Free Fallin. Pre- Lab: The Drag Force and Terminal Velocity. A Bit of History

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1 Free Fallin Pre- Lab: The Drag Force and Terminal Velocity A Bit of History There is perhaps no better sign of a rock star s legendary status than the ability to pull off the one name thing. Think Madonna, Sting, Usher, or Prince (or or whatever). In physics there is only one rock star that can rival the genius of Isaac Newton and Albert Einstein while sporting a single name. That man is Galileo! Galileo s stardom comes from his many talents and discoveries. In Part I of today s lab, we are interested in one such discovery that two objects fall to earth at the same rate regardless of their composition or in Galileo s words: The acceleration of straight motion in heavy bodies proceeds according to the old numbers beginning from one for a ball of one, ten, a hundred, or a thousand pounds will all cover the same hundred yards in the same time. Newton took this idea and ran with it, using a pendulum to perform detailed experiments which showed that an object s response to the gravitational force is independent of the object s mass and composition. This idea later came to be known as the Principle of Equivalence. The Principle of Equivalence is so fundamental that it led Albert Einstein to develop General Relativity, the theory that predicts such crazy phenomena as Black Holes and the Big Bang. Free Fall vs. Real Life One of the first topics in any introductory physics course is free fall. We act like any object moving through the air (whether up, down, left, or right) accelerates downward at 9.8 m/s 2 as a result of gravity. However, we have all experienced wind resistance or drag, so we should know that this is only an approximation. That said, it is a very good approximation as long as the objects we are considering are sufficiently dense and are moving sufficiently slowly. The terms in quotes are admittedly vague. The meaning of sufficient depends on how good you want the approximation to be. In the absence of an atmosphere, we don t have to worry about drag, and the approximation of constant acceleration due to gravity is an incredibly good one. In the first part of this lab, you will see this as you analyze a video of objects falling near the surface of the moon. In the second part of the lab, you will come back to earth and explore how the drag force affects the acceleration of a falling object. The exercises in this Pre- Lab will prepare you to design and analyze your experiments. Introducing Some Equations Before we write an equation for the drag force, let s think a little bit about what its form will be. First, the drag force always opposes motion. Knowing that will allow us to leave the direction out of our equation and write a scalar equation for the drag force. Second, we should expect the drag force to increase with the speed of the object. You have experienced this if you ve ever stuck your hand out of the window of a moving car. In a parking lot, you don t feel much of an effect. On the interstate, though, you will feel quite a force from the air. Third, as the cross- sectional area of the object increases, we 1

2 should expect the drag to increase. If you drop a leaf of paper broad side down, it will fall extremely slowly due to the strong drag force. If you crumple up the same piece of paper into a tiny ball, it will accelerate much more rapidly when dropped. Fourth (and finally) the drag force should depend on the density of the fluid through which the object is moving. Specifically, drag should increase with the density of the fluid. For an example of this, consider your experience moving through air compared to your experience walking through the shallow end of a swimming pool. It s much harder to move through the pool. All of that (and a little more) is contained in our equation for the magnitude of the drag force: F!"#$ = 1 2 CρAv! (Eq. 1) where ρ is the density of the fluid (about 1.23 kg/m 3 for air), A is the cross- sectional area of the object, v is the speed of the object, and C is the so- called drag coefficient, a dimensionless constant that depends on the shape of the object. We will begin our work with this equation by discussing A and C in some detail. The Effect of Size and Shape on Drag Both A and C describe the geometry of the object, but they must not be confused. The drag coefficient (C) is a dimensionless number that depends on the shape of the object. All smooth spheres have the same drag coefficient (0.47) regardless of how big they are. On the other hand, two spheres of different sizes will have different cross- sectional areas (A). The cross- sectional area is not to be confused with the surface area of an object. The surface area of a sphere is given by 4πr!. This is not the cross- sectional area. The cross- sectional are is like the area of the object s shadow. For a sphere, the shadow will have the shape of a circle and its area will be given by A = πr!. PL1. Consider a smooth sphere with a radius of 10 cm. What is its drag coefficient? PL2. Consider a smooth sphere with a radius of 10 cm. What is its cross- sectional area? More Complicated Shapes When we move on to more complicated shapes, determining A and C gets more complicated as well. Let s consider a falling cone. When the cone is falling with the point down, it has a drag coefficient of about 0.5 and a cross- sectional area (A) equal to the area of its circular base. However, if the same cone is falling with the base down (and thus with point up) it will have a larger drag coefficient (close to 1) even though it has the same cross- sectional area. Thus the cone falling with the base down will experience a greater drag force than the cone falling with point down. This should make some sense as you would likely identify the cone falling with the point down as being more aerodynamic. Read This: Here are three more exercises that will prepare you differentiate between the surface area of an object and its cross- sectional area. Consider a cone with a height of 10 cm and a base of radius 4 cm. 2

