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All Textbook Solutions for Physics for Scientists and Engineers with Modern Physics

12P A string that is 30.0 cm long and has a mass per unit length of 9.00 103 kg/m is stretched to a tension of 20.0 N. Find (a) the fundamental frequency and (b) the next three frequencies of possible standing-wave patterns on the string.14P Review. A sphere of mass M = 1.00 kg is supported by a string that passes over a pulley at the end of a horizontal rod of length L = 0.300 m (Fig. P17.15). The string makes an angle = 35.0 with the rod. The fundamental frequency of standing waves in the portion of the string above the rod is f = 60.0 Hz. Find the mass of the portion of the string above the rod. Figure P17.15 Problems 15 and 16.16P 17P 18P 19P 20P The fundamental frequency of an open organ pipe corresponds to middle C (261.6 Hz on the chromatic musical scale). The third resonance of a closed organ pipe has the same frequency. What is the length of (a) the open pipe and (b) the closed pipe?Ever since seeing Figure 16.22 in the previous chapter, you have been fascinated with the hearing response in humans. You have set up an apparatus that allows you to determine your own threshold of hearing as a function of frequency. After performing the experiment and recording the results, you graph the results, which look like Figure P17.22. You are intrigued by the two dips in the curve at the right-hand side of the graph. You measure carefully and find that the minimum values of these dips occur at 3 800 Hz and 11 500 Hz. Performing some online research, you discover that the outer canal of the human ear can be modeled as an air column open at the outer end and closed at the inner end by the eardrum. You use this information to determine the length of the outer canal in your car. Figure P17.22 An air column in a glass tube is open at one end and closed at the other by a movable piston. The air in the tube is warmed above room temperature, and a 384-Hz tuning fork is held at the open end. Resonance is heard when the piston is at a distance d1 = 22.8 cm from the open end and again when it is at a distance d2= 68.3 cm from the open end. (a) What speed of sound is implied by these data? (b) How far from the open end will the piston be when the next resonance is heard?24P 25P 26P As shown in Figure P17.27, water is pumped into a tall, vertical cylinder at a volume flow rate R = 1.00 L/min. The radius of the cylinder is r = 5.00 cm, and at the open top of the cylinder a tuning fork is vibrating with a frequency f = 512 Hz. As the water rises, what time interval elapses between successive resonances? Figure P17.27 As shown in Figure P17.27, water is pumped into a tall, vertical cylinder at a volume flow rate R. The radius of the cylinder is r, and at the open top of the cylinder a tuning fork is vibrating with a frequency f. As the water rises, what time interval elapses between successive resonances? Figure P17.27 29P 30P 31P 32P 33P 34AP 35AP A 2.00-m-long wire having a mass of 0.100 kg is fixed at both ends. The tension in the wire is maintained at 20.0 N. (a) What are the frequencies of the first three allowed modes of vibration? (b) If a node is observed at a point 0.400 m from one end, in what mode and with what frequency is it vibrating?37AP 38AP 39AP Review. For the arrangement shown in Figure P17.40, the inclined plane and the small pulley are frictionless; the string supports the object of mass M at the bottom of the plane; and the entire string has mass m. The system is in equilibrium, and the vertical part of the string has a length h. We wish to study standing waves set up in the vertical section of the string. (a) What analysis model describes the object of mass M? (b) What analysis model describes the waves on the vertical part of the string? (c) Find the tension in the string. (d) Model the shape of the string as one leg and the hypotenuse of a right triangle. Find the whole length of the string. (c) Find the mass per unit length of the string. (f) Find the speed of waves on the string. (g) Find the lowest frequency for a standing wave on the vertical section of the string. (h) Evaluate this result for M = 1.50 kg, m = 0.750 g, h = 0.500 m, and = 30.0. (i) Find the numerical value for the lowest frequency for a standing wave on the sloped section of the string. Figure P17.40 41AP Two speakers are driven by the same oscillator of frequency f. They are located a distance d from each other on a vertical pole. A man walks straight toward the lower speaker in a direction perpendicular to the pole as shown in Figure P17.42. (a) How many times will he hear a minimum in sound intensity? (b) How far is he from the pole at these moments? Let v represent the speed of sound and assume that the ground does not reflect sound. The mans ears are at the same level as the lower speaker. Figure P17.42 43AP 44AP 45AP 46AP Review. A 12.0-kg object hangs in equilibrium from a string with a total length