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Table of Contents
- Introduction…………………………………………………………………………… Page
- Safety in the Laboratory……………………………………………………………….
- Grading………………………………………………………………………………...
- Lab Policies……………………………………………………………………………
- Experiment 1: Collisions and Speed…………………………………………………..
- Experiment 2: Simulation of a Decay…………………………………………………
- Experiment 3: Collisions in Two Dimensions………………………………………...
- Experiment 4: Falling Coffee Filters………………………………………………….
- Experiment 5: Pendulum and Forces………………………………………………….
- Experiment 6: Motion on a Spring…………………………………………………….
- Experiment 7: Springs and Energy…………………………………………………….
- Experiment 8: Falling Object…………………………………………………………..
- Experiment 9: Newton’s Second Law………………………………………………….
- Experiment 10: Force…………………………………………………………………..
- Experiment 11: Modeling………………………………………………………………
- Experiment 12: The Range of a Projectile……………………………………………..
Introduction
The main objective of the laboratory component of the course is to begin to develop your ability to conduct a scientific experiment. This involves many aspects for which you must acquire various skills. In the process, you should become more familiar with the concepts and principles of mechanics. A scientific experiment, in the broadest sense, requires a purpose, acquiring data, analyzing data, and conclusions. The purpose is the driving force behind most of what you do and why. It may involve a question that you want answered or a phenomenon that you want to further understand or investigate.
Acquiring data mostly requires instruments; sometimes, information from your senses is important as well. Therefore, part of the activities each week will be on learning to properly use instruments for acquiring data. You will acquire some data by the use of a computer. See the document Lab Pro and Logger Pro for more information on acquiring data with a computer.
Analyzing data involves many methods that a scientist continually learns and refines. In this course, you will be introduced to a few methods for data analysis. You will use Excel, at times. For more information on the use of Excel in data analysis, see the document Excel Instructions for Physics Lab. Data analysis includes uncertainty analysis, which is an analysis to determine how confident you are in your results. You will also be introduced to some of the methods of uncertainty analysis.
After you have acquired data and analyzed it, you must draw conclusions based on the data and analysis. These conclusions must logically flow from the data and analysis, and some of your conclusions should relate back to the original purpose of the experiment.
Class Sessions The class sessions may be classified as guided inquiry. You will be given the question or phenomenon of interest and a context of why this is important. You will also be given some instructions that will lead you along a line of inquiry. However, you must discuss certain issues with your group members and fill in some of the details. The instructions will not have all the details of what you are to do and why.
The manual is written as a series of activities. Each activity, written in bold, follows a general discussion about the activity. You must read the general discussion carefully, and you must fully understand it before proceeding to the activity. The activities must be done in order; do not skip an activity because you do not understand it.
Feel free to ask your instructor questions. The instructor may answer your question directly or may return a question to you, in order to get you to think about certain issues and to increase your ability to think scientifically.
Grading
The grade is based on three components: the assignments that you write each week, a presentation or exam during the last class of the semester, and participation. Each weekly assignment is worth 100 points, the presentation is worth 200 points, and your participation grade is 10 points per week. Each of these is described below. Your final grade is computed by dividing total earned points by total possible points. The lab grade constitutes part of the overall grade for the course, between 20 and 30% - consult the syllabus for the exact percentage.
Assignments At the end of each experiment in the manual, there are assignments with instructions. These are due the following week. Make sure that you fully understand them before you leave the lab.
You are expected to write at the level of your peers, i.e., someone at the same level as yourself, but who is not aware of what you did in the laboratory. Your writing must also follow scientific logic – all of your statements must logically follow from the data and data analysis. You must give explanations for your statements so that the reader can follow your logic as well. Do not assume that the reader can automatically follow your logic without an explanation.
Presentation or Exam During the semester, you may be asked to design your own experiment to investigate something of interest to you. The experiment will be conducted during the class before the last class of the semester. During the last class you may be asked to present your findings before the class.
In lieu of the presentation, you may be given an exam during the last class of the semester. The exam will consist of two parts, a practical and a written part. During the practical, you will be asked to perform an activity very similar, if not identical, to one that you did during the semester. The written part will consist of questions and problems related to all of the activities during the semester.
Your instructor will inform you, near the beginning of the semester, whether it will be a presentation or exam.
Participation You are required to attend labs on time and to be actively engaged in all activities. If you are tardy, without prior approval from the instructor, beyond five minutes, your participation grade will be reduced systematically up to 30 minutes late, when it will be zero.