3 PL3. What is the surface area of this cone? PL4. If the cone is falling with the point down, what is the cross- sectional area of this cone? PL5. If the cone is falling such that the tip is pointing horizontally (i.e. the plane of the base is vertical) what is the cross- sectional area of the cone? Terminal Velocity A very interesting result of drag is that, if given enough time, an object falling through the atmosphere will reach a constant speed that we call the terminal velocity 1. Terminal velocity is reached when the weight of the object is equal to the drag force on the object. Equation 1 can be used to show that the terminal velocity of an object is approximately given by (Eq. 2) v! = where g is the acceleration due to gravity and m is the mass of the object. Note that the same object can have multiple values for its terminal velocity since A and C can depend on the orientation of the object as it falls. As an example, a sky diver with her body oriented vertically has a much higher terminal velocity (about 200 mph) than when she is in the spread eagle position with her body parallel to the ground (about 100 mph). 2mg CρA In addition, the time, t!, required to reach terminal velocity is approximately given by t! = 2v! g (Eq. 3) Rain Drops A nice example of the difference that drag can make concerns rain drops. Consider a rain drop that has a radius of 2 mm and falls from a height of 1000 m. Further, let us approximate the rain drop as a smooth sphere. The density of water is 1 g/cm 3. PL6. In the absence of drag, how fast would this rain drop be travelling when it strikes the ground? PL7. In real life, the rain drop experiences a considerable drag force. When the drag force is considered, approximately how fast will the rain drop be travelling when it strikes the ground? End of Pre- Lab 1 Some texts call this the terminal speed, which makes more sense because it is a scalar rather than a vector. However, we have chosen to use the more common term in this manual. 3

4 Part I: The Principle of Equivalence You will begin the Free Fall lab by analyzing video of an experiment that tested the Principle of Equivalence. Equipment Computer Hammer and Feather video 1. What Is Dropped Must Go Down It s probably no surprise that if you hold a ball in your hand and then let it go, the ball will fall. It also won t surprise you that if you stand on the moon and drop a ball it will fall to the moon s surface. However, the rate at which the ball accelerates near the surface of the moon is different from the rate at which the ball accelerates near the surface of the earth. In either place the rate at which it falls is dictated by the strength of the local gravitational field. Since the moon has no atmosphere, the drag force is zero. (On earth, the drag force also plays a role. You ll study that in Part II.) The magnitude of the local gravitational field is commonly known as the acceleration due to gravity. Read This: In this part we will use video analysis in Logger Pro to determine the acceleration due to gravity on the surface of the moon. To do, this we will use a video of an Apollo 15 astronaut (Figure 1) who simultaneously released a hammer and a feather on the surface of the moon. Do This: Download HammerAndFeather.mov from the In- Lab Links page of lab website. Do This: Open Logger Pro. To import the video, select Movie from the Insert menu. A window will pop up asking you to select a video. Type Hammer into the search field of this window. Then choose HammerAndFeather.mov.mp4, which should be returned by the search. Figure 1: The Apollo 15 crew Read This: The movie should appear in what is a more- or- less standard video player. The five buttons in the lower left corner of the window have the following functions (from left to right): play, stop, start the video over, go back one frame, go forward one frame. Read This: Before you do a more detailed analysis of the video, let s see that it gives a basic confirmation of the Principle of Equivalence. Do This: Press play and watch the objects fall. Checkpoint 1.1: Does this video appear to support the Principle of Equivalence? For starters, what does the Principle of Equivalence say? Discuss. 4