of L = 5.00 m and a linear mass density of = 0.001 00 kg/m. The string is wrapped around two light, frictionless pulleys that are separated by a distance of d = 2.00 m (Fig. P17.47a). (a) Determine the tension in the string. (b) At what frequency must the string between the pulleys vibrate to form the standing-wave pattern shown in Figure P 17.47b? Figure P17.47 Problem 47 and 48.Review. An object of mass m hangs in equilibrium from a string with a total length L and a linear mass density . The string is wrapped around two light, frictionless pulleys that are separated by a distance d (Fig. P17.47a). (a) Determine the tension in the string. (b) At what frequency must the string between the pulleys vibrate to form the standing-wave pattern shown in Figure P17.47b?49AP 50CP 18.1QQ Consider the following pairs of materials. Which pair represents two materials, one of which is twice as hot as the other? (a) boiling water at 100C, a glass of water at 50C (b) boiling water at 100C, frozen methane at 50C (c) an ice cube at 20C, flames from a circus fire-eater at 233C (d) none of those pairs If you are asked to make a very sensitive glass thermometer, which of the following working liquids would you choose? (a) mercury (b) alcohol (c) gasoline (d) glycerin 18.4QQ A common material for cushioning objects in packages is made by trapping bubbles of air between sheets of plastic. Is this material more effective at keeping the contents of the package from moving around inside the package on (a) a hot day, (b) a cold day, or (c) either hot or cold days?On a winter day, you turn on your furnace and the temperature of the air inside your home increases. Assume your home has the normal amount of leakage between inside air and outside air. Is the number of moles of air in your room at the higher temperature (a) larger than before, (b) smaller than before, or (c) the same as before?1P 2P 3P Liquid nitrogen has a boiling point of 195.81C at atmospheric pressure. Express this temperature (a) in degrees Fahrenheit and (b) in kelvins.5P 6P A copper telephone wire has essentially no sag between poles 35.0 m apart on a winter day when the temperature is 20.0C. How much longer is the wire on a summer day when the temperature is 35.0C?8P The Trans-Alaska pipeline is 1 300 km long, reaching from Prudhoe Bay to the port of Valdez. It experiences temperatures from 73C to +35C. How much does the steel pipeline expand because of the difference in temperature? How can this expansion be compensated for?10P 11P 12P 13P Why is the following situation impossible? A thin brass ring has an inner diameter 10.00 cm at 20.0C. A solid aluminum cylinder has diameter 10.02 cm at 20.0C. Assume the average coefficients of linear expansion of the two metals are constant. Both metals are cooled together to a temperature at which the ring can be slipped over the end of the cylinder.A volumetric flask made of Pyrex is calibrated at 20.0C. It is filled to the 100-mL mark with 35.0C acetone. After the flask is filled, the acetone cools and the flask warms so that the combination of acetone and flask reaches a uniform temperature of 32.0C. The combination is then cooled back to 20.0C. (a) What is the volume of the acetone when it cools to 20.0C? (b) At the temperature of 32.0C, does the level of acetone lie above or below the 100-mL mark on the flask? Explain.Review. On a day that the temperature is 20.0C, a concrete walk is poured in such a way that the ends of the walk are unable to move. Take Youngs modulus for concrete to be 7.00 109 N/m2 and the compressive strength to be 2.00 109 N/m2. (a) What is the stress in the cement on a hot day of 50.0C? (b) Does the concrete fracture?17P 18P An auditorium has dimensions 10.0 m 20.0 m 30.0 m. How many molecules of air fill the auditorium at 20.0C and a pressure of 101 kPa (1.00 atm)?20P 21P 22P In state-of-the-art vacuum systems, pressures as low as 1.00 109 Pa are being attained. Calculate the number of molecules in a 1.00-m3 vessel at this pressure and a temperature of 27.0C.24P 25P 26P 27P 28P The pressure gauge on a cylinder of gas registers the gauge pressure, which is the difference between the interior pressure and the exterior pressure P0. Lets call the gauge pressure Pg. When the cylinder is full, the mass of the gas in it is mi at a gauge pressure of Pgi. Assuming the temperature of the cylinder remains constant, show that the mass of the gas remaining in the cylinder when the pressure reading is Pgf is given by mf=mi(Pgf+P0Pgi+P0)30AP 31AP Why is the following situation impossible? An apparatus is designed so that steam initially at T = 150C, P = 1.00 atm, and V = 0.500 m3 in a pistoncylinder apparatus undergoes a process in which (1) the volume remains constant and the pressure drops to 0.870 atm, followed by (2) an expansion in which the pressure remains constant and the volume increases to 1.00 m3, followed by (3) a return to the initial conditions. It is important that the pressure of the gas never fall below 