So, in this Experiment, I will measure the viscosity of honey using a ball bearing with a known radius and mass, and honey with a known mass & volume . the terminal velocity in this experiment will be measured using the velocity formula v=s/t , by measuring the distances the ball bearings travel and divide them by the time taken for them to pass through that distance (which is measured using the stop clock) .

Variables:

In this investigation there are many variables that will affect the terminal velocity of ball bearings as they pass through honey. These have been considered and the appropriate precautions taken so that their influence can be observed and analysed and the correct conclusions drawn.

Viscosity – any object moving through a viscous fluid is acted on by friction due to the fluid.

Temperature – An increase in temperature decreases the viscosity of a fluid and therefore decreases the friction opposing the motion of the ball bearings. In this investigation the temperature will be the same for all the readings, and this will be monitored using a thermometer.

Gravitational field – The ball bearings fall through the glycerol due to the attraction created by the gravitational field from the Earth. In this investigation, this can be considered constant at 9.8ms-2.

Friction – Whilst friction created by the viscosity of the fluid has already been discussed, friction due to the sides of the tube may also play a part if the vertical tube is narrow. To minimize this, in the investigation, ball bearings were dropped from the centre of the tube so that they were less likely to experience friction due to the sides of the tube.

Distance – In order to make accurate measurements on the terminal velocity of the ball bearings the distance over which their speed is measured must be kept constant and must be selected so that accurate times can be generated. In this investigation, I will divide the cylinder to 4 parts, each is 4cm long to keep the distance traveled equal to all ball bearings.

Material – the surface of the ball bearings must be constant for each radius since this will affect the area exposed to the friction of the fluid opposing the motion. Ball bearings with a smooth shiny surface were selected for this investigation as these were less likely to have a worn surface that would increase the friction independent of radius.

Radius – each ball-bearing’s radius should be measured, because for each certain radius there is a certain terminal velocity, and using this piece of information, we can get to measure the viscosity of honey.

Likely outcomes

The perfect case (theoretically): as the object falls through the fluid, it keeps accelerating until it reaches a point of force balancing [ equilibrium ] . where it reaches its terminal velocity. Mathematically speaking:

Initial Case

Taking the positive sign as the downwards vectors, we have :

?f= positive value because weight;upthrust + viscous drag

As a ? f (Newton’s 2nd law) ; that means we have a positive acceleration .

I.e.: Velocity is NOT constant, but increasing.

Final Case

?f=0 (Object IS in equilibrium), ie: acceleration=0

ie: constant (MAXIMUM) velocity

That means, TERMINAL Velocity is reached in this case.

So typically, the conclusion for this “perfect” case suggests:

Velocity time1 < Velocity time2 < Terminal Velocity (because we had acceleration)

i.e:

Time reading 1 > Time reading 2 > Time reading for Terminal Velocity.

Apparatus

The apparatus selected for this experiment are chosen in order to generate accurate and reliable results.

5 Ball bearings of various radii

Honey

Large wide graduated cylinder

Thermometer

4 Stop clocks

Micrometer

Metric ruler

Marker pen

Magnet

Balance Scale

Method

The set up of the apparatus should be designed to allow the terminal velocity of ball bearings to be determined accurately.

The mass of the graduated cylinder should be measured using

our balance. The measurement should be as exact as possible.

The honey should be poured into the graduated cylinder at a

height which is enough to determine the different lengths the

ball-bearing will travel through. 5cm beneath the top of the

cylinder would be quite alright.

as the graduated cylinder is being divided into several volume

units, the volume of the honey put inside the cylinder can be

measure easily using (mL) units, which are equal to cm3 .

Then the mass of the Cylinder with the honey inside it should be measured also as exact as possible. So that the honey’s mass could be found without losing any drop of it .

The Mass of Honey = Mass of (the Cylinder+Honey) – Mass of the Cylinder

Now by getting the Volume of the honey and the mass of it, the honey’s density could be measured using:

?f = m/V –> preferably kg/m3

Then a metric ruler should be used to produce marks down the length of the tube at 4cm intervals and a thermometer should be placed inside the tube so that the temperature could be monitored.

A micrometer will be used to accurately measure the diameter of a ball bearing and the reading should be recorded.