5 Read This: We will proceed to perform a more detailed analysis of the video with the goal being to determine the acceleration due to gravity on the moon. Checkpoint 1.2: As part of the detailed analysis, you need to tell Logger Pro what the scale of the video is. This will require estimating some distance in the video. Make an estimate of the height or length of something in the video. Do This: Click the button in the bottom right of the video player. This button has three red circles in it:. If you hover the mouse over the button, it reads Enable/Disable video analysis. A new set of buttons should appear to the right of the video. Do This: Now you will use your estimate from Checkpoint 1.2 to calibrate the video analysis software. Look at the toolbar to the right of the video. The fourth button from the top is the Set Scale button and looks like a horizontal ruler:. Click this button. Then click and drag from one end to the other of the object whose height or length you estimated. A new window called Scale will pop up. This window wants to know the length of the line you just drew. The default value is one meter. If you estimated the object s height to be different than one meter, enter your estimated value in the box and then click OK. Read This: Now that the software is prepared, it s time to prepare mentally. Checkpoint 1.3: Which object are you going to track, the hammer or the feather? Checkpoint 1.4: Predict what a plot of the vertical position of the object as a function of time will look like. Make a sketch. Read This: Now the tracking can begin. Figure 2: The scale window is used to calibrate the video. Do This: Use the and buttons to find the frame in the video just as the astronaut drops the two objects. This is the point where you will start tracking the fall of one object. Do This: The second button from the top has a red circle in it:. If you hover the mouse over this button, it reads Add point. Click this button. Then move the cursor over the object of your choice (either the hammer or the feather) in the video player. Click on the object. Read This: When you click on the object, a few things should happen. First, there should be a dot on the video player that indicates where you clicked. Second, the video should have moved forward to the next frame. Third, a plot should have appeared that shows the x and y position of the location that you clicked. These positions are plotted against the time of the video frame. Feel free to resize any of the windows to make your view better. STOP Read This: Sometimes a video can get hidden behind a plot. If at any point, your video seems to have disappeared, try resizing or moving your plot. 5

6 Do This: Click on the object again and again until it has reached the ground. Note that when you are tracking the position of the object, you will want to track the position of the same point on the object. For instance if you are tracking the hammer, you could track the tip of the handle the whole time. Be aware that the quality of the video is very poor! (It is from 1971 after all.) There will be some frames in which you cannot see the object you are tracking. You can always skip a frame without placing a dot on the video using the button. Skipping a frame does not cause any problems. Get as many points as you can. You can undo accidental clicks with the Z shortcut or by selecting a point using and pressing the delete key. Read This: Logger Pro will show two data sets on the same plot. One is the x position as a function of time and one is the y position as a function of time. By clicking on the label of the vertical axis you can change what is shown in the plot. We are interested in the y position of the object as a function of time. Do This: Click the label of the vertical axis and select Y from the pop- up menu. Now the plot should only display one data set. Checkpoint 1.5: Does the plot generated by Logger Pro look like the prediction you sketched in Checkpoint 1.4? What is similar and what is different? What may account for any differences? Discuss. Read This: This plot can be used to determine the acceleration due to gravity near the surface of the moon, which we will call g!""#. Instructions Figure 3: Click the label of the vertical axis to change what data is plotted. on how to fit a curve to data are given in Appendix A. See the Reference page of the lab website for instructions on how to add a title and how to make the plot easy for your TA to read. S1 Synthesis Question 1 (30 Points): Use your plot to determine the acceleration due to gravity on the moon. Explain what you do, and be sure to include your plot in your response. After that, compare your value to the accepted value with a percent difference and briefly discuss what may account for any differences between the two values. (Make sure you give your plot a descriptive title. See the Reference page on the lab website for instructions on using Logger Pro.) 6