0.850 atm so that the piston will support a delicate and very expensive part of the apparatus. Without such support, the delicate apparatus can be severely damaged and rendered useless. When the design is turned into a working prototype, it operates perfectly.A student measures the length of a brass rod with a steel tape at 20.0C. The reading is 95.00 cm. What will the tape indicate for the length of the rod when the rod and the tape are at (a) 15.0C and (b) 55.0C?34AP A liquid has a density . (a) Show that the fractional change in density for a change in temperature T is / = T. (b) What does the negative sign signify? (c) Fresh water has a maximum density of 1.000 0 g/cm3 at 4.0C. At 10.0C, its density is 0.999 7 g/cm3. What is for water over this temperature interval? (d) At 0C, the density of water is 0.999 9 g/cm3. What is the value for over the temperature range 0C to 4.00C?36AP 37AP A bimetallic strip of length L is made of two ribbons of different metals bonded together. (a) First assume the strip is originally straight. As the strip is warmed, the metal with the greater average coefficient of expansion expands more than the other, forcing the strip into an arc with the outer radius having a greater circumference (Fig. P18.38). Derive an expression for the angle of bending as a function of the initial length of the strips, their average coefficients of linear expansion, the change in temperature, and the separation of the centers of the strips (r = r2 r1). (b) Show that the angle of bending decreases to zero when T decreases to zero and also when the two average coefficients of expansion become equal. (c) What If? What happens if the strip is cooled? Figure P18.38 39AP A vertical cylinder of cross-sectional area A is fitted with a tight-fitting, frictionless piston of mass m (Fig. P18.40). The piston is not restricted in its motion in any way and is supported by the gas at pressure P below it. Atmospheric pressure is P0. We wish to find the height h in Figure P18.40. (a) What analysis model is appropriate to describe the piston? (b) Write an appropriate force equation for the piston from this analysis model in terms of P, P0, m, A, and g. (c) Suppose n moles of an ideal gas are in the cylinder at a temperature of T. Substitute for P in your answer to part (b) to find the height h of the piston above the bottom of the cylinder. Figure P18.40 41AP 42AP 43AP 44CP A 1.00-km steel railroad rail is fastened securely at both ends when the temperature is 20.0C. As the temperature increases, the rail buckles, taking the shape of an arc of a vertical circle. Find the height h of the center of the rail when the temperature is 25.0C. (You will need to solve a transcendental equation.)46CP 19.1QQ 19.2QQ 19.3QQ Characterize the paths in Figure 19.12 as isobaric, isovolumetric, isothermal, or adiabatic. For path B, Q = 0. The blue curves are isotherms.19.5QQ 1P The highest waterfall in the world is the Salto Angel in Venezuela. Its longest single falls has a height of 807 m. If water at the top of the falls is at 15.0C, what is the maximum temperature of the water at the bottom of the falls? Assume all the kinetic energy of the water as it reaches the bottom goes into raising its temperature.3P The temperature of a silver bar rises by 10.0C when it absorbs 1.23 kJ of energy by heat. The mass of the bar is 525 g. Determine the specific heat of silver from these data.You are working in your kitchen preparing lunch for your family. You have decided to make egg salad sandwiches and are boiling six eggs, each of mass 55.5 g, in 0.750 L of water at 100C. You wish to take all the eggs out of the boiling water and immediately place them in 23.0C water to cool them down to a comfortable temperature to hold them and peel them. You decide that you wish the mixture of the water and the eggs to reach an equilibrium temperature of 40.0C. Explaining this to a family member, she challenges you to determine exactly how much water at 23.0C you need to achieve your desired equilibrium temperature. Take the average specific heat of an egg over the expected temperature range to be 3.27 103 J/kg C.If water with a mass mk at temperature Tk is poured into an aluminum cup of mass mA1 containing mass mc of water at Tc, where Tk Tc, what is the equilibrium temperature of the system?7P An electric drill with a steel drill bit of mass m = 27.0 g and diameter 0.635 cm is used to drill into a cubical steel block of mass M = 240 g. Assume steel has the same properties as iron. The cutting process can be modeled as happening at one point on the circumference of the bit. This point moves in a helix at constant tangential speed 40.0 m/s and exerts a force of constant magnitude 3.20 N on the block. As shown in Figure P19.8 (page 528), a groove in the bit carries the chips up to the top of the block, where they form a pile around the hole. The drill is turned on and drills into the block for a time interval of 15.0 s. Lets assume this time interval is long enough