ISLE labs for Physics 21X

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1 ISLE labs for Physics 21X General Information: There are 3 types of Investigative Science Learning Environment (ISLE) labs: observation experiments, testing experiments, and application experiments. Observation experiments are intended for students to learn skills such as changing one variable at a time, clearly recording and representing observations, and making accurate observations without mixing them with explanations. This is the first step of the experimental cycle, and allows you to observe phenomena, look for patterns in data, and start to devise explanations (once observations are carefully completed). The next step, once explanations have been devised, is to do testing experiments, which is an independent test students design that will test a hypothesis based on a specific explanation or rule. This helps you practice the skill of making predictions about the outcome of an experiment based on an explanation/rule/relationship. For a testing experiment, you can t just do the experiment and record what happens they must have a predicted outcome based on some explanation and if the outcome of the experiment agrees with the prediction it gives confidence that the explanation may be correct, but if it disagrees, then you know the explanation is incorrect. In order to make this judgment, you also need to apply basic uncertainty calculations. (A guide for understanding uncertainty is included in this packet.) The third type of experiment is the application experiment, where you apply some explanation/rule/relationship that you have tested enough that you think it is good, and you apply it to understand a new situation. Some application experiments require that you determine some unknown quantity multiple ways in order to determine if the methods are consistent, it is necessary to apply basic uncertainty analysis. By performing this sequence of experiments, it is possible to explore and devise a physics relationship, test it, and when you are convinced it is good, apply it to understand a new situation providing you with a complete understanding of the basic physics relationships (equations). By designing your own experiments, it gives you creative control, and assures that you understand the steps that you perform, as they are done by your conscious choice, and not by following instructions or playing around. Lab write-ups: There is one lab write-up per group, each group member must clearly put their name and student ID at the top of the write-up in order to obtain credit. Since the ISLE labs are labs you design yourself, and are intended to help you learn specific skills such as justifying your conclusions, comparing results, and understanding how uncertainty comes into play, it is important that you explain your work carefully in the form of a lab write-up. The lab write-up is done in the lab, there is nothing for you to do before or after the lab period. The write-up is not formal, but does need to be clear. Your lab may have specific questions for you to answer, or specific statements of tasks to complete, in which case you need to answer and document those things in your write-up. The lab will include a general set of bullet points that always need to be addressed within your write-up. Those general bullet points are specific to the type of experiment: observation, testing, or application. The lab reports shouldn't be lengthy or too wordy. They can include complete mathematical equations, complete sentences, bullet points, and/or diagrams. The required points should be addressed succinctly and clearly. Keeping the reports relatively short is good practice for science writing and will save you valuable time.

2 It will also help the reader (your grader) to more efficiently see your main points. Lab grade: There are two requirements for passing the lab. If you do not pass the lab, you do not pass the course. The lab points are also all or nothing. In order to pass the lab, you must attend and conduct all of the labs (you can make up to two labs during week 10, the reserved lab make-up week, in the event you have an excused absence during the term). You must also obtain an average of at least 2/3 s of the possible points for your lab write-ups over the course of the term. Grading: Your TA will grade your lab write-up based on some subset of the items you are asked to include. Each item will be graded out of a possible of 3 points. 0 points means the item is not included in the report. 1 point means the item is included, but incompletely, or incorrectly. 2 points means the item is included, but with some small mistake, or it isn t completely explained. 3 points means it is correct and complete. Roughly 5-9 items will be chosen for grading each week. The general items that are specific to the three types of ISLE experiments (observation, testing, and application), there is a set of rubrics for you and for the TA to use to evaluate your work. You are strongly encouraged to check your write-up with the rubrics while you are doing the lab. That will allow you to have a good gauge for how complete your work is and allow you to improve it before the TA uses those rubrics to grade. Rubrics: The observation, testing, and application rubrics specific to ISLE experiments are given on the next 3 pages. You are encouraged to bring these to lab and use them for self-evaluation as you do the write-up. They are not just used for grading, but are also intended to help you fully understand the skills you are to develop and demonstrate, and to help you assess and improve the quality of your work. Other information: Lab is not for testing your knowledge, but is a place for developing it. Take advantage of the time to explore the physics relationships to your own satisfaction, make sure your work makes sense to you, and use your TA as a resource to make sure you leave the lab with a good understanding of the material. In order to be counted as present for the lab, you must arrive no more than 15 minutes late, and stay until your group has turned in its lab write-up. During the first and the last lab, there may be a conceptual understanding or a similar type of assessment test given to all students. These tests must be completed in order for your lab attendance to count, but they are not graded until after the final grades are submitted. Your specific responses are used only to assess how much students understand from the course, and do not at all factor into your course grade. Your lab TA reserves the right to deduct points from your lab write-up for marginal participation or other reasonable reasons.

3 Ability to design and conduct an observational experiment Scientific Ability Missing Inadequate Needs some improvement Adequate Is able to identify the phenomenon to be investigated Is able to design a reliable experiment that investigates the phenomenon Is able to decide what is to be measured and identify independent and dependent variables Is able to use available equipment to make measurements Is able to describe what is observed without trying to explain, both in words and by means of a picture of the experimental set-up. Is able to identify the shortcomings in an experimental design and suggest improvements No mention is made of the phenomenon to be investigated. The experiment does not investigate the phenomenon. The chosen measurements will not produce data that can be used to achieve the goals of the experiment. At least one of the chosen measurements cannot be made with the available equipment. No description is mentioned. No attempt is made to identify any shortcomings of the experimental design. An attempt is made to identify the phenomenon to be investigated but is described in a confusing manner. The experiment involves the phenomenon but due to the nature of the design it is likely the data will not contain any interesting patterns. The chosen measurements will produce data that can be used at best to partially achieve the goals of the experiment. All chosen measurements can be made, but no details are given about how it is done. A description is mentioned but it is incomplete. No picture is present. Or, most of the observations are mentioned in the context of prior knowledge. An attempt is made to identify shortcomings, but they are described vaguely and no suggestions for improvements are made. The phenomenon to be investigated is described but there are minor omissions or vague details. The experiment investigates the phenomenon and it is likely the data will contain interesting patterns, but due to the nature of the design some features of the patterns will not be observable. The chosen measurements will produce data that can be used to achieve the goals of the experiment. However, independent and dependent variables are not clearly distinguished. All chosen measurements can be made, but the details of how it is done are vague or incomplete. A description exists, but it is mixed up with explanations or other elements of the experiment. A labeled picture is present. Or some observations are mentioned in the context of prior knowledge. Some shortcomings are identified and some improvements are suggested, but not all aspects of the design are considered. The phenomenon to be investigated is clearly stated. The experiment investigates the phenomenon and there is a high likelihood the data will contain interesting patterns. All features of the patterns have a high likelihood of being observable. The chosen measurements will produce data that can be used to achieve the goals of the experiment. Independent and dependent variables are clearly distinguished. All chosen measurements can be made and all details of how it is done are clearly provided. Clearly describes what happens in the experiments both verbally and by means of a labeled picture. All major shortcomings of the experiment are identified and specific suggestions for improvement are made Is able to construct a mathematical (if applicable) relationship that represents a trend in data Is able to devise an explanation for an observed relationship Is able to identify the assumptions made in devising the explanation Ability to construct, modify, and apply relationships or explanations No attempt is made to construct a relationship that represents a trend in the data. No attempt is made to explain the observed relationship. No attempt is made to identify any assumptions. An attempt is made, but the relationship does not represent the trend. An explanation is made but it is vague, not testable, or contradicts the observations. An attempt is made to identify assumptions, but most are missing, described vaguely, or incorrect. The relationship represents the trend but no analysis of how well it agrees with the data is included (if applicable), or some features of the relationship are missing. An explanation is made and is based on simplifying the phenomenon but uses flawed reasoning. Most assumptions are correctly identified. The relationship represents the trend accurately and completely and an analysis of how well it agrees with the data is included (if applicable). A reasonable explanation is made and is based on simplifying the phenomenon. All assumptions are correctly identified. Rubric B

4 Ability to design and conduct a testing experiment (testing an idea/hypothesis/explanation or mathematical relation) Scientific Ability Missing Inadequate Needs some improvement Adequate Is able to identify the hypothesis to be tested Is able to design a reliable experiment that tests the hypothesis Is able to distinguish between a hypothesis and a prediction Is able to make a reasonable prediction based on a hypothesis Is able to identify the assumptions made in making the prediction Is able to determine specifically the way in which assumptions might affect the prediction Is able to decide whether the prediction and the outcome agree/disagree Is able to make a reasonable judgment about the hypothesis Is able to revise the hypothesis when necessary No mention is made of a hypothesis. The experiment does not test the hypothesis. No prediction is made. The experiment is not treated as a testing experiment. No attempt to make a prediction is made. No attempt is made to identify any assumptions. No attempt is made to determine the effects of assumptions. No mention of whether the prediction and outcome agree/disagree. No judgment is made about the hypothesis. A revision is necessary but none is made. An attempt is made to identify the hypothesis to be tested but is described in a confusing manner. The experiment tests the hypothesis, but due to the nature of the design it is likely the data will lead to an incorrect judgment. A prediction is made but it is identical to the hypothesis. A prediction is made that is distinct from the hypothesis but is not based on it. An attempt is made to identify assumptions, but the assumptions are irrelevant or are confused with the hypothesis. The effects of assumptions are mentioned but are described vaguely. A decision about the agreement/disagreement is made but is not consistent with the outcome of the experiment. A judgment is made but is not consistent with the outcome of the experiment. A revision is made but the new hypothesis is not consistent with the results of the experiment. The hypothesis to be tested is described but there are minor omissions or vague details. The experiment tests the hypothesis, but due to the nature of the design there is a moderate chance the data will lead to an inconclusive judgment. A prediction is made and is distinct from the hypothesis but does not describe the outcome of the designed experiment. A prediction is made that follows from the hypothesis but does not incorporate assumptions Relevant assumptions are identified but are not significant for making the prediction. The effects of assumptions are determined, but no attempt is made to validate them. A reasonable decision about the agreement/disagreement is made but experimental uncertainty is not taken into account. A judgment is made and is consistent with the outcome of the experiment but assumptions are not taken into account. A revision is made and is consistent with the results of the experiment but other relevant evidence is not taken into account. The hypothesis is clearly stated. The experiment tests the hypothesis and has a high likelihood of producing data that will lead to a conclusive judgment. A prediction is made, is distinct from the hypothesis, and describes the outcome of the designed experiment A prediction is made that follows from the hypothesis and incorporates assumptions. All assumptions are correctly identified. The effects of the assumptions are determined and the assumptions are validated. A reasonable decision about the agreement/disagreement is made and experimental uncertainty is taken into account. A reasonable judgment is made and assumptions are taken into account. A revision is made and is consistent with all relevant evidence. Rubric C

5 Ability to design and conduct an application experiment Scientific Ability Missing Inadequate Needs some improvement Adequate Is able to identify the problem to be solved Is able to design a reliable experiment that solves the problem Is able to use available equipment to make measurements Is able to make a judgment about the results of the experiment Is able to evaluate the results by means of an independent method Is able to identify the shortcomings in an experimental design and suggest specific improvements Is able to choose a productive mathematical procedure for solving the experimental problem Is able to identify the assumptions made in using the mathematical procedure Is able to determine specifically the way in which assumptions might affect the results No mention is made of the problem to be solved. The experiment does not solve the problem. At least one of the chosen measurements cannot be made with the available equipment. No discussion is presented about the results of the experiment No attempt is made to evaluate the consistency of the result using an independent method. No attempt is made to identify any shortcomings of the experimental design. An attempt is made to identify the problem to be solved but it is described in a confusing manner. The experiment attempts to solve the problem but due to the nature of the design the data will not lead to a reliable solution. All of the chosen measurements can be made, but no details are given about how it is done. A judgment is made about the results, but it is not reasonable or coherent. A second independent method is used to evaluate the results. However there is little or no discussion about the differences in the results due to the two methods. An attempt is made to identify shortcomings, but they are described vaguely and no specific suggestions for improvements are made. The problem to be solved is described but there are minor omissions or vague details. The experiment attempts to solve the problem but due to the nature of the design there is a moderate chance the data will not lead to a reliable solution. All of the chosen measurements can be made, but the details about how they are done are vague or incomplete. An acceptable judgment is made about the result, but the reasoning is flawed or incomplete. A second independent method is used to evaluate the results. Some discussion about the differences in the results is present, but there is little or no discussion of the possible reasons for the differences. Some shortcomings are identified and some improvements are suggested, but not all aspects of the design are considered. Ability to construct, modify, and apply relationships or explanations Mathematical procedure is either missing, or the equations written down are irrelevant to the design. No attempt is made to identify any assumptions. No attempt is made to determine the effects of assumptions. A mathematical procedure is described, but it is incomplete, due to which the final answer cannot be calculated. An attempt is made to identify assumptions, but most are missing, described vaguely, or incorrect. An attempt is made to determine the effects of some assumptions, but most are missing, described vaguely, or incorrect. Correct and complete mathematical procedure is described but an error is made in the calculations. Most assumptions are correctly identified. The effects of most assumptions are determined correctly, though a few contain errors, inconsistencies, or omissions. The problem to be solved is clearly stated. The experiment solves the problem and has a high likelihood of producing data that will lead to a reliable solution. All of the chosen measurements can be made and all details about how they are done are provided and clear. An acceptable judgment is made about the result, with clear reasoning. The effects of assumptions and experimental uncertainties are considered. A second independent method is used to evaluate the results. The discrepancy between the results of the two methods, and possible reasons are discussed. A percentage difference is calculated in quantitative problems. All major shortcomings of the experiment are identified and specific suggestions for improvement are made. Mathematical procedure is fully consistent with the design. All quantities are calculated correctly. Final answer is meaningful. All assumptions are correctly identified. The effects of all assumptions are correctly determined. Rubric D

