About drag force and Reynold's number
In the module, you will learn about how objects may reach terminal velocity. You will be able to:
Demonstrate motion in the presence of drag force
Demonstrate analysis to test a model
Identify variables of measuring the terminal speed of a falling object
Identify the Reynold's number of the falling object and compare with other objects
Note: Please research the types of variables that would be encountered in this or any typical lab experiment (dependent, independent, and controlled). Know the difference!!
And although this is your lecture over drag forces, and what we want the students to do in this part of it especially, is to try to do just a little research. What is the meaning of drag force? >> Looking at simple vector diagrams, some of the terminologies associated with falling objects that deal with drag and so forth and so on. >> We want you to take the time and just, just use the Web if you have to and just look at some basic concepts of track. >> So and starting the discussion, just think of an objectfalling under gravity, just like what you've learned. And general physics one where the net force is just the weight and whose acceleration is just minus 9.8 meters per second squared. >> Well, for our experiment, we're going to begin by assuming that there is an upward force called the drag force, which is proportional to speed squared. >> You can see the streamlines air upward as the object falls under gravity. >> C is the coefficient of drag, is the cross-sectional area, and rho is the density of air.>> So let's, let's see what happens in remarkable thing, where the upward force associated with the drag force equals the force of the gravitational.>> Well, the terminal speed is the speed reached by the falling objectwhen the net force is 0. >> Again, when the gravitational force pointingdown and the drag force pointing up are equal, this means and it's 0 acceleration. And from there we can set these two magnitudes equal and determine theoretically what the terminal speeds. And we see that it's proportional to the square root of the mass. >> So outside, if all else is the same, the coefficient of drag, the cross sectional area, and of course, the density of air. Then what we can see is a plot of V sub t versus m. And when we fitted with a curve, it would look like a square root. Well, beta is just a constant here defined, and you will actually be able to determinewhat that constant is with the trend lines fit. So you will perform a trend line and the, there will be the B sub x to some power. We hopefully it's around 0.5. >> And then a coefficient out front, which would be beta. >> So well that you can actually calculate what the coefficient of drag is.Well, let's look just very briefly at some of the calculus equations that are associated with all this. >> An algebra students are invited. >> You should it, you're not expected to perform calculations dealing with derivatives and integrals. >> But it's awfully nice sometimes just to see how these things are set up. Well, for our drag force, which was proportional to v squared, when I write this equation out in terms of derivatives, the calculus-based students should be able to duplicate this if we define alpha as this constant with a negative sign out front. >> And of course, when you assume V squared, when there is 0 acceleration, you get your terminal speed. Now it's very important to know that it would be nice toderive an equation for V as a function of t. I'm not talking about big T and terminal speed, I'm talking about as a function of time. >> The problem, it's a little bit beyond the scope of geophysics one for us to do that because of the tedious algebra that must be done. >> But that's okay. >> We have an equation here that we can actually test the model with. >> But let's assume instead, we said that yes, there is an upward force and it is proportional to v instead of v squared. Well, we can actually perform the, the mathematics to determine what a velocity versus time equation would look like. Here, gamma is just equal to the negative of some coefficient divided by little m. >> And that coefficient, little b is analogous to capital C. >> And this equation for part B has been use when they're talking about frictional forces of objects falling in a fluid. >> And so here itwouldn't be a bad assumption for a student to say, Well, you know, I would assume perhaps that the upward force will be proportional to v,and that will be at decent assumption to make. >> But now analysis has tocome in and check these things. Of course, you can still get a terminal speed, which is just g over gamma ends with, again, no net acceleration.So let's take a look at that equation. Will fast as just having an upward force as a function of just v instead of v squared. >> We can actually determine what v of t would be with just some integration and some simple algebra. >> And assuming one of the masses use, which we used in this experiment, was placed in there. >> This is what we would see when we tried to fit our data. >> Our data is the blue curve for a falling object with this mass. This is, I believe, three coffee filters. >> Or two. >> And then we try to fit that curve with assuming that the upward force is justproportional to v. You see it's not a very good fit. >> But one of the things that should fortify us is that if you actually did this using a Latin code,assuming the squared, the fit looks a lot better, at least better than s. >> So again, it's beyond the scope, but if you're interested, you should try it,especially if you know lab view, Reynolds number, and flow. Well, let's assume that our just have a picture in our mind of our coffee filters dropping through this tube of air. >> And you see the laminar flow is beautiful order streamlines. >> It's very smooth looking flow, but it looks a little bit more chaotic, so to speak. When the flow is turbulent, it's, it's not so well defined. And so the streamlines doesn't really look smooth like they do in a laminar flow. The Reynolds number actually is a, is a quantity that I want you to research also. And we're dealing with a ratio of inertial forces with shearing forces as this object is falling. >> And it's equation is very simple. We're only really interested in the actual terminal speed,which we can get experimentally for these filters. >> And here are your quantities, which is symbolized here. >> Density here, everything is air. >> Fluid viscosity is really viscosity of air, density of air, and then the diameter of your filter. When we're talking about laminar flow, when you actually calculate this number with the terminal speed and your diameterand the density of air and viscosity of air. >> You can have one or two options here. It could be laminar, it could be intermediate, something in between, or it could be turbulent. >> So when you calculate your Reynolds number, and it also helps to actually drop some filters yourself.>> Try one and then put two together and drop those dual with paper just to get an idea of what, you know, just get a picture of what turbulent flow versus laminar flow might look like. >> And then record your observations and, and a comment where it asks you to enrich spreadsheet. >> It, it'll be fun. Enjoy.
