Extended Practical Investigation
Purpose
The Purpose of this investigation is to explore how the terminal velocity of a sphere
falling through glycerol varies with the temperature of the glycerol and the size of the
sphere.
Introduction
In the early stages of the project it was intended to investigate how the speed of a
sphere falling through glycerol varies with the size of the sphere. However, after
analysis it was decided that the investigation would be more callenging if a second
variable was incorporated. There are many constants that could have been manipulated
such as, amount of glycerol used, distacnce over which times were taken, distance
sphere was allowed to fall before timing was taken and the temperature of the glycerol.
After much consultation it was decided that the temperature of the glycerol should be
varied. Once this had benn incorporated into the investigation some scientific concepts
related to the viscosity of a liquid had to be attained. (Refer to article).
In conducting the experiments an attempt was made to attain results that could, produce
graphs that showed the terminal velocity of a sphere related to the temperature of the
glycerol and the terminal velocity of a sphere related to its size.
Apparatus used
? 600 ml of glycerol (density 1.26/ml. Assay 98.0 - 101.0%)
? Small ball bearings of radius: 3.175mm
3.960mm
5.000mm
6.000mm
7.000mm
? 900 ml measuring cylinder
? Stop watch
? Thermometer
? Some type of heating and cooling device to varie the temperature of the glycerol
? Tweezers
Variables and Constants
The variables that have been used in this investigation are the size of the ball
bearings and the temperature of the glycerol. The constants that have been used in this
investigation are the amount of glycerol used, the size of the measuring cylinder, the
intervals at which time were taken, the distance the sphere was allowed to drop before
times were taken and the number of tests taken.
Method
To begin experimentation the distance over which the sphere accelerates to reach
terminal velocity had to be determined. This was done by systematically varying the
distance over which the sphere was allowed to fall then finding the point at which the
spheres acceleration is zero. It was found that for the sphere to reach terminal
velocity it had to be allowed to fall 6 - 7 centimeters before an accurate, constant
reading could be taken. It was found that the distance needed for a sphere to reach
terminal velocity is only slightly changed when the temperature of the glycerol is varied
(+/- 0.2cm).
To attain that the sphere had reached terminal velocity by varying the distance that the
sphere fell before timing began, the distance was varied from 2cm to 10cm. Starting at
2cm the measuring cylinder was marked at 2cm intervals and times were taken for each
interval. the times taken were analysed to determine if the rate of descent of the
sphere was constant for each reading. To ensure that the sphere had reached terminal
velocity a full 10 cm of descent was allowed.
Using 'Stokes law for the terminal velocity of a sphere falling under gravity' and the
relationship of mg = U + F at terminal velocity the above result is proven. These
calculations can be seen in the results section.
For all experiments room temperature was recorded at 20oc.
The first part of the experiment was to vary the size of the ball bearing but not the
temperature. A sphere of 3.175mm in diameter was dropped from just above water level and
allowed to fall 6 cm before timing began. Once the sphere had fallen the initial 6 cm
timings were taken at intervals along the measuring cylinder every 200ml (10cm). This
experiment was repeated 4 times and an average was taken.
The experiment was then repeated using ball bearings of sizes 3.960mm, 5.00mm, 6.00mm,
7.00mm, 9.00mm. Each individual experiment was repeated 4 times and an average was
taken. All results are shown in the results section.
The second part of the experiment was to vary the temperature of the glycerol but not
the ball bearing size. A sphere of 3.175mm was chosen to be used in all experiments,
due to its extremely slow descent rate. The same procedure as above was used except five
temperatures of 7oc, 12 oc, 15 oc, 17 oc and 20 oc for the glycerol were used.
Results
The averaged results obtained from the experiment are presented in the following tables
and graphs. (For full documentation of all the results obtained refer to appendix 1.)
Size of Sphere Timing Interval No Averaged Results Averaged Velocity Temperature Timing
Interval No Averaged Results Averaged Velocity
3.175mm 1 2 3 1.4371.6001.637 6.178 cm/s 7oc 123 5.5758.1158.095 1.24 cm/s
3.960mm 1 2 3 0.7500.9820.970 10.240 cm/s 12oc 123 2.6504.4754.400 2.24 cm/s
5.000mm 1 2 3 0.5000.6900.680 14.598 cm/s 15oc 123 2.3753.6573.755 2.68 cm/s
6.000mm 1 2 3 0.7850.6550.627 15.600 cm/s 17oc 123 2.1252.9352.977 3.36 cm/s
7.000mm 1 2 3 0.3400.3600.360 27.777 cm/s 20oc 123 1.4801.6021.627 6.17 cm/s
Chart One
Note that there is a reflex error for all the recordings of +/- 0.1 seconds. Also, the
first timing interval cannot be used for any calculations as the sphere has not yet
reached terminal velocity.
This is a graph representing how the velocity of a 3.175mm sphere varies with the
temperature of the glycerol.
This is a graph representing how the velocity of a sphere varies with the diameter of
that sphere.
Analysis of Results
Chart One demonstrates that as expected the terminal velocity of the sphere increases as
the temperature of the glycerol and the size of ball bearings increase. Graphs one and
two visually illustrate this point and it can be seen by the positive gradient shown. It
is interesting to note that the change of velocity with the temperature is signifigantly
greater as the temperature becomes higher (15oc to 20.5 oc). The reason for this is
directly related to the change in viscosity as the temperature is varied. As the
temperature increase the viscosity becomes less and so the sphere is able to move freely
through this less viscous liquid thus having a greater terminal velocity. A chart of
temperatures and their relative viscosities for glycerol is shown in appendix two. A
hypothetical relationship can be developed between velocity and temperature. The shape
of the graph, although not smooth, is a curve and therefore it is reasonable to suggest
that the relationship would invole T to the power of something: ie) v = kTn (where k is
a constant). Thus, Log10v = Log10k + n Log10T, where n takes the gradient value. If a
graph of Log10v vs. Log10T is plotted it may be possible to form a relationship.(Graph
3)
A line of best fit for the above graph gives a gradient of 2.69. Therefore a hypothesis
for the relationship between velocity and temperature is V = kT2.69. Of course for the
results to be most accurate the sphere would ideally have reached terminal velocity when
the times in graph three were taken. An attempt has been made to calcualte the terminal
velocity at 20oc using stokes law and the relationship mg = U + F at terminal velocity so
that it can be compared to the velocity found at this temperature.
THIS GRAPH SHOWS HOW VELOCITY VARIES WITH TIME
Refering to the graph the velocitites of the ball bearings for each temperature are
shown. These results can be proven using Stoke's Law (for a detailed description of
Stoke's Law and other related physics concepts refer to the article), but due word limit
restrictions these calculations have been removed.
From chart one a relationship between the size of a ball bearing and its velocity can
also be formed. Studying graph two it can be seen that there is a gradual curve which
indictes that it is reasonable to suggest that the relationship would once again involve
T to power of something. Therefore a relationship could be formed using a Log-Log graph,
shown below.
Using a line of best fit the gradient can be found as 0.638. Therefore the relationship
between the Log of Velocity vs. Log of Diameter is V = kD0.638. All discrepancies in
calculations for graph five and the same as for graph three.
Difficulties
Difficulties encounted during this investigation are:
? Trying to establish weather the sphere had reached terminal velocity before timing
began.
? Trying to maintain the temperature attained once the glycerol has been heated or
cooled.
? Human errors when timing.
? Human errors in general.
? Transfering the glycerol from the measuring cylinder to bottles without loosing any.
? Trying to hold the ball bearings just above the glycerol without dropping them in.
? Trying to perform as many tests as possible (in an effort to get a more accurate
average) within the time allocated in class.
Although every difficulty was hard work to around, trying to establish weather the
sphere had reached terminal velocity before timing began was the main difficulty
encountered.
Errors
% error in distance = 0.15cm x 100 = 1.5%
10cm 1
% error in time = 0.36s x 100 = 4.4% This is in regard to human error in 8.1
1 responding with the stopwatch.
% error in velocity = 8%
% Error in temperature = 7 x 100 = 32% This allows for a possible increase
20 1 or decrease in temperature whilst the experiment was taking
place or for the chance that the thermometer wasn't calibrated correctly
Error in radius = 1% This accounts for human error in
1 reading the measurements or that the radius' of the spheres used was
not uniform.
% Error in velocity calculations
using Stoke's Law and mg = U + F = 1%
Success of The Investigation
The aim of this investigation was show that the terminal velocity of a sphere falling
through glycerol varies with the temperature and the size of the sphere. From the
results shown I believe that the investigation was a success.
Conclusions
As a result of this investigation it can clearly be concluded that as the temperature of
glycerol increases, viscosity decreases and therefore any sphere falling through the
glycerol will experience an increase in terminal velocity. Also the rate of increase in
velocity is greater as the temperature rises. This is because the less viscous the state
of the glycerol, the more freely the sphere is able to fall. It can also be concluded
that as the diameter of the sphere increases the weight of the sphere increases and
therefore its terminal velocity increases.
Bibliography
De Jong, Physics Two Heinman Physics in Context, Australia 1994
McGraw-Hill Encyclopedia of Physics 2nd edition, 1993
Appendix One
Size of Sphere Test 1 Test 2 Test 3 Test 4 Average
3.175mm 1 1.5802 1.9503
1.940 1.2801.4101.570 1.5501.5401.410 1.3401.5001.630 1.4371.6001.637
3.960mm 1 0.7502 1.0403
1.050 0.7500.9100.910 0.7200.9700.950 0.7801.0100.990 0.7500.9820.970
5.000mm 1 0.5302 0.6303
0.670 0.4400.4800.470 0.5300.7400.610 0.4800.5100.590 0.5000.5900.590
6.000mm 1 0.7402 0.6403
0.580 0.6600.6500.670 0.9600.6600.660 0.7800.6700.600 0.7850.6550.627
7.000mm 1 0.3102 0.3603
0.340 0.3600.3500.370 0.3300.3600.350 0.3500.3700.380 0.3400.3600.360
Temperature Test 1 Test 2 Test 3 Test 4 Average
7oc 1 5.4252 8.0503
8.060 5.9008.2508.150 5.3008.1008.050 5.6008.0608.050 5.5008.0508.060
12oc 1 2.7002 4.5403
4.420 2.8004.6004.700 2.6004.5004.450 2.5004.3004.400 2.7004.5004.400
15oc 1 2.3002 3.6303
3.920 2.3003.6003.800 2.4003.7003.700 2.5003.8003.600 2.3003.5303.920
17oc 1 2.0402 2.8903
3.360 2.0002.9003.000 2.2002.9502.950 2.3003.0002.900 2.0002.8903.060
20oc 1 1.4402 1.6003
1.640 1.5001.6001.650 1.4501.6101.630 1.5301.6001.590 1.4401.6001.640
Appendix Two
This chart demonstrates that as temperature increase there is a signifigant decrease in
the viscosity.
Temp. oc Viscosity cp
-42 6.71x106
-36 2.05x106
-25 2.62x105
-20 1.34x105
-15.4 6.65x104
-10.8 3.55x104
-4.2 1.49x104
0 12,100
6 6,260
15 2,330
20 1,490
25 954
30 629
Physics CAT One
Extended Practical Investigation
Report
Student Number:
Purpose
The Purpose of this investigation is to explore how the terminal velocity of a sphere
falling through glycerol varies with the temperature of the glycerol and the size of the
sphere.
Introduction
In the early stages of the project it was intended to investigate how the speed of a
sphere falling through glycerol varies with the size of the sphere. However, after
analysis it was decided that the investigation would be more callenging if a second
variable was incorporated. There are many constants that could have been manipulated
such as, amount of glycerol used, distacnce over which times were taken, distance
sphere was allowed to fall before timing was taken and the temperature of the glycerol.
After much consultation it was decided that the temperature of the glycerol should be
varied. Once this had benn incorporated into the investigation some scientific concepts
related to the viscosity of a liquid had to be attained. (Refer to article).
In conducting the experiments an attempt was made to attain results that could, produce
graphs that showed the terminal velocity of a sphere related to the temperature of the
glycerol and the terminal velocity of a sphere related to its size.
Apparatus used
? 600 ml of glycerol (density 1.26/ml. Assay 98.0 - 101.0%)
? Small ball bearings of radius: 3.175mm
3.960mm
5.000mm
6.000mm
7.000mm
? 900 ml measuring cylinder
? Stop watch
? Thermometer
? Some type of heating and cooling device to varie the temperature of the glycerol
? Tweezers
Variables and Constants
The variables that have been used in this investigation are the size of the ball
bearings and the temperature of the glycerol. The constants that have been used in this
investigation are the amount of glycerol used, the size of the measuring cylinder, the
intervals at which time were taken, the distance the sphere was allowed to drop before
times were taken and the number of tests taken.
Method
To begin experimentation the distance over which the sphere accelerates to reach
terminal velocity had to be determined. This was done by systematically varying the
distance over which the sphere was allowed to fall then finding the point at which the
spheres acceleration is zero. It was found that for the sphere to reach terminal
velocity it had to be allowed to fall 6 - 7 centimeters before an accurate, constant
reading could be taken. It was found that the distance needed for a sphere to reach
terminal velocity is only slightly changed when the temperature of the glycerol is varied
(+/- 0.2cm).
To attain that the sphere had reached terminal velocity by varying the distance that the
sphere fell before timing began, the distance was varied from 2cm to 10cm. Starting at
2cm the measuring cylinder was marked at 2cm intervals and times were taken for each
interval. the times taken were analysed to determine if the rate of descent of the
sphere was constant for each reading. To ensure that the sphere had reached terminal
velocity a full 10 cm of descent was allowed.
Using 'Stokes law for the terminal velocity of a sphere falling under gravity' and the
relationship of mg = U + F at terminal velocity the above result is proven. These
calculations can be seen in the results section.
For all experiments room temperature was recorded at 20oc.
The first part of the experiment was to vary the size of the ball bearing but not the
temperature. A sphere of 3.175mm in diameter was dropped from just above water level and
allowed to fall 6 cm before timing began. Once the sphere had fallen the initial 6 cm
timings were taken at intervals along the measuring cylinder every 200ml (10cm). This
experiment was repeated 4 times and an average was taken.
The experiment was then repeated using ball bearings of sizes 3.960mm, 5.00mm, 6.00mm,
7.00mm, 9.00mm. Each individual experiment was repeated 4 times and an average was
taken. All results are shown in the results section.
The second part of the experiment was to vary the temperature of the glycerol but not
the ball bearing size. A sphere of 3.175mm was chosen to be used in all experiments,
due to its extremely slow descent rate. The same procedure as above was used except five
temperatures of 7oc, 12 oc, 15 oc, 17 oc and 20 oc for the glycerol were used.
Results
The averaged results obtained from the experiment are presented in the following tables
and graphs. (For full documentation of all the results obtained refer to appendix 1.)
Size of Sphere Timing Interval No Averaged Results Averaged Velocity Temperature Timing
Interval No Averaged Results Averaged Velocity
3.175mm 1 2 3 1.4371.6001.637 6.178 cm/s 7oc 123 5.5758.1158.095 1.24 cm/s
3.960mm 1 2 3 0.7500.9820.970 10.240 cm/s 12oc 123 2.6504.4754.400 2.24 cm/s
5.000mm 1 2 3 0.5000.6900.680 14.598 cm/s 15oc 123 2.3753.6573.755 2.68 cm/s
6.000mm 1 2 3 0.7850.6550.627 15.600 cm/s 17oc 123 2.1252.9352.977 3.36 cm/s
7.000mm 1 2 3 0.3400.3600.360 27.777 cm/s 20oc 123 1.4801.6021.627 6.17 cm/s
Chart One
Note that there is a reflex error for all the recordings of +/- 0.1 seconds. Also, the
first timing interval cannot be used for any calculations as the sphere has not yet
reached terminal velocity.
This is a graph representing how the velocity of a 3.175mm sphere varies with the
temperature of the glycerol.
This is a graph representing how the velocity of a sphere varies with the diameter of
that sphere.
Analysis of Results
Chart One demonstrates that as expected the terminal velocity of the sphere increases as
the temperature of the glycerol and the size of ball bearings increase. Graphs one and
two visually illustrate this point and it can be seen by the positive gradient shown. It
is interesting to note that the change of velocity with the temperature is signifigantly
greater as the temperature becomes higher (15oc to 20.5 oc). The reason for this is
directly related to the change in viscosity as the temperature is varied. As the
temperature increase the viscosity becomes less and so the sphere is able to move freely
through this less viscous liquid thus having a greater terminal velocity. A chart of
temperatures and their relative viscosities for glycerol is shown in appendix two. A
hypothetical relationship can be developed between velocity and temperature. The shape
of the graph, although not smooth, is a curve and therefore it is reasonable to suggest
that the relationship would invole T to the power of something: ie) v = kTn (where k is
a constant). Thus, Log10v = Log10k + n Log10T, where n takes the gradient value. If a
graph of Log10v vs. Log10T is plotted it may be possible to form a relationship.(Graph
3)
A line of best fit for the above graph gives a gradient of 2.69. Therefore a hypothesis
for the relationship between velocity and temperature is V = kT2.69. Of course for the
results to be most accurate the sphere would ideally have reached terminal velocity when
the times in graph three were taken. An attempt has been made to calcualte the terminal
velocity at 20oc using stokes law and the relationship mg = U + F at terminal velocity so
that it can be compared to the velocity found at this temperature.
THIS GRAPH SHOWS HOW VELOCITY VARIES WITH TIME
Refering to the graph the velocitites of the ball bearings for each temperature are
shown. These results can be proven using Stoke's Law (for a detailed description of
Stoke's Law and other related physics concepts refer to the article), but due word limit
restrictions these calculations have been removed.
From chart one a relationship between the size of a ball bearing and its velocity can
also be formed. Studying graph two it can be seen that there is a gradual curve which
indictes that it is reasonable to suggest that the relationship would once again involve
T to power of something. Therefore a relationship could be formed using a Log-Log graph,
shown below.
Using a line of best fit the gradient can be found as 0.638. Therefore the relationship
between the Log of Velocity vs. Log of Diameter is V = kD0.638. All discrepancies in
calculations for graph five and the same as for graph three.
Difficulties
Difficulties encounted during this investigation are:
? Trying to establish weather the sphere had reached terminal velocity before timing
began.
? Trying to maintain the temperature attained once the glycerol has been heated or
cooled.
? Human errors when timing.
? Human errors in general.
? Transfering the glycerol from the measuring cylinder to bottles without loosing any.
? Trying to hold the ball bearings just above the glycerol without dropping them in.
? Trying to perform as many tests as possible (in an effort to get a more accurate
average) within the time allocated in class.
Although every difficulty was hard work to around, trying to establish weather the
sphere had reached terminal velocity before timing began was the main difficulty
encountered.
Errors
% error in distance = 0.15cm x 100 = 1.5%
10cm 1
% error in time = 0.36s x 100 = 4.4% This is in regard to human error in 8.1
1 responding with the stopwatch.
% error in velocity = 8%
% Error in temperature = 7 x 100 = 32% This allows for a possible increase
20 1 or decrease in temperature whilst the experiment was taking
place or for the chance that the thermometer wasn't calibrated correctly
Error in radius = 1% This accounts for human error in
1 reading the measurements or that the radius' of the spheres used was
not uniform.
% Error in velocity calculations
using Stoke's Law and mg = U + F = 1%
Success of The Investigation
The aim of this investigation was show that the terminal velocity of a sphere falling
through glycerol varies with the temperature and the size of the sphere. From the
results shown I believe that the investigation was a success.
Conclusions
As a result of this investigation it can clearly be concluded that as the temperature of
glycerol increases, viscosity decreases and therefore any sphere falling through the
glycerol will experience an increase in terminal velocity. Also the rate of increase in
velocity is greater as the temperature rises. This is because the less viscous the state
of the glycerol, the more freely the sphere is able to fall. It can also be concluded
that as the diameter of the sphere increases the weight of the sphere increases and
therefore its terminal velocity increases.
Bibliography
De Jong, Physics Two Heinman Physics in Context, Australia 1994
McGraw-Hill Encyclopedia of Physics 2nd edition, 1993
Appendix One
Size of Sphere Test 1 Test 2 Test 3 Test 4 Average
3.175mm 1 1.5802 1.9503
1.940 1.2801.4101.570 1.5501.5401.410 1.3401.5001.630 1.4371.6001.637
3.960mm 1 0.7502 1.0403
1.050 0.7500.9100.910 0.7200.9700.950 0.7801.0100.990 0.7500.9820.970
5.000mm 1 0.5302 0.6303
0.670 0.4400.4800.470 0.5300.7400.610 0.4800.5100.590 0.5000.5900.590
6.000mm 1 0.7402 0.6403
0.580 0.6600.6500.670 0.9600.6600.660 0.7800.6700.600 0.7850.6550.627
7.000mm 1 0.3102 0.3603
0.340 0.3600.3500.370 0.3300.3600.350 0.3500.3700.380 0.3400.3600.360
Temperature Test 1 Test 2 Test 3 Test 4 Average
7oc 1 5.4252 8.0503
8.060 5.9008.2508.150 5.3008.1008.050 5.6008.0608.050 5.5008.0508.060
12oc 1 2.7002 4.5403
4.420 2.8004.6004.700 2.6004.5004.450 2.5004.3004.400 2.7004.5004.400
15oc 1 2.3002 3.6303
3.920 2.3003.6003.800 2.4003.7003.700 2.5003.8003.600 2.3003.5303.920
17oc 1 2.0402 2.8903
3.360 2.0002.9003.000 2.2002.9502.950 2.3003.0002.900 2.0002.8903.060
20oc 1 1.4402 1.6003
1.640 1.5001.6001.650 1.4501.6101.630 1.5301.6001.590 1.4401.6001.640
Appendix Two
This chart demonstrates that as temperature increase there is a signifigant decrease in
the viscosity.
Temp. oc Viscosity cp
-42 6.71x106
-36 2.05x106
-25 2.62x105
-20 1.34x105
-15.4 6.65x104
-10.8 3.55x104
-4.2 1.49x104
0 12,100
6 6,260
15 2,330
20 1,490
25 954
30 629
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