# Drag, dimensional analysis, and coffee filters

This is another demonstration that Ben Sill showed me. It is very simple and can be used to demonstrate drag, dimensional analysis, experimental measurement, and data presentation.

Equipment

All you need is a tape measure, a stop watch, a packet of coffee filters and a couple of student volunteers.

Demonstration

Version 1:

1. Give one student a single coffee filter and another student two filters stacked together.
2. Have the students drop the filters at the same time from 5 feet above the ground. The 2 stacked filters will land first.
3. Have the student with the single filter move their release point down a little bit and repeat the simultaneous drop.
4. Keep moving the single filter release point down until the single filter and double filter land at the same time, and measure the release height (it should be about 3.5 feet).

Version 2:

1. Have a student drop a single filter from 5 feet and a second student measure the time taken for it to fall to the floor.
2. Add a second filter to the stack, repeat the drop, and measure the time to reach the floor.
3. Keep adding more filters to the stack and measuring the drop time until you have about 4-5 different drop times (the drop times get so short after about 4 filters that they are too hard to measure).
4. Plot drop time versus number of filters in the stack.

Class discussion

This is a great demonstration (particularly version 2) to use as a basis for discussing measurement errors, and the need to take multiple measurements. There are a lot of sources of measurement error including:

1. Differences in release time in version 1
2. Error in starting and stopping the stopwatch at release and at impact respectively.
3. Deformation of the coffee filter between drops

If you have time you can re-do all the drops a few more times and take the average of the times. However, once you have more than about 4 filters the drop time is small enough that the reaction time error is large compared to the difference in drop time you get from adding an additional filter.

Analysis

There are two basic assumptions in the analysis.

1. The filter accelerates to its terminal velocity very quickly. Therefore, the terminal velocity is approximately the distance fallen divided by the time taken.
2. The drag coefficient, Cd, is constant.

Drag analysis:

If you have covered bluff body drag then you can do the following.

When at its terminal velocity, the filter’s weight is balanced by the drag force. Therefore, you can write

mg=ρU2CdA/2.                  (1)

The time taken for the filters to drop a distance h is

T=h/U=h(ρCdA/2mg)1/2.

The only things that vary are the mass and the drop height (the area is constant as the filters stack very closely together). Therefore, the drop time is given by

T=ch/m1/2                          (2)

where c=(ρCdA/2g)1/2 is an unknown constant.

For version 1 of the demonstration the time taken for the single filter and the double filter to hit the floor is identical and given by:

T=c(5/(2m)1/2)=ch1/m1/2

where h1 is the release height of the single filter and the 5 is the 5 feet drop height for the double filter. Therefore, the height that the single filter must be dropped from to land at the same time as the double filter dropped from 5 feet is given by

h=5(m/2m)1/2=3.5 feet.

For version 2 of the demonstration h is constant so T~m-1/2. Plot the time versus number of filters on both a linear-linear scale and a log-log scale. The linear scale plot should show a non-linear decreasing function. This log-log scale plot should be a line of slope -1/2. Example plots for the linear and log scales are shown below. The use of the log-log scale plot illustrates an alternate method of data presentation. It shows the power-law nature of the relationship between time and mass. You can use this to lead into a discussion with the class  on how to identify power law relationships (log-log scale plots) and exponential (semi-log scale plots) in experimental data.

linear-linear scale plot of time taken to drop 5 feet as a function of the number of nested coffee filters.

log-log scale plot of time taken to drop 5 feet as a function of the number of nested coffee filters. Also shown is a line of slope -1/2 (see equation (2) above).

Dimensional analysis:

If you have not yet covered bluff body drag but have covered dimensional analysis then you can derive the drag force scaling as an application of dimensional analysis. Assuming that the drag is controlled by the flow separation around the filter, then viscosity plays no significant role. In this case, the drag force is a function of the object area, density of air, and the velocity of the object. that is:

D=D(A,ρ,U).

The dimensions of these parameters are

[D]=MLT-2, [A]=L2, [ρ]=ML-3, and [U]=LT-1

There are 4 parameters and 3 independent dimensions so we can only form 1 non-dimensional group. Standard dimensional analysis calculations lead to

Π1=D/ρU2A.

From here you can write the balance of weight (mg) and drag as given in equation (1) and follow the rest of the analysis outlined above. The only thing different from the previous analysis is that the 2 and the Cd are missing from the equations and are replaced by another unknown constant.

don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu).