Experiment 3 Acceleration Due To Gravity Questions How do you compute an object's velocity from its position? How do you measure the acceleration of a mass as it falls under the influence of gravity? What can be gained by plotting graphs? What is the procedure for interpreting the slope of a graph? How do you quantitatively compare an experimental value to an established value? Spark Tape Rod Wood Block Spark Generator oo 50 g Mass Table Clamp Figure 1. Diagram Of The Apparatus Concepts Newton's Law of Gravity describes the gravitational attraction between two objects: Fool by 2 = G -2 G is the universal gravitational constant (= 6.67 x 10-11 N-m2 / kg2). The m's refer to the masses of the two objects and r is the distance between the centers of the two objects. If an object of mass m, falls near the surface of the Earth, it will accelerate according to F = mja. This is Newton's Second Law, which is normally introduced a little later in Physics 1. Let us write a new equation putting the right hand side of Newton's Law of Gravity on the right hand side of a new equal sign and the right hand side of the Newton's Second Law on the other side. Next we divide out the common term and solve for the acceleration. Substituting the values for an object of mass mj on the surface of the Earth gives a very familiar number. Method In this experiment, the student drops a weighted paper tape through a slot cut into a block of wood. Two electrodes are screwed into the block and connected to a high voltage power supply. The power supply causes an electrical spark to jump from one electrode to the other at 3 - 3Perform spelling and grammar checking here and throughout the report. Include page numbers centered at the bottom (in the footer) of each page. B Data - 20 points 1 ) Record ALL measurements on a handwritten or computer printed data sheet. (Do NOT record data in the margins of your lab manual.) 2) Neatness counts. Also, include explanatory notes and phrases. 3 ) Do NOT rewrite or erase from the data sheets. Instead, cross out mistakes. Data sheets are modeled after the research lab notebook that represents a permanent record of an experiment. 4 ) Place original data sheets immediately after the introduction. Do not rewrite the data to save time that is better spent on the Discussion. Have all data sheets signed and dated by the instructor before leaving the laboratory. Record data in tables. Label the top of each column with a heading and proper units. 9 ) Write errors using one significant figure. Use two significant figures when the first digit is '1'. Use the errors to determine the correct number of digits past the decimal to display. 10) If a data table is inappropriate, use short phrases to explain what is being measured. C Data Analysis - 30 points This section contains the results of calculations, sample calculations, and graphs. a) When experiments are qualitative (contain few numbers), this section is omitted and the point value of the Discussion section is increased accordingly. b When the data sheet contains all the calculations and graphs, this section contains only sample calculations. In these cases, do not rewrite what is on the data sheets. C ) If you perform many repeated calculations (with a spreadsheet), write only one sample calculation in full detail. Tabulate the results of the remaining calculations. N Perform calculations correctly and completely using only your data. 3 ) Perform error analysis correctly (see Appendix C). 4) Use the correct number of significant figures in the errors (see Experiment #1 in Lab 1). 5 ) Write so the report is self-contained. Use short phrases to explain each calculation or graph. Do not reference a procedure number. Assume the reader is familiar with the procedure. 6) At a minimum, always calculate a % difference between experimental and theoretical values. However, this is often not sufficient for a true quantitative comparison of results. 7) Usually, calculate d the difference between experiment and theory and also calculate the error in this difference: of to determine if experiment and theory agree, see Appendix C. 8) 9 ) Include correct units with every numerical value or place units in a column heading. It is NOT necessary to format equations using an equation editor. Handwriting the mathematics is fine as long as it is neat and readable. 10) Produce professional looking technical graphs. a) If you plot graphs by hand, use real graph paper (not notebook paper). b) Use half a page for each graph or a full page if the data have three significant figures. c) Make major divisions on each axis a multiple of 2, 5, or 10. d) Label each axis with a title and units and place a title over the entire graph. e) Follow formatting instructions in Appendix D for graphing with Excel. These include making the points black, changing the background from gray to white and making the horizontal grid lines dashed. Appendix - B - 2f) When a straight line is expected between Y and X, draw a best-fit line. If there is some scatter to the points, half the points should lie above the line and half below. Do not connect dots in a zigzag line (don't let computer software do it either). If the procedure was performed incorrectly, it is usually evident when the data are analyzed. Instructors reduce the report's grade for this mistake in this section. D Discussion - 40 points This section contains a table of results and paragraphs discussing the accuracy of the results, the sources of errors, the relevant physics and/ or answers to any questions. 1) Print this section double-spaced to give the instructor room to write comments. First, rewrite qualitative or numerical results, with their errors and units, in a summary table. Do not tabulate every measured quantity that feeds into calculations. Focus on the results. 