6 Electric Force I. CHARGING OBJECTS Application Experiment It i s know n t hat a pl astic r od r ubbed w ith w ool i s ne gatively c harged, and a foam board rubbed with plastic wrap is positively charged. Using this information, design experiments to determine if the following objects are uncharged, positively charged or negatively charged. i. a foam board ii. a foam board rubbed with fur iii. a piece of scotch tape that is first stuck to the top of a table and then pulled off. iv. any other objects at your table Write the following in your notebook: a. Devise a pr ocedure t o determine i f t he obj ects ar e cha rged. Write an outline of your design. b. Draw a labeled diagram of your experimental set-up. c. What other equipment do you need for your design? d. Write down the physical principles you will use to make your judgment. e. Perform the experiment. Record your observations in a table. f. Write your results for the charge on each of the objects listed above g. Check whether your results are consistent with each other. II. CHARGING OBJECTS Testing Experiment Design an experiment to show that the interaction between charged objects can not be explained solely by gravitational interaction. Write the following in your notebook: a. Give an outline of your experimental design. b. What equipment do you need? c. Draw a labeled diagram of your experimental set-up. d. Create a table like the following with the arguments, prediction and evidence for testing your hypothesis. Look at the example provided. Explanation/ Hypothesis Experiment design Predicted outcome If and I do this then Observed outcome And I saw/ But I saw Conclusion (hypothesis supported or not) Therefore III. CONDUCTORS and INSULATORS Testing Experiment A textbook says that metallic objects such as aluminum cans have electrically charged particles that can freely move inside, and that insulating objects such as plastic bottles do not have freely moving charged particles. Design an experiment to test this using any equipment at your table. (cans, plastic bottles, tape, foil, and the objects from experiment 1)

7 Write the following in your notebook: a. Give an outline of your experimental design. b. What equipment do you need? c. Draw a labeled diagram of your experimental set-up. d. Fill out the following table with the arguments and evidence for testing your hypothesis. Look at the example provided in part II of this lab. IV. Observe and Explain Using your experience from part III of this lab, determine if paper is a conductor or an insulator. Tear a piece of paper into small bits and lay them on the lab bench. Charge the foam board and bring it near the paper. a. Conduct any additional experiments to determine if paper is a conductor or an insulator. b. Draw a labeled diagram of your experimental set-up. c. Perform the experiment and record your observations. d. What is the outcome of your experiment? e. Devise an explanation for your observation based on your knowledge about conductors and insulators. (Are there freely-moving charges inside the paper?) V. Oscillating charge Observe and Explain Hang a very small wad of tin foil (using string and a ring stand) within 1 cm of a smooth, charged, metal object (the side of a soda can should work). (Make sure the metal object is isolated.) Put your finger roughly 1 cm on the other side of the wad. Let the wad touch the metal, then try to get it oscillating between the metal and your finger (without you physically swinging it). a. Draw a labeled diagram of your experimental set-up. b. Perform the experiment and record your observations. c. Will the outcome of your experiment depend on the kind of charge located on the can? If you re not sure, test this. d. Devise a detailed explanation for your observation based on your knowledge about conductors, neutral, and charged objects. (Is your finger a conductor?) Discuss your ideas with your instructor.

8 Electric Flux observation experiment: Electric Field At your lab table you have two nail-boards with differing numbers of nails in each, and a wire loop. The nails represent electric field lines. Flux can be visualized as the number of nails (electric field lines) which pass through the loop. Using the nail-boards and the loop, determine the parameters on which flux is dependent. Use the following guidelines: a. Devise procedures for making your observations. b. Draw a clearly labeled diagram of your experimental set-up. c. What assumptions about the objects are you making in your procedure? H ow could these affect the result? d. Perform the experiment and record your observations. e. What parameters does flux depend on, and how did you arrive at this conclusion? Acceleration and Deflection of Electrons - application experiment: In this experiment, you will use a cathode-ray tube (CRT) to accelerate and deflect a beam of electrons. The CRT is a glass vessel which contains: (1) an electron gun that emits electrons into a beam that travels along the tube; (2) a deflection system to change the direction of the beam; and (3) a viewing screen. CRTs are used in oscilloscopes, and the same principle is used in old TV sets and computer monitors. There are 3 stages for the electrons moving along the CRT. 1. The electrons leave a heated filament (approximately from rest) and are accelerated horizontally down the tube a distance a, due to a voltage V a applied across this section of the tube. We assume that the electrons have reached a maximum constant speed just at the end of this distance d, and that the entire acceleration is due to the applied voltage V a. 2. The electrons then enter the deflection region of length b, where they are bent from their horizontal path by a deflection voltage Vb. This deflection is perpendicular to their horizontal motion (either up, down, left or right, when facing the tube). 3. Finally, the electrons enter region c where they travel some distance c at constant velocity with no voltage applied, until they hit the screen. A disassembled tube is available for inspection. The voltages are established by parallel plate capacitors, and therefore have nearly constant electric fields between their plates (E = V/x, where x is the distance between the two plates). In this experiment, you will investigate the mathematical relationship between the deflection of the electron, and the two voltages, V a and V b (separately). You will need to take enough data to plot the relationship, and also work out the theoretical prediction based on your knowledge of electric force and field, and compare your plotted data to your derived equations. PROCEDURE for collecting data: Before turning on the power supply to the CRT, verify, or have your instructor verify, proper connections. A wiring diagram will be provided, and all devices should be correctly wired up ahead of time check with the diagram though, as an incorrectly wired set-up can burn out the tube. The power supply has a standby position that turns on the filament before the high voltage. Leave it in standby for 30 seconds or so before turning on full power. Make sure that the meter switch is set correctly to read the B+ voltage and

