Beginning with the ancient Greek philosophers and scientists, Aristotle thought that a cannon ball would fall to the ground because its natural position is in the Earth (explanation without the discovery of friction)

Mass and Velocity Effects of the Impact of

Spherical Objects with a Planar Fluid Interface

By John Rankin and Ben Winegar

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Introduction: Back to table of contents

Beginning with the ancient Greek philosophers and scientists, Aristotle thought that a cannon ball would fall to the ground because its natural position is in the Earth (explanation without the discovery of friction). This belief relied mainly upon astronomy: the rotation of the stars, moon, and planets. Since then, scientists like Galieo, Sir Isaac Newton, and Albert Einstein, have found a more accurate definition that includes friction, as a means to explain the movement of objects on different surfaces. According to Dr. Lamott, friction force is defined as: "the force created between two contacting surfaces that tend to rub or slide past each other." There are many factors that contribute to friction such as, the texture of both surfaces, weight of the object, the amount of surface area, and the amount of force applied to the object. There are two types of friction. Static friction, the force between two stationary objects, and kinetic friction, the force between two objects when one is moving relative to the stationary object. This is also known as sliding friction.

According to Douglas C. Giancoli, in order to calculate the force of friction the normal force is multiplied with the coefficient of friction. The coefficient of friction is defined as the force of friction divided by the normal force. This concept is illustrated in the following formula: FFr=mSFN and mS=FFr/FN.

Also according to Giancoli, the displacement, showed as s; the initial velocity, shown as u; the acceleration, shown as a; and the final velocity, shown as v are related in the following ways: s=u+av and v²=u²+2as. These are some of the formulas that we will use in our experiment (described later).

When one describes friction, one must address the Newton's Laws of Motion. Dealing with classical mechanics, Newton's first and second law can be summarized in the following formula: F=ma; where F is the force acting upon the body of mass m, and a is the resulting acceleration (Podesta). This formula is widely used and is the base support for much of the calculations concerning force. The F stands mainly for the overall force, which includes kinetic or static friction.

Review of Related Literature: Back to table of contents

Galileo interpreted friction force as a force to slow an object down. He likened friction to pushing or pulling on an object to slowing it down or speeding it up. He then determined the basis that is evident in Newton’s 1st Law of Motion, in which it states that everything “continues . . . in a straight line unless it is compelled to change that state by a new force acting on it,” for example, friction force (Giancoli 67). Another interpretation can be that “every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed on it,” or in more plain language, “things that are standing still tend to stand still, unless you push on them hard” (Gleick 29).

Newton’s 2nd Law of Motion involves acceleration, stating that it is “directly proportional to the net force acting on it and is inversely proportional to its mass. The direction of the acceleration is in the direction of the applied net force” (Giancoli 69). One other source stated that Newton’s 1st and 2nd Laws of Motion can be summarized as: F=ma, or Force equals an objects mass multiplied by its acceleration (Podesta 17). One source notes that the mass, or sometimes called the inertial mass, is the same as the “quantity that measures the ability of an object to produce (or react to) a gravitational field (Davies 70).

Friction has many valuable uses, including keeping a car on a road, stopping a car, or standing something up without it sliding around; yet it can also reduce the efficiency involving many other uses (Sieck).

The coefficient of friction very much depends on the materials of the two items, the block, and the surface, as well as the cleanliness of the contacting surface. It does not depend on the size or shape of the contact surface (Lerner 418).

One must note that friction does exist between two solid surfaces, even if not evident at first glance, as discussed earlier involving the contact areas. This is because even the smoothest looking surface is quite rough when looking at it microscopically (Giancoli 69). When two solids are pushed against each other, contact occurs between the microscopic peaks of the objects.

Our Experiment: Back to table of contents

One concept that scientists have explored, throughout the years, is the depth at which an object will fall into a body of water, when elements such as: speed and the weight of the object vary. Examples of this include: Gemini and Apollo, the first manned space flight. Astronauts needed to be able to calculate and solve problems like this in order to measure the amount of resistance needed (the amount of protection on the pod) in order to withstand the extreme pressure of landing in the ocean at high speeds.

We will try some of these same calculations in our experiment, with a few changes (due to our low budget). Instead of landing pods, we will have Styrofoam balls. And, instead of an ocean, we will use a fish tank. However, we will still take into affect the important elements such as the weight of the object and depth it falls into the water.

In our experiment, we will find the amount of friction water creates, when our Styrofoam balls are dropped into the water at a constant height. We will drill a hole into the top of the ball and put varying amounts of weights in it. This will account for the difference in depth as it relates to the weight of the ball. Differently weighted spheres (Styrofoam balls) enter water at different depths. These depths depend on three main variables: mass, volume, and velocity. In our studies, we will only involve mass and velocity. This will help us further understand the relationship between the three variables and how that relates to the friction of water.

Hypotheses: Back to table of contents

In performing this experiment, we hope to prove the following hypotheses: first, that the mass of the ball would be directly related to the depth it falls into the water. As the weight of the ball increases, the depth of fall would do the same. Second, the acceleration of the ball before it hits the water would decrease as the weight of the ball increases. Third, the friction between the water and the ball will be a constant, but the force of impact will change due to the different masses of the balls. Fourth, that there is some sort of a pattern concerning the relationship between the mass of ball, the density of the water, and the depth that ball falls into the water. Hopefully, this pattern will lead us to some conclusive evidence.

Procedure: Back to table of contents

Materials Used:

1) One 25 gallon fish tank 5) 2 meter sticks

2) 12 Styrofoam balls 6) 6-ft. ladder

3) 20 gallons of water 7) lead weights of various sizes

4) Video camera and video tape 8) felt-tip marker

Set-up:

The set-up for our experiment involved the following events in order: moving random pieces of furniture in Ben’s living room (creating our lab area), finding ladder in Ben’s garage, setting it up in his living room, taking fish tank and filling it with 20 gallons of water, measuring height of water and marking it on glass (every centimeter); in order to provide a ruler-like device to measure the depth of the ball when it fell into the water, setting up the video camera and random lighting for recording purposes, setting down many towels to avoid a wet carpet, and finally measuring two meter sticks up from the water and duct taping them to the ladder to ensure the same height for each drop. The design of our experimental set-up relies upon three main things: creating a set-up which will help us gather the most accurate data, cost effectiveness, and amount of space available for the experiment. We used a clear fish tank to ensure that we could properly measure the depth of the ball drop. We used a 6-foot ladder and only dropped the ball from a height of two meters, because of space requirements (Ben’s mom didn’t want to re-model). And last and most importantly, because we are poor students who were unable to buy anything bigger than a 25-gallon fish tank and some 99 cent lead weights. Another aspect of our set-up includes the video camera, which was used to try to measure the depth more accurately as we will state later in the paper. For visual depiction of the set-up, see drawing located at the end of this paper in the “data and calculations section.”

Gathering Data:

First, we took one styrofoam ball and strategically placed lead weights in it so that it would be equal on all sides, creating a non-lopsided object. Then we weighed the ball and dropped it from a height of 2 meters. Recording this data, we repeated this process four times for the one ball. However, each time we dropped the ball, we had to take into consideration the change in weight due to the water clinging to the pores of the styrofoam. Although the difference was small, we tried to compensate for it by drying the ball off each time, but finally resulted to calculating the average weight of the four trials; we weighed the ball each time directly before the drop (see table and calculations below – not all data is included).

First Ball Only

	Weight of Ball (g)	Depth (cm)

Trial #1	48.8	10
Trial #2	49.4	8.5
Trial #3	49.6	8.8
Trial #4	50.7	11

Averages:	49.625	9.575

entire data file click here

We did this in order to make sure that the impact of the ball would not be altered drastically by the changing weight of the ball. We repeated this process for three more balls, using different amounts of weights for each ball. For each individual trial we measured the depth that the ball fell into the water. While one person dropped the ball, the other would watch how far it would fall into the water and then mark on the glass the exact depth, and then record the data on a piece of paper. To ensure even greater accuracy, we used the video camera to record the ball drop, playing it in slow motion to be sure we were accurate in finding the depth of the fall (still an uncertainly, but trying to lessen the impact of it on our data).

We also spent much of our time attempting to retrieve the styrofoam ball from a very playful dog. We found that the saliva from the dog’s mouth had a minimal effect on the ball drop (not included in one of our original hypotheses, but an interesting bit of data).

Data Analysis/Results: Back to table of contents

After we finished gathering our data from the 16 trials (4 different ball weights), we arranged it in the following format, which made it easier to calculate the averages and plot the graphs:

	Weight of Ball (g)	Depth (cm)

Trial #1	48.8	10
Trial #2	49.4	8.5
Trial #3	49.6	8.8
Trial #4	50.7	11

Averages:	49.625	9.575


Ball 2	82.5	19
	82.6	20
	82.8	19.5
	83.3	16
	82.8	18.625

Ball 3	102.2	24
	102.8	25
	103.6	25.5
	105.2	26
	103.45	25.125

Ball 4	123.5	34
	125.4	36
	127.1	37
	128.1	35
	126.025	35.5

For text file of our data click here

After plotting the graphs, using the average weight and depth of each ball as the x and y-axis, we saw a linear graph, which proved our first hypothesis. Although quite obvious, our first hypothesis states that the weight of the ball is directly proportionate to the depth of the drop into the water. This graph clearly depicts this statement, by showing that the heavier the ball got, the further it fell into the water. Starting with the lightest ball, (averaged at 49.625) which had an average depth of 9.575 centimeters, and ending with the heaviest ball (averaged at 126.025) which had an average depth of 35.5 centimeters (almost to the bottom of the fish tank). See graph for a clear visual depiction of this phenomenon (located in “data/graph section”). To prove our second hypothesis, we used the formula: v²=u² +2as. We calculated the accelerations of each drop and concluded that as the ball’s weight increased, the acceleration decreased (see “data/calculations section” for a clearer understanding). Our third hypothesis is a given because the coefficient of friction will not change when the speed changes. Instead of being a variable in our calculations, it was a constant. Not really sure in what direction we were headed next, we decided to experiment with our data by dividing the weight of each ball by the depth. These calculations concluded with very interesting results. As the ball’s weight continued to increase, this value decreased. We graphed this digression (see in “data/graph section”), and added another hypothesis to our list. This value (a ratio of mass over depth) suggested that once the value reaches 1, the ball would continue to sink without bound to the bottom of the tank (or whatever contains your water). This shows that once the ball’s mass reaches its same value in depth, it will infinitely sink to the bottom. We plotted this data and using skills acquired in the absolute zero lab that we did earlier in the year, we tried to estimate this value by carefully drawing a line which continues past our plotted points (see graph in “data/graphs section”).

Uncertainties: Back to table of contents

There were a variety of uncertainties involved in this project. Some of which we were told to disregard and others we tried to fix or compensate for as the experiment progressed. One of these uncertainties was that we could have made an error when the ball was dropped because it was dropped by hand. We also could have misread the drop when it hit the water – visual perception, human error. The mass of the styrofoam balls could have been decreased since the balls were wet, they could have dripped when they were carried to the takeoff point. Our measurements on the tank could have been off because we were unable to hold the meter stick on the tank while the drop was taking place. We had to go by our markings alone.

Conclusion: Back to table of contents

In conclusion, our experiment proved to be mostly successful. We began with our first two hypotheses, and then as the experiment progressed two more appeared, one of which we were able to answer (the force of impact in relationship to the friction between the water and the ball). The other, we figured out using principles from a previous lab and a graph, but could not come up with a very reliable way to depict this hypothesis using calculations. We believe that in order to accomplish this, we might have to add a third variable to our project (possibly an expansion for next semester): varying heights. Meaning, that we would drop the ball at varying heights instead of just one. This would enable us to find the data to possibly prove our fourth hypothesis with more accuracy.

Links Back to table of contents

The following are links to related sites:

http://www.k12.hi.us/~dnekoba/boatinfo.htm

http://www.inetarena.com/~noetic/pls/dpb.html

http://ns1.wviz.org/TTI/NTTI/Plans/1997/Bouyancy/

http://ame-www.usc.edu/

http://www.phy.ntnu.edu.tw/java/buoyantForce/buoyantForce.html