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Archimedes’ Buoyant Force and the Density of Liquids Purpose: The purpose of this experiment is to use Archimedes’ Principle to explore how gravity, density of fluid, and volume of fluid displaced by an object affect the Buoyant Force on an object placed in a fluid. In addition, Archimedes’ Principle is used to determine the density of a fluid. Materials/Equipment: (See Figure 3 for Assembly) GLX Hooked Brass Metal Cylinder with String High Resolution Force Sensor Lab Jack 100 mL Graduated Cylinder Water or Other Fluid (more than one optional) Ring Stand (or Table Clamp and Rod) and Printer Right Angle Clamp Background/Theory: According to Archimedes’ Principle, there exists a force on an object wholly or partially submerged in a fluid. This force (FB) is equal to the weight of the fluid (WF) displaced by the object and has a direction opposite that of gravity. In other words, if an object is placed in a fluid, then part of that object is occupying a volume that was once occupied by the fluid. The fluid then attempts to push the object back out of that volume, and its push is as strong as the weight of the fluid that once occupied that volume. In this lab, you will be hanging a rod from a force sensor into a graduated cylinder full of a fluid, as depicted in Figure 1. When the rod is positioned like that, there are three forces acting on it: its own weight due to gravity (which we know is its mass m times the gravitational acceleration g, WOBJ=mg), tension in the string from which it is hanging (T), and the buoyant force due to it displacing fluid in the cylinder (FB). Gravity is pulling down on the rod, the string tension is pulling up on it, and the buoyant force is pushing it up. These forces are indicated in Figure 2. Now that we know how the forces are acting on the rod, we can write down an equation describing the balance of these forces. The total combination of the forces must be equal to 0 since the rod isn’t going to be moving (it will be in equilibrium). Taking up as the positive direction, we have: T + FB – mg = 0 (1) Archimedes’ Principle states that the buoyant force is equal to the weight of the fluid displaced, or: FB = WF (2) Tension (T) Force Sensor Rod Buoyant Force (FB) Weight (WOBJ) Figure 1: Diagram of Figure 2: Balance Experiment Setup of forces on the rod Substituting that into equation (1), we have: T + WF – mg = 0 (3) We also know that the weight of an amount of fluid is equal to the fluid’s mass times g, and that the fluid’s mass is equal to the fluid density (ρF) times the volume of fluid, or ρFVF. In this case the volume of fluid is the same as the volume displaced by the rod (VD). All of that is given by: WF = mFg = ρFVDg (4) And substituting into equation (3): T + ρFVDg – mg = 0 (5) or T = mg – ρFVDg (6) So, if tension in the string is plotted as a function of volume of fluid displaced (string tension (T) is y, and volume of fluid displaced (VD) is x), the result should be a straight line with slope -ρFg and y- intercept mg. That is because the general equation of a straight line is y = mx + b where m is the slope and b is the y-intercept. So: T = mg – ρFVDg becomes y = mg – ρFgx (7) Where the slope of the line is -ρFg and the y-intercept is mg. Procedure/Analysis: Setup: 1) Connect the high resolution force sensor to the GLX, and turn the GLX on using the power button on the bottom right of the GLX. 2) Select the Meter option on the home screen of GLX using the arrow and check mark keys. 3) Making sure nothing is attached to the measuring end of the force sensor, press the zero button on the force sensor. The GLX should now be reading 0 Newtons, or something very close to it. 4) Assemble the lab stand, high resolution force sensor, rod, graduated cylinder of fluid, and lab jack as shown in Figure 3. Have enough fluid (about 60mL) in the graduated cylinder so that the cylinder can be completely submerged but without spilling fluid when it is fully submerged. Record the amount of fluid in the cylinder on the Student Data Figure 3: Detailed Experiment Setup Sheet. Measurement: 1) Making sure the rod is not in contact with the fluid, note the force being displayed on the GLX. This is the tension in the string connecting the rod to the force sensor, and should currently be equal to the weight of the rod. It is negative due to the way the force sensor is constructed. Record this value for the force (go ahead and make it positive) and the fluid level in the graduated cylinder in Table 1 (this fluid level should be the same as the value in Setup step 4). 2) Use the lab jack to raise the graduated cylinder until the fluid reaches the second mark on the rod. Again, record the force from the GLX and the fluid level in the graduated cylinder in Table 1. Repeat this process, raising the graduated cylinder such that the fluid level moves up two marks on the rod each time and recording the new force and fluid level until Table 1 is full. Data Manipulation: 1) Recall that the data we needed was the tension in the string and the volume of fluid displaced. You have the tension (the force from the GLX), but you don’t quite have the volume of fluid displaced. You do have the initial fluid level, and the fluid level for each force measurement. As you lowered the rod into the graduated cylinder, the fluid level rose. This is because the rod was displacing more and more fluid. So, the difference between each fluid level you recorded and the initial fluid level should be equal to the volume of fluid displace by the rod for each tension measurement. So, copy the tensions from Table 1 into the first column of Table 2, and then calculate the difference between each fluid level and the initial fluid level. Record that difference (which is the volume of fluid displaced, VD) in the second column of Table 2. 