Download Determining the Drag Coefficient for a Ping Pong Ball

Determining the Drag Coefficient for a Ping Pong Ball
Anthony Stanich
Department of Physics and Earth Space Sciences
University of Indianapolis
May 4, 2011
Abstract
The results of the experiment show that the average drag coefficient is 0.001436236 with a standard
deviation of 0.001039045. The results also show that the average terminal velocity of a ping pong ball in free fall is
0.331375 m/s with a standard deviation of 0.013159312 m/s.
Introduction
This experiment was performed in order to gain an understanding of the properties of a fluid dynamics,
especially drag coefficients. The purpose was also to determine the dependency of mass on the drag
coefficient. In fluid dynamics, the drag coefficient is a dimensionless quantity used to quantify the
resistance of an object in a fluid environment such as air or water.
Description of Experiment and Calculations
A ping pong ball in free fall experiences fluid friction caused by the molecules in air. The force is called
the drag force. The drag coefficient depends on the type of surface and the surface area. To test this force
a ping pong ball was used. The ball was released from a height of 2 meters and recorded using a high
speed video camera. The videos were then analyzed using LoggerPro video analysis.
The force on a moving object due to a fluid is described by equation 1.
Equation 1
Where
FD is the drag force
ρ is the mass density of the fluid for air approximately 1kg/m3
v is the velocity of the object relative to the fluid in m/s
A is the reference area in m2
CD is the drag coefficient dimensionless
Typically, the reference area A for simple shapes, such as a sphere, is exactly the same as a cross sectional
area. For other objects (for instance, a rolling tube or the body of a cyclist), A may be significantly larger
than the area of any cross section along any plane perpendicular to the direction of motion.
In order to formulate a differential equation describing motion of an object falling in the atmosphere near
sea level there are two variables to look at time, t and velocity, v. Newton’s 2nd Law states that the sum of
the forces is equal to the mass times the acceleration, which can be written as the time derivative of the
velocity function and is represented by equation 2.
F = ma = m(dv/dt) net force
Equation 2
The forces acting on an object in free fall are gravity and air resistance. Air resistance always opposes the
motion and is depicted in figure 1.
2
Figure 1
The force of gravity on an object is related to the mass of the object as explained in Newton’s Universal
Law of Gravity and shown in equation 3.
F = mg downward force
Equation 3
The force of air resistance can expressed as equation 4, where ϒ= (1/2)ρvCDA.
F = v upward force
Equation 4
Substituting equations 3 and 4 into Newton’s Second Law (equation 2), equation 5 can be derived.
Equation 5
When the object reaches terminal velocity or the point of maximum velocity, the acceleration of the
object is 0m/s2. This allows equation 5 to be simplified to equation 6 resulting in an equation that relates
the velocity of the object to the drag coefficient.
mg=ϒv
Equation 6
By substituting equation 1 in for the drag force in equation 6, equation 7 is derived.
mg=(1/2)ρv2CDA
Equation 7
Solving equation 7 for CD, equation 8 is derived.
CD=ρv2A/2mg
Equation 8
The first step in the experiment was the construction of the release apparatus. I used a piece of 2in PVC
pipe cut to a 2ft length and placed a hole at 1 in from the bottom and affixed it to the ceiling using zip ties.
In the hole I used a zip tie attached to a string that I could pull from the position of the camera. I started
the camera and then pulled the string waited for the ball to drop out of view of the camera and stopped
recording. After that I used the camera to remove frames before and after the drop where the ball was not
in view. I then uploaded the videos into LoggerPro and used the video analysis function to obtain the time
and position of the ball. I was able to change the settings in the analysis to match the frame speed of the
camera, 420 frames/s. I exported the data to excel and created a graph of the terminal velocity. To do this,
I used the trend line function and removed data at the beginning and end until the R2 value was above
3
0.9900. The slope of the position versus time graph is equal to the terminal velocity at this point. I then
used those terminal velocities to find the drag coefficients. Finally, I graphed the drag coefficients as a
function of mass. In order to change the mass of the ping pong ball, I inject water into the ball in
increments of 1mL to reach a final volume of 10mL. For the last trial I completely filled the ping pong
ball with water.
Analysis of Results
Figures 2 through 13 are the graphs of position versus time for a specific amount of water injected into the
ping pong ball and the data set where the R2 value was greater than 0.9900.
