Use the known relationship between the velocity of a pendulum bob at its lowest point and the vertical distance of fall of the bob to determine the value of g.
Part A. Read the Introduction below. Then complete L131A on WebAssign.
Part B. Equipment set up
An ideal, simple pendulum consists of a point mass suspended from a massless string. The bob swings in a circular arc under the influence of gravity. In the real world, the string has mass and the bob isn't a point. However, it's easy to approach the ideal by using a lightweight string, and a small, dense bob. In this context, small means that the size of the bob is small compared to the length of the string. Dense means that the bob has a large mass to volume ratio similar to that of, say, a metal or even wood. (A wad of paper wouldn't be a good choice.) With choices such as these and a firm support for the string, the pendulum becomes a good device for studying conservation of energy, because non-conservative forces such as air friction and friction in the support do not play a large part over the course of a cycle of the motion.
We'll begin by analyzing the motion of the pendulum in terms of forces and will then tell why conservation of energy is a better method to use for this situation.
The situation is shown in Figure 1 below. The string of length L is pulled to an angle of θ with the vertical and released from rest at point A. The goal is to determine the speed of the bob at point B. The acceleration of the bob isn't constant, so one can't, in principle, use dvat equations. Let's see why.
In Figure 2, we've drawn the forces acting on the bob when in position A. Tension acts toward the center of the circle and weight acts vertically downward. Unlike previous problems we've had, there are two acceleration components. There's a component toward the center of the circle, but there's also a component tangent to the circular path. This latter component causes the bob to speed up if it's headed down or causes the bob to slow down if it's headed up.
In order to write net force equations, we select axes parallel and perpendicular to the bob's velocity vector. In Figure 3, we've resolved the weight force into components along the axes. The net force equations are the following.
Applying Newton's 2nd law to the Fnet,x equation yields ax.
This shows that the acceleration of the bob tangent to the path is non-uniform. The acceleration depends on the angle θ, which is changing. This is the reason we can't apply dvat equations to the solution of the problem of finding the speed of the bob at the lowest point. The dvat equations assume uniform acceleration.
This leaves conservation of energy as the method of choice for the problem. For conservative forces, of which gravity is one, we need know only the initial (A) and final (B) states of the pendulum. We don't need to know the path followed by the bob, because the work done by gravity is independent of the path, and the work done by tension is 0. You'll solve the conservation of energy problem as part of the prelab.
Let's revisit the situation with the series of diagrams below. Figure 4 is the same as the previous Figure 1. Figure 5 adds horizontal line AC in order to define the distance CB that the bob falls vertically from points A to B.
From the application of conservation of energy to the system of the pendulum and the Earth, the speed of the bob at point B is the following. Note that (yA - yB) = CB.
vB = [2g(yA - yB)]0.5
Measuring the magnitude of the maximum velocity
Theoretically, you need to find the instantaneous velocity at the bottom of the swing. In practice, you'll have to settle for an approximation. Consider Figure 6 above. If you measured the time, Δt, for the bob to travel from P to Q (equally-spaced on either side of the vertical), then an approximation to the speed at the lowest point would be: vave = Δx/Δt, where Δx is the straight-line distance from P to Q. The closer together P and Q are, the better your approximation would be, in principle. We have to add the phrase, in principle, because in practice the uncertainty increases as the time interval decreases. Suppose, for example, that Δt is 0.4 s with an uncertainty of 0.1 s in starting and stopping a stopwatch. That's a 25% uncertainty. If a smaller Δx is used so that Δt is, say, 0.2 s, then the percentage uncertainty increases to 50%. So there are two things working against each other here. Theoretically, vave approaches the instantaneous velocity, vB, as Δx decreases; however, experimentally, the measurement becomes highly uncertain. Here is the approach we use to deal with it.
We use a photogate to minimize starting and stopping error. The photogate is placed at the bottom of the pendulum's swing. The photogate starts a timer when the bob enters it and stops the timer when the bob leaves. With this method, timing uncertainties can be reduced to thousandths of a second. Moreover, the distance Δx over which the timing occurs is limited to the width of the bob. Accurate results can be expected with this method. Here are some tips for getting good results: a) Make sure the photogate is positioned at the lowest point of the swing, b) Make sure you know which points of the bob are triggering the photogate on and off. You have to know this, because that gives you your distance, Δx. If the bob were a cube, then Δx would depend on how the cube is turned as it passes through the gate. Therefore, a cube is not a good choice for the bob. A sphere is a better choice, since it doesn't matter if the sphere rotates as it passes through the gate. However, it's important to position the sphere vertically so that the photogate beam has the same elevation as the center of the sphere. That way, you can say that the distance Δx is the diameter of the sphere.
