Holt Physics Problem 9A

Holt Physics Problem 9A BUOYANT FORCE PROBLEM SOLUTION 1. DEFINE 2. PLAN The highest natural concentration of salts in water are found in the evaporating remnants of old oceans, such as the Dead Sea in Israel. Suppose a swimmer with a volume of 0.75 m 3 is able to float just beneath the surface of water with a density of 1.02 10 3 kg/m 3. How much extra mass can the swimmer carry and be able to float just beneath the surface of the Dead Sea, which has a density of 1.22 10 3 kg/m 3? Given: V = 0.75 m 3 r 1 = 1.02 10 3 kg/m 3 r 2 = 1.22 10 3 kg/m 3 Unknown: m =? Choose the equation(s) or situation: In both bodies of water, the buoyant force equals the weight of the floating object. F B,1 = F g,1 r 1 V g = m g F B,2 = F g,2 r 2 V g = (m + m )g = r 1 V g + m g Rearrange the equation(s) to isolate the unknown(s): m =(r 2 r 1 )V 3. CALCULATE 4. EVALUATE ADDITIONAL PRACTICE Substitute the values into the equation(s) and solve: m =(1.22 10 3 kg/m 3 1.02 10 3 kg/m 3 )(0.75 m 3 ) m = (0.20 10 3 kg/m 3 )(0.75 m 3 ) m = 150 kg The mass that can be supported by buoyant force increases with the difference in fluid densities. 1. The heaviest pig ever raised had a mass of 1158 kg. Suppose you placed this pig on a raft made of dry wood. The raft completely submerged in water so that the raft s top surface was just level with the surface of the lake. If the raft s volume was 3.40 m 3, what was the mass of the raft s dry wood? The density of fresh water is 1.00 10 3 kg/m 3. 2. La Belle, one of four ships that Robert La Salle used to establish a French colony late in the seventeenth century, sank off the coast of Texas. The ship s well-preserved remains were discovered and excavated in the 1990s. Among those remains was a small bronze cannon, called a minion. Suppose the minion s total volume is 4.14 10 2 m 3. What is the 98 Holt Physics Problem Workbook

minion s mass if its apparent weight in sea water is 3.115 10 3 N? The density of sea water is 1.025 10 3 kg/m 3. 3. William Smith built a small submarine capable of diving as deep as 30.0 m. The submarine s volume can be approximated by that of a cylinder with a length of 3.00 m and a cross-sectional area of 0.500 m 2.Suppose this submarine dives in a freshwater river and then moves out to sea, which naturally consists of salt water. What mass of fresh water must be added to the ballast to keep the submarine submerged? The density of fresh water is 1.000 10 3 kg/m 3, and the density of sea water is 1.025 10 3 kg/m 3. 4. The largest iceberg ever observed had an area of 3.10 10 4 km 2, which is larger than the area of Belgium. If the top and bottom surfaces of the iceberg were flat and the thickness of the submerged part was 0.84 km, how large was the buoyant force acting on the iceberg? The density of sea water equals 1.025 10 3 kg/m 3. 5. A cannon built in 1868 in Russia could fire a cannonball with a mass of 4.80 10 2 kg and a radius of 0.250 m. When suspended from a scale and submerged in water, a cannonball of this type has an apparent weight of 4.07 10 3 N. How large is the buoyant force acting on the cannonball? The density of fresh water is 1.00 10 3 kg/m 3. 6. The tallest iceberg ever measured stood 167 m above the water. Suppose that both the top and the bottom of this iceberg were flat and the thickness of the submerged part was estimated to be 1.50 km. Calculate the density of the ice. The density of sea water equals 1.025 10 3 kg/m 3. 7. The Russian submarines of the Typhoon class are the largest submarines in the world. They have a length of 1.70 10 2 m and an average diameter of 13.9 m. When submerged, they displace 2.65 10 7 kg of sea water. Assume that these submarines have a simple cylindrical shape. If a Typhoon -class submarine has taken on enough ballast that it descends with a net acceleration of 2.00 m/s 2, what is the submarine s density during its descent? 8. To keep Robert La Salle s ship La Belle well preserved, shipbuilders are reconstructing the ship in a large tank filled with fresh water. Polyethylene glycol, or PEG, will then be slowly added to the water until a 30 percent PEG solution is formed. Suppose La Belle displaces 6.00 m 3 of liquid when submerged. If the ship s apparent weight decreases by 800 N as the PEG concentration increases from 0 to 30 percent, what is the density of the final PEG solution? The density of fresh water is 1.00 10 3 kg/m 3. Problem 9A 99

PRESSURE Holt Physics Problem 9B PROBLEM SOLUTION The largest helicopter in the world, which was built in Russia, has a mass of 1.03 10 3 kg. If you placed this helicopter on a large piston of a hydraulic lift, what force would need to be applied to the small piston in order to slowly lift the helicopter? Assume that the weight of the helicopter is distributed evenly over the large piston s area, which is 1.40 10 2 m 2. The area of the small piston is 0.80 m 2. Given: Unknown: F 2 =? m = 1.03 10 5 kg A 1 = 1.40 10 2 m 2 A 2 = 0.80 m 2 g = 9.81 m/s 2 Use the equation for pressure to equate the two opposing pressures in terms of force and area. P = A F P1 = P 2 ADDITIONAL PRACTICE F 1 F = 2 A1 A 2 F 2 = F 1 A 2 = mg A1 A 2 A1 F 2 = (1.03 10 5 kg)(9.81 m/s 2 2 0.80 m ) 1.4 0 10 2 m 2 F 2 = 5.8 10 3 N 1. Astronauts and cosmonauts have used pressurized spacesuits to explore the low-pressure regions of space. The pressure inside one of these suits must be close to that of Earth s atmosphere at sea level so that the space explorer may be safe and comfortable. The pressure on the outside of the suit is a fraction of 1.0 Pa. Clearly, pressurized suits are made of extremely sturdy material that can tolerate the stress from these pressure differences. If the average interior surface area of a pressurized spacesuit is 3.3 m 2, what is the force exerted on the suit s material? Assume that the pressure outside the suit is zero and that the pressure inside the suit is 1.01 10 5 Pa. 2. A strange idea to control volcanic eruptions is developed by a daydreaming engineer. The engineer imagines a giant piston that fits into the 100 Holt Physics Problem Workbook

volcano s shaft, which leads from Earth s surface down to the magma chamber. The piston controls an eruption by exerting pressure that is equal to or greater than the pressure of the hot gases, ash, and magma that rise from the magma chamber through the shaft. The engineer assumes that the pressure of the volcanic material is 4.0 10 11 Pa, which is the pressure in Earth s interior. If the material rises into a cylindrical shaft with a radius of 50.0 m, what force is needed on the other side of the piston to balance the pressure of the volcanic material? 3. The largest goat ever grown on a farm had a mass of 181 kg; on the other hand, the smallest pygmy goats have a mass of only about 16 kg. Imagine an agricultural show in which a large goat with a mass of 181 kg exerts a pressure on a hydraulic-lift piston that is equal to the pressure exerted by three pygmy goats, each of which has a mass of 16.0 kg. The area of the piston on which the large goat stands is 1.8 m 2. What is the area of the piston on which the pygmy goats stand? 4. The greatest load ever raised was the offshore Ekofisk complex in the North Sea. The complex, which had a mass of 4.0 10 7 kg, was raised 6.5 m by more than 100 hydraulic jacks. Imagine that his load could have been raised using a single huge hydraulic lift. If the load had been placed on the large piston and a force of 1.2 10 4 N had been applied to the small piston, which had an area of 5.0 m 2, what must the large piston s area have been? 5. The pressure that can exist in the interior of a star due to the weight of the outer layers of hot gas is typically several hundred billion times greater than the pressure exerted on Earth s surface by Earth s atmosphere. Suppose a pressure equal to that estimated for the sun s interior (2.0 10 16 Pa) acts on a spherical surface within a star. If a force of 1.02 N 10 31 N produces this pressure, what is the area of the surface? What is the sphere s radius r? (Recall that a sphere s surface area equals 4pr 2.) 6. The eye of a giant squid can be more than 35 cm in diameter the largest eye of any animal. Giant squid also live at depths greater than a mile below the ocean s surface. At a depth of 2 km, the outer half of a giant squid s eye is acted on by an external force of 4.6 10 6 N. Assuming the squid s eye has a diameter of 38 cm, what is the pressure on the eye? (Hint: Treat the eye as a sphere.) 7. The largest tires in the world, which are used for certain huge dump trucks, have diameters of about 3.50 m. Suppose one of these tires has a volume of 5.25 m 3 and a surface area of 26.3 m 2. If a force of 1.58 10 7 N acts on the inner area of the tire, what is the absolute pressure inside the tire? What is the gauge pressure on the tire s surface? Problem 9B 101

Holt Physics Problem 9C PRESSURE AS A FUNCTION OF DEPTH PROBLEM SOLUTION In 1969, a whale dove and remained underwater for nearly 2 h. Evidence indicates that the whale reached a depth of 3.00 km, where the whale sustained a pressure of 3.03 10 7 Pa. Estimate the density of sea water. Given: Unknown: r =? h = 3.00 km = 3.00 10 3 m P = 3.03 10 7 Pa P o = 1.01 10 5 Pa g = 9.81 m/s 2 Use the equation for fluid pressure as a function of depth, and rearrange it to solve for density. P = P o + rgh r = P P o gh r = r = (3.03 10 7 Pa) (1.01 10 5 Pa) (9.81 m/s 2 )(3.00 10 3 m) 3.02 10 7 Pa (9.81 m/s 2 )(3.00 10 3 m) r = 1.03 10 3 kg/m 3 ADDITIONAL PRACTICE 1. In 1994, a 16.8 m tall oil-filled barometer was constructed in Belgium. Suppose the barometer column was 80.0 percent filled with oil. What was the density of the oil if the pressure at the bottom of the column was 2.22 10 5 Pa and the air pressure at the top of the oil column was 1.01 10 5 Pa? 2. One of the lowest atmospheric pressures ever measured at sea level was 8.88 10 4 Pa, which existed in hurricane Gilbert in 1988. This same pressure can be found at a height 950 m above sea level. Use this information to estimate the density of air. 3. In 1993, Francisco Ferreras of Cuba held his breath and took a dive that lasted more than 2 min. The maximum pressure Ferreras experienced was 13.6 times greater than atmospheric pressure. To what depth did Ferreras dive? The density of sea water is 1.025 10 3 kg/m 3. 4. A penguin can endure pressures as great as 4.90 10 6 Pa. What is the maximum depth to which a penguin can dive in sea water? 102 Holt Physics Problem Workbook

5. In 1942, the British ship Edinburgh, which was carrying a load of 460 gold ingots, sank off the coast of Norway. In 1981, all of the gold was recovered from a depth of 245 m by a team of 12 divers. What was the pressure exerted by the ocean s water at that depth? 6. In 1960, a bathyscaph descended 10 916 m below the ocean s surface. What was the pressure exerted on the bathyscaph at that depth? Problem 9C 103

