Worksheet for Exploration 14.1: Floating and Density How can a boat made out of a material more dense than water float? The block has a mass of 0.185 kg (position is given in centimeters). If this block is a cube, what is the density of the block? Note that since it is greater than water (1000 kg/m 3 ) the block sinks as shown in the animation. Restart. Block Density= We reshape the block so that it has the same depth into the screen, but is wider and taller with walls that are 0.21-cm thick. a. When the animation runs, what is the volume of water displaced (the dimension of the water container into the screen that you cannot see is 10 cm)? i. First sketch a free body force diagram for forces acting on the reshaped box. (One of these is the buoyant force). Displaced water: Length= Width= Depth= Volume displaced=
b. Using the density of water (1000 kg/m 3 ), find the mass of the water displaced. Show that it is equal to the mass of the reshaped block. Thus, the block floats. i. Think about how mass, density and volume all relate. mass of water= c. Another way to think about this is that in its new shape the block has an effective density (total mass/total volume) less than that of the water. Divide the mass (0.185 kg) by the new volume to find the new effective density of the block. Effective Block Density= d. How does the effective density compare to the density of water? The weight (mass*9.8 m/s 2 ) of the water displaced (even if the displaced water leaves the container) is equal to the buoyant force on the block. In the case of a floating object, the buoyant force is equal to the weight of the floating object.
Worksheet for Exploration 14.2: Buoyant Force When an object is put into a liquid, it experiences a buoyant force that is equal to the weight of the liquid the object displaces. The force on the wire is given as the block is slowly lowered into the liquid (position is given in centimeters and force is given in newtons). You can change the mass of the block between 0.125 kg and 0.375 kg and the density of the liquid between 500 kg/m 3 and 1000 kg/m 3. The object is in static equilibrium when the clock stops. Restart. a. What is the weight of the block and the tension in the string when the block is in the liquid? Therefore, what is the value of the buoyant force? The buoyant force and the tension in the string (the force on the support wire) act upward and the weight acts down. i. First sketch a free body force diagram for forces acting on the reshaped box. (One of these is the buoyant force). F g block submersed = F Tension =
b. What is the volume of the block in the liquid either the submerged part of the block if the block is partially submerged when you paused it or the entire block if it is completely submerged (the dimension of the block that is into the screen is 5 cm)? Length= Width= Depth= Volume Block= c. What is the volume of the water that is displaced by the block (the dimension of both water containers into the screen is 10 cm)? Verify that this is equal to the answer in (b). Length= Width= Depth= Volume displaced water= d. What is the mass of the liquid displaced? What is the weight of the liquid displaced? Check that this is equal to the buoyant force. mass liquid= e. Pick two different masses and densities and verify that the buoyant force is equal to the weight of the water displaced. i. mass= Density= F g = ii. mass= Density= F g =
Worksheet for Exploration 14.3: Buoyancy and Oil on Water This Exploration will address the buoyant force in more depth (pun intended). Specifically, what happens if we put an object in two "layers" of fluids? Assume the brown block is a cube (position is given in meters and pressure is given in pascals). Restart. Note: The format of the pressure is written in short hand. For example atmospheric pressure, 1.01x10 5 Pa, is written as 1.01e+005. Move the pressure indicator and measure the pressure at the bottom of the wooden block and at the top of the block. a. If the block is a cube, what is the force on the block due to the water (buoyant force)? i. First sketch a free body force diagram for forces acting on the reshaped box. (One of these is the buoyant force). b. What, then, is the weight of the block? What is the density of the block? (Use the symbol F g as the force due to gravity, ie., weight). F g = density=
c. Another method: How much (what percentage) of the block is submerged? Check that the density of the block is that same percentage of the density of water (1000 kg/m 3 ). i. When you determine density use the mass of the block from part b. %Submerged= density= Now consider what would happen if we put the block in an oil with a different density. d. Predict what you expect will happen if we put the block in an oil with a density of 700 kg/m 3. e. Try it. Was your prediction correct? Explain. f. What is the pressure at the bottom of the block and at the top of the block? What is the buoyant force on the block in the oil? P bottom = P top = i. To determine the buoyant force you can use the mass of the block from above, and the density of the oil.
ii. OR you can use the pressure measurements at the top and bottom to help you determine the buoyant force on the block. The two results should agree. iii. How does the buoyant force compare to the weight of the object? Now, suppose the wood block is put in a mixture of water on the bottom with oil on the top (the oil floats on the water and doesn't mix with the water). g. What do you expect will happen? Why? h. Try it. Is more or less of the block submerged in water in this case compared with the block simply floating in water (without oil)? Why? i. One way to look at what happened is to measure the pressures. Find the pressure at the bottom of the block and at the top of the block. P bottom = P top =
j. What is the pressure difference and thus the TOTAL BUOYANT FORCE on the block? P= k. In order for a block to float only in water (with air on top), to get the same pressure difference to support the block, why does the block need to be lower in the water? (Think about the density of air compared with the density of oil and, therefore, the change in pressure with depth in air and the oil.) Another way to look at this is to compare the buoyant forces. l. In comparison with the block floating in water only, has the buoyant force increased, decreased or stayed the same? m. What is the volume of water that the block displaces? i. When in water/air. V water displaced = ii. When in water/oil V water displaced =
n. What is the weight of that water? i. When in water/air. F g diplaced water (i) = ii. When in water/oil. F g displaced water (ii) = o. What is the volume of oil that the block displaces? V oil displaced = p. What is the weight of the displaced oil? F g displaced oil = q. How do those two compare with the weight of the block? i. That is, compare the total weight of displaced material for each case.