Fluid Mechanics with teacher's notes

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$Holt Math Minnesota Test Prep Workbook for Grade 9$

$Math Skills: More Practice$

For a floating object, the buoyant force equals the object’s weight Imagine a cargo-filled raft floating on a lake. There are two forces acting on the raft and its cargo: the downward force of gravity and the upward buoyant force of the water. Because the raft is floating in the water, the raft is in equilibrium and the two forces are balanced, as shown in Figure 9-3. For floating objects, the buoyant force and the weight of the object are equal in magnitude. BUOYANT FORCE ON FLOATING OBJECTS F B = F g (object) = m og buoyant force = weight of floating object Notice that Archimedes’ principle is not required to find the buoyant force on a floating object if the weight of the object is known. The density of an object determines the depth of submersion When an object floats in a fluid, the net force on the object is zero. Relating the net force on the object to the buoyant force, using Archimedes’ principle, gives the following result: F net = 0 = (r fV f − r oV o)g This equation can then be rearranged to show two equal ratios: rf ⎯⎯ = ⎯ ro Vo ⎯ Vf Of course, the displaced volume of fluid can never be greater than the volume of the object itself. So for an object to float, the object’s density can never be greater than the density of the fluid in which the object floats. Furthermore, the ratio of the total volume of a floating object, V o,to the submerged volume of the object, V f, is equal to the ratio of the two densities. If the densities are equal, the entire object is submerged, but the object does not sink. Buoyancy can be changed by changing average density The buoyancy of an object can be changed by changing the object’s average density. For example, a fish can adjust its average density by inflating or deflating an organ called a swim bladder. A fish fills its swim bladder with gas either by gulping air at the surface or by secreting gas from its gas gland into the swim bladder. The ballast tank of a submarine works in much the same way as the swim bladder of a fish. In submarines, compressed air is pumped into the ballast tanks (and water is pumped out) to make the submarine rise to the surface. When the submarine is ready to dive again, air in the tanks is replaced with water, which increases the overall average density of the ship. Copyright © by Holt, Rinehart and Winston. All rights reserved. F B F g Figure 9-3 The raft and cargo are floating because their weight and the buoyant force are balanced. The human brain is immersed in a fluid of density 1007 kg/m 3 , which is slightly less than the average density of the brain, 1040 kg/m 3 . As a result, most of the weight of the brain is supported by the buoyant force of the surrounding fluid. This fluid also serves to absorb shock to the brain during sudden movements of the head. SECTION 9-1 Demonstration 3 Float an egg Purpose Demonstrate the effect of liquid density on the buoyant force. Materials raw egg, demonstration scale, glass jar, water, table salt, measuring spoon, stirring stick Procedure Measure the weight of the egg and record it on the chalkboard. Fill the glass jar about two-thirds full with water. Demonstrate that the egg sinks in the water. Is a buoyant force acting on it? (yes, but less than the egg’s weight) Explain that adding salt to the water will increase its density. Have the class watch the egg as you stir in one spoonful of salt at a time. The egg will start to float but will remain submerged. Ask students to estimate the buoyant force (same as the egg’s weight). Keep adding salt until the egg floats partially above water. Again ask students to estimate the buoyant force (same as the egg’s weight). Explain that the buoyant force balances the egg’s weight in both cases, but as the density of the water increases, the volume of the egg that is immersed decreases. Fluid Mechanics 321 321
SECTION 9-1 Demonstration 4 Lifting weights with a balloon Purpose Demonstrate Archimedes’ principle in gases. Materials large helium balloon, plastic-foam cup, small objects of known weight, string, meterstick Procedure Hang the plastic-foam cup under the balloon and put weights in it to keep the cup on a table. Record the weight in the cup and compare it with the buoyant force exerted by the air on the balloon (weight > buoyant force). Measure the circumference of the balloon with a string and have students calculate its radius (divide by 2π) and its volume (assume it is approximately spherical). Ask students to estimate the buoyant force from the air (r airVg). Remove weight from the cup until it floats just above the table. On the chalkboard, record the weight remaining in the cup. Compare this weight with the calculated buoyant force. ANSWERS TO Conceptual Challenge 1. The buoyant force causes the net force on the astronaut to be close to zero. In space, the astronauts accelerate at the same rate as the craft, so they feel as if they have no net force acting on them. 2. disagree; because the buoyant force is also proportional to g 3. Helium is less dense than air and therefore floats in air. 322 322 Chapter 9 F B F g Figure 9-4 The raft and cargo sink because their density is greater than the density of water. NSTA TOPIC: Buoyancy GO TO: www.scilinks.org sciLINKS CODE: HF2092 1. Neutral buoyancy Astronauts sometimes train underwater to simulate conditions in space. What are some advantages of this kind of training? What are some disadvantages? 2. More gravity A student claims that if the strength of Earth’s gravity doubled, people would be unable to float on water. Do you agree or disagree The apparent weight of a submerged object depends on density Imagine that a hole is accidentally punched in the raft shown in Figure 9-3 and that the raft begins to sink. The cargo and raft eventually sink below the water’s surface, as shown in Figure 9-4. The net force on the raft and cargo is the vector sum of the buoyant force and the weight of the raft and cargo. As the volume of the raft decreases, the volume of water displaced by the raft and cargo also decreases, as does the magnitude of the buoyant force. This can be written by using the expression for the net force: F net = (r fV f − r oV o)g Because the raft and cargo are completely submerged, the two volumes are equal: F net = (r f − r o)Vg Notice that both the direction and the magnitude of the net force depend on the difference between the density of the object and the density of the fluid in which it is immersed. If the object’s density is greater than the fluid density, the net force is negative (downward) and the object sinks. If the object’s density is less than the fluid density, the net force is positive (upward) and the object rises to the surface and floats. If the densities are the same, the object floats suspended underwater. A simple relationship between the weight of a submerged object and the buoyant force on the object can be found by considering their ratio as follows: Fg(object) ⎯⎯ = ⎯ FB ro–V – –g ⎯ rf–V – –g Fg(object) ⎯⎯ = ⎯ FB ro ⎯ rf This last expression is often useful in solving buoyancy problems. with this statement? Why? 3. Ballooning Explain why balloonists use helium instead of pure oxygen in balloons. Copyright © by Holt, Rinehart and Winston. All rights reserved.
Page 1 and 2: Chapter 9 Overview Section 9-1 defi
Page 3 and 4: Section 9-1 Demonstration 1 Volume
Page 5: SECTION 9-1 The Language of Physics
Page 9 and 10: SECTION 9-1 PRACTICE GUIDE 9A Solvi
Page 11 and 12: SECTION 9-2 STOP 326 Misconception
Page 13 and 14: SECTION 9-2 Visual Strategy Figure
Page 15 and 16: SECTION 9-2 Classroom Practice The
Page 17 and 18: Section 9-3 Demonstration 7 Fluid f
Page 19 and 20: SECTION 9-3 Misconception STOP Aler
Page 23 and 24: Section 9-4 Demonstration 10 Temper
Page 27 and 28: Chapter 9 Summary Teaching Tip Ask
Page 29 and 30: 9 REVIEW & ASSESS 12. because the p
Page 31 and 32: 9 REVIEW & ASSESS 42. 7.1 m 43. 17
Page 33 and 34: 9 REVIEW & ASSESS 63. 60.9 m/s 2 64
Page 35 and 36: Chapter 9 Laboratory Exercise NOTE
Page 37 and 38: CHAPTER 9 LAB Boyle’s Law Apparat

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