9 set the voltage to around 350 V. Adjust the focus knob for the best spot size. It sometimes happens at start up that electrons collect on the phosphor screen and don't migrate back to ground. These electrons can repel the beam from the area and make the screen look like it has a burned spot on it. Time will usually create a conducting path and correct the problem, but sweeping the electron beam back and forth across the spot might speed up the process. The deflection voltage V D is obtained from a voltage divider that takes its power from the table outlets. Make sure that the middle terminal (GND) from the table outlet connects to the red terminal on the CRT table power input terminal (GND). Positive and negative table power voltages connect to the black CRT table power input through a reversing switch. This allows positive and negative voltages to be put across the CRT's deflection plates, thereby reversing the direction of the deflecting electric field E x. The actual deflection voltage is measured with a voltmeter connected across the output of the voltage divider and in parallel with the CRT's horizontal deflection plates. Include the following in your report: a. Draw a sketch of the experimental apparatus, making sure each variable you will use is clearly defined in terms of the actual apparatus. b. Write an outline of the mathematical procedure you will use to derive the relationship between the deflection of the electrons and the voltages in the tube. c. Decide what assumptions about the objects, interactions, and processes you need to make to derive the equation. d. For one particular assumption, describe in detail how this assumption would affect the equation and the prediction for the deflection of the electron (for example, will it be deflected more than we predict because we have chosen a particular assumption in deriving the equation)? e. Draw appropriate physical representations for devising the mathematical procedure to solve the problem. f. Complete your derivation and from it, predict the mathematical relationship between the deflection of the electrons and each of the two voltages. g. Explain qualitatively why the relationships you found make sense (for example, if you throw the baseball with more force it will have a higher initial velocity and travel further) h. What are the possible sources of experimental uncertainty when taking your data? How could you minimize them? i. Perform the experiment and record your observations in an appropriate format, including graphing the deflection vs. each of the two voltages (separately). j. What is the outcome of your experiment? k. How does the outcome of your experiment compare with your mathematical prediction? Resolve, or suggest and explain physical reasons, for any discrepancies. l. Suggest specific improvements in the experiment and/or the mathematical derivation.

10 Electric Potential Experimental Apparatus: A tray is filled to a depth of about 1/4" with tap water and two electrodes are placed in the tray. When a potential difference V is placed across the electrodes, an electric field is established in the water. Since the water is conducting and air is not, and the electric current is always parallel to the electric field, virtually all the electric field lines in the water are parallel to the surface of the water. This allows us to treat the water and electrodes as a purely two-dimensional system. Make sure the water is very level before proceeding (any tilt in the tray/table will be a source of error). Observation Experiment: mapping electric fields for a button electrode inside a ringshaped electrode Experimental Method: There is a voltage divider attached to the electrodes in the water, and an amplifier-detector. The voltage divider allows us to tap off voltages from 0 up to the maximum applied voltage V in steps of 0.1V. One lead of the detector is connected to the tap of the voltage divider and the other goes to a probe placed in the water tank. By moving the probe around until the output from the detector-amplifier is a minimum, we can find a point in the water tank where the potential is equal to the potential of the voltage divider tap. Objective: Move the probe in the water and observe that the voltage varies from point to point. Find several points in the tray which measure the same voltage. Try to find a curved path that has the same voltage everywhere. Using your probe, trace out locations (over an arc of at least 90 degrees) where the potential is.2v,.4v,.6v,.8v and 1V. Use a sheet of graph paper to mark the positions and shapes of the electrodes and the voltages you probe. a. Draw a clearly labeled diagram of your experimental set-up. Make sure you represent the important aspects of your experiment. b. List the physical quantities you will measure, and how you will measure them. c. Perform the experiment. d. Graph your data clearly. e. From your graph, what can you conclude about the relationship between distance from the central electrode and the potential around the electrode? f. What assumptions did you make when you devised your relationship? If necessary, review the relationship between equipotentials and electric field lines before doing the following: a. Draw a few representative field lines on your graph paper with a different color pencil (or pen) than you used for the equipotential lines. b. Describe the shape of the electric field around the central electrode, and explain why it should have this shape. c. Where is the electric field the strongest, and where is it the weakest? How do you know? d. Choose a single electric field line. Using Cartesian graph paper, plot the electric potential along this line as a function of distance from the center of the tray. 5-8 well chosen data points are sufficient for this plot. e. Comment on the shape of the graph and write a general statement describing the potential around the central electrode.

11 f. From your data, write a mathematical relationship for the potential around the electrode and compare it to your earlier observation. Is the mathematical relationship what you expected? If so, why? If not, reconcile any differences. Observation experiments: Electric potential and field around different geometries Use the following geometries and make the associated observations: (1).. Point electrodes in water, separated by about 6 inches. (2) Long straight electrodes, separated by about 4 inches. (3) Geometry of part 2 but with a hollow conducting cylinder between the two electrodes. For each of the above geometries: a. Use the probe to map out the equipotentials. b. Represent them on your graph paper. c. Use the equipotential lines to map out the electric field. d. Identify regions of higher and lower electric field, and explain why the electric field has the distribution that it does. e. For the parallel plate geometry, devise a mathematical relationship for both the potential and the electric field between the plates (and not near the edges). f. For the parallel plate geometry, what are your sources of error and how do they impact your results?

12 DC Circuits I Resistance observation experiment: At your table, you have a sheet of resistive paper, and several strips of the paper cut to different sizes. You also have a multi-meter, several metal blocks, and wires. By placing the metal blocks on the ends of the strips of paper, you can measure the resistance of the paper with the multi-meter. D evise an experiment(s) to determine the relationship between the resistance of the paper and the geometry of the paper. (Hint: the paper is essentially 2 dimensional, so how many parameters will you need to test?) Use the following guidelines: a. Devise a procedure for making your observations. b. Draw a clearly labeled diagram of your experimental set-up. c. List the physical quantities you will measure and how you will measure them. d. What assumptions about the objects are you making in your procedure? How could these affect the result? e. Perform the experiment(s) and record your observations both with a brief description and an organized table. You will need to make several measurements in order to find a mathematical relationship between the parameters. f. Represent the data graphically (you will need one graph for each parameter you varied). g. Consider sources of experimental uncertainty. How do they effect your observations? h. Using your graph(s), determine the mathematical relationship between all of the parameters. What parameters does the resistance depend on, and how did you arrive at this conclusion? Voltage across a Resistor application experiment: Using the relationship you found in the previous experiment, devise an experiment to determine how voltage and resistance are related. In order to do this, you will need to keep any other parameters fixed, which in this case is current. You are encouraged to check with your TA to make sure you have accomplished this with your setup before you begin taking data. You have the same equipment as before, plus a 3V battery source. Use the following guidelines: a. Write a brief description of the procedure you will use and make a sketch of your experimental design. b. Decide what assumptions about the objects you need to make to solve this problem. How might these assumptions affect the result? c. List the physical quantities you will need to measure. Describe how you will measure each of them. d. Perform the experiment and record your observations in a table and a graph. Also include a brief description of your observations. e. What are the possible sources of experimental uncertainty? How could you minimize them? A quantity known as conductance, which is simply the inverse of resistance is sometimes more illuminating when used in the place of resistance. Conductance is defined as: G = 1/R a. Re-write your mathematical relationship from the first experiment in terms of conductance. Your TA has an experiment to do with you there is only one device for the whole class, so let your TA know you have reached this point so they can coordinate doing the experiment with you after you have done the following: It is always important to test the limits of a prediction. These include when things get very cold, very hot, very large, or in the case we will be exploring, very small. We will be testing the nanoconductance of platinum. The way that we will accomplish this is to slowly pull apart two lengths of platinum wire. If this is done slowly enough, small nanowires will form between the two lengths of platinum. Below is a

