Coffee Filter Drag

 

Overview

 

The purpose of this laboratory investigation is to observe and determine the effect of air on a falling object.  To do this, you will drop a coffee filter (or stack of filters) and measure the downward motion.  In Part A you will simply use a stopwatch to time filters falling a set distance for several trials.  In Part B you will do a single trial measuring a falling filter by video analysis. 

The experimental result will be compared to theories for air resistance in which the drag is assumed to be either proportional to speed or speed squared.  The air resistance acting on the filter should most closely follow the equation F = k2v2.  However, a reasonable model can often be achieved with F = k1v.  In either case, the constant k in the equation is related to the density of the air, the cross-sectional area of the object, and the degree to which the object has an aerodynamic shape.  The values of the two constants will be different for a given object.

 

Part A – Drag vs. Speed

 

In this part of the experiment you will measure the relationship between drag and speed for an object of a given size and shape.  In the experiment that follows you will vary only the mass of the falling object without significantly changing its cross-sectional area or shape.    

1.      Take a stack of seven coffee filters (do not separate the filters).  Drop the stack of filters from a set height and measure the time for the filters to fall this distance.  Record the distance and time in the data table.

2.      Remove a single filter from the stack and label it with the number seven.  Then repeat the experiment with a stack of six coffee filters.  Record the time.

3.      Remove another filter and label it with the number six and repeat the experiment with a stack of five coffee filters.  Continue like this until you have completed the time column of data.

4.      Use an electronic balance to determine the total mass of each stack that fell in each trial. 

 

 

Part B – Velocity vs. Time

 

In this part of the experiment you will produce a velocity graph by analyzing a video of a falling coffee filter.

A video will be made of a person dropping a coffee filter.  The video will be analyzed using Logger Pro 3 software (not Logger Pro 2) in order to create a set of data showing the x and y components of the object’s position and velocity with respect to time. 

A word about dropping the filter:  it is important that your moving hands do not affect the motion of the filter.  For example, if you pull your hand downward out from under the filter it will tend to get dragged down by the draft created by your hand.  The best technique is to hold the filter with both hands – one on either side – and then move your hands horizontally outward and away from the filter.  This will minimize disturbance of the air directly below the filter.

Making the video

1.      Select a single filter and determine and record its mass.

2.      While one person is dropping the filter, another will be operating the computer and the camera.

3.      Start up the computer but do not yet start Logger Pro 3.

4.      Mount the camera securely so that is pointed horizontally toward the site of the experiment. 

5.      Plug in the camera to a USB port.  Wait for the computer to “recognize” the camera; you should get a message “found new hardware” or something similar.

6.      Once the camera is connected and recognized by the computer start the Logger Pro 3 program.

7.      Under the Insert menu, choose Video Capture…

8.      A “preview” window should appear with a “live” shot from the camera.  Do a “trial run” of dropping the filter and make sure that it will be visible to the camera for its entire “flight”.  It is best if the filter moves in a plane that is perpendicular to the camera’s aim.  Also it is best if the object does not get too near the edge of the camera’s view.

9.      Mount a meter stick vertically in the plane of the object’s motion – this will be used to scale the video images.

10.  Click on Options and enter a capture duration of say 4 seconds.

11.  Check the exposure settings by clicking again on Options, then Camera Settings, then Advanced.  Try an exposure of 1/120 th of a second and adjust the Gain slider to the middle of its range.  (To get a good video of a rapidly moving object you need a short exposure – otherwise the object will appear blurred in each video frame.)

12.  Start the video capture and drop the filter across the camera’s field of view and past the meter stick – the camera will stop recording automatically. 

13.  Close or drag and move the Video Capture window and you should see another window with your video of the dropped filter.  Play the video and make sure that it is satisfactory.  Also try clicking through the video one frame at a time to make sure the moving object is clearly visible in each image.  If you are not satisfied then simply delete your video and repeat the video capture process, making whatever changes are necessary.  (To delete the video select its window and press Delete on the keyboard.)

14.  Once you have a satisfactory video close the Video Capture window.

 

Getting data from the video

1.      Before you do anything else select the video and drag the corners of its window to make it as large as possible – this will allow you to be most precise.

2.      Click on the button at the bottom right of your video’s window to Enable Video Analysis.

3.      Click on the ruler button to set the scale and then click and drag from one end of the meter stick to the other.  A green line will appear superimposed on the meter stick.  In essence you are “telling” the computer that this distance in the image is equal to one meter in reality.  If you make a mistake then simply repeat the process.

4.      Advance through the frames of the movie to just before the point where the filter first leaves the person’s hand.

5.      Click on the red dot and crosshairs button to activate data collection.  At this point you are ready to produce the actual data.  Every time you click with the mouse, the computer will record the crosshair’s position and then advance one frame through the movie. Center the crosshairs on a particular point on the coffee filter and click on it.  You will see a dot appear where you clicked and you should see the movie advance one frame.  Click on the object again and again to produce a set of dots that follow its trajectory – stopping at the last frame before it hits anything. 

6.      Click on the red dot and crosshairs button again to deactivate data collection.

7.      Note:  if you make a mistake you can use the Select tool (button looks like an arrow) to select any point and delete it or drag it to a new position.

8.      Once you are satisfied with the points of the object’s trajectory you may want to move the video out of the way.  One way to do this is to put it on a second page.  Go to the Page menu and choose Add Page … and then choose Blank.  Now you can cut and paste the video onto the blank page.  Notice the page “navigator” in the menu bar.

9.      Now consider your results.  The program automatically produces four sets of data based on your “clicking of the filter”:  x and y position and x and y velocity.  We will ignore any horizontal movement of the filter and therefore you can delete the x values.  Under the Data menu choose Delete Column and then X.  Because the x velocity was based on this, it will also be deleted automatically.

10.  Under the Data menu choose Column Options and Y Velocity and then make the following changes:  name = Velocity; short name = v; Point Protector style = Empty Circle & Display every 1 point.

11.  Next, you will produce a graph of velocity vs. time.  Go to Page 1 and change the graph so that it shows velocity vs. time.  Double click on the graph and give it an appropriate title.

12.  Now add an automatic curve fit.  This is under the Analyze menu.  Choose an equation of type inverse exponential but then click on the Define Function button and modify the equation to read:  y = A*(1 − exp(−C*(t − B))).  Then click Try Fit. 
Note this equation is equivalent to: 

13.  If you cannot get a satisfactory curve fit you may need to select only part of the graph for analysis.  In the end, it is up to you to take the necessary steps to get as good a match as possible between the data and the best-fit curve.

14.  You can use the program to modify just about anything that is shown on the graph.  Make whatever modifications you deem necessary.  Under the File menu use Print Graph to print a copy of the graph for each person in your group.  (For best results do not use the Print button in the tool bar).

15.  Select the data table by clicking on it. Under the File menu use Print Data Table to print a copy of the data table for each person in your group.  (For best results do not use the Print button in the tool bar).

16.  Record on the printed data table the mass of the filter that was used in the video.

Analyses

1.      Complete the data table for part A by computing weight, speed, speed squared, and the values of the “constants” k1 and k2.  The constants are found by assuming that the speed measured is the terminal speed (in which case there is no acceleration). Determine the best values (i.e. mean) for each constant.

2.      Using your determinations of weight and terminal speed construct a graph of drag vs. speed.  (Weight and drag are balanced when falling at terminal speed!)  Use your calculator or computer to determine the best fit using a “power regression” of type y = A xB .  Plot the curve produced by the equation so that it may be compared to the data.  Include the equation on the graph

3.      Construct a graph of drag vs. speed squared.  Use your calculator or computer to determine the best fit using a linear regression.  Plot the line produced by the equation so that it may be compared to the data.  Include the equation and correlation coefficient on the graph.

 

Questions

1.      Starting with Newton’s 2nd Law, write the equation of motion, solve the differential equation, and derive an equation giving velocity as a function of time based on the value of k1 for a single filter and on the mass of the filter dropped in part B.  Substitute known numerical values into this equation and simplify (do not leave in symbolic form).  This equation will not be based on the velocity data from part B.  Now compare this theoretical equation with the best fit equation determined by the computer.  Discuss similarities and differences with the goal of determining how well reality is modeled by such equations.  Make specific comparisons of the various numerical coefficients in the two equations.

2.      The model F = kv2 is more specifically written as  F = ½ CDAr v2, where A = cross-sectional area, r = density of air, and CD = the coefficient of drag – a dimensionless quantity that characterizes the aerodynamic shape of the body.  The more aerodynamic and streamlined the body, the closer the coefficient of drag is to zero.  Use your value of k2 to determine the coefficient of drag for a coffee filter.  Show your work clearly. 
(density of air r = 1.29 kg/m3)

3.      Based on the tables, graphs, etc. which model – speed or speed squared – do you feel most accurately models air resistance acting on a falling coffee filter?  Be specific and support your position.

4.      Discuss error.  What are signs of error in the results, tables, graphs, etc?  And what are the most likely sources of the error that is apparent?

Part A – Drag vs. Speed

 

 

Outer diameter of filters =

 

Drag model:

F = k1v

 

Drag model:

F = k2v2

Distance fallen by filters =

 

 

#

m (g)

wt. (N)

time (s)

speed (m/s)

k1 (            )

speed2 (m2/s2)

k2 (            )

7

 

 

 

 

 

 

 

6

 

 

 

 

 

 

 

5

 

 

 

 

 

 

 

4

 

 

 

 

 

 

 

3

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

mean k1 =

mean k2 =

 

 

A complete report (50 pts) consists of the following items arranged in this order: