Physics 1210 Lab Book Spring 2018 Instructor: Brad Lyke

Transcription

1 Physics 1210 Lab Book Spring 2018 Instructor: Brad Lyke Created by and used with permission from: Dr. Kobulnicky

2 Physics 1210 Experiments

3 Physics 1210 General guidelines for experiment reports 1) Reports should be typed and include tables and graphs, as appropriate, to demonstrate the work and support the conclusions. 2) Reports should include the full names of all persons contributing to the work. 3) Matlab scripts used to make plots or do computations should be included as an appendix 4) There are no particular font or margins or pages requirements 5) A complete report should include: An Abstract stating the main goal, the methods, and the main result or finding. Include numerical results and their uncertainties (if calculated) in this section as well. A short Introduction describing why the experiment is being performed, and the physics concepts in use. A Methods & Data section that describes the experimental setup in both words and with appropriate graphics. This section may also include formulae or derivations needed to demonstrate the objectives of the experiment. The Data section should also include tables of data or derived parameters with appropriate units. An Analysis section interpreting the data. This section may also talk about the precision of the results achieved and the main sources of error or uncertainty. This section should include graphs or figures that help interpret the data. Equations or derivations using basic data to compute other parameters may also be included here. If a lab calls for a derivation or extensive calculation, it will appear in this section. All data interpretation should be in this section (graphs, calculations, etc). A Results & Conclusions section describing what worked well or what could be changed to achieve better results in the future if the equipment or the goals were slightly different. An Appendix (or Appendices), which includes work performed but perhaps not essential to the main body of the report. Things such as Matlab scripts used to make plots should be included in the Appendix. 6) Feel free to include a digital photo of pertinent aspects of your setup or equipment. Drawings are often better as they can be labeled to show sizes, distances, etc. All drawings or photos MUST be original. 7) The text of the report should follow standard English grammar, punctuation and sentence structure. 8) Grading of experimental reports will follow the rubric distributed to the class. 9) An example of a well-written report will be posted to the website.

4 Physics 1210 Experimental Report Grading Rubric Abstract and overall: Does the abstract state clearly the purpose and results of the experiment? Does the report conform to standard English sentence structure and grammar usage? 10% Introduction: Does it contain a brief background of why the experiment is being performed and the relevant physical principles or equations? 20% Methods & Data: Does the methods section show figure(s) illustrating the experimental setup and clearly describe the procedure followed? Are the fundamental relationships explained in equations that stem from fundamental physics principles? Does the data section include tables summarizing the individual measurements, include multiple measurements to reduce random error, as needed, averages are computed, and any needed figures to show the data or its relation to an underlying physical principle or hypothesis? 20% Analysis: Does the analysis show original thought? Does it all (or examples of all) calculations used throughout the experiment? Does it include diagrams for vectors, forces, energy, etc. where those are calculated? 20% Results and Conclusions: Are the results succinctly stated, along with an analysis of the errors or uncertainties and how they affect the final result? This section should also include what was learned from re-doing the experiment after any changes were put into place. Are questions posted in the Experiment Handout answered completely and correctly? 20% Is the work, overall, neat and legible and does it show original thought and understanding (or is the work copied from a friend or a solutions manual?) 10% Poor Excellent