13 cartoon of this situation, note that the size of the nanowire has been greatly exaggerated. Since this happens very quickly we will need different tools to explore conductance on this scale. To this end we will be using an oscilloscope and a second device specifically created for this purpose. An oscilloscope is used to measure electric potentials on a very fast time scale. Before we can make any measurements, we must first discuss how our devices operate. The box used in this experiment has a knob on the front. This knob controls the voltage supplied by the box that will be utilized to measure conductance. This voltage (Vsource) is what is displayed by the multimeter hooked to the box. The voltage out of the box (Vsignal) is measured by the oscilloscope. To read an oscilloscope multiply the number of squares from the zero line the signal is and multiply it by the displayed units per division. There are two knobs for each channel (we are using Channel 1 in this experiment). One knob controls the units of voltage per division. The second knob controls the offset of the channel. This changes where the zero voltage line is. There are two similar knobs for the time axis. Varying these knobs can make measurements more convenient or illuminate details that would otherwise be unobservable. Using these tools, the conductance or resistance of an object can be measured. The relationship used is: G = k (V signal /V source ), where k 10-5 Ω -1. Experiment The goal of this experiment is to observe a platinum nanowire. To accomplish this, the break in connection must happen slowly enough to be recorded on the oscilloscope. The breaks you are looking for will have a constantly decreasing potential and take 30 μs or more to completely go to zero. To most easily observe nanowires, the following settings should be used: Vsource= mv Time Scale = μs Voltage Scale = 100 mv/division Voltage Offset = 2-3 divisions below zero Trigger Level = -200 mv When you think you've observed a slow enough break, press the stop button. Once your TA is satisfied that you are observing the correct phenomena, accurately reproduce your data below, making sure to include units on your axis. b. Is this qualitatively consistent with your previous mathematical model for conductance? c. Are the steps in your data consistent with your mathematical model? If so, why? If not, can you come up with any ways you might resolve the differences between your data and your model? (It may help to calculate the conductance at various steps.) d. Using your graph, calculate the conductance and resistance where the nanowire is the thinnest.

14 e. Before we can apply your mathematical model, you'll need to know a little bit more about the nanowire. The steps in your graph correspond to atoms. The first step is a nanowire has a cross section of a single atom, the second equal step has a cross section of two atoms, etc. The approximate length of a typical atomic width wire is 1 nm, while the cross section is about 0.3 nm by 0.3 nm. The resistivity of platinum is 1.06 x 10-7 Ω*m. Using your mathematical model for resistance, calculate the expected value of this resistance. f. How does the expected value from your previous model compare to the actual measured value? g. Can you account for any differences based on your sources of error? h. Can you account for any differences based on the assumptions of your previous model? ( Does your large scale model for resistance apply on the nanoscale?)

15 DC Circuits II Observation Experiment: 1. Given one piece of wire, a light bulb and a battery, design experiments to make the bulb light in at least two different ways. Also design two circuits in which the bulb will NOT light. Write the following in your lab-report: a) Draw the circuit diagram for each case, and connect the elements according to the circuit diagram. b) Record your observations. c) From your observations, devise a general rule for connecting circuit elements such that a bulb will light. State the conditions under which a bulb will not light. Testing Experiments: 2. Design a n e xperiment t o t est w hether K irchoff s j unction r ule and l oop r ule work. The j unction r ule s tates that t he s um of t he currents e ntering a ny j unction m ust be equal to the sum of the currents leaving that junction. T he loop rule states that the algebraic sum of the changes in potential encountered in a complete traversal of any loop of a circuit must be zero. You must have more than one junction and more than one battery in your circuit. Use at least 4 circuit elements besides batteries and wires. Equipment: Batteries, resistors, light bulbs of di fferent r atings, an ammeter, and a voltmeter. Write the following in your lab-report: a) State the hypothesis you will test. b) Devise an experiment to test your hypothesis. Write an outline for your experimental design. c) Draw a circuit diagram for your experimental set-up. d) Write the mathematical procedure you will use. e) What assumptions are you making? How could these affect the results? f) Make a prediction for current and voltage readings at various points in the circuit (indicate these on your circuit diagram). g) What are possible sources of experimental uncertainty? How could these affect the data? How could you minimize the uncertainties? h) Perform the experiment. Measure and record the currents and voltages at the necessary points in the circuit to test your hypothesis. Make a table if necessary. i) Did the outcome of the experiment confirm your prediction? j) Based on your prediction and the outcome of the experiment, what is your judgment about the hypothesis? Charging and Discharging a Capacitor observation experiment: Devise an experiment for determining the direction and strength (qualitative) of current flow in a circuit while a capacitor is charging and discharging. (To discharge, you simply remove the battery from your circuit.) You have wires, batteries, capacitors, light bulbs,

16 and a compass at your table. Hint: make a simple circuit with one capacitor and some light bulbs. Where do you need to place the light bulbs to test current flow in each part of your circuit? Use the following guidelines: a) Draw a diagram of your circuit. Be sure to indicate any differences in your diagram between when the capacitor is charging and when it is discharging. b) Describe your experimental design. c) Which measuring instruments will you use? How will you use them? Where will you place them in the circuit? d) What assumptions are you making? What limitations are present with your measuring devices? e) Record your observations. f) What did you find from your observations? g) Are your observations clear within the limits of your measuring devices? What can you say that is conclusive? h) Devise a r ule f or w hat happens t o t he c urrent while t he c apacitor i s c harging a nd discharging. Current flow in a Series Circuit with Capacitors testing experiment: Using the information you found in the previous experiment, test the direction and strength (qualitative) of current flow through all wires of a series circuit with more than one capacitor. Do this for both charging and discharging the capacitors. The same equipment is provided, and the same hint applies. Use the following guidelines: k) Devise a procedure to test your hypothesis (based on your work above). Write an outline. l) Draw a circuit diagram depicting your experimental set-up. m) Write down the procedure you will use. n) Make predictions about the current at various points in the circuit during charging and discharging based on the relationship/rule you developed above. o) Perform the experiment and record information about the current in each wire of the circuit. p) Did the outcome of the experiment confirm your prediction? q) Based on your prediction and the outcome of the experiment, what is your judgment about the hypothesis?

