FLUID STATICS II: BUOYANCY 1 Learning Goals After completing this studio, you should be able to Determine the forces acting on an object immersed in a fluid and their origin, based on the physical properties of the object and the fluid; Determine the magnitude of the buoyant force on an object from its volume and the density of the object and the fluid; Determine if an object will float, sink, or be neutrally buoyant in a fluid based on the densities of the object and the fluid; and Use Newton s laws to determine the acceleration of, and forces on, an object in a fluid. A. Archimedes Principle Archimedes Principle states that a fluid will exert an upward force (called a buoyant force) on an immersed object and that the magnitude of that force will equal the magnitude of the weight of the fluid displaced by the object. Block A is released from rest at the center of a tank of water. The block accelerates upward. 1. At the instant the block is released, is the magnitude of the buoyant force on block A greater than, less than, or equal to the magnitude of its weight? Explain 2. When block A reaches the surface, it is observed to float at rest as shown in the diagram on the right. In this final position, is the buoyant force on block A greater than, less than, or equal to its weight? Explain 3. If 90% of block A lies below the surface of the water in its final position, then how does the density of block A compare to the density of water? Explain/show your reasoning. Imagine that block A were released in the center of a tank filled with a fluid that is twice as dense as water. 4. Describe what will happen to block A after it is released. 5. Calculate the percentage of block A that is submerged after block A comes to rest and draw its final position as accurately as possible. Block B is the same size and shape as block A. Block B is neutrally buoyant, which means that when it is released from rest at the center of a tank of water it does not move. Block B s final position is shown at right. 6. How does the buoyant force on block B compare to the buoyant force on block A at the instants they are released from the center of the tank? Explain 7. How does the buoyant force on block B compare with the buoyant force on block A at their final positions? Explain Final Position of Block A 8. Draw the free-body diagrams for blocks A and B at the instants they are released and when they are in their final positions. The lengths of your vector arrows should be qualitatively correct. Make sure your free-body diagrams agree with your answers to questions A5 and A6. Block C is the same size and shape as blocks A and B, but its mass is slightly greater than block B s mass. Block C is released from the center of the tank. A Final Position of Block B B
2 FLUID STATICS II: BUOYANCY 9. Draw the free-body diagram for block C at the instant it is released. The lengths of your vector arrows should be qualitatively correct. 10. Draw block C in its final position in the beaker. Explain 11. Two students are discussing their answers to question 9. Stan drew the diagram shown at right. Stan: Eileen: Since block C is slightly heavier than block B, it will come to rest slightly below where block B is at rest. This is because the buoyant force is slightly less than the weight of the block. Since the buoyant force is smaller than the block s weight, the block will experience a downward acceleration. I think your drawing is wrong. The block should sink all the way to the beaker s bottom. Do you agree or disagree with either or both of the students? Explain 12. Is the buoyant force on block C when it is at the bottom of the tank greater than, less than, or equal to the buoyant force on block C when it is first released from the center of the tank? You can assume there is still a thin layer of water between the bottom of the block and the tank. Explain 13. Your studio instructor will have an aluminum block attached to a spring scale. Measure the weight of the block while it is suspended in air and then slowly immerse it in a tank of water. a. Completely immerse the block in water, but don t let it touch the bottom of the beaker. How does the reading on the spring scale now compare to what you found when the block was suspended in air? Explain why this happens. b. Let the block just barely touch the bottom of the tank, but make sure the string connecting the block to the scale remains taut. Did the buoyant force of the water on the block and/or the tension force of the string on the block change? Explain. 14. Imagine that block C has a weight of 49.0 N. If it came to rest on a scale at the bottom of a tank of water, would the scale read a value that is greater than, less than, or equal to 49.0 N? Explain 15. How would scale s reading change if the tank s water were replaced by oil, which is less dense than water? Explain 16. Point your browser to the PhET simulation Buoyancy (https://phet.colorado.edu/sims/density-and-buoyancy/buoyancy_en.html). The 5.00 kg block of bricks on the right represents block C. Place the block on the scale outside the water to make sure its weight is 49.0 N. Now place the block on the scale in the tank of water. a. What does the scale at the bottom of the tank read? Stan s Drawing of Final Position of Block C b. Click the button below the tank to change the fluid from water to oil. What does the scale at the bottom of the tank now read? c. Explain why the scale at the bottom of the tank has a different reading than the scale outside the tank. d. Explain why the reading of the scale at the bottom of the tank changes when the fluid changes from water to oil, which is less dense than water. C
