Archimedes’ Principle Lab Report

Abstract One main goal of this experiment was to look at how the buoyant force exerted on an object in water affects that object and the other forces acting on it. The idea of buoyant force was illustrated by lowering a golf ball attached to a spring into a bucket of water. Then, the buoyant force was actually calculated for an object that was submerged in water. It was found that as the volume of the cylinder submerged increased, the buoyant force also increased, and the tension force decreased. The second main goal was looking at density of both the fluid (water in this case), and the density of various objects. The density of the water was calculated to be 1.44 g/cm3 using the slope, and 1.23 g/cm3 using the y-intercept. As for the specific densities of different objects, the heavier an object was (while keeping almost the same volume), the greater the density of the object. Introduction Archimedes was an ancient Greek physicist who, upon discovering that the material of something can be determined if its density as well as its displacement in a fluid is known, came up with a principle named after him. This principle states that an object in a fluid experiences an upward buoyant force that is equal to the weight of the fluid displaced. If the buoyant force is greater than the weight, the object will float to the top of the fluid. If the buoyant force is less than the weight, the object will sink to the bottom. If buoyant force is the same as the weight, the object will be completely submerged in the fluid, but will be suspended in the middle of the fluid. The buoyant force is directly related to the density of the fluid, the volume of the object that is submerged in the water, and gravity. This lab demonstrated how the buoyant force changed as an object was further submerged into a fluid, as well as how the mass of an object affected its density. The relationship between buoyant force and the volume submerged as well as how the weight of the object and fluid around it affects the object’s density was confirmed by the end of the experiment. Theory When an object is placed in a fluid, in this case water, there is always a buoyant force acting on that object. The equation for this force can be given by F B=ρ fluidVg (1) where FB is the buoyant force, ρfluid is the density of the fluid, V is the volume, and g is the acceleration due to gravity. In our case, an object was placed in the water, and it remained stationary. In other words, there was no acceleration. This indicates that the forces going up must be equal in magnitude and opposite in direction to the forces going down. The buoyant force in Equation 1 indicates the water pushing up on the object. The only force pointing down would be the weight of the object. Therefore, mobjectg=ρ fluidV ¿ g (2) where mobject is the mass of the object in the water, and the volume, indicated by Vsub, is specific to the volume of the object that is submerged in the water. The above equations demonstrate the density of the fluid. The density of the object in the fluid can be written as 2 ρobject= where give ρobject mobject V (3) is the density of the object in the fluid. Equations 2 and 3 can be combined to ρfluid= ρobject (4) which explains that in order for an object to remain suspended in the fluid, its density must be equal to that of the fluid. In part two of the experiment, the density of water was determined by determining the volume of a graduated cylinder submerged in a beaker as more washers were added to the cylinder. This can be related back to Equation 3. For this specific experiment, mobject=mcylinder+nmwasher (5) where mcylinder is the mass of the graduated cylinder, mwasher is the mass of one washer, and n is the number of washers in the cylinder. On a graduated cylinder, there is the volume of the base as well as the volume reading on the side of the cylinder. This is given by V ¿ =V base+ V side (6) where Vbase is the volume of the base of the cylinder and Vside is the volume reading on the side of the cylinder. By combining these equations, we can get V side= mwasher m n+ cylinder −V base ρ fluid ρfluid (7) which shows the side volume reading as a function of the number of washers that allows us to determine the density of the water. As long as an object is not accelerating, the sum of the forces in all directions will be zero. For this lab, only the vertical forces were considered (horizontal buoyant forces were still present). The vertical forces present were the tension force, buoyant force, and weight force. The buoyant force is pushing up on the object from below, the tension force is going up in this case, and the weight force always points down. This means that F B +T =mobjectg (8) where T is the tension force. Therefore, if we have the mass of the object and the tension force, we are able to calculate the buoyant force. The specific densities of the various objects were also determined in part four of the lab. The