17 Magnetic Force Testing Experiment: Design three experiments to test the right hand rule as used to find the direction of the force that a magnetic field exerts on a moving charge/current. The equation describing this is F = qvxb. No more than two of the experiments can use the cathode ray oscilloscope. (The cathode ray oscilloscope contains a beam of electrons which produce a green dot when they hit the screen). Equipment: Bar magnet, cathode ray oscilloscope, horseshoe magnet, battery, long wire. (If you are not sure about the magnetic field surrounding any of the objects in this lab, use the magnetic field probe first to find this out before using the object in your experiment). a. What rule are you investigating in these experiments? b. Write an outline for each of your experimental designs including labeled diagrams. c. What assumptions are you making? How can they affect your observations? d. Make a prediction for each experiment based on the rule you are testing. e. Which of your experiments best minimizes assumptions and errors? Briefly discuss the merits and limitations of each experiment. f. Conduct each experiment and record your observations. g. Did the outcomes of your experiments match your predictions? If not, list possible reasons. Revise your predictions if necessary. h. Based on your predictions and the experimental outcomes, make a judgment about the right-hand rule. Application Experiment: In this activity we will use the force exerted by a uniform magnetic field to bend a beam of electrons into a circular path. In the apparatus, the electrons are first accelerated through a potential difference V. After acceleration, the beam enters a region in which there is a magnetic field B perpendicular to the velocity of the electrons. Apparatus Notes: The electron beam is produced by an electron gun consisting of a heated cathode (which emits electrons), a focusing grid, and an anode held at a potential V with respect to the cathode. The electrons are accelerated through this potential difference as they move from the cathode to the anode. The electron beam emerges from a small hole in the anode and then enters the magnetic field region. The magnetic field is produced by two circular coils of wire whose separation distance is equal to their radius. This arrangement is called a Helmholtz pair and produces a very uniform magnetic field in the region between the coils. The field is much larger than the Earth s magnetic field, which can be neglected in this experiment. The electrons move in an evacuated tube into which a small amount of inert gas has been introduced; when the electrons collide with the gas atoms, the atoms emit blue light, which makes the path of the electron beam visible. CAUTION: THERE IS A SHOCK HAZARD FROM THE POWER SUPPLIES USED IN THIS EXPERIMENT. The magnetic field due to the Helmholtz coils is

18 B 8µ NI 0 = (4) 125 R where N is the number of turns in each coil, I is the current through the coils, and R is the coil radius. Experimental tips: The circuit has been pre-wired. Set the power supply to standby. This will cause the filament to heat, and its glow should be visible in subdued light. After a couple of minutes, turn on the high voltages and raise the anode supply voltage to about 80 volts. The blue trace from the beam should now be visible. Your goal is to determine the magnetic field of the Helmholtz coil and compare it to the theoretical equation given above. There are considerable sources of error in this experiment (for instance, measuring the radius of the electron path within the magnetic field), so it will be necessary to make measurements with different accelerating voltages rather than relying on just one measurement for comparison. (The mass and charge of the electron are given in your textbook.) a. Draw a sketch of your experimental design. b. Write an outline of the procedure you will use. c. Decide what assumptions about the objects, interactions, and processes you need to make to solve a problem. How might these assumptions affect the result? d. Draw an appropriate physical representation for the situation and use it to devise the mathematical procedure to solve the problem. e. What is your derived mathematical relationship between the accelerating voltage and the magnetic field of the coils? f. What are the possible sources of experimental uncertainty? How could you minimize them? g. Perform the experiment and record your observations in an appropriate format. (Be sure to record the radius of the circle made by the electron beam, the current in the Helmholtz coils, and the accelerating voltage of the electrons.) To make an accurate determination of the vertical diameter of the circle, use the mirror to line up the bottom of the beam with its reflection and then the top. h. Using your derived relationship and experimental measurements, determine the magnetic field of the coil. i. Compare your result to the theoretical value you can calculate for the magnetic field of Helmholtz coils (your TA has the value of N). j. How does the result of your experiment compare to the theoretical value? What are possible reasons for the difference? k. Suggest specific improvements in the experiments.

19 Magnetism and Electric Current Application experiment: Apply your understanding of magnetic force on a current carrying wire and the right hand rule you tested last lab to explain the operation of the simple motor at your table. This motor consists of a coil of current-carrying wire (suspended in such a way that it makes electrical contact with batteries but can still rotate freely) and a strong magnet held near the coil. In order to get your motor started, you may need to first balance the coil so it rotates easily and give it a gentle initial spin. Include the following in your write-up: a. Draw a diagram of the motor and a description of it while it is in operation. b. Determine what positions of the coil with respect to the magnetic field are important for understanding the rotation (what parts of the motion do you need to consider more carefully to explain what is happening)? c. Draw diagrams for each of these positions and apply the right hand rule to find the direction of the force on the coil (will the force be the same everywhere along the coil? How many places along the coil do you need to consider?) d. Using your diagrams and a brief summary, explain the operation of the motor in detail. What causes the coil rotate in the magnetic field? Application experiment: In this experiment, you will use the magnetic field produced by a circular loop of current to determine the magnitude of the horizontal component of the Earth s magnetic field. At a point along the axis and a distance x from the plane of the current loop, the magnetic field pointing along the axis of the loop is 2 µ 0NIR B = (1) 2 2 3/2 2( R + x ) where I is the current in the loop of radius R and N is the number of turns in the loop. At the center of the loop (x = 0), this reduces to the form you are familiar with: µ 0NI B = (2) 2R We will measure the magnetic field of the loop by comparing it with the Earth s magnetic field. When a compass is placed near the loop, the direction of the compass needle will indicate the direction of the total magnetic field (the vector sum of the field of the loop and the field of the Earth). We first align the loop so that the Earth s magnetic field is perpendicular to the axis of the loop as shown in Figure 1. Figure 1 shows a top view looking down on the loop from above. Only the horizontal component of the Earth s field is observed. B loop B total Loop θ B Earth Figure 1

