Prediction/Understanding
If we think about a ball bearing in air, when the ball is released from rest, there is no frictional force F=0. So, let us say that the resultant force is R.
The velocity will increase as long as F<R. The resultant force is R-F.
Terminal velocity is reached when F=R. There is no acceleration. R-F = 0
Friction forces will oppose the motion of the falling ball. Friction will increase as the ball falls. When the ball is released from rest, the initial speed is zero, there is no friction force and the resultant force is R.
Acceleration = resultant force
Mass
So, R = mg = g
m m
So, in air the acceleration due to gravity is g (9.81m/s²)
However, when the ball has gained velocity v, there is an opposing frictional force F.
So, at this point:
Acceleration = resultant force
Mass
So, R - F = mg - F = g - F
m m m
The acceleration is now less than g.
As the velocity increases, so does the frictional force. Until F=R, at this point acceleration is zero. The ball now falls at a continuous speed. This is the maximum velocity (i.e. the terminal velocity). The forces are balanced, so this depicts Newton’s first law of motion.
Apparatus
These are the required apparatus for the investigation, as shown in the diagram:
• Large measuring cylinder
• Steel ball bearings of various radii
• Magnet (on a string)
• Glycerol (1 mol/dm³)
• Stopwatch
• Ruler
• Elastic Bands
• Micrometer
• Electric Balance
• Thermometer
Experiment
The apparatus was set up as shown above. The temperature was taken at the beginning of every experiment, as a precaution mentioned before. The tube was filled to five centimetres below the top, to avoid splashing, with glycerol. Five centimetres below the top of the glycerol was marked with an elastic band around the tube, to signify the point to start timing. Ten centimetres below that, another elastic band was placed to mark the point to stop timing.
The initial marker was placed five centimetres below the top of the glycerol, so that I had enough time to drop the ball and start the stopwatch.
Before I began the actual experiment, I decided to do some trial runs, to see which size balls would be best to use. I had to find a ball that would reach terminal velocity a short distance, and I also had to find a ball that would fall slowly enough to time accurately.
I chose three different ball bearings and measured the time it took to fall ten centimetres. The trial was done several times so that an average could be taken.
The micrometer was used to accurately measure the diameter of the steel ball bearings. The ball bearing was held above the glycerol and released. Once the ball bearing had fallen to the bottom they were retrieved by using a magnet on a length of thread.
The results were as follows:
The tables show the time taken in seconds for the ball to fall ten centimetres, over four different attempts. I had some other ball bearing, but they were much bigger and so fell much faster.
From these results I concluded that the ball bearings of radius 3.18 mm were to be used for the experiment. They travelled sufficiently slowly to be timed accurately and should also reach terminal velocity over a small distance.
The readings that were taken were used to calculate average time taken and average speeds for this experiment.
Other measurements were recorded to be used throughout the investigation. The density of the glycerol and ball bearings were determined. Weighing an empty glass beaker, and then measuring it again with 100ml of glycerol measured the density of the glycerol.
Problems
The tube in this experiment is very narrow, and so there was probably extra friction created from the sides of the tube. This is known as edge effect. I tried to minimise this by dropping the balls in the centre of the tube, but I think it would be better to repeat the experiment with a much wider tube.
If I were going to use a different tube, I would look for a longer tube, so that I can take more readings for different distances. For example, time taken to travel 0-10 cm, 10-20 cm etc… With the original tube, I am not sure that the balls reached their terminal velocity.
Another problem that I noticed was that every time I retrieved the ball bearings with the magnet, I also removed a small amount of glycerol. Over several retrievals, a considerable amount of glycerol may have been removed. If the volume changes enough, then the final results will be too varied, as density of fluid is dependant on volume.
Apparatus
The Apparatus was set up as before, but a taller and wider tube was used. As this was not a measuring cylinder, the ruler was used to make out 10 cm intervals. These intervals were, as before, marked with the elastic bands.
Experiment
The experiment was carried out as it was before. Though, this time, time readings were taken at 10 cm intervals, between 0-40 cm inclusively.
The experiment was undertaken three times. This allowed any anomalous results to be spotted immediately so that further repeats could be taken. With several results, it also allows averages to be calculated that reduce the extent of any anomalies and therefore increase the accuracy of any the conclusions made from these results.
From these results we can see a few abnormalities in experiment t2. Experiments t1 and t3 seem very similar. I think that we would have a more realistic average if we ignore t2 and take averages from t1 and t3. In experiment t2, the balls may have drifted to the side of the tube and so may have experienced additional friction forces, which would generate these unusual timings.
You can see from the graph that the ball bearing is falling at a fairly constant speed. But, the ball is slowing down. This must mean that the terminal velocity has already been reached. The ball must have reached a maximum speed from rest and then decelerated to a constant velocity. At a constant velocity, acceleration is zero, therefore the balls have reached their terminal velocity.
I carried out a few more experiments with different sized balls (similar diameter to 3.18 mm, as we have already stated that much bigger balls fall too fast to time accurately), so that we can draw a chart to show the relationship between terminal velocities and radius.
Conclusion
All the graphs show a linear trend line. So, if we count the average speed as the terminal velocity, we can show the following relationship.
If we look at the relationship between radius and terminal velocity, we can see that there is a definite correlation, as radius increases, terminal velocity increases.
My first prediction was that terminal velocity was proportional to radius squared.
This graph clearly shows a straight trend line that passes through the origin. This proves that the terminal velocity is directly proportional to the radius squared.
This theory was proven with a line of best fits, but it is clear from the graphs that the points lie above and below the line, which means that there are inaccuracies. The inaccuracies lie in the timing of the balls. If I were to do this experiment again, I would look at trying to make the readings more accurate.
Talk about experiments with varying temperature
Risk Assessment
Further Reading