Take two pieces of notebook paper. Wad one up and leave the other unblemished. Release the one you compressed into a ball and watch it fall. Take two hands and release the other. What was the first thing you noticed? How much longer did it take the ball of paper to reach the ground versus the one that floated about before it hit?
page1image60018240
In this section, we used coffee filters to study drag force. You saw in the video how to analyze a falling object in which drag forces affect the velocity vs. time curve. Figure 1 illustrates a typical drop. The filters are held beneath a motion detector. Upon release, the student’s fingers are not to move, as the detector will record those motions as well. Running parallel to the drop axis is a two-meter stick. The detector is activated, and then the student drops. The motion is detected until it reaches the floor.
Activity
Open up an Excel spreadsheet.
Fig. 1 A typical student drop of the object
page1image57921184 page1image60019392
Insert a textbox and place it in the upper left corner of the sheet. In the box, list all of the variables, based on your research with dropping coffee filters (with cone upwards) and paper at home as well as Figure 1, in performing an experiment measuring the velocity vs. time curve of dropping a different number of filters (up to four). Again, refer to the picture. Clearly label the type of variable it is (dependent, independent, controlled,..). Make sure you neatly label things forclarity and presentation.
Insert another textbox. List all randomerrors that should be considered whendoing a proper analysis.
Insert a final textbox. In this box,describe your observations when dropping one or more filters, the paper ball, and just the paper. Put things in context with what we have discussed in this module.
Rename the tab, Discussion, and save the file.
Create a new tab. Name it terminal. Figure 2 is the graph for velocity vs. time for 1-5 coffee filters. Notice how the speed
2.5 2 1.5 1 0.5 0
velocity vs. time
page1image38689344
0 0.5 1 1.5 2 2.5 time (s)
Fig. 2 Velocity vs. time graph for various masses (number of filters).
velocity (m/s)
reaches a value that remains approximately the same. For the dark-blue curve, we could not reach the terminal speed unless we increased the height of the drop (something you would NOT have done in the lab itself).
- In Column A1, label it mass (kg). In B1, label it terminal speed (m/s). Expand the column to show the full label. Place the data from the table below into Columns A and B (starting from row 2) respectively.
page2image60135808 page2image60129664
Mass (grams) 0.0009 0.0018 0.0027 0.0036
Terminal speed (m/s) 0.89
1.48 1.72 1.98
page2image60142528 page2image60131584 page2image60130624page2image60131008 page2image60142912 page2image60126720page2image60129024 page2image60120576 page2image60123264page2image60107648
Plot terminal speed vs. mass. Label the graph! Cation: When you plot the mass, make sure it is in SI-base unit of kilograms!
Right-click on the data. Select Trendline and chose the power fit. Paste equation (blow up the font) on the graph. Note the coefficient out-front and the power of the fit.
Using a density of air to be 1.21kg/m3, the diameter of the bottom part of the filter as 0.15m (assume circular cross-section), and the power fit of your Trendline equation, calculate the drag coefficient. Solve for it first (see video) and then plug in the values.
Insert a textbox and ADDRESS these question about this analysis.
Based on your list of random errors, are the results for the drag coefficient and powerof the fit reasonable? Explain. Hint: For the coefficient, google drag coefficients andresearch off of the internet.
Say something about your choice of controlled variable. Why is this critical for thisexperiment?
Add a final tab. Call it Reynolds. Calculate the Reynold's number using a viscosity of airas 1.81E-05 kilograms/(meters-seconds), the density of air (see above), the diameter as0.15 m, and, from the data, 0.89 m/s.
Insert a textbox. Comment on your value. How is this value compared to a skydiver or theblood flowing to your brain? Compare your value also to a butterfly and a Boeing jet.
Save the file and upload.