specific density of an object is the ratio between the density object and the density of the water. This is given by (9) 3 s object= ρ object w object = ρwater wsystem− wwater where sobject is the specific density of the object, wobject is the weight of the object, wwater is the weight of the water, and wsystem is the weight of the water plus the weight of the object. The densities of these objects can also be determined using the equation ρobject= mobject V object (10) and these values can be compared to the specific densities of the corresponding objects. Finally, the error in our measurements can be calculated. A percent error calculation can be done for the density of water because it has a known value. The percent error is given by Error= | ActualValue−TheoreticalValue |∙ TheoreticalValue (11) and allows us to see how far off our experimental values were from the theoretical value. A percent difference calculation can also be done using the equation Difference= Difference BetweenValues ∙ AverageofValues (12) to see how different the two experimental values for the density of water are from each other. Procedure 4 Figure 1. Apparatus for part one of the experiment. Shows a golf ball attached to the spring and a bucket filled with water. In part one of the experiment, a golf ball attached to a spring was lowered into a bucket of water. The setup is seen in Figure 1. The golf ball was added to the end of the spring and allowed to stretch all the way out. The golf ball was then slowly lowered into the water until it was fully submerged. Observations about the various forces acting on the golf ball were recorded. Figure 2. Apparatus for part two of the experiment. Shows a beaker filled with water and a graduated cylinder filled with washers in the beaker. 5 Figure 2 shows the apparatus for part two of the lab. The beaker was filled with water and a graduated cylinder was placed inside. Initially, ten washers were placed inside the graduated cylinder, and the volume reading on the outside of the cylinder was recorded (corresponded to where the water was on the outside of the cylinder). The washers were not all exactly the same weight. So, the initial ten washers were placed on a balance and this mass was divided by ten to give us an average mass for one washer. Washers were added, one by one, to the cylinder, with the volume being recorded each time. This was continued until the graduated cylinder touched the bottom of the beaker. Figure 3. Apparatus for part three of the experiment. Shows a piece of wood attached to a force sensor dangling over a beaker full of water. The third part of the lab required us to look at the buoyant force exerted on an object. A piece of wood was attached to a force sensor, as shown in Figure 3. The force (in this case, the tension force) was calculated using the PASCO Capstone software on the computer. The tension force was first calculated when the piece of wood was not submerged. The piece of wood was then submerged one centimeter into the beaker filled with water and the tension force was recorded again. This was repeated until four centimeters of the wood was under the water. These tension forces were used to calculate the buoyant forces used for analysis. 6 Figure 4. Apparatus for part four of the experiment. Shows a cube attached to a string dangling over a beaker full of water on a balance. Part four of the experiment can be visualized in Figure 4. The specific densities of various cubes and a rock were determined. The mass of the water without any object in it was first found. Then, an object was attached to a string and lowered into the water until all of the object was submerged. The final mass was recorded. This was done for each of the objects in order to determine the specific density of the objects. For the cubes, the density was also determined by finding the mass and dividing it by the volume. These calculations can be compared to the specific densities of the same objects. Sample Calculation and Results In part two (part one did not require any calculations, only observations), the density of water was calculated. As washers were added to the graduated cylinder, the volume on the side of the cylinder was recorded, as seen in Table 1. The volume submerged as a function of the number of washers added was then graphed in Graph 1. According to Equation 7, the slope of this graph will allow us to calculate the density of the water. V side mwasher = n ρfluid g cm = = ρfluid (13) g cm By looking at Equation 7 more closely, the value for the density of water can be found in more than one place. It is also possible to calculate our experimental value for the density of water using the y-intercept. 7 y−intercept = cm = g ρfluid mcylinder −V base ρfluid − (14) g cm cm = Because we now have two different experimental measurements for the same value, a percent difference calculation can be done with the two numbers. This will allow us to determine how different these two values are from each other. ( = Difference BetweenValues ∙ Difference= AverageofValues ( g − cm g cm g + cm g cm )∙ (15) = ) This value indicates that these values differ by a factor of 15.73%. Finally, these values can be compared to the actual value for the density of water (which is 1 g/cm3) by doing a percent error calculation. The value determined using the slope will be used for this example. | | ActualValue−TheoreticalValue Error= ∙ TheoreticalValue = | | g g − cm cm ∙ g cm = (16) The other percent error was calculated to be 23%. These errors can be contributed to the errors explained later. For part three, we were required to determine how the buoyant force of an object changed as more of its volume was submerged in water. The data for this section is compiled in Table 2. The PASCO Capstone software gave us a reading for the tension outside of the water, which was equal to the weight force. As the object was submerged, the tension force changed, we were able to calculate the buoyant force by rearranging Equation 8. F B=m objectg−T = N− N= N (17) The above example was done for when the object was one centimeter below the water (a volume of 2.63 cm3). This value and the other buoyant force values were graphed in Graph 2. This allowed us to see that as the volume of the object submerged increased, the buoyant force also increased. The specific densities and the densities of different objects were then calculated in part four. The specific density was calculated for all objects, whereas the density was only calculated for the cubes. This is because the rock has an irregular shape, and the volume cannot be directly measured. The rock will be used for the specific density calculation. (18) 8 ρ object w object = s object= = ρwater wsystem−wwater ( ( g)( g)( m )−( s m ) s g )( m ) s = g cm The densities of the cubes were calculated using Equation 10, where the volume of the cube was equal to the product of the length, width, and height of the cube. mobject g g (19) = = ρobject= V object cm cm Table 3 shows all the specific densities and the densities of the various objects. The general trend found was that as the mass of the object increased, the density increased. Discussion and Conclusions In the first part of the lab, we were simply lowering a golf ball attached to a spring into a bucket filled with water. It was observed that the golf ball sank, which indicates that its density is greater than that of water. It was also observed that as the ball was lowered further and further into the water, the lighter and lighter the golf ball seemed to be. This is due to the buoyant force. Figure 5 illustrates what happens to an object as it is lowered into a fluid. In our case, before the object touches the fluid, the tension force is equal to the weight of the object. Once the object is in the fluid, a buoyant force becomes applied to the object. As more of the object’s volume is submerged in the fluid, the buoyant force increases while the tension decreases in order for both the forces to add and equal the magnitude of the weight force (the object is not accelerating). This is why the golf ball appeared to be lighter. The tension force provided by us holding the spring decreased as the ball got deeper into the bucket of water. This figure can also be used to demonstrate the forces exerted on the objects in Figure 5. Illustration of the forces acting on the various objects parts three and four of the experiment. The only over time as more of their volume was submerged in water. difference between the two parts was that in part three, the tension force was given by the force sensor, whereas in part four the tension force was provided by us as we held the string. In part two of the experiment, an experimental value for the density of water was determined. According to Equation 7, there are two ways to calculate this value after the data was graphed; using the slope and using the y-intercept. The value obtained by using the slope was 1.44 g/cm3. With the y-intercept, the density was calculated to be 1.23 g/cm3. Each of these values’ calculated percent error were 44% and 23%, respectively, when compared to the theoretical value of 1 g/cm3. These errors can be explained shortly. When comparing them to each other, the percent difference was 15.73%. It is hard to say which method is better than the other. However, by looking at the values, I would say that using the y-intercept was the better method only because the value obtained that way was closer to the expected value. 