3) Second, write a very brief paragraph describing the physics of the experiment. Three sentences are enough. 4) Next, list all the sources of experimental error that did occur. Do not list hypothetical errors that may or may not have occurred. 5 ) State the category each error belongs to. The categories are: random error in the measurement tool or process, systematic error in the measurement tool or process, intrinsic random error in the quantity itself and intrinsic systematic error in the quantity itself. 6) State how these errors affect your results. For example, for a systematic error, would it tend to increase or decrease your numerical result? 7) State which measurement has the largest error. State whether this error accounts for most (or all) of the total error. Mention if any of the errors are negligible. 8 ) 9 ) Answer questions posed in the manual and place them where they fit into the flow of thought. State whether the entire experiment was successful. Success is not always measured by a small percent difference between theory and experiment. Usually, the difference between theory and experiment must be less than the error in this difference, see Appendix C. 10) Explain discrepancies greater than experimental error by referring to physical effects ignored in the theory, apparatus or the procedure. E Conclusion - 5 points 1 ) In two sentences, summarize the entire experiment. The introduction describes the measurement goal and your conclusion states whether you reached this goal. 2 ) w State whether the experiment was a success by comparing experimental and theoretical results. Include a one-sentence evaluation of the degree to which your lab partner(s) contributed to conducting the experiment. This valuable, brief comment will help the instructor arrange and populate lab groups in the future. Appendix - B - 3Procedure Part 1 The Spark Tape Method of Determining g 1) Participate with the instructor and your fellow students in a discussion of the Method section. 2) Use the long meterstick (2m) to measure the distance from the floor to the ceiling. Adjust the height of the block containing the electrodes so it is half this distance above the floor. This maximizes the amount of data you can collect on a spark tape. Tear off about 1.4 meters of spark tape. Place a foam pad on the floor to catch the weight. 60 3) Adjust the frequency of the sparker to its highest setting. All the sparkers have a maximum setting of 60 sparks per second. Safety Note: A shock from these sparkers can be painful. Do NOT touch the wires or electrodes while sparks are generated. 4) Have one lab partner stand on the table top and thread the spark paper through the vertical slot in the wooden block. A second lab partner views the extended spark tape from one direction. A third can view the tape from a direction 90 degrees from the first viewer (and trigger the sparker). In this way, one can be certain the spark tape is vertical. Conduct an "experiment run" by dropping the weighted spark tape through the slot. Start the sparker before you release the tape. This eliminates the chance of the weight falling and no spark record being recorded. Examine the line of burn marks to see if it is satisfactory. If not, repeat the trial. 5) Exchange roles and so each lab partner climbs onto the table and release the tape for a second trial and third trial. Once good tape records are obtained for each member of the group, turn the sparker OFF and begin analyzing your data as follows. 6) Using masking tape, secure the spark tape to the table and place the 2m-long meterstick edgewise along the line of sparks. Don't line up the zero of the meterstick at a burn mark since the ends of metersticks are notoriously untrustworthy. 7) Ignore the first large black burn mark and record the positions of the burn marks (not the distances between them) into a blank spreadsheet. When necessary, estimate these positions to some fraction of a millimeter. Associate a time value of zero to the first position as in the following example data table. O A B C D E Your Name Your Partner Today's Date PHY 2091 - XX Time (S) Position (cm) Inst. Time (s) V-Bar (cm/s) 4 0.0 = A4 + 0.5/60 = (B5 - B4) / (C5 - C4) 5 = A4 + 1/60 6 7 3 58) Complete all the calculations outlined in the example data table. Note the following: a) A worksheet will help you understand why, in column C, you add half of one sixtieth of a second to the time coordinates in column A. This gives the time when the weight's instantaneous speed was equal to the average speed computed in column D. b) Select all active cells and set the display precision to at least 5 digits past the decimal. c) Because the formulas in the average velocity column depend on the value of the next cell down, your last average velocity will be 'garbage'. If you delete this velocity and time coordinate, it will affect the next data pair above it. To get around this problem, manually enter the second to last value for the velocity into the spreadsheet, then delete the last average velocity and its time coordinate. d) Insert a row just below the column labels. Where applicable, use this extra cell to display the equation used in that column. There are two ways to do this in Excel. One way is to enter a blank space in the cell, then use Edit, Copy and Paste to put the text of the equation after the blank space. The second way is to use the FORMULATEXT (cell reference) command. 