Holt Physics Problem 9D BERNOULLI S EQUATION PROBLEM SOLUTION 1. DEFINE 2. PLAN The widest road tunnel in the world is located in California. The tunnel has a cross-sectional area of about 4.00 10 2 m 2. On the other hand, the Three Rivers water tunnel in Georgia has a cross-sectional area of only 8.0 m 2. Imagine connecting together two tunnels with areas equal to these along a flat region and setting fresh water to flow at 4.0 m/s in the narrower tunnel. If the pressure in the wider tunnel is 1.10 10 5 Pa, what is the pressure in the narrower tunnel? Given: A 1 = 8.0 m 2 A 2 = 4.00 10 2 m 2 r = 1.00 10 3 kg/m 3 v 1 = 4.0 m/s P 2 = 1.10 10 5 Pa Unknown: P 1 =? Choose the equation(s) or situation: Because this problem involves fluid flow, it requires the application of Bernoulli s equation. P 1 + 1 2 rv 1 2 + rgh1 = P 2 + 1 2 rv 2 2 + rgh2 3. CALCULATE The flow of water is horizontal, so h 1 and h 2 are equal. P 1 + 1 2 2 rv 1 = P2 + 1 2 2 rv 2 To find the speed of the flowing water in the wider tunnel, use the continuity equation. A 1 v 1 = A 2 v 2 Substitute this equation for v 2 into Bernoulli s equation. P 1 + 1 2 2 rv 1 = P2 + 1 2 r A 1 v 1 A 2 2 Rearrange the equation(s) to isolate the unknown(s): P 1 = P 2 + 1 2 2 rv 1 A 2 1 A2 1 Substitute the values into the equation(s) and solve: P 1 = 1.10 10 5 Pa + 1 2 (1.00 10 3 kg/m 3 )(4.0 m/s) 2 2 8. 0 m 2 4.00 10 2 m 1 2 P 1 = 1.10 10 5 Pa + (8.0 10 3 N m/m 3 )(4.0 10 4 1) P 1 = 1.10 10 5 Pa (8.0 10 3 Pa)(0.9996) P 1 = 1.10 10 5 Pa 8.0 10 3 Pa P 1 = 1.02 10 5 Pa 104 Holt Physics Problem Workbook

4. EVALUATE The pressure of the water increases when it flows into the wider tunnel. The speed of the water s flow decreases, as indicated by the continuity equation. v 2 = (4.0 m/s)(8/400) = 8 10 2 m/s ADDITIONAL PRACTICE 1. The Chicago system of sewer tunnels has a total length of about 200 km. The tunnels are all at the same level and vary in diameter from less than 3 m to 10 m. Consider a connection between a wide tunnel and a narrow tunnel. The pressure in the wide tunnel is 12 percent greater than the pressure in the narrow one, and the speeds of flowing water in the wide and the narrow tunnels are 0.60 m/s and 4.80 m/s, respectively. Based on this information, find the pressure in the wide tunnel. 2. A certain New York City water-supply tunnel is almost 170 km long, is all at the same level, and has a diameter of 4.10 m. Suppose water flows through the tunnel at a speed of 3.0 m/s until it reaches a narrow section where the tunnel s diameter is 2.70 m. The pressure in the narrow section is 82 kpa. Use the continuity equation to find the water s speed in the narrow section of the tunnel. Then find the pressure in the wide portion of the tunnel. 3. An enormous open vat owned by a British cider company has a volume of nearly 7000 m 3. If a small hole is drilled halfway down the side of this vat when it is full of cider, the cider will hit the ground 19.7 m away from the bottom of the vat. How tall is the vat? 4. The tallest cooling tower in the world is in Germany. Consider a water pipe coming down from the top of the tower. The pipe is punctured near the bottom, which causes the water to flow from the puncture hole with a speed of 59 m/s. How high is the tower? Assume the pressure inside the pipe is equal to the pressure outside the pipe. 5. A water tower built in Oklahoma has a capacity of 1893 m 3 and is about 66.0 m high. Suppose a little hole with a cross-sectional area of 10.0 cm 2 is drilled near the bottom of the tower s water tank. At what speed does the water initially flow through the hole? Assume that the air pressure inside and outside the tank is the same. 6. The Nurek Dam, in Tajikistan, is the tallest dam in the world its height is more than 300 m. Suppose the difference in water levels on the opposite sides of the dam is 3.00 10 2 m. If a small crack appears in the dam near the lower water level, at what speed will the water stream leave the crack? Assume the air pressure on either side of the dam is the same. 7. The world s largest litter bin has a volume of more than 40 m 3 and is 6.0 m tall. If this bin is filled with water and then a hole with an area of 0.16 cm 2 is drilled near the bottom, at what speed will the water leave the bin? Assume that the water level in the bin drops slowly and that the bin is open to the air. Problem 9D 105

Holt Physics Problem 9E THE IDEAL GAS LAW PROBLEM SOLUTION 1. DEFINE 2. PLAN A hot-air balloon named Double Eagle V traveled a record distance of more than 8000 km from Japan to California in 1981. The volume of the balloon was 1.13 10 4 m 3. If the balloon contained 2.84 10 29 gas particles that had an average temperature of 355 K, what was the absolute pressure of the gas in the balloon? Given: V = 1.13 10 4 m 3 N = 2.84 10 29 particles T = 355 K k B = 1.38 10 23 J/K Unknown: P =? Choose the equation(s) or situation: To find the pressure of the gas, use the ideal gas law. PV = Nk B T Rearrange the equation(s) to isolate the unknown(s): P = Nk BT V 3. CALCULATE 4. EVALUATE ADDITIONAL PRACTICE Substitute the values into the equation(s) and solve: P = P = (2.84 10 29 particles)(1.38 10 23 J/K)(355 K) 1.13 10 4 m 3 1.23 10 5 Pa The pressure inside the balloon is about 20 percent greater than standard air pressure. This pressure corresponds to the higher temperature of the air in the balloon. The hot air s temperature is also nearly 20 percent greater than 300 K. 1. The official altitude record for a balloon was set in 1961 by two American officers piloting a high-altitude helium balloon with a volume of 3.4 10 5 m 3. Assume that the temperature of the gas was 280 K. If the balloon contained 1.4 10 30 atoms of helium, find the absolute pressure in the balloon at the maximum altitude of 35 km. 2. The estimated number of locusts that made up a swarm that infested Nebraska in 1874 was 1.2 10 13. This number was about 7000 times the total human population of Earth back then and about 2000 times the total human population today. It is, however, only a few billionths of the number of molecules in a liter of gas. If a container with a volume of 1.0 10 3 m 3 106 Holt Physics Problem Workbook

is filled with 1.2 10 13 gas molecules maintained at a temperature of 300.0 K, what is the pressure of the gas in the container? 3. In terms of volume, the largest pyramid in the world is not in Egypt but in Mexico. The Pyramid of the Sun at Teotihuacan has a volume of 3.3 10 6 m 3. If you fill a balloon to this same volume with 1.5 10 32 molecules of nitrogen at a temperature of 360 K, what will the absolute pressure of the gas be? 4. A snow palace more than 30 m high was built in Japan in 1994. Suppose a container with the same volume as this snow palace is filled with 1.00 10 27 molecules. If the temperature of the gas is 2.70 10 2 K and the gas pressure is 36.2 Pa, what is the volume of the gas? 5. Suppose the volume of a balloon decreases so that the temperature of the balloon decreases from 280 K to 240 K and its pressure drops from 1.6 10 4 Pa to 1.7 10 4 Pa. What is the new volume of the gas? 6. The longest navigable tunnel in the world was built in France. Suppose the entire tunnel, which has a cross-sectional area of 2.50 10 2 m 2,is filled with air at a temperature of 3.00 10 2 K and a pressure of 101 kpa. If the tunnel contains 4.34 10 31 molecules, what is the volume and length of the tunnel? 7. A balloon is filled with 7.36 10 4 m 3 of hot air. If the pressure inside the balloon is 1.00 10 5 Pa and there are 1.63 10 30 particles of air inside, what is the average temperature of the air inside the balloon? 8. In 1993, a group of American researchers drilled a 3053 m shaft in the ice sheet of Greenland. Suppose the cross-sectional area of the shaft is 0.040 m 2. If the air in the shaft consists of 3.6 10 27 molecules at an average pressure of 105 kpa, what is the air s average temperature? 9. The cylinder of the largest steam engine had a radius of 1.82 m. Suppose the length of the cylinder is six times the radius. Steam at a pressure of 2.50 10 6 Pa and a temperature of 495 K enters the cylinder when the piston has reduced the volume in the cylinder to 3.00 m 3. The piston is then pushed outward until the volume of the steam in the cylinder is 57.0 m 3. If the pressure of the steam after expansion is 1.01 10 5 Pa, what is the temperature of the steam? Problem 9E 107