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.DifficultiesDifficulties 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.BibliographyDe Jong, Physics Two Heinman Physics in Context, Australia 1994
McGraw-Hill Encyclopedia of Physics 2nd edition, 1993Appendix OneSize 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 6.71×106
-36 2.05×106
-25 2.62×105
-20 1.34×105
-15.4 6.65×104
-10.8 3.55×104
-4.2 1.49×104
0 12,100
6 6,260
15 2,330
20 1,490
25 954
30 629