5 Experiment 0 - Numerical Review Name: 1. Scientific Notation Describing the universe requires some very big (and some very small) numbers. Such numbers are tough to write in long decimal notation, so we ll be using scientific notation. Scientific notation is written as a power of 10 in the form: m x 10 e where m is the mantissa and e is the exponent. The mantissa is a decimal number between 1.0 and and the exponent is an integer. To write numbers in scientific notation, move the decimal until only one digit appears to the left of the decimal. Count the number of places the decimal was moved and place that number in the exponent. For example, 540,000 = 5.4 x 10 5 or, in many calculators and computer programs this is written: 5.4E5 meaning 5.4 with the decimal moved 5 places to the right. (Do not write 5.4E5, that is for calculators and computers only). Similarly: = x = 4.2 x x10 2 = x 10 4 You get the idea. Now try it. Convert the following to scientific notation. Decimal Scientific Decimal Scientific ,000,000, x x Arithmetic in Scientific Notation To multiply numbers in scientific notation, first multiply the mantissas and then add the exponents. For example, 2.5x10 6 x 2.0x10 4 = (2.5x2.0) x = 5.0x To divide, divide the mantissas and then subtract the exponents. For example, 6.4x10 5 / 3.2x10 2 = (6.4 / 3.2) x = 2.0x10 3. Now try the following: 4.52x10 12 x 1.5x10 16 = 9.9x10 7 x 8.0x10 2 = 1.5x10-3 x 1.5x10 2 = 8.1x10-5 x 1.5x10-6 = 1.5x10 32 / 3.0x10 2 = 8.0x10-5 / 2.0x10-6 = Be careful if you need to add or subtract numbers in scientific notation. 4.0x x10 5 = 4.2x10 6 since 4.0x10 6 = 4,000, x10 5 = + 200, x10 6 = 4,200,000 Practice: Estimate how many shoes there are in the world. Use scientific notation, and some basic rough-guess numbers to produce an estimate.

6 3. Converting Units Often we make a measurement in one unit (such as meters) but some other unit is desired for a computation or answer (such as kilometers). You can use the tables in the Appendix of your textbook to find handy conversion factors from one unit to another. Example: You have meters. How many kilometers is this? There are 1000 m/km. Because kilometers are larger than meters, we need fewer of them to specify the same distance, so divide the number of meters by the number of meters per kilometer and notice how the units cancel out and leave you with the desired result. Another way to think about this operation, is that you want fewer km than m, so just move the decimal place three to the left since there are 10 3 m per kilometer. Or, if the new desired unit is smaller, and you expect more of them, then multiply. For example, how many cm are there in 42 km? Use the information in the appendix of your text to convert the following. 2 year = s 1000 feet= m 50 km = m 3x10 6 m = cm 52,600,000 km = m 3450 seconds = minute 6.0x10 18 m = mm 600 hours = days 5.2x10 12 kg = g 365 days = s 99 minutes = hr 1200 days = yr

7 4. Angles and Trigonometry Science and engineering is filled with examples where we need to use trig functions to determine angles or sides of triangles, or to compute the projection of one vector onto another. Solve for the unknown side or angle in the following triangles as review and practice. 3.5m φ 2.1m θ θ δ β γ α 5. Vector Addition Vectors allow us to specify directions in two or three dimensions by expressing a direction as the sum of direction along two or more axes. In two dimensions, we let be the unit vector in the X-direction and is the unit vector in Y-direction, and then r is the vector sum of the X- and Y-components. See the first example below for an instance of vector addition, and then complete the two vector addition problems, drawing the individual vectors and the total vector in each case, following the example.

8 6. Measurement and Uncertainties Very few measurements are direct measurements. Length, perhaps, is a direct measurement, when one uses a well-calibrated comparison tool of a standard length. Most measurements, such as mass and temperature are indirect; they depend on intermediate measurements and apparatuses and a subsequent calculation. For many students it comes as a surprise that absolutely exact measurements are impossible. If we weigh a small piece of material on a balance, a typical result could be grams. This is, however, only an approximation to the true weight, just as the value is only an approximation to the number π. A more sensitive balance would give a more accurate number. This is true of all measurements. Measurements always are imprecise, that is, there is some inherent uncertainty (we use the word uncertainty rather than error in most cases, as error implies a mistake) in the measurement, no matter how careful we try to be. In any kind of science or engineering, getting the right answer is usually the easy part; calculating how certain you are of that answer, i.e., what is the uncertainty on your answer, is the hard part, and an important part. The uncertainties reflect both the precision of the measurement/measurer and the accuracy of the instrument. The degree of precision with which an observer can read a given linear scale depends upon the definiteness of the marks on the scale and the skill with which the observer can estimate fractional parts of scale division. In many instruments of precision, the linear scale is provided with some sort of vernier, which is a mechanical substitute for the estimation of fractional parts of scale divisions. Its use requires skill and judgment. The degree of accuracy is determined by how close we can expect to be to the true or actual value. For instance, when we measure the length of a small object, we should expect that a meter stick will give a less accurate answer than a micrometer, provided that both instruments have been calibrated well. A common way of increasing the accuracy of a measurement done with an instrument of a given precision is to repeat a measurement many times under identical circumstances and then build an experimental average.