7 Part II: Stop Dragging Me Down The Story Long before any astronaut dropped anything on the moon, physicists were dropping things on Earth. Sometimes you must wonder whether the whole science was created as a way for them to explore their curiosity, or simply as a way for them to disguise their clumsiness. That is a question we will probably never answer, so let s stick to things we can answer, like describing what happens to an object as it falls to the floor. Equipment Webcam Vernier motion sensor Coffee filters (four different sizes) Paper clips Digital balance Ruler 2. The Drag Force on Coffee Filters The drag force makes the Equivalence Principle difficult to observe on earth, but the drag force is interesting in its own right. Given the equipment listed above, you are going to perform an experiment to test how things mass or cross- sectional area affects the acceleration of a falling object near the earth s surface. Though these quantities have no importance on the moon, they play a crucial role in how things happen on earth due to our life- giving atmosphere. STOP Read This: Your lab group must complete one of the following two Synthesis Questions. That is, you must complete either Synthesis Question 2A or Synthesis Questions 2B. If you complete both questions, only the first one that appears in your report will be graded. Equipment Notes: You can track your falling object using either the motion sensor or the webcam. Here are some helpful notes that you should read very carefully. Motion Sensor: The motion sensor will produce position and velocity plots of a falling object. Plug it into one of the LabQuest DIG ports, open Logger Pro, and press Record. Webcam: If you decide to record a video with the webcam, detailed instructions are in Appendix B. Remember that videos occasionally get hidden behind the plots in Logger Pro. Paperclips: You should have 10 paperclips. If you want more, ask you TA. Coffee Filters: Do NOT flatten the coffee filters. Try to keep them shaped like a coffee filter. Please do not waste coffee filters. The large ones especially are expensive. However, do not put used coffee filters back with the new ones. 7

8 Read This: Here are some helpful tips for analyzing plots made with the motion sensor or webcam. In addition to fitting curves to data, Logger Pro can provide average values or instantaneous values. To find an average value, click and drag the mouse over a portion of the plot that you would like to average. A blue rectangle indicates the portion of the plot that you have selected. Then click This will calculate and display the average value of the data over the range that you have selected. To find an instantaneous value quickly, use Examine:. S2a Synthesis Question 2A (70 Points): Devise, execute, and analyze an experiment that tests the question, How does varying the mass of a coffee filter affect its terminal velocity. In addition to answering the question, here are a few requirements: Collect data with at least five different masses. Include one plot that you use to explain how you determined the terminal velocity of a coffee filter. (See the Reference page on the lab website for instructions on how to produce and analyze a plot in Logger Pro.) Include another plot that shows a curve fit to your five or more data points. (See Appendix C and Appendix D for a few tips about plotting non- linear data.) Compare the results of your experiment to the outcome that Equation 2 would predict. S2b Synthesis Question 2B (70 Points): Devise, execute, and analyze an experiment that tests the question, Do coffee filters of various sizes all have the same drag coefficient? In addition to answering the question, here are a few requirements: Current Research Determine the drag coefficient for all four sizes of coffee filter. Include one plot that you use to explain how you determined the terminal velocity of a coffee filter. (See the Reference page on the lab website for instructions on how to produce and analyze a plot in Logger Pro.) Discuss the plausibility of your values for the drag coefficients. Estimate the uncertainty in your values for the drag coefficients by considering how repeatable your experiment is for a single coffee filter. Read This: In your experiment, you used known properties of a fluid to learn about an object that is falling through that fluid. However, you could just as easily learn about a fluid by letting an object with known properties fall through it. This is exactly what the Ultrasonics Group in the Wash U Physics Department does in some of its experiments, using methods that are very similar to what you may have used here. By recording a video of a sphere or a cylinder falling through a fluid, these researchers are able to determine the viscosity of the fluid which can then be used in further studies. For more about the Ultrasonics Group, check out the links on the lab website. Time to Clean Up! Please clean up your station according to the Cleanup! Slideshow found on the lab website. 8

9 Appendix A: Fitting a Curve in Logger Pro This is an excerpt from the Making Plots in Logger Pro (Manually) which can be found on the Reference page of the lab website. Check out that document for additional information on fits. Fitting a Curve to the Plot To fit a curve, click the button that looks like this: which can be found near the top of the Logger Pro window. A new window will pop up giving you lots of curve fitting options (Figure 4). In this case, the points seem to follow a straight line, so I selected Linear as my mathematical model. Figure 4: The curve fit window has lots of options. Way more than Excel! In other cases, you might want to select a different mathematical model. You can even create your own model, though this will never be required of you. After selecting the model, you need to click Try Fit. Usually the fit will happen without a problem. In the event that the fit does not work, simply click Cancel and try again. If that doesn t work, make sure that you have not inadvertently selected only a small part of your data. Single- clicking the plot should cause the entire plot to be selected. Do that and try again. 9

