Abstract
The aim of the experiment was to determine the relationship between the diameter of a steel ball and its terminal velocity in glycerol. Four balls with different diameters were put in a transparent cylinder full of glycerol. The velocity of the ball was calculated based on the time taken for the ball to pass from one mark on the cylinder to another. The results showed that an increase in the diameter of the steel ball leads to an increase in terminal velocity. Besides, the results show that the larger the diameter of the steel ball is, the faster the ball passes through glycerol. Furthermore, the experiment revealed that squared diameter is directly proportional to terminal velocity. Since the variation in the diameter of the ball is equivalent to the variation in the ball’s mass, it can be deduced that mass of the ball is directly proportional to its terminal velocity in glycerol.
Introduction
This experiment was intended to examine the relationship between the diameter of a particle and its terminal settling velocity while moving in a fluid. For an object undergoing free fall, the only force acting on it is the force due to gravitational pull (Hewitt, 2006, p.71; Kirkpatrick & Francis, 2010). Thus, the object moves with an acceleration of approximately 9.8ms-2. If two objects with different masses are allowed to fall to the ground at the same time, they will accelerate towards the ground with the same velocity that is equal to the value of the velocity due to the force of gravity. Regardless of the difference in their weight, the two objects will land on the ground at the same time. This can be explained by the fact that when the objects are released, they both have an initial velocity of zero which increases as they accelerate towards the ground and have the same acceleration. Therefore, they experience a similar change in speed throughout the flight.
Unlike for an object moving in a vacuum, an object moving through a fluid experiences various forces. An object falling vertically through fluid encounters two main forces: its weight and the fluid resistance (fluid drag) that acts in the opposite direction of the object (Cassidy, Holton, & Rutherford, 2002). Initially, the object encounters no resistance. However, as the object moves further down the fluid, the magnitude of air resistance increases until it becomes equal to the weight of the object. At that point, the resultant external force acting on the object becomes zero. This can be explained using Newton’s second law of motion
F=ma.
F=W-D=0
Where F=net force, W= weight, and D= drag
Consequently, the object begins to move with a constant velocity. This velocity is referred to as the terminal velocity (Pople, 2001, p.28). The air resistance or the drag force that causes the terminal velocity is determined by various factors that include the following: density of the fluid, the velocity of the object, the surface area of the object, and the drug co-efficient among other factors. The velocity of the object is the most significant factor influencing the magnitude of the resistance force of the fluid.
When two objects with different masses are dropped through a liquid, the heavier object will reach the bottom of the liquid faster than the lighter one. The explanation for this phenomenon is that when the two objects are dropped, both the objects experience resistance forces attributed to the liquid and acting in the opposite direction. However, as the objects move down the liquid, the liquid drag increases in both cases. However, it becomes equal to the weight of the object with smaller mass faster than the heavier object. In other words, the lighter object achieves its terminal velocity faster than the heavier object. The heavier object, therefore, continues to move down the liquid longer before the liquid drag can be equal to its weight. Once this happens, it achieves its terminal velocity. At the time the lighter object attains its terminal velocity, the heavier one continues at a higher velocity until it too attains its terminal velocity. Therefore, the terminal velocity of the heavier object is higher than that of the lighter one. This explains why the heavier object reaches the bottom of the liquid faster than the lighter one.
The purpose of this experiment was to determine how the diameter of a steel ball is related to its terminal settling velocity. The following hypotheses were formed:
If the diameter of the steel ball is increased, then the terminal velocity of the ball will increase.
If the diameter of the ball is increased, then the ball will move through the liquid faster.
It was predicted that the larger the diameter of the steel ball is, the larger the velocity of the ball becomes and the quicker the ball moves through the liquid.
Experimental
In the experiment, four sets of steel balls were used: A, B, C, and D. a transparent cylinder filled with glycerol was also used in the experiment. While performing the experiment, the diameter of the four sets of balls were first measured using a micrometer screw gauge. Each set contained four balls of similar size and diameter. The diameter of the balls in set A, B, C, and D were found to be 9.89mm, 6.89mm, 5.89mm, and 4.89mm respectively. Next, the balls were dropped into the transparent cylinder containing glycerol one-by-one and the time take for each ball to drop from one marker to another measured. The first marker was placed some distance below the surface of the liquid to allow the balls to achieve terminal velocity before reaching the first marker. The terminal velocity was then calculated by dividing the distance between the first and the second marker by the time taken for each ball to move from the first to the second marker.
Results and Discussion
The terminal velocity for the respective balls was found to be as shown in table 1 shown below:
The time taken for each ball to move from the first marker to the second one is shown in table 2 below:
The relationship between the ball diameter and the terminal velocity was determined using a graph. The graph showing the relationship between the ball diameter and terminal velocity s shown in figure 1 below:
Figure 1: Graph showing the relationship between the ball diameter and terminal velocity
A graph of terminal velocity versus squared diameter was also plotted. The graph is shown in figure 2 below:
Figure 2: Graph showing the relationship between the ball diameter squared and terminal velocity
The Reynolds number when each ball fell through glycerol was calculated as shown below:
ReBall A
ReBallA=1261*0.00989*0.23841.763=1.6864
ReBall B
ReBall B=1261*0.00689*0.12781.76=0.6308
ReBall C
ReBall C=1261*0.00589*0.095751.763=0.4034
ReBall D
ReBall D=1261*0.00489*0.069311.763=0.2424
The graph shown in figure 1presents the relationship between the diameter of the steel ball and the terminal velocity. The gradient of the graph is positive (0.0344). A positive gradient implies that the variable on the x-axis is directly proportional to the variable on the y-axis. Consequently, the diameter of the steel ball is directly proportional to its terminal velocity when it moves through glycerol. In other words, the larger the diameter of the steel ball is, the larger the terminal velocity becomes. Since the gradient of the graph is greater than zero, it follows that an increase in the diameter of the steel ball by one unit leads to an increase in terminal velocity 0.0344 times the diameter.
Similarly, the graph of terminal velocity versus squared diameter shown in figure 2 has a positive gradient (0.0108). This implies that the squared diameter of steel ball is directly proportional to the terminal velocity of the ball when put in glycerol. The two graphs show that steel ball diameter influences its terminal velocity. The result of this experiment further shows that the balls with larger diameters spent shorter time to move through the glycerol than the balls with smaller diameters. The errors were identified in the experiment. The errors could have been caused by faulty instruments and the changes in temperature of glycerol, thus affecting the resistance and other pertinent properties.
Conclusion
Since the results found that diameter is directly proportional to the terminal velocity as had been predicated, the hypothesis is accepted. Since all the balls were made of the same material, the variation in diameter is univalent to the variation in mass. Therefore, it can be deduced that mass of the steel ball is directly proportional to its terminal velocity in glycerol. The experiment was conducted successfully since all the objectives were achieved. Besides, the students gained many insights into various concepts of the topic under study.
References
Hewitt, P. G. 2006. Conceptual physics. San Francisco, Pearson Addison WesleyBottom of Form
Pople, S. 2001. Physics for higher tier. Oxford, Oxford University Press.
Kirkpatrick, L. D., & Francis, G. E. 2010. Physics: a conceptual world view. Belmont, CA, Brooks/Cole Cengage Learning.
Cassidy, D. C., Holton, G. J., & Rutherford, F. J. 2002. Understanding physics. New York, Springer. http://www.books24x7.com/marc.asp?bookid=16303.
