POSTER PRESENTATION OF A PAPER BY: Alex Shved, Mark Logillo, Spencer Studley AAPT MEETING, JANUARY, 2002, PHILADELPHIA

Damped Harmonic Oscillation Using Air as Drag Force

Spencer Studley Alex Shveyd Mark Loguillo

Santa Rosa Junior College Department of Engineering & Physics 1501 Mendocino Ave. Santa Rosa, CA 95401

Supervising Instructor

Younes Ataiiyan

Introduction

An object moving through fluid encounters a resistive force. The nature of this resistive force depends on the speed of the object. At low velocities the drag force is due to the friction between the moving object and the fluid. At high velocities the drag force is mainly due to the fluid pressure exerted on the moving object. The velocity of the moving object, v, combined with the physical properties of the fluid is used to define a dimensionless quantity, called the Reynolds number written as 1: R = (vD/). In this equation,  is the density of the fluid,  is the viscosity of the fluid and D is the effective length or diameter of the moving object. At low Reynolds numbers, the drag force is linearly proportional to the velocity of the object, FI =-bv and at high Reynolds numbers, the drag force is 2 proportional to the square of the velocity, FII=-bv . The oscillatory motion of a mass attached to a spring would be a damped oscillation due to the drag force exerted by air on the moving object. The damping force is typically assumed to be linearly proportional to the velocity of the object2. The equation of the motion can be described as: dx 2xd bkxbvkxF  m  x dt dt 2 where K is the spring constant and X is the displacement of the object from its equilibrium position. By solving the above differential equation, the solution is found to be in the

 b t form of: )(  eAtX 2m t   )cos( 2 2 2  b  k  b  where  o        2m  m  2m 

 b t )(  eAtX 2m t   )cos(

Figure 1: Typical damped harmonic oscillation.

Procedure

During the initial part of this experiment a spring was suspended from a rod, approximately 2 meters from the ground. A motion sensor was connected to a computer and placed directly underneath the spring. The axis of the spring was aligned as accurately as possible with the center of the motion sensor. Figure 2: Schematic presentation of the experimental set-up. The mass of a thin circular disk with a diameter of 5 centimeters was measured on an analytical balance. Additional mass was added to the plate until the total mass was125 grams  1gram. The circular plate was then suspended from the bottom of the spring, above the motion sensor. The spring was retracted downward a distance of 10 centimeters from the equilibrium position and released. The moment the spring was released and allowed to oscillate, the motion sensor was activated through the computer and displacement-time data was collected and stored. In the second trial the spring was retracted a distance of 20 centimeters downward and released again as the motion sensor was activated. The same procedure was repeated, for the final trial, with the circular disk retracted to a distance of 30 centimeters downward. The release of the disk from multiple displacement distances resulted in varying initial velocities, enabling us to investigate the air drag at several different velocities. The above procedure was repeated for thin circular disks with diameters of 10cm, 15 cm and 20 cm while maintaining a total mass of 125 +/-1 grams for each disk. In the secondary stage of the experiment the spring constant was determined using Hooke's law. This was accomplished by suspending various masses from the spring and measuring each vertical displacement from the equilibrium position. The spring constant was determined from the slope of the of weight versus the displacement curve and was found to be 1.60 +/- 0.01 N/m. The displacement-time data was fit to the equation:

 b t 3 )(  eAtX 2m t   )cos( using a program called CurveExpert version 1.37. This process was performed in two different stages. In the first stage, the entire range of the collected data was used for the curve fitting. In the second stage, selected ranges of collected data corresponding to a specific velocity range was used for the curve fitting.

