Pranav Verma: (2006)
The relationship between velocity and drag for paper cups of varying mass falling in air at terminal velocity is found to be proportional to velocity to the power of 2.0 to within ±10%. The drag coefficient is shown to be 0.52±0.02.
Research Question
What is the relationship between drag force and velocity for
paper muffin cups falling in air?
Hypothesis
A power law with an exponent of two is expected to be a good approximation to the relationship between drag force and velocity for paper muffin cups falling at terminal velocity, because the drag force in turbulent flow is given by 1/2cd rAv2, and at terminal velocity the drag force is equal to mg.
At terminal velocity ...
... where cd is the drag coefficient, r .is the fluid density, A is the cross sectional area, and v is the terminal velocity.
Explanation
When a ball passes through a fluid under conditions of laminar flow, Stoke's law applies and the drag is found to be proportional to the velocity (3). Laminar flow occurs in air only for very low velocities. At higher velocities turbulent flow occurs and the drag is then found to be proportional to velocity squared(4). The velocity range for the transition from laminar to turbulent flow is predicted by a Reynold's number calculation.
Reynolds number(5) is a defined quantity that for a sphere is given by....
... where r is the fluid density, d is the diameter of the sphere, v is the velocity and h. is the viscosity of the fluid.
Paper muffin cups falling in air
The viscosity of air at STP is 1.83x10-5 Nsm-2 and the density is 1.26 kg/m3. The muffin cups have a diameter of ~5 cm and a terminal velocity of ~1 m/s.
Reynolds number for falling paper cups in air (taking them to be approximately spherical) is given by....
Turbulent flow typically occurs when Re is above ~1000 and laminar flow is usually established when Re is less than 1. It is expected that muffin cups will fall in turbulent flow.
Paper muffin cups were dropped 5.2 meters. The terminal velocity was found by timing the descent with a stop watch Muffin cups are convenient because they are of the same size and mass (to within ± 0.01 g). They stack well and the falling mass can be increased in steps without altering the size and shape. At terminal velocity the drag force is equal to the weight.
Data
Repeated drops were performed, with the number of cups varying from one to nine. The times were measured with stopwatches. The acuracy of the data obtained was found by timing each drop independently with three watches and listing half the range as the likely error in Table 1.
Analysis
The data is plotted in Graph 1 at right. A power law with an exponent of 1.8 has been fitted to the data points in Logger Pro. with an automatic curve fit. Since the drag force at terminal velocity (mg) is proportional to the mass, the vertical axis can be calibrated as drag force. Graph 2 shows that the drag force is proportional to velocity squared to within errors of ± 5%.
|
Terminal velocity calculations When drag force is proportional to velocity squared, in turbulent flow, terminal velocity is given by....  If the paper cup was a smooth sphere the coefficient of drag (cd) would be 0.50. The density of air (r), is 1.26 kg/m3, given that Bangkok is approximately at sea level and the radius of the cup was 3.25 cm. |
Entering the terminal velocity relationship into Logger Pro allows calculated terminal velocities to be plotted against the mass. Entering trial values of the drag coefficient into the calculation allows the drag coefficient to be found.
Graph 3 is plotted with two values of drag coefficient, 0.50, 0.54. The drag coefficient is 0.52±0.02.
The relationship between the terminal velocity of muffin cups and the drag force acting upon them is found to be a power law with an exponent of 2.0 to withi ± 5%. (Graph 2) as expected. The relationship is not exact, but it confirms that the cups are falling in the turbulent flow region, and Stokes' law does not apply.
The drag coefficient is found from Graph 3 to be wo ithin errors, the drag coefficient for a smooth sphere which is 0.50.
Applications
Skydivers experience a drag force as they fall. At terminal velocity the drag force is equal to their weight (mg). The terminal velocity for single divers in the stable "belly-to-Earth" position is approximtely 35±5 m/s(1). The illistration below shows two divers in tandem. Because they are stacked (like the muffin cups used in this investigaton) the terminal velocity of the pair will be increased from that of a single diver by 20-30%.
|
A stack of two skydivers  Image from ... www.btinternet.com/~duncan.jauncey/skydiving.html |
Paper cups produced consistent results. They remained stable while falling due to their symmetry and flared edges. The long (5 meter) drop as necessary to ensure that the cups were falling at terminal velocity for terminal velocity over most of the distance. In future, terminal velocity could be determined by filming the descent over a measured distance above the floor so that the time taken to reachterminal velocity and any slowing of the descent as the cup reaches the floor would not affect the data.
Suggestions for further work
Several paper cups could be linked together to determine whether terminal velocity is affected by the number of cups in a formation. When sky diving, large formations are more stable (safer for novices). It would be interesting to compare the terminal velocity of a single diver (cup) with that of a formation.
|
Fig 3 A skydiving group performing a jump in formation. |
References
(1)Chang: 2006: Corking
(3)Chang and Jacobs Aug. 2005: Terminal velocity
(4)Sajjanhar, Gaurav 2001: Damped oscillations
(5)physics@isb, Stokes' law
