Drag Force on Coffee Filters

Drag Force on Coffee Filters

Purpose: To study the relationship between air drag forces and the velocity of a falling object.

Equipment:

• 1 Computer with Logger Pro software
• 1 Lab pro
• 1 Motion detector
• 9 Coffee filters
• 1 Meter stick.

Introduction: 

     When an object moves through a fluid, such as air, it experiences a drag force that opposes its motion. This force generally increases with velocity of the object. In this lab, we are going to investigate the velocity dependence of the drag force. We start by assuming the drag force, FD, has a simple power law dependence on the speed given by the formula:

FD= K*(v)^n

(where the n-th power is going to be determined by our experiment)

     In this lab, we will investigate drag forces acting on a falling coffee filter. Because of the large surface area and low mass of these filters, they will reach terminal speed soon after being released.

Procedure: (from lab handout)

     You will be given a packet of nine nested coffee filters. It is important that the shape of this packet stays the same throughout the experiment so do not take filters apart or otherwise alter the shape of the packet. Why is it important for the shape to stay the same?

It is important to maintain the shape of the packet because drag force is proportional to the 1/4 cross-section area of the object. If we change the shape of the object (packet of coffee filters), we would change the cross-section area giving us an inaccurate result for drag force. We would not keep consistency.

1. Login to your computer . Start the Logger Pro software, open the Mechanics folder and the Graphlab file. Do not forget to label the axes of the graph and create an appropriate title for the graph. We set the data collection rate to 30 Hz.
2. We place the motion detector on the floor facing upward (move any objects around it because may cause reflection) and hold the packet of nine filters at a minimum height of 1.5 m directly above the motion detector. Use the meter stick to measure that distance from motion detector to coffee filters. Start the computer collecting data, and then release the packet. What shoud the position vs time graph look like? Explain
   

   The curve decreases pretty fast at first. Then, at some point, the curve turns to be linear (constant slope).
3. Verify that the data are consistent. If not, repeat the trial. Examine the graph and using the mouse, select (click and drag) a small range of data points near the end of the motion where the packet moved with constant speed. Exclude any early or late points where the motion is not uniform.
4. Use the curve fitting option from the analysis menu to fit a linear curve (y=mx+b) to the selected data. Record the slope (m) of the curve from this fit. What should this slope represent?              The slope of this graph will represent the terminal speed of the object. Slope of position vs time graph is equal to the velocity of the object. In this case, terminal speed.
5. Repeat this measurement at least 4 more times, and calculate the average velocity. Record all data in an excel data table.
6. Carefully, remove one filter from the packet and repeat the procedure in parts 2,3,4,5 for the remaining packet of 8 filters. Keep removing filters one at a time and repeating the above steps until you finish with a single coffee filter. Print a copy of one of your best x vs t graphs that show the motion and the linear curve fit to the data for everyone in our group (Graph only).
7. In Graphical Analysis, create a two column data table with packet weight (number of filters) in one column and average terminal speed ([v]) in the other. Make a plot of packet weight (y-axis) vs terminal speed not velocity (x-axis). Choose appropriate labels and scales for the axes of your graph. Be sure to remove the "connecting lines" from the plot. Perform a Power law fit and record the power, n, given by the computer. Obtain a printout. Check the % error between your experimentally determined n and the theoretical value before you make a printout. 
8. Since the drag force is equal to the packet weight, we have found that dependence of drag force on speed. Write equation 1 above with the value of n obtained from your experiment. Put a box around this equation. Look in the section on drag forces in your text and write down the equation given there for the drag force on an object moving through a fluid. How does your value of n compare with the value given in the text? What does the other fit parameter represent? Explain. 
   
   
   Data and Results:
   
   
   Graph from one of our trials:

   

   Position vs. Time Graph
   slope = 2.420 m/s; terminal speed = 2.420 m/s

   
        
   
   
        We must use the Power Law Fit in order to find the best fitting curve for our graph. The equation is Y= 1.48X^(2.17) where X is equal to terminal speed and Y is equal to the number of coffee filters. We find n to be equal to 2.17. We then had to compare this value of n to the actual value of n given in our book. Actual value of n=2. Our value of n ended up being very close to the actual value of n. The equation given by our book is FD= 1/4Av^n, where n = 2.
   
   

   

   Number of Filters vs. Terminal Velocity Graph
   w/ Power Law Fit Curve

   
   
   Question:  How does the value of n compare with the value given in the textbook? What does the other fit parameter represent?
        We find that n = 2.17 from our power curve fit. This value is almost identical to the actual value of n (n = 2) given to us by our textbook . FD=1/4Av^2 where FD is a force (in this case, mass*acceleration due to gravity), so FD=mg=1/4Av^2; we also found that Y= 1.48X^2.17 (X=terminal speed, Y=number of coffee filters). So, Y*mg= 1/4AX^2 (m = the mass of each coffee filter).
   Conclusion:

        In this lab, we study the relationship between air drag forces and the velocity of a falling object. We can draw some conclusions from our graph and data related to drag force and velocity. When an object starts falling down, its velocity increases. However, there is certain point where its velocity will be constant. At this point, velocity is equal to drag force. We call this velocity terminal velocity. In our graph, we would expect some portion of it to be parabolic shaped and another portion to be linear (where velocity is constant). We know from our equation FD= 1/4Av^2 that as velocity increases, drag force increases as well. As drag increases, acceleration decreases. Some causes of error may be that as the coffee filters fell down, their shapes are easily able to be changed, and since drag force is related to the cross-section area of the object, if shape changes, the drag force will change as well; rounding values is also one source of error that most certainly affected us; and the number of filters vs. the average speed, because since the coffee filters are so light, that they might have been affected by wind, which would cause some unwanted horizontal velocity.