Course Project. Physics I with Lab

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1 COURSE OBJECTIVES 1. Explain the fundamental laws of physics in both written and equation form 2. Describe the principles of motion, force, and energy 3. Predict the motion and behavior of objects based on physical laws 4. Demonstrate the principles of motion, force, and energy in simulated labs BACKGROUND INFORMATION Create an experiment that demonstrates and allows exploration of a physical principle Design the experiment and measurement techniques to minimize measurement uncertainty Explain the experiment s development and solutions to technical issues Analyze the experimental data in terms of known physical concepts and generalize the results to derive a functional law Describe the function of the experiment, the independent quantities under investigation, the measurement technique, and the results PROJECT INSTRUCTIONS Complete your project by following the steps below. Please also refer the Physics with Lab Final Project Lab Manual for specific instructions on completing the final steps. 1. Choose a physical relationship to examine experimentally. You may choose one of the examples from the "Suggested Experiments section of lab manual provided below, or you may design a similar experiment to suit your own tastes and available equipment, but the experiment should relate to one or more concepts studied during the course so far. 2. Design and build an experiment that demonstrates the physical property you have chosen. You will need to perform multiple measurements under similar or identical conditions, so your experiment should be designed to allow you to reproduce the positions, velocities, and orientations of various elements consistently many times. Perform a series of measurements where you vary one quantity to be investigated and record the outcomes.

2 3. Analyze the recorded data. First convert your raw measurements, usually distance and/or time, into the physical quantities relevant to your property. Then generalize the series of measurements to extract the underlying physical law. 4. Create your lab report. To complete your project you will need to document all of your findings in a lab report. To understand how to set up your lab report you will need to follow the directions in the Physics with Lab Final Project Lab Manual (see below). There, you will find items that should be addressed in your lab report. LAB MANUAL INSTRUCTIONS For your final project, you will perform a laboratory experiment to illustrate a physical law or relationship. This manual will help you through the process of designing, operating, and documenting your experiment. At the end of the project, you will hand in a lab report. This report should document everything you've done in the project, present all of your recorded data, and describe your analysis and your findings. A typical lab report will have the following sections: Introduction Describe the chosen problem Experiment Discuss how you designed the experiment including what choices you made to control uncertainties One or more figures or photographs clearly showing the experimental setup and the layout, with relevant parts, distances, etc. labeled Data collection Describe how a typical experimental trial was conducted Discuss how external factors may affect your measurements A record, usually in table form, of each experimental trial, including the independent variable and all relevant measurements at each step If any experimental trials are to be excluded from the final analysis, discuss the reasoning Data analysis Show how the measured quantities (length, time, etc.) are converted into the relevant physical quantities (energy, momentum, force, etc.). This may require diagrams and several steps of calculation.

3 Show how the uncertainties are calculated One or more graphs showing the main physical effect being studied Graphical analysis deriving the underlying physical law Results Summarize the experiment and the main findings SUGGESTED EXPERIMENTS Here are some suggested experiments to consider. Moments of inertia How does the moment of inertia of a cylinder depend on the mass of the contents? This problem is discussed in Section 10.4 of the textbook (see pages ). By rolling a can down an inclined plane, you can measure its moment of inertia. If you fill the can with different materials with different masses, you can study how the moment of inertia varies as a function of mass. For an ideal solid cylinder with radius r, the correct formula is How do your results compare to expectations? I = mr2 2 Variations The equation above is valid for solid cylinders. Do you expect the results to be the same if you fill the can with liquids instead? You can perform the same experiment with multiple cans to vary the radius instead. Or use a different shape instead of a cylinder. Can you think of another way to measure the moment of inertia instead of rolling a can down a slope? What about applying a known torque and measuring the angular acceleration instead? Or using the gravitational potential energy of as separate mass to somehow start the can spinning and then measure the resulting angular velocity? Other possibilities Magnets exert forces on other magnets or on certain metals.

4 Can you measure the strength of a magnetic force as a function of the distance between the two objects? You'll need to find a way to measure rather small forces without letting the objects fly out of control. Or come up with your own... Anything that interests or excites you can be made into a good experiment. Just make sure that you choose something that can be repeated and have one good independent variable to control. If you want to demonstrate conservation of momentum, you can't just collide two objects together once, show that the equations balance, and claim success. You need to show a systematic trend. Repeat the collision many times with varying speed of one of the objects and show that, no matter what the initial momentum, the final momentum is the same. EXAMPLES AND GUIDANCE FOR EXPERIMENTAL UNCERTAINTY When designing and operating your experiment, you'll need to pay close attention to uncertainty. We want the experiment to test real physical effects, and not be swayed by data that is mis-measured. Two main classes of uncertainty need to be considered: external factors and measurement uncertainty. 1. External factors The outcome of a good experimental trial should be determined by only one element: the independent variable you are studying. If there are any extraneous factors that can affect the result, your data will be difficult to interpret accurately. When designing and operating your experiment, be sure to keep external factors in mind. For example, if your experiment uses a ball rolling down a plane, pay close attention that the ball doesn't slide instead. Sliding and rolling are different types of motion and that will give you different results. If the ball does slide during one trial, make a note and consider excluding this data point from your final analysis. If the ball consistently slides during many trials, you may need to redesign the experiment to avoid this, perhaps by using a different angle of incline for the plane or by using different materials.

5 Try also to reproduce the starting condition of the apparatus identically for each trial. Release the ball at the same place each time, perhaps by marking a start line on the plane. Make sure that the plane hasn't moved or tilted between trials. Anything that might affect the rolling motion of the ball has to be controlled. Also, mention in your lab report how your experiment controls such factors. Discuss whether any external influences might still exist and how significantly they might affect your data. 2. Measurement uncertainty Measurements made in the real world are imperfect and subject to uncertainty. The smaller the uncertainty on a given measurement, the more precise the measurement is. This will be more useful in distinguishing between alternative physical theories. When quoting any measurement, it is essential to give both the measured quantity and its uncertainty, so that the reader can easily see the precision of the measurement. A crude balance might give the mass of a projectile as (50±5) g, while a more sophisticated scale might measure (49.3±0.1) g for the same object. The precision of a measurement also dictates how many significant figures to show. It doesn't make sense to quote any digits smaller than the measurement uncertainty. A measuring device often limits uncertainties. A typical ruler has markings for centimeters and millimeters. Under ideal circumstances, you might be able to estimate the length of an object to within one-fifth of one division on the ruler, or 0.2 mm. If the object is irregularly shaped, however, your precision might be limited to 1.0 mm, or even worse, and it is up to you to estimate this. One trick, which may help estimate uncertainty in difficult cases, is to repeat the measurement several times. In this case, the mean value of the set is a good estimate of the true value, while the standard deviation of the set is an estimate of the uncertainty. Recall the definition of standard deviation: 3. Dealing with measurement uncertainty s= 1 N 1 ( x i x) 2 An important part of designing an experiment is trying to minimize uncertainty. One way to minimize uncertainty is to compare the physical quantities that need to be measured to the measuring apparatus.

6 Imagine that you are asked to measure the thickness of a sheet of paper and are given a standard ruler, with millimeter markings, so that the uncertainty might be 0.2 mm. On the left, measuring one sheet is nearly impossible with the markings on a standard ruler. On the right, measuring 500 sheets is much easier with the same ruler, since the quantity being measured is now much larger than the uncertainty in the markings. If you actually try this measurement, you'd probably measure the thickness to be (0.2±0.2) mm. This is not a particularly precise value; it has 100% uncertainty! The trick is to measure the thickness of many sheets of paper together, say 500, which have a total thickness of (48.7±0.2) mm. Dividing by 500 sheets gives the thickness of one sheet as (0.0974±0.0004) mm, a much better result. Trying to measure a large object relative to the intrinsic uncertainty in the ruler is always easier. Note that these measurements all rely on the assumption that all sheets of paper are essentially the same thickness; if they are not, combining them together will spoil the measurement. Be aware of this when designing your experiments. Similar tricks work for other measurements as well. A stopwatch is not very precise when trying to measure an event lasting a fraction of a second, but if that event repeats periodically and you can measure 10-to-20 repeats over several seconds, the measurement becomes much better. 4. Data analysis Once you have recorded raw data, you need to convert these measurements into the physically important variables. Doing this should be straightforward using the techniques you've learned during the course. With your experimental apparatus, you can easily measure masses, distances, and time intervals, which you can then combine to calculate energies, momenta, and forces as needed.

7 Imagine that we wish to test conservation of energy by rolling a toy car down a hill from several different heights. We can measure the velocity of the car at the bottom of the hill by timing its passage between two marked points a measured distance apart. By measuring time and distance, we calculate velocity: v= x 2 x 1 t 2 t 1 then kinetic energy: KE= 1 2 mv 2 4. Propagation of uncertainties When converting these quantities, we need to consider how the uncertainties on the original measurements affect the calculations. Here are the most common rules for propagating uncertainties. Imagine that we have independently measured two quantities, x and y, with their respective uncertainties denoted Δx and Δy. Addition or subtraction: uncertainties add in quadrature. This means that, if z=x+y, the uncertainty on z is given as: δ z= (δ x) 2 + (δ y) 2 ñ Multiplication or division: the fractional uncertainties add in quadrature. If z=x/y, the uncertainty on z is given as: δ z z = ( δ x x ) 2 + ( δ y y ) 2 ñ Exponentiation: the fractional uncertainty is multiplied by the exponent. If z=x 3, then:

8 δ z z = 3 δ x x With these rules in mind, we can propagate uncertainties through nearly every calculation. The textbook gives a slightly different rule for propagating uncertainties through multiplication: that the fractional uncertainties add normally, not in quadrature. The two formulas are equivalent when the uncertainties are small, but, if the uncertainties are significant, you should use the one given here. Consider again the example given above of determining the kinetic energy of a car by timing its passage between two points. Assume that we first measure the mass of the car to be (81±1) g and that we measure the distance between our starting and ending points to be (50.0±0.5) cm. Before starting the experiment, we perform several test measurements with the stopwatch to estimate our timing uncertainty as ±0.08 s. Here are the raw data for several trials: Trial # Measured time (seconds) ± ± ± ± ±0.08 For the first trial, we measured (0.60±0.08) s to cover (50.0±0.5) cm, so the calculated velocity is 83.3 cm/s. To get the uncertainty, we use the rule for division that fractional uncertainties add in quadrature:

9 δv v = ( δ x 2 x ) + ( δt 2 t ) = ( ) + ( ) = Multiplying by 83.3 cm/s, we have v = (83±11) cm/s. Converting velocity to energy follows a similar strategy. First we square the velocity that, according to the rule of exponents, doubles the fractional uncertainty: δv v = δ(v2 ) v 2 = So for the first trial we have v 2 =(6900±1800) cm 2 /s 2. Multiplying this now by the mass of the car requires the same rule as for division: δ(mv 2 ) = mv ( δ m 2 2 m ) + ( δ( v2 ) ) 2= v ( ) = Now we have mv 2 =(560000±150000) g cm 2 /s 2. Multiplying by the constant one-half applies to both the value and the uncertainty, so that KE = (280000±75000) g cm 2 /s 2 = (0.0280±0.0075) J

10 Repeating this calculation for the other four trials gives: Trial # Measured time (s) v (cm/s) KE (g cm 2 /s 2 ) ± ±11 280,000±75, ± ±14 350,000±105, ± ±21 530,000±200, ± ±33 830,000±380, ± ±45 1,100,000±590,000 Finally, note how the uncertainty on the measured time is much larger than the uncertainty on the measured length or mass. If this is the case, you can safely neglect these terms in your calculation and just consider the uncertainty on the time. Or, if possible, redesign the experiment slightly to reduce the uncertainty on the measured time. Note also how the uncertainty on time measurement affects the uncertainty on the energy of trial #5 so much more than for trial #1. This can make analysis of the data much more complicated. If you discover such an effect when recording your data, think about how you could redesign the experiment to reduce the uncertainty. In our example, one good idea might be to measure over a longer distance. At first, we measured velocity over a distance of 50 cm, but, once we see such large uncertainties, we could repeat trials #4 and #5 measuring over 75 or 100 cm instead.

11 5. Graphical analysis The simplest way to analyze a trend in the data is to plot the calculated quantity against the independent variable. In the example above, assume that the car was released at heights of 3, 5, 7, 10, and 15 cm during the five trials. In that case, the plot looks like this, with the uncertainties shown as error bars: We can clearly see the trend of increasing kinetic energy with increasing height. If we apply conservation of energy and recall that gravitational potential energy is defined as PE = mgh, we can see that our data is perfectly consistent with expectations, which are shown as a blue line overlaid on the same data:

12 Imagine now that we wanted to use this experiment to actually measure something rather than just confirming what we already know about gravity. Assume that we know PE = mgh but do not know the value of the gravitational constant g. We can use our data to measure g instead. Our data should lie along a line with a slope of mg when plotted against h. We therefore attempt to draw a line through all of our data points. Since the data has uncertainties, more than one line will meet this criteria:

13 Note that any line between the two dashed lines here will pass through all of our data points within their error bars. Taking the slopes of these two outermost lines and dividing by the mass of the car, we find that the gravitational constant g is between 880 and 1150 cm/s 2, or g = (1020±150) cm/s 2 This calculation would be a primary result of our experiment. It is also worth noting that the derived value is consistent with the true value of 980 cm/s 2. If we had not obtained something close to the correct value, we would need to investigate further and explain what had gone wrong with the data. While this type of graphical analysis can be done by hand, many popular software packages and even some pocket calculators automate the process of finding the best line or curve to fit a set of data. This process is usually called linear regression.

14 PROJECT SUBMISSION 1. A title page is not required for project submissions. Because evaluators do not see student names when reviewing student work, it is important that students not include any personal identifiers in their project submissions. 2. Save your document as a PDF file. In Microsoft Word, you can use the Save As option to select PDF as your file format. 3. If your project requires a video, you should post the video to a free video hosting site like or one of the other free webhosting websites. The following website maintains a list of video hosting sites; In the written materials that you submit as part of the assignment, you should include the title of the video and a link for the faculty member to use to grade your submission. 4. Upload the PDF file in your course. 5. Your assignment will not be returned to you so keep a copy for your files.