AP Laboratory Investigation
The coffee filters and air resistance.
Materials: Pasco motion sensors with a computer interface and DataStudio Software;
large coffee filters, timers, metersticks.
Purpose: To take a closer look at the fluid friction force in a real example of a falling
coffee filter.
Introduction:
We have learned a little bit about the friction force that appears when a solid object
moves through a fluid. We know that this friction force is dependent on the velocity. If an
object of mass M falls under the influence of gravity and a drag force Fdrag , we may write
Newton Second Law as:
M g - Fdrag = M a
-
here a is the acceleration and Mg is the gravity of the object
Fdrag = b vn
-
(1)
(2)
here the coefficient b is a constant that depends on the geometry of the object and
v is the velocity of the object. For objects moving slowly through the air the
power n is close to 2
By substituting (2) into (1) we obtain:
M g - b vn = M a
(3)
As the object falls from rest, the velocity increases until the drag force and the weight are
equal in magnitude. The acceleration is then zero, and we have:
M g = b ( vf )n
(4)
This relation can be rearranged to obtain:
Vf = ( M g / b )1/n
(5)
Now by taking the logarithm of both sides of equation (5) we obtain:
ln ( vf ) = 1/n ( ln M + ln g – ln b ) (6)
This relation may be rewritten as
ln ( vf ) = 1/n ( ln M ) + C,
(7)
where C = 1/n ( ln g – ln b ) = a constant.
We can now determine the exponent n of the velocity if we measure a variety of terminal
velocities for a variety of falling masses. A log-log plot of the terminal velocity as a
function of the mass yields a linear plot with a slope of 1/n.
Procedure / Analyzing the data:
- We will measure the terminal velocity in two different ways (using a motion
sensor, and directly measuring the height H and the time off fall):
- Set a motion sensor on the floor and drop the coffee filters from the ceiling level
(we need to have as large a distance as we can). Run the experiment and drop the
filters (in sets of one, two, three, four, and five). Read the terminal velocity from
the computer graph and tabulate results. Do the experiment at least 4-5 times for
each set and average the results.
- For the same sets, time the drop, also (at the same time or afterwards). Do it 4-5
times for each drop. Tabulate each trial and average the results for each set.
Measure the height H, from the ceiling to the floor.
- Approximate the terminal velocity to the average velocity of the fall ( = H / t )
- Question: Would this approximation get better or worse with increasing the
number of filters that you are dropping?
- Plot, in both cases, terminal velocity as a function of the number of filters. Why
not necessarily the mass of the filters?
- The final 2 graphs should give you the n values. Are they about the same? How
does the approximation of the terminal velocity to the average velocity affects
the result of n? Is it larger or smaller than the real value? Discuss.
A full lab report is required. It should contain all parts of the lab report (in
addition, your results discussion should contain the answers to the above
questions. Discuss other sources of uncertainty, also.