The velocity of the falling ball bearing will be measured while the ball bearing reaches terminal velocity, as the velocity of the ball bearing can be simply measured through: speed = distance/ time. Moreover, during constant speed, the net forces acting on the ball bearing is zero, and the following equation is valid and can be applied:
Upthrust (U) + Viscous Drag (F) = Weight (W)
4πr3ρfluidg / 3 6πrηv = 4πr3psteel g / 3
r is the radius of the sphere, ρfluid is the density of water, ρsteel is the density of steel, and v is the terminal velocity at which the sphere travels at. We could now rearrange the formula to find the viscosity, µ:
F = W - U
6πrηv = 4πr3ρfluidg / 3 - 4πr3ρsteel g / 3
(Cancel out π and r from each side, multiply each side by 6 and divide each side by 2):
9ηv = 2r2ρsteel g - 2r2ρfluidg
(take 2r2g as the common factor of the right hand side and make η the subject):
η = 2r2 g (ρsteel - ρfluid) / 9v
With this equation, we are now able to calculate the viscosity of glycerol using the falling ball viscometer.
To investigate how temperature affects viscosity of a fluid.
It has been predicted that the viscosity of the fluid will increase at higher temperatures. This is because at higher temperatures, more heat energy will be given to the molecules in the fluid. Thus molecules will move and vibrate faster. This increase in movement is not sufficient for the substance to change state, however causes greater gaps between molecules, therefore molecules will be more spread out. The density of the fluid will thus decrease. As there are fewer molecules per unit volume, the viscous drag that resists the downward movement of the ball bearing will be less, as there are fewer molecules per unit volume to block the movement. Consequently viscosity decreases.
Observations will be taken at the following temperatures: 20, 30, 40, 50, 60 and 70 degrees Celsius. For each temperature, five results will be taken, the average between them will be obtained. It has not been planned to take experiments below room temperature, as that would require using ice and will further complicate the experiment and lengthen the duration. It is thought not to be necessary in doing so.
The viscosity of the glycerol will be obtained by measuring the velocity of the ball bearing. By using Stokes law:
η = F/6πrv
where η is the coefficient of viscosity of the fluid, F is the force opposing the direction of the ball bearing, and r is the radius of the ball bearing, it can be seen that viscosity is inversely proportional to the velocity of the ball bearing. Thus, by realising the relationship between temperature and velocity, the relationship between temperature and viscosity can be obtained.
The apparatus which will be needed are:
- A stopwatch to measure the speed of the ball bearing.
- Glycerol which will be used as the medium in which the ball bearing flows.
- A long plastic tube in which to contain the glycerol fluid.
- A stand and clamp to hold on to the plastic tube.
- A ball bearing.
- A metre ruler to set the distance that the ball bearing moves.
- A magnet to help take out the ball bearing.
- A calliper to measure the radius of the ball bearing.
- A thermometer to measure the temperature of the glycerol.
Equipment will be set up as shown in figure below.
Glycerol is heated in tubs placed in hot water baths at different temperatures, at 50°C, 70°C and 90°C. However, the temperature of the glycerol may not be at the temperature of the hot water baths. Thus, the temperature of each tub of glycerol is measured. The tub of glycerol which is at a higher temperature than, but closest to the desired temperature is used. The temperature of the glycerol is closely monitored until it drops to 2°C higher than the desired temperature, when it is carefully poured into the long plastic tube carefully using a funnel. This is due to taking into account that the temperature of glycerol will drop during the experiment. During the preliminary tests, it is found that on average during the experiment, the temperature of the glycerol drops by about 4°C. By starting at +2°C, to -2°C, of the desired temperature, avoids giving biased results.
Prior to taking measurements, the distance travelled by the ball bearing will be marked. 0.3m will be marked from the bottom of the plastic tube, when the bottom of the ball bearing is seen to pass through this line, timing will start. When the bottom of the ball bearing touches the bottom surface of the plastic tube, the stopwatch will be stopped. As the terminal velocity of the ball bearing needs to be measured in this experiment, another line is drawn 0.1m above the 0.3m line. This is where the glycerol will be poured up to. This 0.1m of glycerol should be sufficient for the ball bearing to reach terminal velocity before timing starts at the 0.3m line.
During the experiment, the 1cm diameter ball bearing will be used. Preliminary tests found that this ball bearing drops through the glycerol at a lower speed than larger ball bearings. This maybe due to having a lower weight in relation to the upthrust and viscous drag. This gives more accurate results as the timing is initiated by the operator, there is a greater percentage uncertainty if the ball bearing drops too quickly. The ball bearing used will not be released from air, as the ball bearing will accelerate quicker in air and thus may enter the glycerol at a higher speed than the terminal velocity in glycerol. Consequently, the ball bearing may have to take a longer distance to reach terminal velocity in the glycerol where the limited distance allowed may not be sufficient. Moreover, if the ball bearing is released in air, a splash may occur when the ball bearing enters the glycerol, this may cause a mess and will be time consuming to having to constantly clean the bench after each experiment.
The ball bearing will be held stationary at the 0.4m line in the glycerol with a magnet outside the plastic tube. To release the ball bearing, the operator moves the magnet away from the tube. However, this causes the ball bearing to drop while having contact with the wall of the plastic tube. This may cause more resistant force acting upwards on the ball bearing as there is friction between the ball bearing and the wall of the plastic tube. This cannot be avoided if the ball bearing is released while it is submerged in glycerol. However, this will happen in each experiment, to make the test as fair as possible.
Each test will be repeated four times to gather a total of five times for each temperature of glycerol. The experiment will start at the highest temperature, 70°C. Glycerol at room temperature will be stirred thoroughly into the hot glycerol in a separate container when the lower desired temperature is needed, and the new glycerol will be poured back into the plastic tube to the 0.4m line.
Making a fair test:
To ensure that results are unbiased. All variables, except for the temperature of glycerol, will be kept constant.
The type of fluid used needs to be the same. Therefore, only glycerol will be used. As different fluids have a different viscosity coefficients. Thus, at the same temperature, different fluids will have a different viscosity.
The concentration of glycerol will be kept constant, by using the same glycerol throughout the experiment. As the concentration of fluids directly affects its viscosity due to difference in density. If it has a higher density, there would be more particles in a given volume. Therefore, the ball bearing must move through a larger amount of particles, which provides more resistance, and thus viscous drag. If the concentration of glycerol is reduced, the density of the fluid is reduced. Thus, the viscosity will be reduced.
Errors and accuracy:
During this experiment, instrumental and human errors exist. Human errors may occur in measuring the distance travelled by the ball bearing. A parallax error of ±0.005m will exist during the measuring of the distance using a ruler. Moreover, the marker used to draw the lines is 3mm thick, this gives a further ±0.003m uncertainty.
In addition, error will occur in using the stopwatch. It cannot be ensured that the operator starts and stops the stopwatch at the very instant the ball bearing passes the line or touches the bottom of the plastic tube. There is an estimated error of ±0.1s on each end. This therefore gives a total of ±0.2s.
Instrumental errors will also occur. The metre ruler used measures to the nearest millimetre. . Therefore there is a ±0.001m uncertainty on the ruler used on both ends altogether. Moreover, errors will occur in measuring the time in which the ball bearing covers the given distance. The stopwatch used measures to the nearest ±0.01 seconds. Thus, it will have an error of ±0.01 seconds.
Error in recording time: 0.2s + 0.01s = ±0.21s
Error in measuring distance travelled: 0.005m + 0.003m + 0.001m = ±0.009m
Error in calculated speed (speed = distance/time): 0.009 / 0.21 = ±0.043m/s
Now in order to work out the viscosity of glycerol, we needed to work out its density. It was obtained by measuring the mass of 100cm3 in a cylinder using a digital balance. The mass of the honey came out to be 139.5g (or 0.1395kg). Therefore the density of the honey turns out to be:
Density = Mass / Volume
= 0.1395kg / 0.00010m3
1400kg m-3 (2 s.f.)