RESULTS

The values of b for each disk and various initial displacements were determined directly from the parameters obtained by the curve fittings. It should also be noted that theoretically, the b parameter can be calculated from

2 2 2  b  k  b   o       , where  is determined from the curve-fit. As  2m  m  2m  shown in the NOTE section below, the calculated b using this method results in a very large possible error, due to the close proximity of 0 and b/(2m) values. In the first attempt, the curve fitting was done on the entire range of the collected data for all the situations described in the procedure section. Plots of b versus disk area for all three disks and for different displacements are shown in Figure 3. By generating the velocity-time graphs from measured distance-time data, the overall range of the velocities for all the situations was found to be between 1.3 to 0.3 m/s. Using this range of velocities and  = .0013 g/cm3 for air, =1.8x10 -4 g/(cm.S), the range of the Reynolds number was found to be in the 5x104 to 300 range. It has been shown4 that for a thin disk moving through a fluid at high velocities, corresponding to the Reynolds numbers of greater than 3x104 , the drag force is proportional to the square of the velocity according to : 2 FD = ½(CD  A V ) where CD, called the drag coefficient, is constant and equal to 1.12 for a high Reynolds numbers. In the above equation,  is the air density and A is the area of

b vs plate area (full data curve fit)

0.0250

y = 0.6885x - 0.0001 0.0200 R2 = 0.9975 FOR X=30 cm y = 0.6433x - 0.0004 R2 = 0.9946 FOR X=20 cm

0.0150 b (Kg/s)

0.0100

y = 0.3391x + 0.001 R2 = 0.9857 FOR X=10 cm

0.0050

0.0000 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

plate area (m^2) the disk.

Figure 3: Plots of b versus disk diameter for various displacement values. The values of b were found by curve fitting to the entire range of collected data for each case.

To separate the analysis into a high velocity region and a low velocity region, separate b values were obtained by curve fitting the data corresponding to the velocities above 1 m/s and below 0.3 m/s, respectively. This analysis was performed for the displacement of X=30 cm data only. Plots of these new b values versus disk diameters for the high and low velocity regions are shown in figure 5. An ongoing investigation aimed at finding a correlation between b and the reported value of CD = 1.12 is in progress.

CURVE FITTING AT DIFFERENT SPEED RANGES 0.6 curve A , Hig h sp eed

0.4 curve B , Full Range

curve C, low speed 0.2

0 0 20406080100 -0.2

-0.4

t(Sec.)

Figure 4: Results of the curve fittings to the low velocity range, high velocity range and the full range for the 20 cm diameter disk oscillation from a 30 cm displacement.

Variation of b with plate area (partial curve fit)

0.04

0.03 High speed region y = 3.5175x + 0.0055 R2 = 0.9991

0.02 b(Kg/s)

0.01 Low Speed y = 1.6815x + 5E-05 R2 = 0.9949

0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 Plate Area (m2)

Figure 5: Plots of b determined from curve fitting to the low velocity range and high velocity range data. These values are determined from a 30cm displacement for each of the disks.

REFERENCES:

1- J.K. Vennard, R.L. Street, Elementary Fluid Mechanics, 6th ed. Page 283, John Wiley and Sons, 1982. 2-R.A. Serway, R.J. Beichner, Physics for Scientist and Engineers, 5th. Ed., P408, Sounders College Publishing, 2000. 3-CurveExpert v. 1.37. www.ebicom.net/~dhyams/cvxpt.htm 4-same as 1, page 631.

2 k  b  NOTE: Error calculation for b from     m  2m k  m 0125..kg  9 80  mg  s2  N k   160. x 0. 766m m  k  k k x m  x m  mg  g  k    x   m  x2   x    m    m   0125..kg  9 80 2     980. 2     s     s   N  k  2 0. 001m  0.. 001kg  0 01  0. 766m   0. 766m  m        

b

k bm2  2 m  b  b  b bm k   m k 

             k  1 k 12m b    2  2  m   k        m k m   k   k     2     22      m   m   m                N  N       160.  160. 1  m 1 m 1 2   1   N   20125..kg 351 s  1  b    2 351. s  0001. kg     001.   006. s   0125. kg  N 0125. kg N  m  N    160.   160.   160.   m 1 2   m 1 2   m 1 2    351. s    351. s    351. s   0125. kg   0125. kg   0125. kg  kg  b 01. s