10 Appendix B: Creating a Video With the Webcam Working with the equipment you are given, you might want to create a video that you can analyze in Logger Pro. To acquire such a video, follow these steps: 1. Plug your webcam into one of the USB ports on the back of your computer or in the keyboard. 2. In Logger Pro, under the Insert menu, select Video Capture 3. A window titled Video Capture will pop up. You will likely see a video of yourself. The default webcam is the one that is built in to the computer. We would rather use the one you just plugged in to the USB port. Click the Show Settings button. A new window will pop out of the side of the Video Capture window (Figure 5). Figure 5: Click the Show Settings button to select the correct webcam and set the resolution. 4. Click on the Video Input drop- down menu and select Live! Cam Chat HD. 5. Click on the Video Resolution drop- down menu and select 640 X Now you can click Hide Settings. 7. Make sure that everything is working as it should be. In the Video Capture window, click the Start Video Capture button. Wait a couple of seconds and click Stop Video Capture. 8. A new window should pop up. This is a more- or- less standard video player that will display your new video. The five buttons in the lower left corner of the window have the following functions (from left to right): play, stop, start the video over, go back one frame, go forward one frame. Play around with these buttons until you understand what they do. Then delete this video by clicking on it and pressing the rectangular delete key on the keyboard. (For whatever reason, the square delete key doesn t work!) 9. At this point you and the camera are ready to record a nice video. Pressing the Start Video Capture button in the Video Capture window will start the recording. When you are finished recording press the Stop Video Capture button. 10. You can track objects in the video just as you did with the Apollo 15 video. You can also track multiple objects in the same video by clicking the icon with the red and green circles selecting Add Point Series. 10 and

11 Appendix C: Plotting Non- Linear Data Linear fits are really easy to analyze. In fact, there is a common trick physicists use to turn non- linear data into a straight line. Consider an experiment where we drop a ball from a known height and use a stopwatch to measure how long it takes to fall to the ground. We repeat this from several different heights with a goal of determining the acceleration due to gravity. It can be shown that the time (t) is related to the height (h) by t = 2h g where g is the acceleration due to gravity. We gather data and plot the time as a function of height. The resulting plot is nonlinear. We could fit a square- root curve to the data and find the acceleration due to gravity. Unfortunately, we would have to define the function ourselves. This is not an insurmountable problem by any means, but there is a trick that most physicists would turn to first. If we square both sides of the equation above, we come up with the following: t! =!! h Now we don t have that clumsy radical! Do you see that plotting t! as a function of h should result in a straight line with a slope of!!? It looks just like y = mx + b where t! is like y,!! is like m, h is like x, and b is equal to zero. The resulting plot is linear. It s often easier to analyze the linear plot both qualitatively and quantitatively. Keep this in mind as you plot data that might fit the model of v! =!!"!"#. 11

12 Appendix D: Square Root Fits in Logger Pro Introduction As described in documents on the Reference page of the lab website as well as Appendix A, you can fit a curve to data in Logger Pro by clicking. Often in this course you ll use a linear fit. However, there are many other mathematical models you can use. If you search this list, you will see many functions that you are familiar with. Square Roots There are some other common functions, though, that appear to be missing. One such function is the square root. There are two easy ways to fit a square root curve to data in Logger Pro. One method is to use the variable power curve. Select this model and then set the value of the power to one- half (0.5). A second method is to use the power curve. When Logger Pro performs a fit with this model, the power is one of the fitting parameters. If a square root curve is a good model, then the power of the fit will naturally be found to be close to 0.5. Additionally, data that can be modeled as a square root can be modeled as a parabola when you switch the x- and y- axes. Fitting a parabola to your data is easy, but such a plot may be less clear to the reader. 12