9 ) Plot a graph (a scatter plot) of velocity on the y-axis and instantaneous time on the x-axis. See Appendix D for instructions on how to plot a graph with Excel. Then, fit the data using linear regression (a trendline in Excel). Appendix D also provides instructions on how to use Excel to add a trendline and calculate the error in the slope and y-intercept. Do all of these tasks and display the equation of the fit on the graph. 10) Before you print, use the commands Insert and Textbox to add a textbox to the graph. Enter the errors in the slope and y-intercept in the box. The + symbol is displayed by pressing SHIFT, OPTION and PLUS keys. 11) Excel uses blue dashed lines to indicate the page breaks in a spreadsheet. Show these dashed lines by executing a File, Print... command and then click Cancel. Then arrange the spreadsheet so everything fits onto one page. Reminder: Make certain you have the names of all partners, today's date and the course and section number on every hand- written or computer-generated data sheet. Print a copy of the data sheet and graph for all members of the group. 12) Before you leave, have your instructor sign all the sheets. As stated in the Syllabus, this procedure is required for every experiment. For Your Discussion 13) Identify the physical meaning of the slope of your speed versus time graph. Then, calculate a percent difference between your experimental value of g and the given value many years ago. of 9.792 ms:2. A U.S. Coast and Geodetic survey measured this value in Melbourne - 614) Discuss the precision of your graph. You can consider this experiment a success if the error in your experimental number is large enough to bracket the given value. Explicitly show whether this is true in your report. 15) Discuss sources of random and systematic error in your report. Also, categorize sources of error according to whether they are intrinsic errors or errors in the measuring tool. Include your spark tape in your lab report. Part 2 The Picket Fence Method The picket fence is a narrow acrylic sheet. Black stripes of constant width are printed on it at regular intervals. The picket fence is dropped vertically through a photogate. The photogate has an infrared diode mounted inside a U-shaped plastic housing. The gate's other arm has a photo-diode to detect the beam of invisible, infra-red light. The data acquisition board measures the time that passes while the beam is blocked by a passing object. The object's average speed is then the width of the object divided by the time it took for the object to pass through the gate. Other timing schemes are available too. However, we will use one that makes this part of the experiment easy to conduct. 1) Connect the photogate to one of the LabPro's DIGI/SONIC input jacks. Connect the LabPro's USB cable to one of the computer's USB port. Also connect an AC/DC adapter to the LabPro and plug the adapter into an AC power outlet. There is no ON button for the LabPro. A series of beeps indicates the LabPro is working correctly. 2) Screw a threaded rod into the threaded nut in the photogate and clamp this rod into a right-angle clamp on a ring-stand. Position the photogate over the table high enough so the picket fence will clear the gate before your partner catches it. 3) Place a large flat foam pad beneath the gate on the tabletop. When you run trials, catch the picket fence before it's end strikes the pad. The acrylic can break easily. 4) Run the data acquisition program Logger Pro by clicking on the icon in the Dock that resembles a vernier caliper. The data acquisition electronics should automatically identify that you have connected a photogate to the unit. a) Pull down the Experiment menu and go to Setup Sensors > Show All Interfaces b) Click on the picture of the photogate for another pull-down menu. Make sure Motion Timing is checked. Then click on Set Distance or Length... c) Make sure the Vernier Picket Fence is in the pop-up menu. Then click OK. d) Close the sensor set-up window by clicking the red button in the upper left. 5) Click the green, triangular arrow to start a trial. With the software set to 'picket fence" mode, it should automatically calculate the acceleration due to gravity in the right-hand column of the data table. 6) Run several trials and then record all the values for what you consider to be your * best' trial. This would seem to be an easy matter but there is an important, subtle requirement 3 - 7that must be met to collect quality data. Question: What is this requirement? Use some ingenuity to optimize the quality of the data you collect. 7) Unplug the AC/DC adapter from the AC outlet and quit Logger Pro (in the Logger Pro menu). - 8Niall Harris Javier Perez, Abdullah Alalhasaniyyah 6/2/20 PHY 2091-E4 Time(s) Position(cm) Inst. Time(s) V-Bar (cm/s) =A4 + 1/60 =A4 + 0.5/60 =(85-B4)/(C5-C4) 0.00000 1.00000 0.00833 48.00000 0.01667 1.80000 0.02500 72.00000 Velocity vs Time 0.03333 3.00000 0.04167 78.00000 0.05000 4.30000 0.05833 90.00000 600.00000 0.06667 5.80000 0.07500 114.00000 Error in slope = $6.50 0.08333 7.70000 0.09167 132.00000 500:00000 Error in y intercept =+1.81 0.10000 9.90000 0.10833 150.00000 0.11667 12.40000 0.12500 156.00000 400,00000 0.13333 15.00000 0.14167 174.00000 0.15000 17.90000 0.15833 198.00000 0.16667 21.20000 0.17500 210.00000 24.70000 0.19167 216.00000 Velocity (cm/s) 300,00000 0.18333 0.20000 28.30000 0.20833 246.00000 200.00000 0.21667 32.40000 0.22500 258.00000 0.23333 36.70000 0.24167 276.00000 100.00000 0.25000 41.30000 0.25833 282.00000 0.26667 46.00000 0.27500 306.00000 0.00000 0.28333 51.10000 0.29167 318.00000 D.0080 05000) 10000.1500().20000 25000 30000 35000).4000@) 4SOO( 50000 0.30000 56.40000 0.30833 336.00000 Inst time (s) 0.31667 62.00000 0.32500 354.00000 0.33333 67.90000 0.34167 366.00000 0.35000 74.00000 0.35833 