Measuring the vertical distance of fall
One method of measuring the vertical distance of fall is simply to prop a meter stick up vertically from the floor. Measure the distance from the floor to one point of the bob, say the bottom, and measure the distance from the floor to the same point of the bob when it's in position A. The absolute value of the difference of the two positions gives yA - yB.
Another method is to measure the angle θ shown in Figure 2. If L is the distance from the point of support D to the center of the bob, then the line segment CB, which equals (yA - yB), is also equal to L(1 - cosθ).
Measuring vertical distances directly using a meter stick is the more accurate of the two methods. It's much more difficult to get a good angle measurement than to measure two vertical distances from the floor. Here's why. With the protractor, you have to make alignments with both the vertical and with the string. It's very easy to make an error of a degree or two with either of these measurements. That may not sound like much but consider how the vertical position of the bob at point A would change if the angle of the string were changed by a few degrees. With a string length of half a meter, a deviation of a few degrees could change the vertical position by 1 to 2 cm. Again, that may not seem like much. If, however, the distance of fall is, say, 10 cm, that's a 10 to 20% error.
From your lab kit
The teacher will provide a vernier calipers (for measuring the diameter of the ball).
Equipment Setup and Design
Download and watch this QuickTime movie to see how to set up the experiment. Then set up your own equipment.
Here is a list of important points about the experimental set up that will help you in obtaining good data. These points are also mentioned in the video.
The setup is part of the design, but there's more to the design than just the setup. The design has to do with selecting the independent and dependent variables and deciding how to determine the relationship between them. From past experiments you had experience in determining a relationship following these steps:
In this experiment, the independent variable is the height (yA - yB) through which the bob falls, and the dependent variable is the speed, vB, of the bob at the bottom of the swing. Since a linear relationship is not expected, the process of fitting the data will require a re-expression of one of the variables. This will be discussed further down.
Selecting values of the independent variable
In order to get a smooth curve that will provide a good fit to the data, you need to have a sufficient number of values of the independent variable. At the lower limit, you won't get a curve from less than three values, but three values aren't sufficient for a good fit. Ten values would be fine, but you have to consider how much time you have to collect data. Five or six values is a typical compromise. Having made that decision, then you have to consider how to spread values of the independent variable over the domain. The domain, in this case, would be distances of fall from a little more than 0 to the length of the string (The latter would be a 90° angle of release.). For a quadratic relationship, you'll need to space values closer together near the 0 end of the domain and then spread them further and further apart as you approach the maximum. The reason for this is that the function changes fastest near the origin. An example of a spread of values of the distance of fall would be 2, 5, 10, 20, 35, 65 cm. Of course, you shouldn't try to replicate these values exactly. Note that the highest value in the series is about the distance that would be available under a typical desk.
Testing for reproducibility
Take five trials for each value of the independent variable in order to evaluate the reproducibility of the measurement.
Having set up and designed the experiment, you're ready to take data. Download and print this pdf of a table to record your original data. Record in pen as usual for original data. You'll submit your original data in advance of the report due date.
At this point, submit the Logger Pro file and a scan of your original data page to the WebAssign assignment, L131D.
Now that you've collected your data, you're ready to do the calculations. In order to streamline this process, download this Excel spreadsheet which does the calculations automatically as you enter the data from your original data table. Read the instructions on the spreadsheet carefully. These are in the orange cells. It's your responsibility to check the calculated results to make sure that you're using the spreadsheet correctly and that it's calculating correctly. We recommend doing all the calculations for one of the release heights with a hand calculator as a check.
Rename the file L131C-lastnamefirstinitial.xls for submission with the rest of your report, which will be a Logger Pro analysis file.
You'll do the analysis in Logger Pro. Since a non-linear relationship is expected, you'll need to re-express the independent variable in order to linearize the relationship. Now here's what to do.
Do the following in a text box in a Logger Pro.
In the text box, write a conclusion in which you describe the method of the experiment and the analysis and present your results.
Submit your Excel table and your Logger Pro analysis file to WebAssign assignment L131.
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