A Micrometer is a device which enables you to measure small thicknesses to a high degree of precision. The picture below shows its parts, to measure the diameter of a ball-bearing with a micrometer screw gauge, screw up the micrometer using its ratchet until its jaws are fully closed. Note the reading on the thimble at the datum line- usually this will be zero. If there is a reading apart from zero, then this is the zero error and needs to be recorded to and allowed for. Then screw the thimble open , and put the ball-bearing (the object), and then screw up the thimble until it comes in direct contact with the object, then use the ratchet and screw it up until u hear clicking sounds. Record your measurements from above the datum line, below the dative line and the thimble; then add/subtract the zero error if there is any.

Then the ball bearing should be put on a balance so that its mass can be measured. Though the ball bearing is made of steel, i.e.: its density can be taken from a table, but I would rather measure its density myself, and then compare it to the table’s value.

r = d/2

?s = m/V

?s = m/[(4/3)r3 ?]

?s = 3m/4r3?

The ball bearing should then be held at the surface of the honey and released. The time taken for the ball bearing to travel the first 5cm below the surface would not be measured, I will start measuring once it reaches 5 cm, the measurement should be taken by using the 4 stop clock, each at the corresponding distance once the ball-bearing becomes perpendicular to the viewer’s eye, to provide accuracy; then those measurements should be recorded in a table.

Explanation : each stop clock would be used to measure 4cm distance, the first would measure from the 1st point to the 2nd , once I stop the clock at the 2nd point, I start the other clock, and stop it at the 3rd point, and start the other, and so on.

The ball bearing will then be retrieved from the honey using a magnet dragged up the side of the tube and repositioned at the surface of the honey.

This should be repeated five times to generate three readings at each distance, to allow averages to be taken and the extent of any anomalies to be reduced.

Once the measurements for time will become constant over two consecutive distances, that will mean that terminal velocity had been reached.

The Table below will be used to record the readings.

Diameter =

Distance (m)

0.16-0.12

0.12-0.08

0.08-0.04

0.04-0.00

Time (s) 1

Time (s) 2

Time (s) 3

Mean Time(s)

Terminal Velocity = Distance/Time = (4cm) / time

The means of the three readings should be taken for each of the distances the ball-bearing was dropped in, and then take the mean of the last two readings which should represent the terminal velocity.

After Finishing that part of the experiment, the experiment should be repeated with another 4 ball bearings of different radii , and that will help into getting 4 other Terminal Velocity readings , after which a graph could be plotted from where the viscosity could be measured. Details are provided below.

Radius (mm)

Terminal velocity (ms-1)

Graph:

A Graph should be plotted of terminal velocity against radii such that :

Upthrust + viscous drag = weight

4/3 ?r3 ?f g + 6?r?? = 4/3 ?r3 ?s g

2/3 r2 ?f g + 3r?? = 2/3 r2 ?s g

3r?? = 2/3 r2 ?s g – 2/3 r2 ?f g

? = 2 r2 g (?s – ?f ) / ( 9 ? ) ———————- Special

? = [2 g (?s – ?f) / (9?)] r2 ———————- 1

which implies an equation of a line in a general form of :

y= m x + c

where y: ? <Terminal Velocity> , x: r2 <radius squared> , c=0 and m is the gradient

Note: c=0 because the relationship is “directly proportional” and goes through the origin, that means “y” is NOT elevated any “c” units above the origin.

Gradient = m = 2 g (?s – ?f) / (9?)

==> ? = (2 g (?s – ?f) / 9 ) / m ———————- 2

Considering the dependant variables are ?s, ?f, and we have “g” as a constant ; then the independent variables then will be r2, ?, ? .

From formula 1 , we can conclude that :

r2 ? ? as [radius squared] value increases, terminal velocity increases .

I can then calculate the viscosity from the graph as follows :

Gradient= m= ?y/?x = ??/?r2 from Formula 2 we can say that :

?= (2g(?s – ?f) / 9) x ?r2 /??

Considering a constant velocity; r2 ? ? ; this mean the line that is plotted on the graph should pass through the origin.

Safety Considerations:

I believe this experiment is completely safe, as no sharp edges or hot liquids are used. Clamps and stands could be used as an extra precaution to prevent the cylinder from getting knocked over the table.

Honey is (obviously) a non-poisonous material, so it can be eaten off the hands safely, but I would suggest not to eat any honey due to its direct physical contact with the Lab’s apparatus.

Basic Calculations:

5 Ball Bearings were used, for a quick reference, they were labeled B1, B2, B3, B4, and B5.

B1

B2

B3

B4

B5

Diameter (mm)

12.98

11.76

9.50

7.92

5.98

Radius (mm)

6.49

5.88

4.75

3.96

2.99

Mass (g)

8.963

6.685

3.517

2.049

0.886

Densities calculated as follows:

?s = 3m/4r3?

Then the “Mean” of the calculated densities was taken :

__

Mean = x = ?readings/ no. of readings

B1

B2

B3

B4

B5

Mean

Density(g/cm3)

7.83

7.85

7.83

7.88

7.91

7.86

Density of Steel = ?s = 7.86 x 103 kg/m3 ( + 0.05×103 )

Mass of Graduated Cylinder = 432g

Mass of (Cylinder+Honey)= 1143g

Mass of Honey= Mass of (Cylinder+Honey) – Mass of Graduated Cylinder

= 1143g – 432g = 711g

Volume of Honey = 503 cm3

Density of Honey = ?f = 1.41 x 103 kg/m3

This Figure represents the points at where my measurements were taken

Readings:

Diameter = B5 ( 5.98mm)

Distance (cm)

16-12

12-8

8-4

4-0

Time 1

4.10

4.63

4.81

4.81

Time 2

4.13

4.87

4.89

4.95

Time 3

4.00

4.57

4.83

4.81

Mean

4.08

4.66

4.84

4.86

Time taken by the object at Terminal Velocity = (4.84 + 4.86)/2 = 4.85s

Terminal Velocity= Distance/Time = 4 x 10-2 / 4.85 = 0.82 x 10-2 m/s + 0.005 x10-2

Diameter = B4 ( 7.92mm)

Distance (cm)

16-12

12-8

8-4

4-0

Time 1

2.85

3.27

3.61

3.62

Time 2

2.87

3.06

3.29

3.24

Time 3

2.77

2.98

3.39

3.55

Mean

2.83

3.10

3.43

3.47

Time taken by the object at Terminal Velocity = (3.43 + 3.47)/2 = 3.45s

Terminal Velocity= Distance/Time = 4 x 10-2 / 3.45 = 1.16 x 10-2 m/s + 0.005 x10-2

Diameter = B3 ( 9.50mm)

Distance (cm)

16-12

12-8

8-4

4-0

Time 1

1.69

1.90

1.90

1.87

Time 2

1.59

1.81

1.88

1.86

Time 3

1.84

2.22

2.06

2.12

Mean

1.71

1.97

1.95

1.95

Time taken by the object at Terminal Velocity = (1.95 + 1.95)/2 = 1.95s

Terminal Velocity= Distance/Time = 4 x 10-2 / 1.95 = 2.05 x 10-2 m/s + 0.005 x10-2

Diameter = B2 ( 11.76mm)

Distance (cm)

16-12

12-8

8-4

4-0

Time 1

1.19

1.39

1.50

1.52

Time 2

1.22

1.34

1.46

1.43

Time 3

1.32

1.62

1.51

1.55

Mean

1.24

1.45

1.49

1.50

Time taken by the object at Terminal Velocity = (1.49 + 1.50)/2 = 1.495s

Terminal Velocity= Distance/Time = 4 x 10-2 / 1.495 = 2.676 x 10-2 m/s + 0.005 x10-2

Diameter = B1 ( 12.98mm)

Distance (cm)

16-12

12-8

8-4

4-0

Time 1

0.95

1.02

1.19

1.26

Time 2

1.28

1.07

1.21

1.20

Time 3

0.99

1.26

1.20

1.18

Mean

1.07

1.12

1.20

1.22

Time taken by the object at Terminal Velocity = (1.20 + 1.22)/2 = 1.21s

Terminal Velocity= Distance/Time = 4 x 10-2 / 1.21 = 3.31 x 10-2 m/s + 0.005×10-2

Notes ; Observations:

As is noticed from the above readings, I had some anomalous turning points (which I have marked in red), those are due to human error and inaccurate timings, more explanations will be noted in my evaluation.

A contradiction is noticed, where the object should accelerate to reach terminal velocity, my objects decelerated to achieve terminal velocity. However, I considered that once two consecutive readings are similar, the object has reached terminal velocity, and that’s at the last two readings of each ball-bearing at each time. More will be discussed later on in this investigation.

Terminal Velocities Table :

Radius (mm)

2.99

3.96

4.75

5.88

6.49

Radius2 (mm)

8.9401

15.6816

22.5625

34.5744

42.1201

Terminal Velocity (ms-1)

0.82 x 10-2

1.16 x 10-2

2.05 x 10-2

2.676 x 10-2

3.31 x 10-2

Viscosity Calculations:

Measuring the Viscosity using the mean of the sets of measurements

by using ? = 2 r2 g (?s – ?f ) / ( 9 ? ) (our Special formula , as mentioned under the “Graph” paragraph). We can calculate the viscosity, as follows:

B5:

? = 2 r2 g (?s – ?f ) / ( 9 ? )

? = 2 x 8.9401 x 10-6 x 9.8 x (7.86 – 1.41) x 103 / ( 9 x 0.82 x 10-2 )

? = 15.3145kgm-1s-1

B4:

? = 2 r2 g (?s – ?f ) / ( 9 ? )

? = 2 x 15.6816 x 10-6 x 9.8 x (7.86 – 1.41) x 103 / ( 9 x 1.16 x 10-2 )

? = 18.9892 kgm-1s-1

B3:

? = 2 r2 g (?s – ?f ) / ( 9 ? )

? = 2 x 22.5625 x 10-6 x 9.8 x (7.86 – 1.41) x 103 / ( 9 x 2.05 x 10-2 )

? = 15.4599 kgm-1s-1

B2:

? = 2 r2 g (?s – ?f ) / ( 9 ? )

? = 2 x 34.5744 x 10-6 x 9.8 x (7.86 – 1.41) x 103 / ( 9 x 2.676 x 10-2 )

? = 18.1485 kgm-1s-1

B1:

? = 2 r2 g (?s – ?f ) / ( 9 ? )

? = 2 x 42.1201 x 10-6 x 9.8 x (7.86 – 1.41) x 103 / ( 9 x 3.31 x 10-2 )

? = 17.8745 kgm-1s-1

Radius (mm)

2.99

3.96

4.75

5.88

6.49

Radius2 (mm)

8.9401

15.6816

22.5625

34.5744

42.1201

Viscosity(poise)

15.3145

18.9892

15.4599

18.1485

17.8745

And by taking the mean of those five observed readings, we can reach a more accurate value of viscosity; thus

Viscosity of honey is :

Mean = (15.3145+18.9892+15.4599+18.1485+17.8745)/3 = 17.15732 poise + 10%

Measuring the Viscosity using the gradient from the graph

From the Graph on Page 11 (Graph2), the gradient was calculated by drawing two lines, the vertical one shows a difference in the y-axis and the horizontal one shows a difference in the x-axis , and then dividing ?y/?x where we would get the value of the gradient, and as I said earlier; viscosity could be found using the following formula:

? = (2 g (?s – ?f ) / 9 ) / m where m: gradient

the measured values on the graph were :

?y: 2.4 x 10-2

?x: 3.0 x 10-5

m: 8.0 x 102

Viscosity:

? = ( 2 x 9.8 x (7.86 – 1.41) x 103 / 9 ) / (8.0 x 102)

? = 17.56 poise + 10%

The Viscosity of Honey is

17.56 poise

Note: The Second way is more accurate than the first way, because it takes an over-all value given on a shape which is accurately drawn; hence, I will consider the value 17.56 poise as the viscosity of honey, and this would be the value I conclude.

Conclusion

The Viscosity of Honey is

17.56 poise + 10%

Radius (mm)

2.99

3.96

4.75

5.88

6.49

Terminal Velocity (ms-1)

0.82 x 10-2

1.16 x 10-2

2.05 x 10-2

2.676 x 10-2

3.31 x 10-2

These results (above) have been plotted in Graph 1 in order to be able to examine the relationship between the radius of ball bearings and the terminal velocity.

The graph shows that as the radius increases, the terminal velocity also increases. The upward slope of the graph indicates a power relationship such as

? ? radius2

To see if this relationship is present a further graph has been plotted (see Graph 2). This graph clearly shows a straight line of best fit that passes through the origin. This proves that the terminal velocity is proportional to the radius squared.