9 For part three, the buoyant force on an object was calculated based on the volume of the object submerged. As the volume of the object that was submerged in the water increased, the buoyant force increased. This is due to the variables in Equation 1. The fluid does not change (so the density stays the same) and the value of gravity is always constant. The volume submerged is directly proportional to the buoyant force. Therefore, it makes sense that by increasing the volume submerged, you also increase the buoyant force if the other variables remain constant. This data can be seen in Graph 2, and the trend can be visualized by observing the positive slope of the graph. The last part of the lab had us measure the specific densities and the densities of various objects. To calculate the specific densities, Equation 9 was used. According to the equation, you would think that the answer would be 1 g/cm3 for each object. However, by looking at the calculations in Table 3, the values are not 1g/cm3 are even the same number. This is because not all of the weight of the object is recorded when the object is not touching any part of the container. Next, the densities were calculated for the objects (except for the rock because of its irregular shape) by finding the mass of the object and dividing it by its volume. All of the values for calculating density in this manner were less than each object’s corresponding specific density calculation. The values are different because the specific density is the density of the object in relation to the density of the surrounding fluid whereas the density just looks at the density of the object. This could also be contributed to some error, as explained shortly. The trend found for the densities was that as the mass increased, the density also increased. This makes sense because, according to Equation 10, the mass of the object is directly proportional to the density of the object. By keeping the volume the same (each cube’s side lengths only ever different by a factor of plus or minus 0.01 cm), the only variable affecting the object’s density was its mass. One source of error in our measurements could be that when we were calculating the volume submerged of the graduated cylinder with the washers, we only took into account the volume reading on the side of the cylinder. The volume of the base of the cylinder was calculated, but it was not added to the side volume to get the total volume submerged. This could be why our values for the density of water had large percent errors. Another error possibly contributing to these percent errors was the fact that all the washers were not the exact same mass. An average of ten of the washers was taken, but this is not as accurate of a measurement as it could be. Ways to improve this would be to either weigh all of the washers and take an average weight of all of them (not just ten of them), or to weigh each washer individually. For part three of the experiment, it was difficult for us to calculate the tension because our force sensor was not working properly. It would jump between numbers and we ended up choosing the number that the force sensor stayed on the most, which may not have been the correct number. This could cause our buoyant force calculations to be off. Finally, the specific density calculations could contain some error. When the objects were submerged in the water, the reading on the balance fluctuated due my hand not keeping the object perfectly steady. I kept my hand as steady as I could until the balance stopped fluctuating. However, this number still may not have been accurate for the weight of the system. 10 Mass of Cylinder: 27.85 g Number of washers 10 11 12 13 14 15 16 17 Density of Water Mass of Washer: 2.56 g Volume of Base: 3.97 cm3 Volume Submerged (cm3) 36 38 40 42 44 46 47.5 47.7 Table 1. Volume of the cylinder submerged compared to the number of washers added. As more washers were added, the cylinder was submerged further. Buoyant Force versus Volume Submerged Tension on Spring Scale (N) Volume Submerged (cm3) 0.15 0 0.12 2.63 0.06 5.26 0.03 7.89 0 10.52 Buoyant Force (N) 0 0.03 0.09 0.12 0.15 Table 2. Tension reading on the force sensor compared to the volume of the cylinder submerged. The buoyant force for each corresponding submerged volume was calculated. Name Slag PVC Nylon Acrylic Copper Brass Steel Aluminum Description Rock Gray White Clear Red-Brown Gold Black Silver Density of Objects Specific Density (g/cm3) 3.23 1.88 1.40 1.56 12.12 9.44 8.29 2.93 Density (g/cm3) N/A 1.34 1.10 1.17 8.88 8.39 7.79 2.80 Table 3. Specific densities and densities of various objects. The more mass an object had, the greater its density was. 11 Volume Submerged as a Function of Number of Washers Volume Submerged (cm3) 60 50 40 30 20 10 0 9 10 11 12 13 14 15 16 17 18 Number of Washers Graph 1. Volume submerged as a function of the number of washers added to the cylinder. The black line indicates the best fit line for the data. The calculated value for the density of water was 1.44 g/cm3 with the slope and 1.23 g/cm3 with the y-intercept. Buoyant Force Versus Volume Submerged 0.16 0.14 Buoyant Force (N) 0.12 0.1 0.08 0.06 0.04 0.02 0 0 2 4 6 8 10 12 Volume Submerged (cm^3) Graph 2. The buoyant force applied to the object as the volume submerged changed. The black line indicates the best fit line for the data. As the volume submerged increased, the buoyant force increased. 12