The controlled experiment
You were introduced to the simple pendulum in L101. In that lab, a pendulum was used to help you learn how to reduce uncertainty in timing. In this lab, you'll actually learn about the pendulum itself and what what things influence its period. You'll also learn how to determine functional relationships between variables.
One common type of laboratory investigation is to determine the relationship between physical variables. For example, consider a simple pendulum which is composed of a compact weight (bob) that is hung from a string attached at its upper end to a fixed support. Suppose the goal of the investigation is to determine which variables influence the period of the pendulum, that is, the time for the bob to execute one complete cycle over and back. Some variables whose influence one could investigate include the length of the string, the mass of the bob, and the angle from the vertical at which the string is released. The latter three variables are termed independent variables, because one selects their values in carrying out the experiment. The period is termed the dependent variable, because its value may depend on the values of the independent variables.
In order to determine how each of the independent variables may influence the period, one needs an experimental design in which only one of the independent variables is changed at a time while the others are held constant. In this way, if an influence (or lack thereof) is found on the period, one can be fairly confident that the independent variable that was changed is the variable that influenced (or didn't influence) the period. Such an experiment is called a controlled experiment. By the way, the reason we said fairly confident is because there's always the possibility in dealing with the natural world that there are variables that the experimenter has overlooked and has not controlled. For the case of the pendulum, for example, suppose you carried out the experiment in an elevator that was moving up and down between floors. You would discover some strange results and find it difficult to reach conclusions about how the independent variables affected the period. That's because the elevator accelerates and decelerates and, as it turns out, such motion influences the period of a pendulum. Of course, it's not likely that you would do this experiment in an elevator and, if you did, you would probably guess that the motion of the elevator influenced the results. That's part of being a competent scientist. However, even if you're competent, you can still overlook variables. An example might be your location on the surface of the Earth. Location, in fact, influences the period, although the effect is so small that one typically doesn't notice it. But if you were making very precise measurements, you would have to take the effect into account.
The typical way to carry out an experiment to investigate the possible influence of an independent variable on the dependent variable is to measure the value of the dependent variable for several values of the independent variable. A graph is then made of the dependent variable vs. the independent variable in order to visually represent the relationship between the variables. Graphical analysis is then used to determine the functional form of the relationship. In L103, you learned how to determine the relationship between the distance traveled by sound and the elapsed time by plotting a graph, drawing a line of best fit, and determining the equation of the line. So you already have some experience with the process of graphical analysis. In this lab, you'll do the graphical analysis using software called Logger Pro. You'll also learn how to deal with non-linear relationships.
From your lab kit:
Set up and Protocol
Allot 90 - 120 minutes for the set up and data collection.
While the equipment set up for this lab is simple, attention to the following instructions will help you in obtaining good data.
Fixing the point of support
From the set up considerations described above, you can see that the simple pendulum really is simple. Yet, there's much physics and experimental technique to be learned from it. By the way, later in the course, you'll study another kind of pendulum which is a bit more complex in the physics but even easier to set up. That's a pendulum that has its mass distributed along its length instead of being concentrated at the end. The concentration of mass at the free end is what makes the simple pendulum simple.
You should already have your stopwatch or stopwatch app ready to go. The method of timing the period will be the one that you used in Part 2 of L101. In order to decrease uncertainty, you'll time 10 consecutive cycles. Remember to let the pendulum swing away from you and return once before starting the watch. For each set of independent variables (length, mass, angle), you'll take 5 trials of these measurements of 10 cycles. By doing so, you'll then be able to calculate the percentage mean deviation to get a handle on the timing uncertainty. This method was discussed in Part 1 of L101.
A protocol in science is a way of doing something. In this lab, for example, the protocol is that of a controlled experiment, discussed earlier. The initial set up with the longest length of string, a mass of 10 washers, and an angle of release of 10o from the vertical will be considered the control. When you test the influence of mass on the period, you'll decrease the number of washers by 2 for each set of 5 time trials, but you'll keep the angle of release and length constant. When you test the influence of angle on the period, you'll return the 10 washers to the canister. Then you'll do sets of 5 time trials for angles in 10o increments while keeping the length constant at its original value. When you test the influence of length on the period, you keep the mass at 10 washers and always release the pendulum from 10o.
Now it's time to start taking data.
Download and print this sheet: Data record. This is where you'll record your original data. You'll scan and submit your original data to WebAssign L105D in advance of the final report. Title your file L105D-lastnamefirstinitial.pdf. By submitting your data before the report, this gives the instructor the opportunity to review your data in the event that you need to retake some of it.
You won't be allowed to submit the final report until the instructor has reviewed and approved your L105D file.
Regarding data recording:
Use pen and remember the rule for recording data: Don't obliterate original data. If you think data is wrong, cross it out with a single line and write the new value beside the old.
When to ignore data: There are some situations where you would be justified in not recording data in your data table. Here are some examples:
These are situations where there's an obvious mistake or lapse in procedure and you immediately realize that the data is compromised. Don't, however, let yourself be tempted into playing games with the timing, for example, ignoring readings that don't correspond to some preconceived idea of what you think the time should be. We mention this, because we've seen students do it in an effort to reduce the variation in their results, thinking perhaps that this will somehow get them a better score. It won't; it's a poor scientific practice.
Part A. Period of the Pendulum vs. Mass of the Bob
The method is divided into three parts, one part for each of the independent variables. In this part, you'll test the influence of mass. Your controls will be the string at its longest length and the angle of release from the vertical at 10o. Take 5 time trials of 10 consecutive cycles for 10, 8, 6, 4, and 2 washers.
Part B: Period of the Pendulum vs. Angle of Release
Your controls will be the string at its longest length and a mass of 10 washers. You'll test the influence of the angle of release. As the pendulum swings through consecutive cycles, you'll notice that the maximum angle reached by the string decreases. If the period depends on the angle, then one may reasonably wonder what value of the angle the measured period corresponds to. In order to help address this situation, note the angle when you start timing and the angle when you stop timing. Since it's difficult to read angles as the pendulum is swinging, a precision of 1o in your readings is sufficient.
Change the initial angle in 10o increments starting with 10o and ending at 60o. If you have sufficient working space and want to use increments of 15o, thereby increasing the maximum angle to 85o, that's fine.
As the angle increases, the string experiences greater tension due to increased speed of the bob. Therefore, there will be more of a tendency for the string to slip through the tape. Check it frequently for slippage.
Part C: Period of the Pendululm vs. Length
Your controls will be an angle of 10o and a mass of 10 washers. In order to make it easier to measure the length of the pendulum, you may remove the protractor at this point and visually estimate the angle of 10o. This method is acceptable due to the fact that small angles have an almost imperceptible influence on the period. You may have noticed this while taking data in Part B. An error of a few degrees in your angle estimate isn't going to matter.
Start with the longest length and select the intermediate lengths so that the final value of length is about 10 cm.
For each set of 5 time trials, calculate the mean, deviations, mean deviation, and percentage mean deviation. This is a tedium that no one need endure in the age of spreadsheets. If you don't know how to use formulas in spreadsheets, it's a good skill to have. We're not requiring that at this point, though, as there are other priorities. You can learn to use spreadsheet formulas at a later date. This time, we're providing a spreadsheet calculator. Download it here. Here are some things to take note of.
You'll submit your completed spreadsheet with your final report. The filename is L05C-lastnamefirstinitial.xls.
Analysis and Interpretation
You'll use Logger Pro for the graphical analysis. For each part, you'll make a graph of Period vs. Independent Variable and then use the graph to reach conclusions about the relationship between the variables. The process will be similar to what you did in L103 for the distance traveled by sound versus the elapsed time. However, Logger Pro will automate the process of plotting points and fitting the data. Also, in one case, you'll need to re-express a variable in order to linearize a relationship.
Part A. Period of the Pendulum vs. Mass of the Bob
In this part, we'll present in detail the steps for using Logger Pro to plot a graph of Period of the Pendulum vs. Mass of the Bob. In Parts B and C, we'll expect to carry out similar processes with less guidance. In the instructions below, we'll use the convention that titles in Logger Pro are in boldface and labels that you enter are in italics.
Completing the Data Table
Note about error bars: Error bars provide a visual display on the graph of the uncertainty in the data. One can display both vertical and horizontal error bars for the uncertainties in the vertical and horizontal variables. For the current experiment, we chose only to display vertical error bars to represent the uncertainty in the period. These show as vertical lines, capped with short horizontal bars, that extend above and below the data point an amount equal to the absolute uncertainty. While you used %MeanDev to generate the bars, these values are automatically multiplied by the corresponding Periods to obtain the absolute uncertainties in seconds. If you can't see the error bars on the graph, they may be too small to appear. That simply means you have low uncertainty.
Completing the Graph
Interpretating the Graph
Now we'll show you why we had you select the manual scaling options that we did. At the top of the screen, single-click the symbol . Note that the graph auto scales. Now the error bars seem much bigger. The data also appear to have a trend. But examine the vertical scale. It shows only a small range of periods. Effectively, auto scaling has magnified any errors and trends out of proportion. This is a deceptive display. Of course, tricks like this are used all the time in advertising. You have to be wary of them. Always examine the scales. For the present case, in order to provide a more appropriate visual representation, we choose to scale from 0. In fact, we'll make this a general practice in this course unless we need to magnify a portion of a graph for closer examination. In order to return to the previous scaling, type CTRL-Z on a PC or CMD-Z on a Mac.
Now here's something else you may notice. Are the error bars about the same size as the point symbols? That's typical in this experiment. It's also a good thing if your uncertainties are small enough that the size of the point symbol represents the uncertainty.
In order to avoid confusion, label your work on the Interpration page with the same section headings and numbers as used in these instructions. Your first entry will be under the title Part A and will be numbered 3.
This experiment produces a null result in that no dependence of period on mass is found. This, in itself, is an extremely important result. Being null doesn't make the result unimportant. The lack of dependence on mass is actually a manifestation of one of the foundational ideas of physics, the equivalence of gravitational and inertial mass. We'll say more about this in later chapters after you've studied gravitational force and Newton's 2nd Law. However, you're probably already aware of another manifestation of this equivalence principle, namely, that objects of different mass falling side-by-side in the absence of forces other than gravity fall at the same rate (acceleration).
While this is the end of the story of Period vs. Mass for this experiment, it's not necessarily the end of the story for all time to come. The principle of the equivalence of inertial and gravitational mass is something determined experimentally. Experiments much more sensitive than the one you did have been done to test the principle. This is not to say that even more sensitive experiments in the future wouldn't show that there could be a non-equivalence. That's called doing science.
This completes the analysis and interpretation for Part A. There's no need to prepare a matching table for your fit, because a null relationship was determined.
Part B. Period of the Pendulum vs. Angle of Release
Completing the Data Table
You'll need to create a new data set. Here's how.
Completing the Graph
Interpreting the Graph
This completes the analysis and interpretation for Part B.
Part C. Period of the Pendulum vs. Length
Completing the Data Table
Completing the Graph
Re-expressing the Data to Linearize the Relationship
Examine your graph of Period vs. Length. At first sight it may appear linear; however, you can quickly dispense of that idea by doing a linear fit. If you consider also that you would expect the period to fall rapidly to 0 for lengths below 0.1 m, you can see that there's a pronounced curvature. Recall the third problem of WebAssign L105 which involved the same relationship. In that case, the Period was expected to depend on the square root of the Length. There's a reason based on the theory of the pendulum to expect such a dependence. However, it's too soon in your study of physics to discuss that theory. Instead, we'll use this situation to give you the opportunity to practice the process of re-expression and linearization. Expecting that the relationship between Period and (Length)1/2 is linear, a new graph must be plotted with (Length)1/2 as the independent variable. Follow the steps below to do that.
Interpreting the Graph
Application: Using the Relationship
The process you carried out in which you collected data and then used it to determine a relationship is called inductive reasoning or induction. The nature of this process is that it begins with specific information (in this case, data) and concludes with a general relationship that can be used to predict values that were not part of the original data set. For example, you could use the equation of fit to calculate what length a pendulum must have for the period to have a particular value. This process of predicting something specific from a general relationship is called deductive reasoning or deduction. Scientists use induction to discover relationships, and they use deduction to make predictions based on known relationships. These processes are part and parcel of the scientific method.
Do the following on the Interpretation page of your Logger Pro file.
You've been carrying out a quantitative error analysis while you've been working on the data and analysis, so there's no need to do anything more for that.
In a section labeled Conclusion on the Interpretation page, summarize what you did and what you found out in this lab. Review the goals first, as these provide direction. You should also indicate in the Conclusion whether you met the goals.
Submitting your report
Review the rubric to ensure that you've complete all parts.
Open assignment L105 in WebAssign. Note the first question, which is a self assessment of the completeness and formatting of your Logger Pro file. Check your work carefully to make sure it's complete and clearly presented. Then upload your Logger Pro file, named L105-lastnamefirstinitial.cmbl, as well as your calculation file, named L105C-lastnamefirstinitial.
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