Evaluation

In order to assess the validity of the statement above, I have calculated the uncertainties in the measurements and plotted these on Graph 2 as error bars.

From the graph it is clear that the uncertainties obtained in this investigation are large. This is due to variation of readings for time for each distance traveled. Whilst the line of best fit on the graph shows a distinct trend, the error bars make it clear that the errors are too large to be sure.

To reduce these errors, a more accurate method for measuring the speed of the ball bearings should be found. This might be done in several ways:

Increasing distance – by increasing the distance over which the time taken by the ball bearings is measured, reaction times associated with using a stop clock would be less significant and accuracy would be increased. However, increasing this distance would decrease the accuracy of finding the exact point at which terminal velocity was reached.

Smaller radius – it is clear from the results that a smaller radius of ball bearing decreases the speed at which it travels. By using overall much smaller ball bearings throughout the experiment, the time measurement could be taken more accurately.

Apparatus – the experiment could be redesigned so that apparatus such as light gates could be used to provide an accurate method to obtain data for the speed of the ball bearings.

As predicted , from the “Likely outcomes” back in my plan, every speed-time graph should show that from rest, the ball bearings accelerate to reach a maximum speed which becomes constant (terminal velocity).

But, on my sheets, each speed-time graph showed that from rest, the ball bearings reached a maximum speed which then decreased until a constant speed (terminal velocity) was reached. What should of happened is that from rest the ball bearings accelerate until frictional forces balance and the resultant force = w – F = 0.

The unusual trend found in my results is consistent but the large inaccuracies associated with the experiment may have led to its presence. These inaccuracies arose from:

Measurement of distance – The measurement of distance is accurate to ï¿½ 1mm because each reading is accurate to ï¿½ 0.5mm. this would probably only have contributed a small error in the investigation and not produced the unusual trends present on the graphs.

Measurement of radius – the accuracy of the radius measurements of ball bearings is a ï¿½0.01mm because each reading is accurate to ï¿½0.005mm. This measurement should have been accurate enough to determine the relationship between v and r effectively.

Measurement of time – Digital stop-clocks can give reading precise to within ï¿½0.01seconds. But human error makes readouts accurate to only around ï¿½0.10s. Combined with errors associated with reaction times in starting and stopping the stopwatches, this is a likely source of a large proportion of the uncertainty in this investigation.

Density Concentration Changes – After I set the apparatus on one evening in the LAB, I had to do the experiment the next day because of the time this experiment consumes; So in regard to that, I have noticed, when I came to do the investigation, that the honey’s colour has become darker at the bottom of the tube, which I believe means that it has settled down over-night , so dense particles got downwards, while lighter particles stayed at the top of the cylinder. This has certainly changed the gradient of Density/Viscosity that I am dealing with, and I suggest that the unusual trends found from my results (especially the one referring to a deceleration to reach Terminal Velocity) has to have a relation to these changes. My evaluation to this situation was that as the particle went downwards, the number of heavy particles increased, the density increased; hence the upthrust increased, which resulted in decelerating the ball-bearing until it reached its terminal velocity at the denser part of the honey.

Variation of temperature – Though a thermometer should have been used in this investigation to monitor the temperature, I didn’t use one. That is because I considered that the temperature would stay constant if the experiment was done within a 2-hour period in the same Lab-room.

Friction – Frictional forces acting from the sides of the cylinder may have had a significant effect on the ball-bearings by opposing their motion. This may have produced anomalous results that gave the unexpected trends on the graphs. To eliminate this in a future investigation, a much wider cylinder should be used. Also tweezers can be used to hold the ball-bearing and emerge it within the honey (at the centre) before letting it fall, this will let the ball-bearing bypasses the force affected on it by the surface tension of honey as well as preventing it from slipping towards the sides of the cylinder.

The most unusual trends/anomalies were found at the ball-bearings with the big radii, and this is because those ball-bearings had high terminal velocities, i.e.: they could move so fast downwards which made the measurements more inaccurate; and this supports my idea of using ball-bearings with smaller radii to achieve more accuracies, as it reduces the rate of the human error when starting/stopping/switching the stop-clocks.