9 6a Experimental Averages The first step in quantifying and evaluating an experimental result is to establish a way to reduce random error by building an average of repeated experimental readings. The purpose of the averaging is to improve the knowledge about the actual quantity. Thus, we expect that the average is a better approximation of the actual (true) value than a single measurement. We express that confidence by rounding it to a better precision (more digits) provided that we do have a statistic that allows for that improvement. Find the arithmetic average (or mean) velocity and acceleration for the following sets of data. Mean vel [m/s] Acc [m/s 2 ] vel [m/s] Where N is the number of measurements. 6b. Uncertainties and Weighted Means Sources of uncertainty (or error) are many, but they are divided into two classes: accidental (random) and systematic error. By using precise instruments, the accuracy of the value we extract can be increased. It is our task to determine the most accurate value of a quantity and to work out its actual accuracy. The difference between the observed value of any physical property and the unknown exact value is called the error of observation. Random Errors are disordered in their incidence and variable in their magnitude, changing from positive to negative values in no ascertainable sequence. They are usually due to limitations on the part of the observer or the instrument, or the conditions under which the measurements are made, even when the observer is very careful. One (somewhat silly) example is if you are trying to weigh yourself on a scale but the building itself is vibrating due to an earthquake, leading to a great variety of results. Random errors may be partially sorted out by repeated observations. Sometimes the measurement is too large, sometime it is too small, but on average, it approximates the actual value. Systematic errors may arise from the observer or the instrument. They are usually the more troublesome, for repeated measurements do not necessarily reveal them. Even when known they can be difficult to eliminate. Unlike random errors, systematic errors almost always shift the observed value away from the actual value. In other words they can add an offset to the measurements. One example of a systematic error is if you are trying to weigh yourself, but you are wearing clothes, so the results is systematically larger than your actual weight. Or perhaps the scale is calibrated too high or too low.

10 There are all kinds of systematic errors. As another example, let s take a look at a hypothetical sequence of values made for the gravitational acceleration on earth: 9.78, 9.81, 9.81, 9.79, 12.5, 9.80 [m/s 2 ]. It seems quite possible that some mistake was made in recording 12.5 and it is reasonable to exclude that value from further analysis. It represents an obvious systematic error. There is no absolute limit for which we may assume that the above is the case. For our undergraduate lab we want to keep records of all data and exclude outliers only if they are off the average of the remaining data by 100% or more and only if we have just one outlier. Sometimes it is possible to estimate the uncertainty associated with each measurement. For example, If you try to count the number of shoppers that pass through the entrance to Wal Mart in any 10 minute interval, you'd be able to make a pretty accurate count if it's 1 a.m., and people are just trickling in. Your uncertainty would be quite small. On the other hand, if you try to count the shoppers at 6 a.m. the day after Thanksgiving, you're likely to make more counting mistakes and have a larger uncertainty on each count. So what's the average number? In this case, you want to compute an average that gives more weight to data that are more reliable and less weight to data that are deemed to have larger uncertainties. The way to do this is to compute a weighted average. Most often, we use the inverse-square of the uncertainties as the weight. If the uncertainty on a measurement i is σ i, then the weight is w i =(1/σ i ) 2. Compute the arithmetic average and the weighted average of the following set of measurements. vel [m/s] Uncertainty σ [m/s] Weight w Where x i are the individual measurements and w i are the weights on each measurement. Note that if all the uncertainties are the same (or all the weights are the same) then the weighted average just reduces to the simple arithmetic average. What is the weighted average for these data? The simple arithmetic average? Describe in your own words the effect of using weights? Describe what would happen if all of the weights were identical:

11 6c. Significant Digits Often calculations will yield numbers with large numbers (perhaps infinitely many!) decimal places. Not all of these decimal places are significant in the sense that they communicate reliable information about the accuracy with which the quantity in question may actually be known. For example, if you take a board which you measure to be cm long and cut it into 3 pieces, you find that 121.3/3 yields centimeters. It makes no sense to quote the result to more than one decimal place since you only knew the length of the board to 1 decimal place (presumably plus or minus 0.1 cm) to begin with. The rule of thumb is: Multiplication & division: cite only as many significant figures as the measured number with the smallest number of significant figures. Addition & subtraction: cite as many decimal places as the measured number with the smallest number of decimal places. Each digit counts as a significant figure except leading zeros or trailing zeros without a decimal point. Number # of significant digits Calculation Result (in sig figs) 23 two 3.24 [3 sig figs] x 2.07 [3 sig figs] two 3.2 [two] x [four] three 5.55 [three] / 3.3 [two] two 1.05 [three] [four] 4510 three [ ] [ ] 4501 four [ ] [ ] 0.01 one 1005 [ ] x 231 [ ] 0.2 one 1000 [ ] x 40 [ ] 0.20 two [ ] x 40. [ ] three [ ] x 40.0 [ ]

12 6d. Experimental Error and Data Scatter Another step in quantifying and evaluating an experimental result is to establish a way to describe the scatter or dispersion in the data due to random error. The first way to build such a measure of data dispersion is called the standard deviation, defined as σ where N is the number of data points, is the arithmetic average, and x i is each of the individual data points. Find the mean and the standard deviation for the data set in the table below: = σ = vel [m/s]

13 6e. Comparing experimental results to theoretical expectations The goal of every physics or engineering experiment is to test the theory which predicts certain outcomes for the experiment. The way we achieve this is to build a reliable experimental value based on averaging data and to characterize it by an experimental error. This is then compared to the theoretical value. Consider the following experimental data set consisting of time measurements and velocity measurements for a particle traveling in a straight line. t [s] v [m/s] Dist. [m] Mean Distance: Standard Deviation of Distance: Suppose now that the theoretical distance from the theoretical speed (2.750 m/s) and theoretical time (0.110 sec) gives a distance: d (m) = v (m/s) x t (s) = (m/s) x (s) = m. The percentage error is defined as: Compute the % error. Compare the theoretical value to the measured value. Are these two values within one standard deviation? If the uncertainties (errors) are distribution in a normal or Gaussian manner, we expect that 68% of the time (in other words, in 68% of such experiments if we repeated the whole experiment), the theoretical value and the measured value will differ by less than 1 standard deviation. 95% of the time the theoretical value and the measured value will differ by less than 2 standard deviations!

14 Distance Distance Distance Distance Distance Lab Motion - Experiment 1 Purpose: Learn to use the motion detector; understand position-time, velocity-time, accelerationtime graphs. Estimated time: 70 minutes for Part I, 40 minutes for part II. 1. Log into the computers and open the computer software called Vernier Software and star LoggerPro, then <file> <open> Experiments Additional Physics RealTimePhysics Mechanics Open the module Distance (L1A1-1a). 2. Starting about 2 meters from the motion detector, walk toward the motion detector at a slow pace. Graph the distance-time graph qualitatively. 3. Now start about ½ meter away from the motion detector and walk away at the same pace. Graph qualitatively the distance-time graph that results. 4. Now start at least 2 meters from the motion detector and walk quickly toward it and graph the result. Next start near the motion detector and walk quickly away from it and graph the result. 5. Now try starting near the detector, walk slowly away for 2 s, stand still for 2 s, walk quickly away for 2 s, stand still for 1 s, and walk quickly toward the detector for 3 s. First draw the expected position-time graph and then try it!

15 Velocity Velocity Velocity Velocity Distance Distance Distance Distance Distance Distance 6. Within your group talk about how you would make each of the following distance-time graphs. Then have your instructor watch as a randomly selected person demonstrates one; Make notes to yourself how to do each part below. 7. Now think about velocity-time plots. Open L1A2-1 Velocity Graphs and graph: You may get smoother plots by changing the detector sample time (under "Data" - "Data Collection" ) to 10/s. Walking slowly toward the detector Walking slowly away from the detector Walking quickly toward the detector Walking quickly away from the detector

16 Velocity (m/s) Distance (m) Velocity 8. Talk about with your group and sketch a velocity-time graph if you were to walk toward the detector slowly for 2 s stand still for 1 s walk quickly away for 2 s walk slowly away for 2 s Try it out and verify your prediction. 9. Open Velocity from Position (L1A3-1). Study the position-time graph below and sketch quantitatively the corresponding velocity-time graph. Then try it out. Did it match? When each person can do this, demonstrate it for your instructor and have them initial. They may want to ask you things like How can you tell from a position-time graph that you are moving at a constant speed? or How does the position time graph change if you move faster?

17 Velocity (m/s) Distance (m) Velocity (m/s) Distance (m) Now have your instructor draw a velocity-time graph and your group tries to predict the positiontime graph. Then perform the motion and graph it with the motion detector Your instructor will ask something like On the basis of just the velocity-time graph, can you tell where you end up? Can you tell how far you've moved? If so, how can you tell? 10. As a group come up with a way to make an object accelerate, an object for which you can measure its motion using the motion detector (suggestion: rolling an object down a slope tends to work better than dropping an object). You may get better graphs by increasing the sample rate to 30/s. Discuss and sketch what do you expect the velocity-time and position-time graph to look like for your experiment. Dropping objects onto the motion detector can damage them, so be careful, or come up with some method that does not involve dropping When you have agreed on an experimental approach, describe it to your instructor for their ok and have them initial.

18 Velocity (m/s) Distance (m) Describe briefly your experiment below. Test your plan by using the motion detector to make a position-time and a velocity-time plot. Print out your actual v-t and x-t plots and affix them below (or save a jpeg and it to your whole group). You can use the cursor to measure points on your graph fairly precisely How might you measure the acceleration by using these plots? Show below how you compute the acceleration and then show your instructor.

19 Acceleration (m/s 2 ) Velocity (m/s) Distance (m) Use the software module called "Speeding Up (L2A1-1)" to re-perform your experiment and show the acceleration-time plot along with the v-t and x-t plot. Print out and affix these below and annotate them to describe what you are seeing. You may need to limit the sample time to just a fraction of a second and plot that part How did your computed acceleration compare with the graphed one here? Show how you computed the acceleration.

20 Acceleration (m/s 2 ) Velocity (m/s) Distance (m) Given the following acceleration-time curve, try to predict the v-t and x-t curve. Do you have to assume that your object starts from rest? Do you have to assume that your object starts at x=0? Assuming that the objects starts at v=0, x=0, find the final velocity and the final position. Show how you do this. Instructor initial at end of section.

21 Part II 13. Devise an experiment to measure g. Show whether g is the same or different for a massive object or a less massive object. Also show whether g is the same acting over a big distance versus a small distance. There are many ways to do this. You need not even use any fancy equipment. Concentrate on simplicity and accuracy. When you have a plan, describe it to your instructor for approval and initial. Then go conduct your experiment. Be sure to ask if you would like equipment or tools that you don't see available in the room. Perform your experiment as many times as you like to obtain results that you trust. Collect data carefully as you will need to write up a formal experimental paper describing your purpose, your method, your data, and your results. Turn in a report on your experiment using the provided example Experimental Report as a template. Include a Matlab graph of the position of your dropped object versus time as it falls from the roof. Include a digital photo of pertinent equipment or events in your experiment. Include an estimate of the % error in your measurement of g. Be sure to include details of all of your equipment used. Feel free to ask for advice. It is possible to be very precise with your measurement of g if sufficient attention to detail and measurement is achieved! Practicing your method ahead of time can significantly reduce measurement errors. Please turn in the rubric on pg. 4 at the beginning of the manual with your report.

