/Cole 3. According to statistical theory, if any measurement of a given quantity has a deviation greater than 3sn–1 from the mean of that quantity, it is very unlikely that it is statistical variation, but rather is more likely to be a mistake. Calculate the value of 3sLn1 . Do any of your measurements of length have a deviation from the mean greater than that value? If so, calculate how many times larger than sLn1 it is. Do any of your measurements of the length appear to be a mistake, and, if so, which ones? 4. For the width measurements calculate 3sW n1 . Do any of your measurements of width have a deviation from the mean greater than that value? If so, calculate how many times larger than sW n1 it is. Do any of your measurements of width appear to be a mistake, and, if so, which ones? Name ________________________________________ Date __________________________ Period _________ The Conservation of Momentum Find the Lab  In your web browser, go to www.gigaphysics.com, then go to Virtual Labs, and then click Conservation of Momentum.  If someone else used the computer for this lab before you, click New Experiment. This will ensure that you have your own unique cart data when you do the experiment. Part I: Measure the Carts  To find the length of the purple cart, use your mouse to drag the cart over the caliper in the upper left corner of the lab. Convert the length to the SI unit of meters, then record your result in the table below. Repeat for the green cart.  Find the masses of the carts by dragging each one in turn over the electronic balance in the upper right corner. The balance reads in grams, so convert each mass to the SI unit of kilograms, then record your data. Mass of purple cart Length of purple cart Mass of green cart Length of green cart  These measurements will stay the same as long as you don’t refresh the screen or click the button to start a new experiment. If you don’t complete the lab if one sitting and have to load the lab page again, the lengths and masses will change. If this happens, you will need to measure them again and use the new values for the remainder of the lab. Part II: Determine the Carts’ Velocities  Select “same direction” from the Carts’ Direction menu and “inelastic” from the Collision Behavior menu.  Click Start Carts to put the carts in motion. The red numbers you will soon see tell you how many seconds it took each cart to pass through that photogate. If you lose track of which photogate is measuring which cart, notice the purple and green arrows labelling each; a half purple/half green arrow is used when both carts were stuck together as they passed through. You can also click Start Carts if you want to watch the collision again.  Record your times in the data table at the top of the next page. Also copy the lengths from part I. Be sure to add the lengths of the two carts when the carts are stuck together.  Calculate each cart’s velocity and enter it in the table as well. 1 Elapsed time Length Velocity Purple cart before collision Green cart before collision Carts stuck together after collision Part III: Calculating Momentum  Use the fact that momentum equals mass times velocity to calculate the momentum of each cart. Remember to add the masses when the carts are stuck together. Mass Velocity (from part II) Momentum Purple cart before collision Green cart before collision Carts stuck together after collision  Calculate the total momentum of the two carts before and after the collision. Purple cart’s momentum Green cart’s momentum -------------------- ---------------------- Total momentum Before collision After collision  You should find that the total momentum before and after the collision is identical (at least to within rounding errors.) If you don’t, you should find out what went wrong and correct it before you complete the next part. Part IV: The Elastic Collision  This time, set the Carts’ Direction to opposite and the Collision Behavior to elastic. Repeat the same steps as in part II and III. (The data table is at the top of the next page.)  When you calculate the velocities and momenta, signs matter. Make sure that carts that are moving to the left have negative velocities. If you lose track of which direction the carts were going for each photogate, you have the arrows to help you, and you can click Start Carts to watch the collision again. 2 Elapsed time Length Velocity (with sign!) Mass Velocity Momentum Purple cart’s momentum Green cart’s momentum Total momentum Purple cart before collision Green cart before collision Purple cart after collision Purple cart before collision Purple cart before collision Green cart before collision Purple cart after collision Purple cart before collision Before collision After collision Part V: One More Case  Repeat the experiment once more, this time with any combination of Carts’ Direction and Collision Behavior you have not used already. Record which settings you use, then complete the calculations as before. Carts’ Direction ___________________________ Elapsed time Collision Behavior _________________________ Length Velocity (with sign!) Purple cart before collision Green cart before collision Purple cart after collision Purple cart before collision 3 Mass Velocity Momentum Purple cart’s momentum Green cart’s momentum Total momentum Purple cart before collision Green cart before collision Purple cart after collision Purple cart before collision Before collision After collision Part VI: Conclusions What did you notice about the total momentum before the collision and the total momentum after the collision in each of the above cases? ____________________________________________________________________________________________________________ ____________________________________________________________________________________________________________ ____________________________________________________________________________________________________________ The principle you should have noted in the previous question is called conservation of momentum. What do you think it means to say something is conserved in the context of physics? ____________________________________________________________________________________________________________ ____________________________________________________________________________________________________________ ____________________________________________________________________________________________________________ Do you think there is any combination of conditions in this lab under which momentum would not have been conserved? Explain your answer. ____________________________________________________________________________________________________________ ____________________________________________________________________________________________________________ ____________________________________________________________________________________________________________ Learning physics? Teaching physics? Check out www.gigaphysics.com. © 2016, Donovan Harshbarger. All rights reserved. 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