Florida Institute of Technology © 2020 by J. Gering Experiment 6 Newtons’ Second Law Questions What must be true in Newton’s Second Law (N2) if the object in question moves at a constant velocity? Similarly, what must be true in N2 if the object accelerates? What are the customary rules for drawing a Free Body Diagram (FBD)? What is the value of drawing an FBD? If two objects are in contact with each other, what does Newton’s Third Law (N3) dictate should be evident when FBDs are drawn of the two objects? Concepts Newton’s First Law is the also known as the law of inertia: an object at rest tends to stay at rest and an object in motion tends to stay in motion. The second half really only applies to the special case of straight line, constant velocity motion. One example is the motion of a ball thrown from one astronaut to another inside a space station. Newton’s Second Law (N2) is a statement of cause and effect. It states any object will undergo an acceleration that is proportional to the vector sum of all the forces that act on the object. As with all physical laws, this relationship is an experimental (empirical) result. ! ! (1) ∑ Fi = ma N2 places acceleration (change in velocity) at the center of the analysis. In contrast, Aristotle’s teachings held that any motion implies a force acting on the object. Certainly, it requires a strong push to start a stalled car moving and to keep it moving. But friction (another force) makes the continued pushing necessary. In the absence of friction, when the initial push ends, the idealized car would continue to move at constant speed in a straight line. Do not treat mass multiplied by acceleration as if it were a force. Mass multiplied by acceleration is the effect not the cause. The net force (always, initially on the left side of the equation) is the WHY the object moves. Mass multiplied by acceleration is HOW the object moves. Consequently, forces have the units of Newtons. Mass multiplied by acceleration has equivalent units: kg m/s2 but we never call the units of mass multiplied by acceleration a Newton. In physics, equivalence is different from being the same thing. Method In this experiment, students use an Atwood’s machine to accelerate two different hanging masses. See Fig. 1. Here, two different masses hang from two pulleys, see Fig. 1. 6 - 1 Florida Institute of Technology © 2020 by J. Gering m M Figure 1. The Atwood’s Machine A photo-gate is mounted around one pulley (not shown). It is used to measure the motion of the pulley’s spokes. The data acquisition software then calculates the acceleration of the string and hence the masses. We will assume massless and frictionless pulleys. Newton’s Second Law predicts ⎛ M −m ⎞⎟ ⎟g (3) a = ⎜⎜ ⎜⎝ M + m ⎟⎟⎠ This equation can be derived in class. To do so, one draws free body diagrams of each mass and applies N2. The key is to choose a direction for positive motion and then apply it throughout the derivation. For example, if up is chosen to be positive, then the block of mass M in Fig. 1 will have a negative acceleration. So, a minus sign must be placed in front of the ma term in N2 for the more massive weight. Procedure 1) Arrange the apparatus so the heavy table clamp is near the edge of the table. Screw a threaded rod into each pulley. In one case, use the threaded rod to also mount a photogate around the pulley. Clamp both pulleys to the cross bar so a string passing over them will move free and clear of the edge of the table. 2) To set up the software, click on the Experiment menu and then click on the command Set Up Sensors > Show All Interfaces. An image of the LabPro should appear with a photogate visible as the sensor. Click on the image of the photo-gate and a pop-up menu should appear. Click on the Set Distance or Length… command. This will bring up another selection menu. Choose the option labeled Ultra Pulley (10 Spoke) In Groove. a) Use a total collection time more than the time it takes for the weight hanger to fall. This way you will not be rushed to complete a run. b) Set the data rate to at least 100 points per second. Place a foam pad beneath the hanger you plan to allow to descend. 6 - 2 Florida Institute of Technology © 2020 by J. Gering c) Steady the weight hangers before each run to minimize swinging. Check the alignment of the pulleys to reduce friction. d) If the pulleys rotate when the mass hangers are empty, place paper clips or part of a paper clip on one of the hangers to balance the hangers. Also use this method to determine how much mass it takes to overcome the friction (and rotational inertia) of the pulleys. Measure and record the small added mass. What type of error does this procedure quantify? e) Remove the small added mass and place slotted masses on one of the weight hangers. Make one of the hanging masses 10 to 20 grams greater than the other. Using Logger Pro, press the green collect button and then release the masses. Make sure the hanging masses fall vertically and do not swing from side to side. Also make sure the string rides in the pulley groove and does not slip out of the groove. 3) Measure the mass of each hanging weight (including the weight hanger) on a triple beam balance. 4) Examine the distance, velocity and acceleration graphs. The acceleration graph should display a fairly flat constant value for a short time period. Use the Examine command in the Analyze menu to highlight this constant acceleration. Use the Statistics command in the Analyze menu to compute an average acceleration over this time interval. Record an average and a standard deviation. 5) Perform a second trial to ensure repeatability. 6) Write the name of everyone in the group, the section number and today’s date on the graph using the Text Annotation command in the Insert menu. Print one copy of your best acceleration graph for each lab partner. For the Lab Report 1) Compute the percent error in this ‘experimental’ acceleration. 2) Compare your experimental and theoretical values by calculating a percent difference between them. Is this percent difference smaller that the percent error you found above? If so, the two accelerations agree within the limits of random error. Which type of random error is largest here: error in measurement or intrinsic random error in the acceleration? Is a systematic error present? What physical effect(s) cause(s) these sources of error? 3) For 5 points of extra credit, derive Equation (3) from Newton’s Second Law. For credit to be awarded this work must be written by hand, not typed. 6 - 3 Florida Institute of Technology © 2020 by J. Gering This page has been left blank intentionally. 6 - 4 Florida Institute of Technology © 2020 by J. Gering Appendix B The Laboratory Report The Laboratory Report 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, 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. Appendix - B - 1 Florida Institute of Technology © 2020 by J. Gering A) Cover Page 1) 2) Center all text left to right on a separate page. List in this order: a) the course and section number b) the number and title of the experiment c) your name d) the date the experiment was performed e) the date the report was submitted, f) your partner’s name and g) your instructor’s name. B) Introduction - 5 points 1) 2) 3) 4) 5) 6) 7) 8) Do not discuss learning objectives. They do not belong in this style of report. State what you measured, calculated and are comparing your results to. Mention any deviations from the manual’s procedures. Now is the time for all good men to come to the aid of their country. 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. Perform spelling and grammar checking here and throughout the report. Include page numbers centered at the bottom (in the footer) of each page. C) 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. 5) Do not rewrite the data to save time that is better spent on the Discussion. 6) Have all data sheets signed and dated by the instructor before leaving the laboratory. 7) Record data in tables. Label each column with a heading and proper units. 8) Write errors using one significant figure and two when the first digit is ‘1’. 9) 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. Appendix - B - 2 Florida Institute of Technology D) © 2020 by J. Gering Data Analysis - 30 points 1) This section contains the results of calculations, sample calculations, and graphs. a) If you plot graphs by hand, use real graph paper not notebook paper. b) When experiments are qualitative (contain few numbers), this section is omitted and the point value of the Discussion section is increased accordingly. c) 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. d) If you perform many repeated calculations (with a spreadsheet), write only one sample calculation in full detail. Tabulate the results of the remaining calculations. 2) 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 percent 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: 𝜎d to determine if experiment and theory agree, see Appendix C. 8) Include correct units with every numerical value or place units in a column heading. 9) 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. f) 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). 11) 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. Appendix - B - 3 Florida Institute of Technology E) © 2020 by J. Gering Discussion - 40 points 1) 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. 2) Print this section double-spaced to give the instructor room to write comments. 3) 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. 4) Second, write a very brief paragraph describing the physics of the experiment. Three sentences are enough. 5) Next, list all the sources of experimental error that did occur. Do not list hypothetical errors that may or may not have occurred. 6) 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. 7) State how these errors affect your results. For example, for a systematic error, would it tend to increase or decrease your numerical result? 8) 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. 9) Answer questions posed in the manual and place them where they fit into the flow of thought. 10) 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. 11) Explain discrepancies greater than experimental error by referring to physical effects ignored in the theory, apparatus or the procedure. F) 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. 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. 2) 3) Appendix - B - 4 Alyami 1 PHY 2091 Section: 03 Experiment 6:Newton’s Laws - Dynamics 10/30/2017 Mohammad Alyami Lab partners: George Tsirimokos and Nick Fornadel GSA: Sakhee Bhure Alyami 2 Introduction This experiment was divided into two parts. The first part was The Atwood’s Machine where there are two weights one is heavier than the other, when releasing the heavy weight it drops and lifts the lighter weight and acceleration is measured and compared to 𝑀−𝑚 the acceleration theoretical value using 𝑎 = 𝑀+𝑚 𝑔. In the second part, Centripetal Force where we measure the necessary force to keep the object moving in a uniform circular motion. Alyami 3 Data Analysis Part1: 𝑀−𝑚 First the theoretical acceleration was measured using the equation 𝑎 = 𝑀+𝑚 𝑔 as follows: 𝑎= 148.6 − 130 (9.792) = 0.653 𝑚/𝑠 2 148.6 + 130 Then the percent error of the two masses difference: 2 2 √𝜎𝑀 + 𝜎𝑚 √0.052 + 0.052 = ∗ 100 = ∗ 100 = 0.38% 𝑀−𝑚 148.6 − 130 %𝜎𝑀−𝑚 After that, the percent error of the two masses sum: %𝜎𝑀+𝑚 2 √𝜎𝑀2 + 𝜎𝑚 √0.052 + 0.052 = ∗ 100 = ∗ 100 = 0.025% 𝑀+𝑚 148.6 + 130 From those two equations we found the total error: 2 2 %𝜎𝑀−𝑚 = √%𝜎𝑀−𝑚 + %𝜎𝑀+𝑚 = √0.382 + 0.0252 = 0.38% 𝑀+𝑚 Next, the percent error of the theoretical acceleration was measured using those values as shown: 2 %𝜎𝑎 𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 = √%𝜎𝑀−𝑚 + %𝜎𝑔2 = √0.382 + 0.0052 = 0.38% 𝑀+𝑚 From the percentage of acceleration the error quantity was found which was: 𝜎𝑎 𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 = %𝜎𝑎 𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 0.38 ∗ 𝑎𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 = ∗ 0.653 = 0.0025 100 100 Then discrepancy was found in lab: 𝑑 = |𝑎𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 − 𝑎𝑒𝑥𝑝𝑒𝑟𝑒𝑚𝑒𝑛𝑡𝑎𝑙 | = |0.653 − 2.178| = 1.524 𝑚/𝑠 2 %𝑑 = 𝑑 𝑎𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 ∗ 100 = 233% Alyami 4 %𝜎𝑎𝑒𝑥𝑝 = 𝜎𝑎 𝑒𝑥𝑝 𝑚𝑒𝑎𝑛 ∗ 100 = 0.4379 ∗ 100 = 20% 2.178 %𝑑 > %𝑎𝑒𝑥𝑝 233% > 20% Considered a failure because the experimental value is way greater than the theoretical value. Extra credit for deriving the equation 𝑎 = 𝑀−𝑚 𝑀+𝑚 𝑔 is attached to the report at the end. Part2: 𝑇𝑎𝑣𝑔 = 6.274 2𝜋𝑟 Then the velocity of the bob was calculated using this equation 𝑣 = (𝑇 𝑎𝑣𝑔 2 ) : 2 2𝜋(0.17) 𝑚 𝑣=( ) = 0.17 6.274 𝑠 Then the error of the radius and percent error: 𝜎𝑟 = ± 𝑑 0.005 = = 2.5 ∗ 10−4 2 2 𝜎𝑟 2.5 ∗ 10−4 %𝜎𝑟 = ∗ 100 = ∗ 100 = 1.47% 𝑟 0.017 Next is the Time average error and velocity percent of error: 𝜎𝑇 = 0.1 ∗ 100 = 1.59% 6.274 %𝜎𝑣 = √%𝜎𝑟2 + %𝜎𝑇2 = √1.472 + 1.592 = 2.16% Further more the frequency was calculated by dividing the number of revolutions by the average time: Alyami 5 𝑓= 𝑁 10 = = 1.59𝑠𝑒𝑐 𝑇𝑎𝑣𝑔 6.274 Then the error of the frequency taken from the excel sheet Standard deviation: 𝜎𝑓 𝑒𝑥𝑝 = 0.018𝑠𝑒𝑐 %𝜎𝑓 𝑒𝑥𝑝 = 𝜎𝑓 0.018 ∗ 100 = ∗ 100 = 1.13% ≤ 2.0% 𝑓𝑎𝑣𝑔 1.59 After that, to calculate the experimental Force applied in this experiment: 2 𝐹𝑒𝑥𝑝 𝑚 2𝜋𝑟 = ( ) = 4𝜋 2 𝑚𝑟𝑓 2 = 4𝜋 2 (0.3749)(0.17)(1.59)2 = 6.36𝑁 𝑟 𝑇𝑎𝑣𝑔 2 + %𝜎 2 + 2%𝜎 2 = √0.0092 + 1.592 + 2 ∗ 1.132 = 2.17% %𝜎𝐹 𝑒𝑥𝑝 = √%𝜎𝑚 𝑟 𝑓 𝜎𝐹 𝑒𝑥𝑝 = %𝜎𝐹 𝑒𝑥𝑝 100 ∗ 𝐹𝑒𝑥𝑝 = 2.17 ∗ 6.36 = ±0.138𝑁 100 Next is calculating the theoretical force for this experiment: 𝐹𝑡ℎ𝑒𝑜 = Ms 𝑔 = 0.57 ∗ 9.81 = 5.59𝑁 2 %σF theo = √%σm 2s + %σ2g = √8.77 ∗ 10−3 + 0.0052 = 0.052% σF theo = %σF theo 0.052 ∗ Ftheo = ∗ 5.59 = 2.9 ∗ 10−3 100 100 Discrepancy calculations as follows: d = |Ftheo − 𝐹𝑒𝑥𝑝 | = |5.58 − 6.36| = 0.78𝑁 σd = √σF 2theo + σF 2exp = √0.1382 + 0.00292 = ±0.138N %σd = 𝜎𝑑 0.0138 ∗ 100 = ∗ 100 = 1.76% 𝑑 0.78 d is larger than the error of d which means no agreement within the limits of the experimental error. Alyami 6 Discussion Part1: %𝑑 %𝜎𝑎𝑒𝑥𝑝 𝑎𝑒𝑥𝑝 𝑎𝑡ℎ𝑒𝑜 %𝜎𝑎 𝜎𝑎 𝑡ℎ𝑒𝑜 233% 20% 2.178 0.653 𝑚/𝑠 2 0.38% 0.0025 Part2: 𝑣 %𝜎𝑣 𝑓 %𝜎𝑓𝑒𝑥𝑝 𝐹𝑒𝑥𝑝 %𝜎𝐹𝑒𝑥𝑝 𝜎𝐹𝑒𝑥𝑝 %𝜎𝐹𝑡ℎ𝑒𝑜 𝜎𝐹𝑡ℎ𝑒𝑜 𝑑 𝜎𝑑 %𝜎𝑑 0.17 m/s 2.16% 1.59sec 1.13% 6.36N 2.17% 0.14N 0.052% 5.59N 0.78N 0.138 1.76% If an object moves at a constant velocity, the acceleration must be equal to zero as stated in Newton’s second law. Similarly, if an object accelerates, it has to be in the same direction as the force acting on it. In Free Body Diagrams (FBD) are usually represented by a box and arrows of the force direction coming out from the center of the box or the object. The sizes of the arrows represent the magnitude of the force. Newton’s third law dictates that an action-reaction force pairs have to be evident in the FBDs of two objects in contact with each other. There was intrinsic systematic error in the results of the velocity using the photo gate of the Logger Pro. Also, there was random error in Alyami 7 measurement when measuring the radius of the centripetal force machine. Moreover, random intrinsic error air friction when dropping the weights in part one Atwood’s machine. Alyami 8 Conclusion In conclusion, the experiment was inaccurate and unfortunately failed. The discrepancy was significantly larger than the error of it, which does not make an agreement within the limit. We worked in a group of three while the experiment was supposed to be for a group of two but there were no other student to make a pair. As a result, the work was more divided into 2 parts, George and Nick had most the experiment done and I had the calculations ready and helped setting up the logger pro. Variable Units m g Value 151.10 201.00 9.792 Error 0.100 0.100 M g g m/s2 m/s2 0.005 0.7478 aexp 1.2910 Variable Units Error m g Value 61.20 71.20 9.792 0.050 0.050 M g g m/s2 0.005 aexp m/s2 0.675 0.202
Purchase answer to see full attachment