Title.PDF

9 October 2015
Fluid Dynamics
PHYS-2010 & PHYS-2011
Cheyenne N Reed
MIDDLE TENNESSEE STATE UNIVERSITY

PHYS 2011-008 Cheyenne Reed
1
r
h
THE TRUTH OF FLUID RESISTANCE
INTRODUCTION
Description:
This experiment is designed to study the effects of resistance on objects with varied cross-
sectional areas, while falling through a clear liquid. The distance in which they fall through this liquid
will remain constant, three times will be recorded for each object with a stop watch, and two different
liquids will be used as a means to inspect the concept of fluid resistance thoroughly.
Purpose:
The purpose of this experiment is to understand the force exerted by a fluid as an object falls
through it, how the object’s size relates to this force, and how the object reaches a terminal velocity
falling through this liquid. To understand how the viscosity may affect the time it takes for an object to
reach terminal velocity and how the cross-sectional area may change this time. The questions we are
hoping to answer with this experiment are: Does resistance change with different levels of viscosity?
Does size really matter when it is faced with resistance? Is terminal velocity equal for each area in
different viscose liquids? Is terminal velocity equal for all objects?
EXPERIMENT, MATERIALS, AND MEASURING DEVICES
Detail of Experiment:
Equations that will be used during this experiment:











vis
VisFormula
a
acbb
v
tyTermVeloci
vACF
DragForce
RvF
VisForce
VgBF
rceBuoyancyFo
term
cDD
:
2
4
2
1
:
6
:
:
2
2   maF
sSumofForce

2
DF
mgFg 
BF F
y
x
Objects:
 Poker Chip: Coin 1
 Token: Coin 2
 US Quarter: Coin 3
 US Dime: Coin 4
The experiment uses two 1.5 L bottles attached together with tape/hot glue, one having both end
pieces cut off and the other only having the top removed. The distance of the drop was measured with a
ruler, and the time was measured with a stop watch. A mark 1/8th
of an inch above the drop distance is the
actual height of the liquid, this distance was determined so that the “coins” would be dropped below the
surface. Being dropped below the surface of the water means that we did not have to account for surface
tension. This experiment has no x-axis based forces therefore the only forces that need to be summed in
this case are the y-axis based forces of buoyancy, drag, viscosity, and gravity, taking the positive direction
to be the upward direction. The drop distance for this experiment, through the liquids, is 12”, or 30.48 cm,
this is kept fairly constant due to a mark being placed on the top bottle, the bottle also has a radius of 2”,
or 5.08 cm. The “coins” were dropped 3 times each per liquid, this allows for a more accurate time
measurement.
Uncertainties related to this experiment include time, distance, radius, area, and mass. These will be
calculated and accounted for in the final conclusion. The uncertainty for mass is set at .00005 kg to allow
for dings and loss of material from the objects. The uncertainty for time is set at .25 s to allow for human
error in response time when stopping the stop watch. The uncertainty for radius is set at .005 m to allow
for human error when reading measurements. The uncertainty for distance is taken to be .001 m allowing
for human error. The uncertainty in area and terminal velocity will be calculated.
δ (m)= .00005 kg δ (A) = ?
δ (t) = .25 s δ (h) = .001 m
δ (r) = .005 m δ (vterm) = ?

3
DATA
Table 1.1: Coin Measurements
Object: Mass: Diameter: Radius: Area:
2
r Volume:
hr2

Coin 1 0.0114 kg 0.03255 m 0.01628 m 0.00083 m2
2.7E-6 m3
Coin 2 0.0085 kg 0.02778 m 0.01389 m 0.00061 m2
2.0E-6 m3
Coin 3 0.0056 kg 0.02381 m 0.01191 m 0.00045 m2
8.1E-7 m3
Coin 4 0.0023 kg 0.00873 m 0.00437 m 0.00006 m2
3.4E-7 m3
Table 1.1 contains the mass, diameter, radius area and volume for each coin used during this experiment.
This data was collected with a scale, a ruler and a calculator, using the given equation for area. These
measurements will be used again later in the calculation section of this report.
Table 1.2: Fluid Dynamics
Fluid: Density: Viscosity: Viscosity Coe: Volume: hr2

Water 1000 kg/m3
1.0 m2
/s 1.002 P 0.0024711 m3
Canola Oil 915 kg/m3
44.7 m2
/s 0.411 P 0.0024711 m3
Table 1.2 contains the liquid constants that are associated with the liquids used during this experiment, these
constants include: density, viscosity, viscosity coefficient; volume is also included and was found using a
calculator.