2) The only problem left with the data is that it is not all in SI units – you need to convert VD from mL to m3. One mL is one cm3, and there are 106 cm3 in one m3 (100cm*100cm*100cm). Therefore, to convert mL to m3, you need to divide each measurement in mL by 106 (or 1,000,000). Record these converted values in the third column of Table 2. Plotting and Analysis: 1) Now it is time to record and plot the data using the GLX. Begin by turning the GLX off and disconnecting the force sensor. Then turn the GLX back on and select the Table option on the GLX home screen. Press the F3 button, check that the New Data Column option is selected, and then press the check mark button. Then create another new data column by pressing F3 and selecting the New Data Column option again. Press the ESC button until you see a dash mark selection box on the table. Using the arrow keys, place this box on the first row of the first column. 2) With the dash mark box on the first row of the first column, press F2. You should see a blinking cursor. Type the first tension value from Table 2 into the box, then press the check mark button. The cursor should move horizontally into the first row of the second column. Here type the displaced volume from the third column of Table 2 that corresponds to the tension you just entered, then press the check mark button. Now the cursor will jump to the second row of the first column. Continue entering data into the GLX until the first and third columns of Table 2 (tension and displaced volume in m3) are recorded in the first and second columns of the GLX, respectively. 3) After you’ve entered the last value, press the check mark button so that the cursor is in a blank box, then press the ESC button. Make sure you still see all of your data on the screen, and then press the home button. Now select the graph option on the GLX home screen. Part of your data will now have been graphed, but not all of it. To fix this, press the check mark button. The label for the y-axis of the graph should now be highlighted. Use the left and right arrow keys a few times to see how to navigate between the x and y axis labels (there should be an upper and lower choice on the y-axis and just one choice on the x-axis). Make sure the x-axis option is highlighted, and then press the check mark button. A list of options will pop up, including Data 1 and Data 2, which were the two columns of data you entered in the table. Data 1 is the tension force, which should be on the y-axis label, and Data 2 is the one that should be on the x-axis. Select Data 2, and then press the check mark button. Now you have a plot of string tension versus volume of fluid displaced in units of Newtons and m3, respectively. 4) With the graph now properly constructed (it should be a diagonal line which falls going from left to right). Now you need to do a linear fit to this graph. Press F3 to access the Tools menu, and then select the linear fit option. You’ll then be able to determine which points the fit uses by moving the circular cursor on the graph with the left and right arrow keys. Press the right arrow key until all the points are being used. The slope, y-intercept, and various other properties of the fit are displayed below the graph. Record the slope on the Student Data Sheet in Table 3. Then press F4 and select the Print option to print your graph. 5) Given that the slope of the line should be equal to -ρFg, calculate ρF (the density of the fluid) using g = 9.8 m/s2. Record this value in Table 3, and then calculate the percent error with respect to the known density of the fluid. For example, if your fluid was water, its density is 1000kg/m3. The percent error is given by the formula: Experiment al Known ErrorPercent * 100 (8) Known For the water example, Experimental would be the density you just calculated by the dividing the slope by –9.8m/s2, and Known would be 1000kg/m3. Once you have calculated the error, record it in Table 3 as well. This error represents how far your measured value for density was from the correct value. For example, if you got a percent error of 100%, then your value was exactly twice what it should have been. Percent error is an easy way to describe how your value compares to the expected values. Time permitting or according to teacher instructions: Repeat the experiment using a different fluid. Then answer the questions on the Student Data Sheet. When you have completed the experiment, turn off the GLX. Make sure you don’t need any more data from it, because your data will be erased when you turn it off. When prompted, Do Not Save the file. Put your equipment away as directed by your teacher. Student Data Sheet Name: ____________________ Partner’s Name(s): ________________________ Period: _____________ Date: ________________ Initial Fluid Level/Volume (mL) _______ Table 1 Table 2 Tension (N) Fluid Level/Volume Tension (N) Volume Displaced Volume (mL) (mL) (Fluid Level – Displaced Initial Level) (m3) Table 3 Slope of Linear Fit (kg/m2s2) Calculated Fluid Density (kg/m3) Percent Error in Density Questions: 1. Explain in your own words what the graph indicates. What are possible sources of error in the data? 2. When floating while swimming, would you expect to float higher out of the water in fresh water or salt water? Explain. 3. Given a ship of weight W which is capable of displacing a volume V of water, write the mathematical expression that describes the maximum load L the ship can carry before sinking. Explain what other factors might influence this in actual practice as the ship sails to various ocean and river ports. 4. Icebergs float even though they are made of water - explain why this happens. Calculate the fraction of an iceberg that is above ocean level using the density values: Sea Ice 910 kg/m3, Sea Water 1030 kg/m3.