Your participation grade will also be reduced if, after the instructor warns you, you continue to not be actively engaged.
Policies
We will adhere to the following policies :
- You are required to bring something, on which you may store computer files. We recommend a USB memory device. Diskettes are not recommended. Any files saved on the computer will be deleted by the time you to come to lab the following week.
- If needed, you must bring your own one- foot ruler and protractor.
- You will be deducted participation points if you have not done pre- lab work before attending lab. You may not have pre- lab activities every week.
- Labs may be made- up for approved absences. Consult your instructor to agree on a make-up schedule. However, if you miss the make- up date without approval, then you cannot make up the lab, and you will receive a zero for that week.
- You will be deducted points for late work – 10% up to one school-day late, 20% between one and two school-days late, etc.
- You cannot pass the course if you get less than 60% for the lab. In this case, your grade is F, even if you pass all exams.
- We adhere to the College’s policy on academic dishonesty. Academic dishonesty includes falsifying data, fudging any analysis, the use of someone’s data or work without approval, plagiarism, etc. This behavior is governed by rules on academic dishonesty set by Spelman College. Consult the Student’s Handbook. You may obtain a copy from the Academic Dean’s office.
To allow more precise measurements of time, your instructor might now give you a device called a photogate that connects to a computer interface box called the Lab Pro. The computer uses software called Logger Pro to communicate with the Lab Pro. The Lab Pro contains the electronics necessary to connect to a variety of measuring devices.
The instantaneous speed is estimated by measuring the time it takes for the cart to move a small distance close to the point where the speed is required. It is convenient to tape part of an index card on the cart, so that the index card passes through the photogate to start and stop the timing.
Read the appropriate sections in the document “Lab Pro and Logger Pro”, and familiarize yourself with using Logger Pro, Lab Pro, and a photogate to measure the time required for the index card to pass through the photogate. Logger Pro has several ways of using photogate measurements; take the time to ensure that your way gives the appropriate numbers.
When a scientist reports a measurement, she must also report how reliable that measurement is. For instance, the law firm needs to know how reliable the speeds are that you determine, since a more reliable speed allows the lawyers to make a stronger case for a particular monetary award. Scientists generally report measurements as a range. For instance, you might tell the lawyers that the speed was between 20 mph and 26 mph, which is a fairly broad range reflecting a less reliable value. On the other hand, if you tell the lawyers that the speed was between 22 mph and 24 mph, then the value is more reliable. By convention, the report is usually made as a “best” estimate with a ± value that indicates the range of likely values. For instance, in the two cases above the speeds would be reported as 23 ± 3 mph and 23 ± 1 mph. The number after the ± is generally called the uncertainty.
When you make measurements, sometimes you can easily repeat the measurement under the same conditions. Then, you can compute an average and standard deviation as your instructor discussed with you earlier (when you were first estimating the speed of a cart). This procedure would work for measuring a fixed distance, for instance. Sometimes you cannot easily repeat the measurement. For instance, when you measure the time for the card to pass through the photogate, you cannot easily push the cart again in exactly the same way and repeat the measurement. In such a case, you must estimate how accurate the measurement is, rather than calculating it from repeated data. One factor to consider is, how small of a change could you actually detect with the measuring device?
Discuss in your group how you might estimate the uncertainty of your time measurements. Discuss your conclusions with your instructor.
Now measure and record the appropriate quantities that will allow you to determine the speed of the cart at a particular point in its
motion. For each measured quantity, estimate and record its uncertainty.
There is now an issue in uncertainty analysis. You should have estimated uncertainties on the time and distance. However, how is the uncertainty on the speed determined, where the speed is estimated by dividing the distance by the time? The speed is a calculated quantity; it is not measured directly. To determine the uncertainty on the speed, one has to ‘propagate the uncertainty on the measured quantities through the calculation’. The simplest way to do this is to compute the maximum and minimum value of the speed based on the uncertainties on the time and distance. For example, suppose the time were 1.43 ± 0.02 seconds (possible values ranging from 1.41 s to 1. s), and the distance were 4.27 ± 0.05 cm (possible values ranging from 4.22 cm to 4.32 cm). The “best” value of the estimated speed would be 4.27cm/1.43 s = 2.986 cm/s. Based on the uncertainties, the maximum that the calculated speed could be is found from the maximum distance and minimum time to get 4.32cm/1.41s = 3.064 cm/s; while the minimum calculated speed is based on the minimum distance and maximum time for 4.22cm/1.45s = 2.901 cm/s. The result is that the speed has a value between 2.901 and 3.064 cm/s, where the best value is assumed to be 2.986 cm/s. The situation is pictured below.