4
Table 1.3: Total Time Traveled Measurements
Fluid: Coin 1 Coin 2 Coin 3 Coin 4
Water TT1: 2.21 s T1: 1.74 s T1: 1.47 s T1: 0.89 s
TT2: 2.32 s T2: 1.62 s T2: 1.43 s T2: 0.95 s
TT3: 2.25 s T3: 1.71 s T3: 1.52 s T3: 0.93 s
TTave: 2.26 s Tave: 1.69 s Tave: 1.47 s Tave: 0.923 s
Canola Oil TT1: 3.5 s T1: 2.74 s T1: 2.17 s T1: 1.47 s
TT2: 3.54 s T2: 2.78 s T2: 2.15 s T2: 1.46 s
TT3: 3.6 s T3: 2.75 s T3: 2.16 s T3: 1.51 s
TTave: 3.55 s Tave: 2.76 s Tave: 2.16s Tave: 1.48s
Table 1.3 contains the collected time data for the four separate coins, as well as in their separate liquids.
Averages were taken for the three drop trials and placed at the bottom of each coin column in their
respected liquid rows.
Table 1.4-Time Increment Measurements (H2O)
Object Time Measurements
T1 T2 T3 T4 T5 T6 T7 T8
Coin 1 .65 1.1 1.3 1.45 1.49 1.57 1.78 2.26 s
Coin 2 .38 .4 .61 .77 .87 1.09 1.25 1.84 s
Coin 3 0.28 s 0.34 s 0.4 s 0.48 s 0.53 s 0.51 s 0.77 s 1.47 s
Coin 4 0.41 s 0.46 s 0.53 s 0.6 s 0.63 s 0.63 s 0.66 s 0.923 s
Table 1.4 contains all eight time measures for each separate coin, this table is for water specifically. T1 is
the shortest distance, while T8 is the farthest distance.

5
Table 1.5-Time Increment Measurements (Oil)
Object:
Time Measurements
T1 T2 T3 T4 T5 T6 T7 T8
Coin 1 1.82 2.01 2.44 2.57 2.63 2.77 2.85 3.55
Coin 2 1.57 1.63 1.69 1.77 1.82 1.8 2.06 2.76
Coin 3 0.97 s 1.03 s 1.09 s 1.17 s 1.22 s 1.2 s 1.46 s 2.16 s
Coin 4 0.967 s 1.02 s 1.09 s 1.16 s 1.19 s 1.19 s 1.22 s 1.48 s
Table 1.5 contains the time measures for each separate coin, T1 being the shortest distance and T8 being
the longest distance. This table is for oil.
Table 1.6-Distance Increment Measurements
Object:
Distance Measurements
H1 H2 H3 H4 H5 H6 H7 H8
Coin 1 0.1048 m 0.1548 m 0.2048 m 0.2148 m 0.2348 m 0.2548 m 0.2748 m 0.3048 m
Coin 2 0.1048 m 0.1548 m 0.2048 m 0.2148 m 0.2348 m 0.2548 m 0.2748 m 0.3048 m
Coin 3 0.1048 m 0.1548 m 0.2048 m 0.2148 m 0.2348 m 0.2548 m 0.2748 m 0.3048 m
Coin 4 0.1048 m 0.1548 m 0.2048 m 0.2148 m 0.2348 m 0.2548 m 0.2748 m 0.3048 m
Table 1.6 contains the exact measures marked to drop the coin from for each time measure above.
ANALYSIS
We took eight height measurements and marked them on the bottle so that the drop heights would
be kept roughly the same for all four coins in both liquids. We recorded the times for each coin to drop
from the various heights and put the data in excel so that the orientation of the graph would be a Distance
vs. Time graph. The results are as follows:

6
Graphs for Water:
Full Time Graph: Last 3 Points:
Coin 1 (Poker Chip):
Results: The graph we acquired from our data plot shows that we have a deceleration in our coin
through the liquid, water in this case. Coin 1 does not reach terminal velocity before it hits the bottom
of the bottle. The last three data points should show a straight line to represent a constant velocity,
however these points still have a slight curve to them.
Coin 2 (Token):
Results: The graph we acquired from our data plot shows that we have a deceleration in our coin
through the liquid, water in this case. Coin 2 did not reach terminal velocity before it hits the bottom of
the bottle. These last three data points have a very linear orientation, which could be taken as a
constant velocity; however, terminal velocity was never reached.

7
Coin 3 (Quarter):
Results: The graph we acquired from our data plot shows that we have a deceleration
in our coin through the liquid, water in this case. Coin 3 reaches terminal velocity for an
instant before it hits the bottom of the bottle.
Coin 4 (Dime):
in our coin through the liquid, water in this case. Coin 4 reaches terminal velocity before it
hits the bottom of the bottle, but speeds up again.

8
Graphs for Oil:
Full Time Graph: Last 3 Points:
Coin 1 (Poker Chip):
in our coin through the liquid, oil in this case. Coin 1 does not reach terminal velocity before
it hits the bottom of the bottle.
Coin 2 (Token):

9
Coin 3 (Quarter):
Coin 4 (Dime):

10
MATHEMATICAL MODEL
Excel calculates the slope using
t
d
v


 where our theoretical calculations take into account the
forces acting on the coins.
Terminal Velocity Calculations:
a
acbb
v
VgmgBFmgc
rFb
ACFa
FFBFmg
maF
term
DD
D
2
4
6
2
1
0
2












)(2
))((4)(
2
4
6
))()()(0036(.
2
1
2
1
0
2
2
D
D
term
term
DD
D
F
BFmgFFF
v
a
acbb
v
VgmgBFmgc
rFb
Aa
ACFa
FFBFmg
maF


















11
Table 1.7- Calculated Measurements for Terminal Velocity (Water)
Table 1.7 contains the components for the quadratic formula, used to calculate terminal velocity in water.
Table 1.8- Calculated Measurements for Terminal Velocity (Oil)
Table 1.8 contains the components for the quadratic formula, used to calculate terminal velocity in oil.
Object: Coin 1 Coin 2 Coin 3 Coin 4
Liquid: Water
a 0.005 0.004 0.003 0.0003
b 0.31 0.26 0.22 .08
c 0.085 0.064 0.047 0.019
vterm .3 m/s .25 m/s .22 m/s .23 m/s
Liquid: Canola Oil
a 0.004 0.003 0.002 0.0003
b 0.13 0.11 0.09 0.03
c 0.09 0.07 0.05 0.02
vterm 0.21 m/s 0.65 m/s 0.56 m/s 0.67 m/s

12
059.0)(
0085.
00005.
)(
)(
)(
2
2
2



C
C
C
mFu
kg
kg
mFu
m
m
mFu

36.0)(
01389.
005.
)(
)(
)(
2
2
2



C
C
C
rFu
m
m
rFu
r
r
rFu

Fractional Uncertainties:
Table 2.1- Mass Uncertainty
Fu(m)
044.0)(
0114.
00005.
)(
)(
)(
1
1
1



C
C
C
mFu
kg
kg
mFu
m
m
mFu

088.0)(
00567.
00005.
)(
)(
)(
3
3
3



C
C
C
mFu
kg
kg
mFu
m
m
mFu

22.0)(
00227.
00005.
)(
)(
)(
4
4
4



C
C
C
mFu
g
kg
mFu
m
m
mFu

Table 2.1- Radius Uncertainty
Fu(r)
307.0)(
01628.
005.
)(
)(
)(
1
1
1



C
C
C
rFu
m
m
rFu
r
r
rFu

42.0)(
01191.
005.
)(
)(
)(
3
3
3



C
C
C
rFu
m
m
rFu
r
r
rFu

14.1)(
00437.
005.
)(
)(
)(
4
4
4



C
C
C
rFu
m
m
rFu
r
r
rFu

Height Uncertainty:
033.0)(
3048.
01.
)(
)(
)(



hFu
m
m
hFu
h
h
hFu


13
148.0)(
69.1
25.
)(
)(
)(
2
2
2



C
C
ave
C
tFu
s
s
tFu
t
t
tFu water

091.0)(
76.2
25.
)(
)(
)(
2
2
2



C
C
ave
C
tFu
s
s
tFu
t
t
tFu oil

Table 2.3- Time Uncertainty (H2O)
Fu(twater)
111.0)(
26.2
25.
)(
)(
)(
1
1
1



C
C
ave
C
tFu
s
s
tFu
t
t
tFu water

170.0)(
47.1
25.
)(
)(
)(
3
3
3



C
C
ave
C
tFu
s
s
tFu
t
t
tFu water

271.0)(
923.0
25.
)(
)(
)(
4
4
4



C
C
ave
waterC
tFu
s
s
tFu
t
t
tFu

Table 2.4- Time Uncertainty (Canola Oil)
Fu(toil)
07.0)(
55.3
25.
)(
)(
)(
1
1
1



C
C
ave
C
tFu
s
s
tFu
t
t
tFu oil

116.0)(
16.2
25.
)(
)(
)(
3
3
3



C
C
ave
C
tFu
s
s
tFu
t
t
tFu oil

169.0)(
48.1
25.
)(
)(
)(
4
4
4



C
C
ave
C
tFu
s
s
tFu
t
t
tFu oil

Table 2.5- Terminal Velocity Uncertainty H2O
Fu(vterm)
342.)(
)(3.)(
)(
)(
4
1



v
rFuv
v
v
vFu
C
C



285.)(
)(25.)(
)(
)(
4
1



v
rFuv
v
v
vFu
C
C



251.)(
)(22.)(
)(
)(
4
1



v
rFuv
v
v
vFu
C
C



262.)(
)(23.)(
)(
)(
4
1



v
rFuv
v
v
vFu
C
C




14
Table 2.6- Terminal Velocity Uncertainty Canola Oil
Fu(vterm)
239.)(
)()(
)(
)(
4
1



v
rFuvv
v
v
vFu
C
C



741.)(
)()(
)(
)(
4
1



v
rFuvv
v
v
vFu
C
C



638.)(
)()(
)(
)(
4
1



v
rFuvv
v
v
vFu
C
C



764.)(
)()(
)(
)(
4
1



v
rFuvv
v
v
vFu
C
C



Table 2.7- Area Uncertainty
Fu(A)
000946.)(
)()(
)(
)(
4
1



A
rFuAA
A
A
AFu
C
C



000695.)(
)()(
)(
)(
4
1



A
rFuAA
A
A
AFu
C
C



000513.)(
)()(
)(
)(
4
1



A
rFuAA
A
A
AFu
C
C



000068.)(
)()(
)(
)(
4
1



A
rFuAA
A
A
AFu
C
C



GENERAL CONCLUSION
Our experiment did not wield the results we had hoped for, with only two of the eight trials
actually reaching terminal velocity; there are a few causes for this, height constraints and human error
being the main culprits. The coin did not fall perfectly flat through the liquids during the different trials,
which could either speed up or slow down the object; height constraints were put in place to make
portability easy, sadly though this worked against us. The perfect height for this experiment would have
been one meter, however this would have been impossible to fill and test properly. Using engineering
based formulas took a lot of variables that I didn’t understand out of the picture. Simplifying the forces to
buoyancy, drag, force of gravity and the force due to viscosity. The calculations were easy once
everything was in its proper units, even though it made everything very small numerically. The difference
between viscosity and density when it comes to liquids is really interesting. Oil is not as dense as water,
yet it is much thicker and causes objects to fall slower. Our experiment put this idea into perspective, it
even showed us that the cross-sectional area plays a large role in the terminal velocity of an object, not to
mention its shape. Each shape has a different drag coefficient, for a coin with a slightly turbulent surface,
like ours, is .0036. Resistance does change with viscosity levels, the higher the viscosity the higher the
resistance. The terminal velocity for each object is unique to that object, because volume of the object and
area play a large part; you could have objects with the same volume but different areas and therefore have
a different terminal velocity. It wasn’t a perfect experiment, but despite that we were still able to
understand the concepts and answer all of our questions.

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