384.00000 For Graph: 0.36667 80.40000 0.37500 402.00000 Slope(cm/s Y-intercept 0.38333 87.10000 0.39167 414.00000 952.4514 41.88978 0.40000 94.00000 0.40833 438.00000 Errors 6.500504 1.813713 0.41667 101.30000 0.42500 450.00000 0.998744 4.881393 0.43333 108.80000 0.44167 450.00000 21468 27 0.45000 116.30000 0.45833 474.00000 511539.5 643.3559 0.46667 124.20000 0.47500 497.90042 90- to\fAppendix B The Laboratory Report Indivisual - Per- person . Each week you will make measurements and use theories to calculate results from data. The lab report summarizes this effort and discusses how consistent the results were and how close those results came to a predicted value. Precision is the technical term for consistency. Accuracy means how correct' the results were. Usually, we determine precision by calculating a standard deviation but sometimes an educated estimate determines the error. Next, we combine (or propagate) these errors to obtain an error in the experimental result. We determine accuracy by subtracting the experimental and theoretical values and by comparing this difference to the propagated error in this difference. Sometimes it is not possible or practical to make a theoretical prediction. Then we use different methods to obtain two experimental results. In these cases, we compare the two experimental results in the same way and settle for a certain degree of precision instead of accuracy. The format of the report is a reduced version of the standard technical report written by engineers and applied scientists in industry. This format is different from papers published in scientific journals. This style of report does NOT contain an abstract, a theory section, an equipment list or a written procedure (since a detailed procedure is provided in this manual). These omissions focus effort on the experimental results and keep the student workload reasonable. Reports are graded out of 100 points and contain the sections listed in the following outline. The point distribution may differ for qualitative experiments. The report can contain everything in the outline and still not make sense. Reasons include not understanding the physics, mistakes in performing the experiment, calculation mistakes or poor writing. Lab instructors may use abbreviations to indicate which items in the outline are lacking or faulty. For example, A1, A2, A3 refer to the first three points in the Introduction. So, if an instructor writes D7 -3 it means three points were deducted because the largest source of error was not identified in the discussion. When a lab report contains many mistakes, the instructor may not indicate the point deduction for each problem but write an overall score based on experience. This is acceptable practice; however, instructors must provide feedback to the student so he or she can improve. If a student feels feedback is lacking, he or she should see the Laboratory Director as soon as possible. The following outline provides the requirements for the report. Cover Page Centered left to right on a separate page, this sheet lists: the course and section number, the number and title of the experiment, your name, the date the experiment was performed, the date the report was submitted, your partner's name and your instructor's name. A Introduction - 5 points 1 ) Do not discuss learning objectives. They do not belong in this style of report. 2) In your own words, state what you measured, what you calculated and what you are comparing your results to. 3 ) Mention any deviations from the manual's procedure. Do NOT rewrite the procedure. Write a brief introduction (three sentences are plenty). The audience for your report is another student who must perform the same experiment. Write clearly and avoid using the first person. Appendix - B - 1regular time intervals. (Hence, the power supply is referred to as a sparker.) This leaves a small burn mark on the paper as it falls under the influence of the Earth's gravity. After a successful drop, students measure the position of the burn marks and then calculate the continuously increasing speed of the paper tape. After generating a graph of speed versus time, the slope of the resulting straight line is calculated. This slope is equivalent to the local acceleration due to gravity. Figure 1 displays a drawing of the apparatus. Figure 2 is a (not to scale) drawing of the paper tape after a successful drop. Notice that the distance between each successive burn mark increases with time. x(1) x(2) x(3) x(4) x(5) x(6) *(7) Figure 2. Also, note that in physics, words are defined very specifically. Position is the location of an object relative to an arbitrary fixed origin. Distance is the length of a straight line path between two points and is independent of any coordinate system. For example, if the sparker produces 60 sparks each second, then the succession of dots on the tape will show how the position of the weight (and the spark tape) changes every 60th of one second. You will measure these positions (given the symbols x(i), where i = 1, 2, 3, ...) and record this data into a computer spreadsheet. Next, you will use the spreadsheet to compute the quotient: Ar / At, where Ar = x(i+1) - x(i) and Ar = ((i+1) - I(i). When the division is performed you obtain a number which shows how far the object fell in any one time unit during the time interval At. This quotient, shown in Equation (1), is the average speed. distance time interval ( 1 ) Lastly, you will associate these average speeds with the specific time instant when the falling mass had that particular instantaneous velocity. Graphs of position and velocity as a function of time will also be plotted using the spreadsheet program. 3