Position vs Time for 0mL H2O Added
1.2
x = -0.3201t + 1.3336
R² = 0.9931
Position (m)
1
0.8
0.6
0.4
0.2
0
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Time (s)
Figure 2 Graph of terminal velocity for no water added,
Position (m)
Position vs Time for 1mL H2O Added
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
x = -0.3291t + 1.1002
R² = 0.9915
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Time (s)
Figure 3 Graph of terminal velocity for 1mL of water added
4
Position vs Time for 2mL H2O
Added
x= -0.3199t + 1.0972
R² = 0.9911
Position (m)
1
0.8
0.6
0.4
0.2
0
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Time (s)
Figure 4 Graph of terminal velocity for 2mL of water added
Position vs Time for 3mL H2O
Added
x = -0.3254t + 1.1539
R² = 0.9909
Position (m)
1
0.8
0.6
0.4
0.2
0
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Time (s)
Figure 5 Graph of terminal velocity for 3mL of water added
Position (m)
Position vs Time for 4mL H2O Added
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
x = -0.3405t + 1.1254
R² = 0.9932
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Time (s)
Figure 6 Graph of terminal velocity for 4mL of water added
5
Position vs Time for 5mL H2O
Added
x = -0.3212t + 1.091
R² = 0.9933
Position (m)
1
0.8
0.6
0.4
0.2
0
1
1.5
2
2.5
Time (s)
Figure 7 Graph of terminal velocity for 5mL of water added
Position vs Time for 6mL H2O
Added
x= -0.3377t + 1.1024
R² = 0.9928
Position (m)
1
0.8
0.6
0.4
0.2
0
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Time (s)
Figure 8 Graph of terminal velocity for 6mL of water added
Position (m)
Position vs Time for 7mL H2O
Added
x = -0.3203t + 1.1138
R² = 0.9918
1
0.8
0.6
0.4
0.2
0
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Time (s)
Figure 9 Graph of terminal velocity for 7mL of water added
6
Position vs Time for 8mL H2O
Added
yx= -0.3179t + 1.1051
R² = 0.9912
Position (m)
1
0.8
0.6
0.4
0.2
0
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Time (s)
Figure 10 Graph of terminal velocity for 8mL of water added
Position vs Time for 9mL H2O
Added
x= -0.3409t + 1.1279
R² = 0.9925
Position (m)
1
0.8
0.6
0.4
0.2
0
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Time (s)
Figure 11 Graph of terminal velocity for 9mL of water added
Position vs Time for 10mL H2O
Added
x = -0.343t + 1.1369
R² = 0.9978
Position (m)
1.00
0.80
0.60
0.40
0.20
0.00
1
1.5
2
2.5
Time (s)
Figure 12 Graph of terminal velocity for 10mL of water added
7
Position vs Time for Full of H2O
Added
Position (m)
1
x= -0.3605t + 1.1546
R² = 0.991
0.8
0.6
0.4
0.2
0
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
Time (s)
Figure 13 Graph of terminal velocity for full of water added
Figure 14 is a table that lists the volume of water added in mL, the mass of the ball in kg, and the terminal
velocity in m/s from the graphs.
Volume of H2O
(mL)
0
0.96
1.9
2.96
3.94
4.93
5.98
6.91
8.16
8.89
9.99
21.87
Average
Standard
Deviation
Mass of Ball
(kg)
0.00181
0.00277
0.00371
0.00477
0.00575
0.00674
0.00779
0.00872
0.00997
0.0107
0.0118
0.02368
Terminal Velocity
(m/s)
0.3201
0.3291
0.3199
0.3254
0.3405
0.3212
0.3377
0.3203
0.3179
0.3409
0.343
0.3605
0.331375
0.013159312
Figure 14 Values of volume, mass, and terminal velocity for each trial.
The Grubbs Test determines if a data point is an outlier, equation 9. If the calculated Grubbs value is
greater than the Grubbs table value, the data point is an outlier.
Equation 9 The Grubbs test formula
8
The Grubbs calculated value for the highest and lowest velocities are 2.213 and 1.024, respectively. The
table value for N=12 is 2.412. Therefore there are no outliers in the data set. Figure 15 shows the
relationship between terminal velocity and mass. The data shows that there is a slight positive correlation
between mass and terminal velocity, meaning that as the mass goes up so does the terminal velocity.
Terminal Velocity (m/s)
Mass vs Terminal Velocity
0.37
y = 0.0017x + 0.3175
R² = 0.5647
0.36
0.35
0.34
0.33
0.32
0.31
0
5
10
15
20
25
Mass (g)
Figure 15 Graph of the relationship between mass and terminal velocity.
Using equation 8, the drag coefficients were found. The results for the drag of each mass are shown in
figure 15. The radius of the ball is 0.021m and the cross sectional area is 0.00139m2.
Mass
0.00181
0.00277
0.00371
0.00477
0.00575
0.00674
0.00779
0.00872
0.00997
0.0107
0.0118
0.02368
Average
Standard
Deviation
Drag
Coefficient
0.004001522
0.002763811
0.001949786
0.001569095
0.001425275
0.001081991
0.001034802
0.000831629
0.000716503
0.00076772
0.000704756
0.000387937
0.001436236
0.001039045
Figure 16 Results for the drag coefficients for each mass
Figure 17 shows the relationship between mass and the drag coefficient. The graph shows that there is a
definite correlation between mass and the drag coefficient. The coefficient appears to decrease
exponentially as mass increases.
9
Drag Coefficients as a Function of
Mass
Drag Coefficients
0.005
0.004
0.003
0.002
0.001
0
0
0.005
0.01
0.015
0.02
0.025
Mass (kg)
Figure 17 Relationship between mass and drag coefficients
Conclusions
The results of the experiment show that the average drag coefficient is 0.001436236 with a standard
deviation of 0.001039045. The results also show that the average terminal velocity of a ping pong ball in free fall is
0.331375 m/s with a standard deviation of 0.013159312 m/s. The results show that the terminal velocity of the
ping pong ball has a slight correlation with increasing mass. The results also show that the drag coefficient has a
definite exponential decay relationship with mass.
References
Edward & Penny. Differential Equations. 4th Ed.
Fowles & Cassiday. Analytical Mechanics. 7th Ed.
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