22 Lab Projectiles Experiment 2 In this experiment, your team will fire a projectile at a given random angle to hit an intended target accurately. Your instructor will show you the experimental setup. In brief, the rules are: 1) You pick the muzzle velocity (one, two, or three clicks of the spring-loaded cannon). 2) You will be given a standard cannonball. 3) You may test fire the cannon with any launch angle, but the final angle will be given to you. 4) You pick the target location and the launch location. 5) You may test fire your cannon as many times as you like on your tabletop without letting the cannonball hit the tabletop or the floor. 6) The projectile must be fired from the table and land on the floor. 7) When you fire for real, you only get one shot. If you fail to hit the paper target on the ground a new angle will be given to you and you will have to recompute the range. We will use carbon paper underneath a target piece of paper to record the distance from the intended landing site. 8) Think and measure carefully. The most accurate groups often land within 3 cm of the target location!! When you have a strategy for computing the landing location, discuss your intended launch plan and your plan to compute the landing location with your instructor and have them initial. Make sure that everyone in your group can explain the procedure that will be used. Launch angle 1: Launch angle 2: Instructor initials: Instructor initials: Report guidelines 1) Be sure to include in your report a diagram of the experimental setup with any necessary measurements and other details, such as masses, distances, etc. 2) Include a set of calculations (can be handwritten if done neatly) that shows your theoretical target location and how you arrived at this number. What other things did you have to measure to estimate your intended target location? No guessing allowed. Make careful measurements and document everything! 3) Include a Matlab plot that shows the landing location as a function of θ, where θ is the launch angle above the horizontal. You will only be given one θ for your real launch, but your graph will nicely show how distance varies with θ given your fixed values of launch velocity and initial height. 4) After your experimental test firing, you will have a chance to assess your results and fix/remedy any mistakes that you can identify. If something went wrong, include an analysis of where the problem occurred. Then, document any changes you made and reperform the experiment to show that you have caught and fixed any mistakes. Equipment Spring-launched cannon, standard cannonball, meter sticks, paper, carbon paper, c-clamps. Please turn in the rubric at the beginning of the manual with your report.

23 Lab Springs Experiment 3 Part I Measure a spring constant. Make several measurements with tools you already know as you stretch the spring several different distances. Make a Matlab plot to show the force required as a function of distance and fit a reasonable looking function to your data. Take as much data as you need to get a reliable result, using the best practices that you already know. Hints: 1) The more data points you have the better the Matlab fit will be. 2) Remember to measure an unweighted length. 3) Do not use many similar masses. Try using some very large and very small masses. When you have your data, show your instructor your method and have them initial. Integrate this curve to show how much energy is stored in the spring and make a plot of energy stored in your spring versus distance. This involves doing an integral. Show the calculations in the Analysis section of your experiment report. If you want a challenge, I'll show you how to do a numerical integral in Matlab (optional). The report for this experiment only covers part I. Part II is not part of the experiment. Something for part II will be included at the end of the report, however. Equipment: Large spring, rod, rod clamps, standard mass set, meter sticks, laptop (provided in lab) Part II Devise a workout. The guidelines are that your workout must 1) Consume at least 400 Calories (1 Calorie = 4186 J) of work for one individual person, you!!! (If this seems too easy because you're a superstar, you are welcome to go for more Calories.) 2) Have a peak power output of at least 300 W for some 30-sec duration or longer (or go for more W if you are a superstar.) 3) Consist of at least 3 and not more than 6 activities from the following lists 4) Where needed, adopt the mass of a 70 kg person (or use your own mass if you wish). 5) Muscles are only about 33% efficient, meaning that the Calories required are actually about three times greater than the actual mechanical work achieved. Compute your mechanical work in the strict physics sense, and the multiply by a factor of 3 to find the Calories used. Group I These are activities are relatively easy to figure out the work required. Pick most or all of your activities from this list. A. Lifting weights (either free or simple machine weights). You can do several of these in your workout (e.g., bench, curls, leg press, etc) but it counts as one activity. B. Climbing stairs (or stepping repeatedly onto a box) or the stair climber machine. C. Squats or similar (note, you do as much work going down as up, why? Also with pushups, etc.) D. Pushups, pull-ups, or similar E. Walking/running (ask for help as the work you do here is mostly against gravity; on average, walking 1 mile burns about 110 Calories). Look for information on how to calculate this on the internet. Calories burned are dependent on walking/running speed.