20 Experimental tips: Connect the circuit needed to send current into the coils, but do not connect the power terminals until your instructor has checked your circuit. CAUTION: To keep the loop from overheating, place the switch in the neutral (center) position when you are not making a measurement. Place the compass at the center of the loop when there is no current flowing. Turn the loop until its axis is perpendicular to the direction of the Earth s magnetic field, as shown in Figure 1. Use this as your initial reference position. a. Draw an appropriate physical representation for the situation. b. Write an outline of the procedure you will use (read the rest of the bullet points if you are not sure yet of what experiment to perform). c. Decide what assumptions about the objects, interactions, and processes you need to make to solve a problem. How might these assumptions affect the result? d. What are the possible sources of experimental uncertainty? How could you minimize them? e. Vary the current in the loop and observe the variation in the angle of the compass needle. Record several currents and the respective compass angles. Try reversing the direction of the current as well, and taking more data. f. Use the equations given for the magnetic field of the loop and the data taken to calculate the average value of the Earth s magnetic field. g. Compare this to the known value of the earth s magnetic field, do your values agree within the range of experimental uncertainty? If not, explain any discrepancies. Observation experiment: In this experiment we will use a compass to map the magnetic field of a small magnet. Place the magnet so that its axis is horizontal. Place the compass about 10 cm (measured from the center of the compass to the center of the magnet) from the N pole of the magnet along the axis of the disk. Be sure the axis of the magnet is in the same horizontal plane as the compass. Assuming the compass needle represents the direction of the field of the magnet, draw a vector on Figure 2 to indicate the direction of the field at that location. Keeping the distance between the center of the magnet and the center of the compass the same, make similar observations of the direction of the field at intervals of 30 o around the circle. At each location draw a vector that indicates the direction of the field. N S 10 cm Figure 2 (top view looking down on magnet)

21 a. Does the Earth s field have a significant effect on your observations? Explain your answer. b. Suppose the large circle in the above figure represents the surface of the Earth. Mark the approximate location of Corvallis. Assuming that the field directions you have drawn can be taken to represent the direction of the Earth s magnetic field at its surface, estimate the angle of the magnetic field direction above or below the horizontal at Corvallis. Observation experiment: The magnetic field of the magnet can be approximated as that of a magnetic dipole. One expectation for a dipole field is that it falls off along the axis of the magnet like 1/r 3. Another expectation based on a dipole field is that the field at a distance r along a line through the center of the magnet perpendicular to its axis is half as large as the field along the axis at the same distance r from the center. Devise experiments to test these two predictions, and explain your experiment in your lab write up. Do your results agree with the predictions? Justify your conclusions.

22 Induction These experiments purposefully give less instruction. You will need to (with the help of the rubrics) decide what information you need to include in your report to show a clear scientific process, and justified conclusions. Testing experiments with magnets, coils and a galvanometer: Equipment: Magnets, galvanometer, coils of wire, connecting wires, batteries, a resistive light bulb, and other miscellaneous objects. a. Devise a method to test the following statement: The only way to induce a current in a coil is to move a magnet near the coil. (Think of what you should try to do to test a statement that says The only way ). Write a brief outline of your design. Make a prediction, perform your experiment, record the outcome and make a judgment about the statement. b. Your friend thinks that the only way to induce a smaller current in the same coil is by moving the magnet slower. Devise an experiment to test your friend s statement. (Again, think of what you need to do to test a statement that says will always induce ). Write a brief outline of your design. Make a prediction, perform your experiment, record the outcome and make a judgment about the statement. Observation experiment: Your task in this part is to determine the relationship between the direction in which the galvanometer needle deflects, and the direction in which the magnet is moved. First, you will have to determine the direction in which the galvanometer needle deflects for a certain direction of current. Construct a circuit with the galvanometer, a battery and a light bulb acting as a resistor (no magnet). Draw the circuit diagram. Record the direction of current flow in the coil, and the direction in which the galvanometer needle deflects. Next, use Lenz s law to predict what should be the direction of current in the coil when the magnet is moved towards the coil with the north pole facing the coil. Then use your results from the previous experiment to predict the direction in which the galvanometer needle should deflect. Explain how you reached this conclusion. Draw pictures that show the orientation of the magnet with respect to the coil, the direction of motion of the magnet and the direction of magnetic field. Perform the experiment, record the outcome and revise your reasoning if necessary. Hint: Think of how the flux through the coil changes when the magnet moves, what should be the direction of the induced magnetic field according to Lenz s law, and what should be the direction of the induced current which will lead to that induced field (use right hand rule). Come up another way to move the magnet so that the galvanometer needle deflects in the same direction. Explain your procedure. Perform the experiment. Record the orientation of the magnet, the way you moved the magnet and the direction in which the needle deflects (use pictures or words to record these).

23 Invent your own investigation: a) Write a question that you want to investigate using the equipment you h ad in this lab. You can use additional equipment that is commonly available in the lab. b) Write a br ief s ummary of t he e xperiment you w ould l ike t o pe rform t o a nswer t he question. Check with your TA if you need additional equipment, and try to perform the experiment. What is the outcome? Observation experiments: Play with the other objects at your table and explain briefly to your TA either how the device (or effect) works (or can be explained).

24 EXPERIMENTAL UNCERTAINTIES No physical quantity can be measured exactly. One can only know its value with a certain range of uncertainty. This fact can be expressed in the standard form X ± ΔX. This expresses the experimenter's judgment that the "true" value of X lies between X - ΔX and X + ΔX. Uncertainties of measurements 1. Instrumental uncertainties Every measuring instrument has an inherent uncertainty that is determined by the precision of the instrument. Usually this value is taken as a half of the smallest increment of the instrument scale. That is 0.5 millimeter is precision of a ruler, 0.5 sec is precision of a watches etc. Instrumental uncertainties are the easiest one to estimate, but unfortunately they are not the only source of the uncertainty in your measured value. You must be a very skillful and lucky experimentalist to get rid off all other sources and to have the measurement uncertainty equal to the instrumental one. 2. Random uncertainties Sometimes when you measure the same quantity you can get different results each time you measure it. That happens because different incontrollable factors affect your results randomly. This type of uncertainty, random uncertainty, can be estimated only by repeating the same measurement several times. For example if you measure the distance at which cannonball hits the ground, you could get different distances every time you repeat the same experiment. For example, say you took three measurements and obtained 50 m, 51 m, and 49 meters. To estimate the absolute value of the random uncertainty you first find the average of your measurements: X = (50m + 51m + 49m) / 3 = 50 m. You then estimate approximately how much the values are spread with respect to this average in this case we have a spread of about ΔX = 1 m. That is, our measurement of the distance was X = 50 m ± 1 m. 2ΔX Most cannonballs will get into the range from X-ΔX to X+ΔX. 0 X Thus multiple trials allow you to find the average value and to estimate the uncertainty range. 1

25 3. Effect of assumptions Assumptions inherent in your model may also contribute to the uncertainty of the desired quantity. For example, you wish to find the speed of the ball moving on the floor. You are assuming that a ball moves along a straight line while in fact the surface of the floor in bumpy and the bumps contribute significantly to the distance that the ball covers and thus decrease the speed that that you calculate. Repeating the measurement will not let you get rid off the bumpiness of the floor and will contribute to the uncertainty with which you will determine the value of the speed. This type of uncertainties is not so easy to recognize and to evaluate. First of all, you have to determine the sign of the effect, i.e. whether the assumption increases the value of the quantity, decreases it or affects it randomly. Then try to estimate the size of the effect. For example, you measure the diameter of the baseball assuming it is perfect sphere. However, the real size of the ball may differ by 1 2mm if you measure in different dimensions. This difference will determine the uncertainty of your measurement. It is difficult to give strict rules and instructions on how to estimate uncertainties in general. Each case is unique and requires thoughtful approach. Be ingenious and reasonable. Comparison uncertainties If you are comparing the uncertainties in the values of two quantities then by analyzing the absolute uncertainty ranges you cannot tell which of the measurements is more accurate because the units of the measured quantities are different. How can we decide then which quantity has a larger uncertainty? Here we need to compare relative uncertainties, the ratio of the absolute uncertainty and the quantity itself ΔX / X. This may be expressed as a fraction or as a percentage by multiplying the ratio by 100%. The sizes of the circles at the picture are determined with different accuracy. The absolute size of the soft edge (9 units) is the same for both blue circles. However the large blue circle (90 units) looks sharper than the small one (30units). That happens because we compare relative uncertainties, which are 10% for the large circle and 30% for the small one. Note: Common sense and good judgment must be used in representing the uncertainties when stating a result. Consider a temperature measurement with a thermometer known to be reliable to ± 0.5 degree Celsius. Would it make sense to say that this causes a 0.5% uncertainty in measuring the boiling point of water (100 degrees) but a whopping 10% in the measurement of cold water at a temperature of 5 degrees? Of course not! (And what if the temperatures were expressed in degrees Kelvin? That would seem to reduce the relative uncertainty to insignificance!). However in most calorimetry tasks value of interest is not temperature itself but only the change of the temperature or the temperature difference. 2

26 Reducing uncertainties The example with the circles shows a way to reduce the relative uncertainty in your measurement. The same absolute uncertainty yields the smaller relative uncertainty if the measured value is larger. Suppose you have a bob attached to a spring and want to measure time for it to oscillate up and down, back to its starting position. If you are using a watch to measure some time interval, the absolute uncertainty of the measurement will be 0.5 s. If you now measure the time needed for the bob to go up and down one, you get 5 s. This means that you have a relative uncertainty of 10% in the time measurement. What if you measured the time for 5 oscillations instead? Say you measure the time for 5 oscillations and get 25 s. The instrumental uncertainty is still 0.5 s! The relative uncertainty in your measurement of the time interval is now: time relative uncertainty = (0.5 s/25 s) *100% = 2% By measuring the time for a longer time period, you have managed to reduce the uncertainty in your time measurement by a factor of 5! Of course you should not forget about the obvious way of reducing relative uncertainties by minimizing absolute uncertainty with better design, decreasing effect of assumptions or increasing the accuracy of instrument if it is possible. Uncertainty in calculated value It is important to estimate data uncertainties because uncertainties propagate through the calculations to produce uncertainty in results. Consider now the following example. Suppose you know the average mass of one apple m with the uncertainty Δm. If you want now to calculate the mass of the basket of the 100 apples you will get the value M ± ΔM = 100m ± 100Δm. It means that relative uncertainty of calculated value M remains the same as the relative uncertainty of the single measured parameter m ΔM / M = Δm / m. If you have more than one measured parameter, estimating uncertainty of the result may be more complicated. However, if one of your sources of uncertainty is much larger than the others (comparing relative uncertainties!) then you can neglect other sources and use the weakest link rule. Weakest link rule The percent uncertainty in the calculated value of some quantity is at least as great as the greatest percentage uncertainty of the values used to make calculation. Thus to estimate uncertainty in you calculated value you have to: 1. Estimate the absolute uncertainty in each measured quantity used to find calculated quantity. 2. Calculate the relative percentage uncertainty in each measured quantity. 3. Pick out the largest percentage uncertainty. This will be the percent uncertainty in your calculated quantity. Comparable uncertainties In case you have measured values with comparable uncertainties the rules are more complicated: they depend on the type of the mathematical relationship you use for calculation. In the most frequent case of multiplying two values their relative uncertainties add. One of the consequences of this rule is that the raising to the second power doubles the relative uncertainty and the raising to the third power triples it. Thus, in the 3

27 example above, the relative uncertainty in the calculated volume of the baseball will be three times larger then the relative uncertainty in the measured radius. Why do you need to know uncertainty? Is the measured value in agreement with the prediction? Do the data fit the physical model? Are two measured values the same? You cannot answer these questions without considering the uncertainties of your measurements. Indeed, how can you compare two values if the difference between them is smaller than the uncertainty in their measurement? Which bunch of grass is higher? You cannot tell this because their heights are determined with the uncertainty larger than the height difference. If you cannot tell which of two values are larger you can claim them the same. Thus to make judgment about two values X and Y you have to find the ranges where these values lie. If the ranges X ± ΔX and Y ± ΔY overlap you can claim the values X and Y the same within your experimental uncertainty. Summary When you are doing a lab and measuring some quantities to determine an unknown quantity: Decide which factors affect your result most. Wherever possible try to reduce effects of these factors. Wherever possible, try to reduce uncertainties by measuring longer distances or times etc. Decide what the absolute uncertainties of each measurement are. Then find the relative uncertainties of each measurement. If one uncertainty is much larger than the others you can ignore all other sources and use this uncertainty to write the value of the uncertainty of the quantity that you are calculating. Find the range where your result lies. Make a judgment about your results taking into the account the uncertainty. 4