Concepts & Calculations

Note: Each of these problems consists of Concept Questions followed by a related quantitative Problem. The Concept Questions involve little or no mathematics. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem-solving technique.

| 97 | Concept Questions A gold prospector finds a solid rock that is composed solely of quartz and gold. (a) How is the total mass mT of the rock related to the mass mG of the gold and the mass mQ of the quartz? (b) What is the relationship between the total volume VT of the rock, the volume VG of the gold, and the volume VQ of the quartz? (c) How is the volume of a substance (gold or quartz) related to the mass of the substance and its density?
Problem The mass and volume of the rock are, respectively, 12.0 kg and . Find the mass of the gold in the rock.

| 98 | Concept Questions A meat baster consists of a squeeze bulb attached to a plastic tube. When the bulb is squeezed and released, with the open end of the tube under the surface of the basting sauce, the sauce rises in the tube to a distance h, as the drawing shows. It can then be squirted over the meat. (a) Is the absolute pressure in the bulb in the drawing greater than or less than atmospheric pressure? (b) In a second trial, the distance h is somewhat less than it is in the drawing. Is the absolute pressure in the bulb in the second trial greater or smaller than in the case shown in the drawing? Explain your answers.
Problem Using for the atmospheric pressure and 1200 kg/m3 for the density of the sauce, find the absolute pressure in the bulb when the distance h is (a) 0.15 m and (b) 0.10 m. Verify that your answers are consistent with your answers to the Concept Questions.

| 99 | Concept Questions A hydrometer is a device used to measure the density of a liquid. It is a cylindrical tube weighted at one end, so that it floats with the heavier end downward. It is contained inside a large “medicine dropper,” into which the liquid is drawn using a squeeze bulb (see the drawing). For use with your car, marks are put on the tube so that the level at which it floats indicates whether the liquid is battery acid (more dense) or antifreeze (less dense). (a) Compared to the weight W of the tube, how much buoyant force is needed to make the tube float in either battery acid or antifreeze? (b) Is a greater volume of battery acid or a greater volume of antifreeze displaced by the hydrometer to provide the necessary buoyant force? (c) Which mark is farther up from the bottom of the tube? Justify your answers.
Problem The hydrometer has a weight of and a cross-sectional area of . How far from the bottom of the tube should the mark be put that denotes (a) battery acid () and (b) antifreeze ()? Check to see that your answers are consistent with your answers to the Concept Questions.

| 100 | Concept Questions Water flows straight down from an open faucet. The effects of air resistance and viscosity can be ignored. (a) After the water has fallen a bit below the faucet, is its speed less than, greater than, or the same as it was on leaving the faucet? (b) Is the volume flow rate in cubic meters per second less than, greater than, or the same as it was when the water left the faucet? (c) Is the cross-sectional area of the water stream less than, greater than, or the same as it was when the water left the faucet? Give your reasoning.
Problem The cross-sectional area of the faucet is , and the speed of the water is 0.85 m/s as it leaves the faucet. Ignoring air resistance, find the cross-sectional area of the water stream at a point 0.10 m below the faucet. Make sure that your answer is consistent with your answers to the Concept Questions.

| 101 | Concept Questions Consult Multiple-Concept Example Example 15 to review the concepts upon which this problem depends. Water flowing out of a horizontal pipe emerges through a nozzle. The nozzle has a smaller radius than does the pipe. Treat the water as an ideal fluid. (a) What does the fact that the pipe is horizontal imply about the pipe’s elevation y above the ground? (b) Is the speed at which the water flows in the pipe greater than, less than, or the same as the speed of the water emerging from the nozzle? (c) The emerging water is at atmospheric pressure. Is the absolute pressure of the water in the pipe greater than, less than, or the same as the atmospheric pressure? Explain your answers.
Problem The radius of the pipe is 1.9 cm, and the radius of the nozzle is 0.48 cm. The speed of the water in the pipe is 0.62 m/s. Determine the absolute pressure of the water in the pipe. Verify that your answer is consistent with your answers to the Concept Questions.

| 102 | Concept Questions A ship is floating on a lake. Its hold is the interior space beneath its deck and is open to the atmosphere. The hull has a hole in it, which is below the water line, so water leaks into the hold. (a) How is the amount of water per second (in m3/s) entering the hold related to the speed of the entering water and the area of the hole? (b) Approximately how fast is the water at the surface of the lake moving? Justify your answer. (c) What causes the water to accelerate as it moves from the surface of the lake into the hole that is beneath the water line? Explain.
Problem The effective area of the hole is and is located 2.0 m beneath the surface of the lake. What volume of water per second leaks into the ship?

| 103 | Concept Questions A hot-air balloon is accelerating upward under the influence of two forces, its weight and the buoyant force. For simplicity, consider the weight to be only that of the hot air within the balloon, thus ignoring the balloon fabric and the basket. (a) How is the weight of the hot air determined from a knowledge of its density and the volume V of the balloon? (b) For a given volume V of the balloon, does the buoyant force depend on the density of the hot air inside the balloon, the density of the cool air outside the balloon, or both? Provide a reason for your answer. (c) Draw a free-body diagram for the balloon, showing the forces that act on it. How is the upward acceleration of the balloon related to these forces and to its mass?
Problem The hot air inside the balloon has a density of , and that of the cool air outside is . What is the acceleration of the rising balloon?

| 104 | Concept Questions Two hoses are connected to the same outlet using a Y-connector, as the drawing shows. The hoses A and B have the same length, but hose B has the larger radius. Each is open to the atmosphere at the end where the water exits. Water flows through both hoses as a viscous fluid, and Poiseuille’s law [] applies to each. In this law, P2 is the pressure upstream, P1 is the pressure downstream, and Q is the volume flow rate. (a) For hoses A and B, is the value for the term the same or different? (b) How is Q related to the radius of a hose and the speed of the water in the hose? Account for your answers.
Problem The ratio of the radius of hose B to the radius of hose A is . Find the ratio of the speed of the water in hose B to that in hose A.

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