FLUID STATICS II: BUOYANCY 3 Three blocks (A-C) of identical mass and volume are suspended from identical strings, with tensions T A -T C, respectively. Blocks A and B are underwater, as shown in the diagram below. Block C is immersed in oil. Blocks A and C are at the same depth below their respective surfaces. All the blocks are at rest. A water B C oil 17. Rank, from largest to smallest, the magnitudes of the tension forces (T A, T B, and T C ). Explain Imagine a 10 cm 3 ice cube is floating in a beaker of water. Ice has a density of 900 kg/m 3 = 0.9 g/cm 3 and water has a density of 1000 kg/m 3 = 1.0 g/cm 3. 18. What is the mass of the water displaced by the ice cube? Explain/show your work. 19. What is the volume of water displaced by the ice cube? Explain/show your work. 20. The ice cube melts after a long time. Will the water level in the beaker rise, fall, or stay at the same location? Explain Now imagine that there is a stone of mass 5.0 g and density 1.2 g/cm 3 sitting on top of a 10 cm 3 ice cube floating in a beaker of water. 21. What is the mass of the water displaced by the stone and the ice cube? Explain/show your work. 22. What is the volume of water displaced by the stone and the ice cube? Explain/show your work. 23. The ice cube melts after a long time. What volume of water is displaced by the water from the melted ice cube? Explain/show your work. 24. What is the volume of water that is displaced by the stone after the water melts? Explain/show your work. 25. Will the water level in the beaker rise, fall, or stay at the same location? Explain your reasoning. B. Weighing Dinosaurs 1 We can use Archimedes s Principle to estimate the masses of extinct animals. You have an anatomically accurate model of the dinosaur Pachyrhinosaurus lakustai. It is a 1:40 scale model, which means it is 1/40 th of the height, 1/40 th of the width, and 1/40 th of the length of a real Pachyrhinosaurus. You also have a beaker of water and a spring scale. 1 This procedure is taken from R. M. Alexander, Dynamics of Dinosaurs and Other Extinct Giants (New York, NY: Columbia University Press, 1989) and R. M. Alexander, Mechanics of posture and gait of some large dinosaurs, Zoo. J. of Linn. Soc. 83, 1-25 (1985).
4 FLUID STATICS II: BUOYANCY 1. Suspend your dinosaur model from the spring scale. What is the weight of the model? 2. Slowly lower the dinosaur (still attached to the spring scale) into the beaker of water. What does the spring scale read when the dinosaur is completely submerged but not touching the bottom? 3. Draw the free-body diagram for the dinosaur when it is attached to the spring scale and fully submerged in the water. The lengths of your vector arrows should be qualitatively correct. 4. Determine the volume of the model dinosaur. Explain/show how you arrived at your answer. 5. Given that you have a 1:40 scale model, what was the volume of a real Pachyrhinosaurus? Show all of your work. 6. Crocodiles are one of the closest living relatives of dinosaurs. Nile crocodiles (Crocodylus niloticus) have densities approximately equal to the density of water (1000 kg/m 3 ). Assuming dinosaurs had the same density, what was the mass of a full-grown Pachyrhinosaurus? Show all of your work. C. Swim Bladders Many fish have an organ called a swim bladder, which a fish can inflate or deflate in order to remain at a certain depth without swimming. A goldfish (Carassius auratus) is neutrally buoyant in an aquarium. 1. If the goldfish expands its swim bladder, does it move up, move down, or stay at the same depth in the aquarium? Explain 2. If a goldfish has a mass of 10 g and a density of 1080 kg/m 3 when its swim bladder is completely deflated, then to what volume must it expand its swim bladder to remain suspended in freshwater of density 1000 kg/m 3? How many cm 3 larger is this compared to the volume of the fish with its swim bladder completely deflated? Show all of your work. 3. Below is a picture of a goldfish with swim bladder disease. The fish can no longer control the volume of its swim bladder. Is this fish s swim bladder too large or too small? Explain This poor goldfish has swim bladder disease and is no longer able to control its buoyancy. (Photo courtesy of the AllExperts website 2 ) 2 http://en.allexperts.com/q/fish-1472/2011/12/swim-bladder-disorder-lionhead.htm
FLUID STATICS II: BUOYANCY 5 4. Your studio instructor will show you a partially inflated balloon that is neutrally buoyant at a certain depth in a container of water. Your instructor will then push the balloon down to a greater depth in the water. Predict whether the balloon will rise, sink, or remain stationary once it reaches this new depth. Explain D. Bringing It All Together To assess your understandings of some of this studio s key ideas, your group must answer the following questions together without help from the instructors or other groups. An object immersed in a fluid always experiences an upward buoyant force and Archimedes Principle allows us to determine the magnitude of that force. Many fluids questions can be answered using Newton s laws, as long as you include the buoyant force. Questions D1 and D2 refer to the following situation: Three blocks (D-F) of identical volume are placed in a tank of water. The masses of the blocks are unknown. Blocks D and E are suspended from strings. The tension in Block D s string is T D = 5 N and the tension in Block E s string is T E = 7 N. Block F is floating, as shown below. The blocks are at rest. D E F 1. Rank from largest to smallest, the magnitudes of the buoyant forces acting on Blocks D-F. Explain 2. Rank, from largest to smallest, the masses of Blocks D-F. Explain King Hiero II suspected that his gold crown was not pure gold (ρ gold = 19,300 kg/m 3 ). He asked Archimedes to determine if the crown was pure gold without damaging the crown. Archimedes suspended the crown by a rope in a tank of water (ρ water = 1000 kg/m 3 ). The tension in the rope was 30 N. The crown s volume was 0.0002 m 3. 3. Was the crown made of pure gold? Explain/show your work,