Now viscosity can be worked out:
µ = 2r2 g (psteel - pfluid) / 9v
= [2 x 0.0052m x 9.81Nkg-1 (7700Nsm-2 - 1400Nsm-2)] / [9 x v (ms-1)]
= 3.09015/9v Nsm-2
= 3.09015/9 x 1/v
= 0.34335/v Nsm-2
Interpreting and Evaluating
In the graph showing terminal velocity of ball bearing at different temperatures of glycerol, it can be observed that velocity increases with an increase in temperature. All of the points are close to the line of best fit. The points that are slightly more deviated from the line of best fit are the ones at a temperature of 40 and 50 degrees Celsius. As the deviation is not large, it is most probably due to experimental errors.
As all the points are close to the line of best fit, the graph suggests that temperature is proportional to velocity. From Stokes Law: η = F/6πrv, viscosity is inversely proportional to the velocity. Thus, as temperature is proportional to velocity, viscosity must decrease proportionally to an increase in temperature. As temperature increases, viscosity decreases proportionally.
This can be explained by my theory explained in the planning section. As temperature is increased, more heat energy is supplied to the molecules in the fluid. Thus, the molecules move faster, and become more spread out. Therefore, the number of molecules per unit volume is decreased. Thus, there are fewer molecules to resist the downward movement of the ball bearing, decreasing the viscous drag of the fluid. As viscous drag is decreased, the velocity will increase.
There are, however, limitations to my results. Firstly, there are the experimental errors. This is due to both human errors and instrumental errors. The main error that I had was due to the measurement of time. I could not make sure that I measured the time that ball bearing covered 0.3 metres because there would certainly be a time lag for me to see the ball bearing land on the bottom of the plastic tube and record the time. There is also a ±0.01s error in the stopwatch used.
Another limitation is the fact that I have used a narrow plastic tube in which to conduct the experiment. Stokes Law works under the condition that there is plenty of open space around the ball bearing. However, in using a narrow tube, there is not a lot of space around the ball bearing. Resistance against movement of the ball bearing will increase when it is closer to the sides. This will make my results less accurate. Moreover, I was not provided with the proper equipment to release the ball bearing while submerged in water without touching the wall of the plastic tube. This caused friction between the ball bearing and the plastic, which caused further inaccuracy.
I have also made the assumption that the ball bearing reaches terminal by the time it reaches the 0.3m line. However, this cannot be proven with our limited knowledge and equipment provided. But I believe that the error produced as a result of this should not be very large, as the ball bearing reaches terminal velocity very quickly.
I think that I have conducted this experiment accurately within experimental limitations. I have tried my best to ensure that all the variables apart from temperature was controlled, and I think that my results should be accurate enough to draw a reliable conclusion.
Next time, I could eliminate the error of using a narrow tube by containing the fluid in a wider container that would allow much more space around the ball bearing. Thus, error due to increased resistance at the side of the tube would be eliminated. Therefore, results obtained would be more accurate.
I can also increase the distance that the ball bearing has to travel. As a result, the percentage of time the ball bearing is travelling at terminal velocity is increased. Thus, the inaccuracy due to the time when the ball bearing was accelerating would be less significant in my results. Therefore, the margin of error would be decreased, thus increasing the accuracy of my results. In addition, sets of phototransistors can be used next time to eliminate the operator errors I caused by timing the fall of the ball bearing with the naked eye and by hand.
In the future, I would like to see how concentration of the fluid will affect its viscosity. I will change the concentration by mixing it with water. I think that as the density of water is much lower than glycerol fluid, the higher concentration of the fluid, the greater the viscosity.
A r t h u r C h a n 1 2 L