0.085 0.
2.901 2.986 3.
There are differences of 0.085 cm/s and 0.088 cm/s between the best values of the speed and the maximum and minimum. The propagation of the uncertainty results in a speed of 2.986 ± 0.087 cm/s, where 0.087 is the average of these differences. The uncertainty indicates that the speed is only known to three significant figures, so it should be reported as 2.99 ± 0.09 cm/s.
Now this method of propagating the uncertainty can be applied to the calculation of speed using your data.
Use the propagation of uncertainty method above to determine the uncertainty on the speed of the cart that you measured. Show your result to the instructor.
Now that you understand how to measure speed, it is time to fulfill your consulting contract.
In your group, discuss a plan of action for your subsequent measurements, keeping in mind the information requested by the law firm. Discuss your plan with the instructor. Make the measurements. Plan on doing your calculations and other analysis outside of lab.
Experiment 2: Simulation of a Decay
While the lawyers are reviewing your paper, the firm’s intellectual property division has contacted you regarding a patent infringement case. The client, who is the defendant in the case, wishes to challenge the validity of a patent that involves decaying atoms. The client claims that the patent should not have been issued, since the ideas were well- known to anyone with “ordinary skill in the art”. The firm has turned to your group to help demonstrate the state of knowledge about decays, with focus on general principles that have been tested by experiments.
For our purposes, we will define a decay as a process where an object ‘changes’ into two or more parts. An explosion can be thought of as a decay. An energetic atom changing into a less energetic atom and a photon is another example, although less familiar.
Before lab, write down a few more examples of decays.
One objective of physicists is to develop general principles about processes of nature such as decays. These principles are formulated with concepts such as momentum. A possible question of investigation is ‘what princip le can be stated about momentum in decay processes?’.
Before lab, think of a possible principle involving momentum that is valid for decays, based on your reading of the first few chapters of Unit C.
To investigate decays, we will simulate a decay using carts on a smooth track. The carts are placed side by side with a compressed spring between them. When a release mechanism is touched, a spring is relaxed so that the two carts fly apart in opposite directions on the track. Your instructor will demonstrate this for you. The idea here is that the ‘two-cart object’ decays into two carts.
Become familiar with the simulation of a decay using two carts. Draw a schematic of the experimental set up in your lab notes; indicate direction of motion on the schematic.
To study decays as they relate to momentum, you must be able to measure the momentum of a cart. Recall that momentum is a vector quantity, so that it is measured by measuring its magnitude and direction. Since the carts are confined to move on the track, direction is easy to measure. Thinking of the vector component along the track, you can assign one direction as positive and the other as negative. The magnitude of the momentum of the cart is determined by measuring its mass and its instantaneous speed.
Up to this point, you have learned the techniques of timing with a photogate, measuring the speed, and estimating and propagating the uncertainty. You should be ready to perform an experiment on decays.
Simulate a decay with two carts, and make appropriate measurements of momentum so that you will be able to formulate or test a general principle concerning momentum about decays.
Assignment
Write a paper about decays. The first paragraph should be about a possible general principle that involves momentum. The remaining part of the paper should be about the experiment, the results, and conclusions. Remember that you are writing to lawyers who want to understand whether principles about decays are “obvious” to persons with “ordinary skill in the art”.
Plot each pair of x and y columns simultaneously as scatter plots (no lines). Give each series a useful name (such as “Purple ball before collision”). Make sure that you have both horizontal and vertical grid lines. See Excel Instructions for Physics Lab , if you need help on how to plot several set of point on the same coordinate system.
Print the graph. In order that the graph is a representation of the actual collision drawn to scale, make sure that the vertical and horizontal scales are consistent. For instance, if 1 cm on the paper vertically is equivalent to 10 cm in the actual collision, then 1 cm horizontally must also be equivalent to 10 cm. One way to do this is to make things look right on the screen by adjusting the axes and the size of the graph. Then select the graph and on the menu bar choose File→Page Setup→Chart****. Select the option Scale to fit page****.
The graph, with the plotted points, constitutes a motion diagram. You may use the motion diagram to get an arrow representation of certain quantities.
Use the motion diagram to get pictorial representations of the momenta. Note that you will need measurements and calculations not explicitly asked for—think carefully about how momentum relates to velocity, which in turn relates to displacement. Pictorially check whether the momentum appears to be conserved when added as a vector.
Assignment
The lawyers, if pleased with yo ur work, may want to call you as an expert witness in the field of momentum and collisions. To this end, write a line of questioning (including your answers) that a lawyer would use to convince a jury that momentum is a conserved vector quantity. Of course, you must back up your answers with experimental results. Also, you should anticipate questions that might be asked by an opposing lawyer during cross-examination. Finally, discuss how your work might be useful in an actual case.
Experiment 4: Falling Coffee Filters
Usually in science, we begin an investigation by formulating a question that we would like to answer. The question should be as specific as possible, so that it will lead to a well- formulated experiment. For example, the question “how fast does an object fall?” is vague. It is not clear where in the motion it is referring to, and how and where the object was put in motion. However, the question “how does the time taken for a dropped object to hit the ground from a certain height depend on its weight?” is very specific and it leads to a clearly defined experiment. Part of the objectives of the laboratory component of the course is to develop skills for doing science.
Before lab, qualitatively observe some falling objects, and formulate two questions.
Let us begin the study of falling objects by investigating the latter question above. To do this, we need to be able to measure weight and time. Let us start with time.
Use a stopwatch to measure the time taken for a coffee filter to fall 2 meters. Each person in the group must do this independently.
Before proceeding with the relationship between the time to fall 2 meters and the weight, there is an issue about measurement that needs to be understood. If you and your partner did not get the same result, then there is a problem since results in science must be reproducible. The problem, however, may not be a mistake on your part. The difference in results could be due to factors inherent in the measurement technique.
List all the factors that could cause errors in the timing measurement that you just performed.
These factors are called sources of uncertainty. These sources generally fall into two categories. Systematic sources cause the same error every time. For instance, if a clock reads 2:05 when the time is actually 2:00, and the clock is running at the correct speed, then every time read with that clock is 5 minutes late. Other sources of uncertainty result from random errors that are different every time. In these cases, repeated measurements will fall within a certain range of numbers instead of always being the same “correct” number. The random uncertainty can be estimated, and it is recorded as a ± on the result of the measurement. For example, a measurement of 1.46 ± 0.03 seconds indicates an uncertainty of 0.03 second, and most repeated measurements (about two-thirds) should fall within the range 1.43 – 1.49 seconds.
Categorize each of the factors that you listed as systematic or random. Give a reason for each. Fo r the systematic errors, discuss what you might do to eliminate or account for the error in your experiment and analysis. Discuss your conclusions with your instructor.
In this course, we will use the standard deviation to determine the uncertainty when the measurement is repeated. However, you must compare the standard deviation with the estimated uncertainty and use the larger for the uncertainty. When the measurement is not repeated, then use the estimated uncertainty. The uncertainty also will enable you determine the significant figures. For example, an average value of several measurements of 3.42 ± 0.12 shows that the hundredths place is not significant since its value can be any number as indicated by the uncertainty. On the other hand, the tenths place is significant because you know it is 3, 4, or 5. In other words, the range of values including the uncertainty is 3.30 to 3.54; you can place limits on the value of the tenths digit but not on the value of the hundredths digit. The uncertainty first appears in the tenths place. The unit place is known for certain – it is 3. Thus, the average value has two significant figures, and it should be reported as 3.4 ± 0.1.
Now that you understand about measuring the time and determining the uncertainty, you may proceed to study the relationship between the time and the weight. Assume that the weight of each coffee filter is identical.
Measure the time taken for coffee filters to fall 2 meters as a function of the numbe r of coffee filters. Do not take data when the number of coffee filters is four. It is probably easier to start with at least twelve coffee filters and remove one at a time as you make the measurements. Record your data neatly in tables. Finally, you s hould have data that shows the time with uncertainty versus the number of coffee filters. Show your data to your instructor.
We would like to determine the relationship between the time and the weight. It is easier to observe the trend in the data if it is plotted. It is also more convenient for the reader to present the data graphically rather than numbers in a table.
Use Excel to plot the time to fall 2 meters versus the number of coffee filters. (If one plots A vs. B, then A is on the y -axis.) Include error bars, a title for each axis, and a title for the graph. See the document Excel Instructions for Physics Lab. Observe the data points and choose a type of function (for instance, a sine function or a straight line function) that may describe the data. Show the graph to the instructor, and discuss your type function with the instructor.
Once you have chosen a type of function that may describe the data, the next step is to determine the equation of a function of that type that goes through the data points (or comes within the uncertainty). For example, suppose you chose a quadratic type function
(a parabola). The general form of the function is a x^2 + bx + c. Each set of parameters
a , b , and c determines a specific parabola. The question is, what are the values of these parameters so that the graph of the parabola comes within the uncertainty of each data point? One way to answer this question is to graph the parabola on the same axes as the
data points, and change the values of the parameters until you see the parabola come within the uncertainty of the data points.
Write a general form of the function type you chose with certain parameters. Arbitrarily choose values for the parameters and graph the function on the same axes as the data points. Change the parameters until you find the best values, so that the function best describes the data. Does your chosen type of function actually seem to describe the data? If not, you may need to choose a different type of function.
After you have decided that the function is reasonable and found the best parameters, see how much you can change each parameter so that the function is still a good description of the data; this procedure allows you to estimate the uncertainty of the parameters.
Rather than visually observing how close the function comes to the data points, there is a more mathematical way of doing it. This involves a method called the Least Squares method. The idea is to find the sum of the square of the vertical displacement between each data point and the function at the same x value. The set of parameters that yield the least sum are considered to be the best values. Again, as before, the square is used to eliminate adding numbers where some are positive and some are negative. If the i th data point is ( xi , yi ), the value of the function is f ( xi ), and the uncertainty in yi is σ i , then the
expression to be minimized is
2
1
=
N −
i i
f xi yi σ
, where N is the number of data points.
The numerator is the distance between the data point and the function. (The division by the uncertainty is from statistics, and the detailed reason is beyond our scope. Roughly speaking, this division means that data with small uncertainty contribute more to the sum, hence the more precise measurements are weighted more strongly, as they should be since they are more reliable.) This expression is called χ^2 , pronounced chi-squared.
Use the method above to determine the best values for the parameters and compare to the values you obtained fro m the previous method.
At this point, you have collected data on the falling of objects and obtained a mathematical expression that describes the data. One further question concerns the validity of this expression. For example, using the method above, you could fit any function to the data points and obtain the best values for the parameters. We have assumed that the type of function is good for the data points that we have. Another concern is the certainty of the values for the parameters. The uncertainty on the values for the parameters is a topic that we will discuss further in a later experiment, as well as how to determine the ‘goodness’ of the fit.
Once we have this expression, what is its importance? It can be used as a predictive tool.
Experiment 5: Pendulum and Forces
Your group has received another consulting contract with the law firm. The case involves a carnival ride in which the riders are essentially swung on a large pendulum. The firm represents the plaintiffs, who claim that they were subjected to excessive force that varied too much during the ride. The ride has since been dismantled, but the lawyers are hoping to find video tape shot by visitors to the carnival. They want you to explore the use of video analysis to analyze the force acting during the motion of a pendulum, to answer the question “how does the force vary with time and position for a pendulum?”
Using a fairly long string, set up a simple pendulum. Set the pendulum into motion and use the webcam to record a few swings. Use Logger Pro to analyze the video and plot the x and y coordinates versus time.
To determine the forces acting as a function of time, you need the acceleration. Logger Pro does allow the computation of derivatives such as velocity and acceleration from the data, but small random errors in the position data lead to large fluctuations in the computed derivatives. You will, therefore, fit your position data to some function of time, then analytically find the derivatives needed to determine the acceleration.
In your group, decide on what type(s) of function to use to describe the data of x versus time and of y versus time. Do the fits and determine the parameters of the best fits. Discuss the results with your instructor.
Use calculus to find the analytical derivatives of the functions needed to determine the acceleration. Create graphs of the force components, the magnitude of the force, and the direction of the force, all versus time.
You have made graphs versus time, but sometimes it helps to see what the force is as a function of position. In other words, for different positions of the ball you can indicate the direction and magnitude of the force.
Graph by hand y versus x for one swing (this is then a motion diagram). Make sure that the vertical and horizontal scales are the same. At each point, draw to scale the force vector corresponding to that point, with the appropriate magnitude and direction. (The previous sentence involves many steps.)
Assignment Write a paper that answers the question “how do the forces vary with time and position for a pendulum?” Be sure that the paper includes sufficient evidence to support your
conclusions. Also comment whether this pendulum exerts excessive force on the object. You will need to define what is meant by excessive force.