24 F. Shuttle relay (suicides; where you run back and forth, changing your kinetic energy many times) Group II These are activities it is a challenge to compute the work/power for. Pick at most one activity from this list. If you use a machine at a gym that calculates Calories burned, cite the machine. A. Rowing machine with variable resistance B. Stationary bike with variable resistance C. Elliptical trainer D. Swimming E. Ask about others that you may want to invent. First sketch out your workout plan and have your instructor initial to approve the basic plan. Instructor initials: After the end of the full experimental report for part I, include a short section for part II. Show calculations for each of your proposed activities to demonstrate how much work you do in each activity, and your average power during the activity. Show explicitly how the peak power in W is achieved. Summarize your workout in a table showing the activity, the work done, the average power, and the time of each activity. This short writeup is separate from the spring report in part I. It is not a full experimental report. A couple paragraphs and a couple tables (with some calculations) should be enough. If you actually do your workout, on your honor, add in your report how easy or hard, doable or undoable each phase exercise was and how you would modify your workout based on what you experienced. For 5 extra credit points, get at least 2/3 of your group to go do your workout together. Again, on your honor. I trust you. Mention it in your short writeup mentioned above. Please turn in the rubric at the beginning of the manual with your report.

25 Lab Engineering Experiment 4 Purpose: Design a ramp/spring/friction system roughly as pictured below and as demonstrated by your instructor. The goal is to come as close as possible to a target standing on the track without hitting them. Spring launcher Firing pin Wood block w/ mass (M) Target θ L Cart Rules: 1) The slope angle,, must be significant greater than 5 less than 50. 2) You must incorporate some substantial fiction into your cart. You can accomplish this by added mass to the block. 3) L should be larger than about 0.3 m. 4) You get only one real shot. You may not test your apparatus even on a level surface. 5) Measure the spring constant at two different click settings. The recommended way is to push with the force probe and integrate the F-x curve to obtain the energy stored in the spring. 6) You need to measure µ k of the block with the added masses. Note that µ k is dependent on the material of the track AND the block, so this must be found experimentally. The complete write-up should include showing how you get a single function for L in terms of the other variables, M, µ k, k, x (compression distance), and, whose values you will choose, within the given constraints. Also include raw data and plots of how you measure the crucial parameters like k and µ k. For A-level credit also estimate how accurately you can measure each of these things. In other words, come up with an uncertainty on M, µ k, k, x,. Represent these as Δ M, Δ µ, Δ k x, Δ. I will show you how to estimate the uncertainty on L, Δ L, given the uncertainties on each of these. As part of your analysis, discuss not only were you successful in coming close to the target without hitting them, but was your actual travel distance within one standard deviation (1 Δ L ) of your intended target distance. Tips: 1) Put into practice all the things you know about making good measurements of friction, of spring constants, etc., because the quality of your result will depend on the ability to measure accurately the quantities on which L depends. 2) Buff/polish the track surface to make sure that the coefficient of friction is the same everywhere. 3) Note that the spring cannon is still compressed some very small distance x 0, even when the cannon is fully released. The total energy stored in the cannon is really 1/2 k (x 1 +x 0 ) 2 where x 1 is the distance that you compress the spring to fire it. Hopefully x 0 is small and can be neglected. Is this true?

26 4) You must add mass to the block to obtain sufficiently high frictional values. Complete a report on this experiment, giving details of your preparations, your calculation, and your results, along with what you learned and how you later modified your apparatus or calculation to ultimately make it work the way you intended. Once you have set up your spring/cart/block/target system you should take before and after photos of the experiment to make measuring travel distance easier. Equipment: Spring cannon, 2m track, wheeled cart with masses to add, wood block, meter stick, firing pins for the cannon, target object, bricks to elevate one end of the track, force probe. Please turn in the rubric at the beginning of the manual with your report. To use the force probe: - Connect the force probe to the lab laptop. - Open Logger Pro. - Set the force probe to ±50 N - Click * Force = in Logger Pro above the data table. - Click the force meter. - Click Zero - The spring launcher at one and two clicks is less than 50 N. At three clicks most are above 50 N, so the reading with the force probe cannot be trusted.

27 Lab Inertia Experiment 5 Purpose: Measure the moment of inertia for a metal ring two different ways and compare these different methods to a moment of inertia derived using calculus methods. Method I: Roll it down a slope. Include in your write-up a Matlab plot of computed moment of inertia, I, versus velocity at the bottom of the slope, v, where the maximum possible value of v is the speed you'd expect if you just dropped the object. Also make a plot of I versus, the ramp slope. Does this plot tell you what moment of inertia you would measure if you recorded the same time/velocity down the slope but with a different slope angle? Interpret this plot and see if it makes sense in the extreme limits. To do these two plots you will need to roll the ring down slopes of different angles. Try more than two angles (90 is straight down, do not test this angle). Describe your process to the instructor and have them initial. Instructor initial: Inertia ring Slope Method II: Use the rotating apparatus pictured below to try to spin up the ring. You will have to spin up and measure the gray platter first. Then add the ring and repeat the experiment. The moment of inertia for the ring is the difference of the two computed values. θ Inertia ring Pulley Rotational table w/ platter String Hanging Mass Method III: Use calculus to derive the moment of inertia for a ring. Find the general expression first, then use your measurements of mass and inner/outer radii. After you have computed I for each method, find the % difference between Method I and Method III (III is the theoretical value) and the difference between Method II and Method III (again, III is the theoretical value). Remember to discuss in your Analysis whether Method I or II is more accurate and why that might be.

28 Write up the report in the standard style describing your experiments. Note that two different setups require two different diagrams in the Methods section. Include the calculus derivation of your expression for moment of inertia. For Methods I/II decide which variable is known LEAST accurately, that is, which one is responsible for creating the largest error in your result? Equipment: Large metal ring, rotational table, standard masses, pulleys, wood plank (slope), bricks to set slope angle, vernier calipers, meter stick, string, rotational platter, stop watch. Please turn in the rubric at the beginning of the manual with your report.

29 Lab Raft Experiment 6 Construct a raft from the specified materials to hold a number of pennies that you will predict. The winning team must hold the most pennies and ALSO correctly predict the number of pennies it will hold, within the errors. In your report, give your single equation for N, the number of pennies that the raft will support without sinking. This equation will be in terms of any other important measurable variables. Be sure to measure EVERYTHING (don't assume!) Also, estimate uncertainties on each of the other parameters in your equation and compute an uncertainty on the number of pennies the raft will hold, Δ N, using the same method as Lab 4. For example, you don't know the boat volume with infinite precision...so let Δ V be the uncertainty on the volume of your boat in cm 3. Do you claim to know the volume of your boat to Δ V =1 cm 3, Δ V = 0.5 cm 3, Δ V = 0.1 cm 3? How about the mass of a penny; do you know that to Δ M =1 g or better? In other words, estimate the level of uncertainty on your ability to measure the key variables you need to measure to predict N. Equipment: A big vat of some liquid, balsa wood sheets for each group, knife, hot glue, lots of pennies Please turn in the rubric at the beginning of the manual with your report.