for conduction within the steel to bring it all to a uniform temperature. Furthermore, assume the steel objects lose a negligible amount of energy by conduction, convection, and radiation into their environment. (a) Suppose the drill bit cuts three-quarters of the way through the block during 15.0 s. Find the temperature change of the whole quantity of steel. (b) What If? Now suppose the drill bit is dull and cuts only one-eighth of the way through the block in 15.0 s. Identify the temperature change of the whole quantity of steel in this case. (c) What pieces of data, if any, are unnecessary for the solution. Explain. Figure P19.8 9P How much energy is required to change a 40.0-g ice cube from ice at 10.0C to steam at 110C?11P 12P In an insulated vessel, 250 g of ice at 0C is added to 600 g of water at 18.0C. (a) What is the final temperature of the system? (b) How much ice remains when the system reaches equilibrium?14P One mole of an ideal gas is warmed slowly so that it goes from the PV state (Pi, Vi) to (3Pi, 3Vi) in such a way that the pressure of the gas is directly proportional to the volume. (a) How much work is done on the gas in the process? (b) How is the temperature of the gas related to its volume during this process?(a) Determine the work done on a gas that expands from i to f as indicated in Figure P19.16. (b) What If? How much work is done on the gas if it is compressed from f to i along the same path? Figure P19.16 A thermodynamic system undergoes a process in which its internal energy decreases by 500 J. Over the same time interval, 220 J of work is done on the system. Find the energy transferred from it by heat.18P A 2.00-mol sample of helium gas initially at 300 K, and 0.400 atm is compressed isothermally to 1.20 atm. Noting that the helium behaves as an ideal gas, find (a) the final volume of the gas, (b) the work done on the gas, and (c) the energy transferred by heat.(a) How much work is done on the steam when 1.00 mol of water at 100C boils and becomes 1.00 mol of steam at 100C at 1.00 atm pressure? Assume the steam to behave as an ideal gas. (b) Determine the change in internal energy of the system of the water and steam as the water vaporizes.A 1.00-kg block of aluminum is warmed at atmospheric pressure so that its temperature increases from 22.0C to 40.0C. Find (a) the work done on the aluminum, (b) the energy added to it by heat, and (c) the change in its internal energy.In Figure P19.22, the change in internal energy of a gas that is taken from A to C along the blue path is +800 J. The work done on the gas along the red path ABC is 500 J. (a) How much energy must be added to the system by heat as it goes from A through B to C? (b) If the pressure at point A is five times that of point C, what is the work done on the system in going from C to D? Figure P19.22 (c) What is the energy exchanged with the surroundings by heat as the gas goes from C to A along the green path? (d) If the change in internal energy in going from point D to point A is +500 J, how much energy must be added to the system by heat as it goes from point C to point D?23P A concrete slab is 12.0 cm thick and has an area of 5.00 m2. Electric heating coils are installed under the slab to melt the ice on the surface in the winter months. What minimum power must be supplied to the coils to maintain a temperature difference of 20.0C between the bottom of the slab and its surface? Assume all the energy transferred is through the slab.Two lightbulbs have cylindrical filaments much greater in length than in diameter. The evacuated bulbs are identical except that one operates at a filament temperature of 2 100C and the other operates at 2 000C. (a) Find the ratio of the power emitted by the hotter lightbulb to that emitted by the cooler lightbulb. (b) With the bulbs operating at the same respective temperatures, the cooler lightbulb is to be altered by making its filament thicker so that it emits the same power as the hotter one. By what factor should the radius of this filament be increased?26P (a) Calculate the R-value of a thermal window made of two single panes of glass each 0.125 in. thick and separated by a 0.230-in. air space. (b) By what factor is the transfer of energy by heat through the window reduced by using the thermal window instead of the single-pane window? Include the contributions of inside and outside stagnant air layers.28P Gas in a container is at a pressure of 1.50 atm and a volume of 4.04 m3. What is the work done on the gas (a) if it expands at constant pressure to twice its initial volume, and (b) if it is compressed at constant pressure to one-quarter its initial volume?30AP You have a particular interest in automobile engines, so you have secured a co-op position at an automobile company while you attend school. Your supervisor is helping you to learn about the operation of an internal combustion engine. She gives you the following assignment, related to a simulation of a new engine she is designing. A gas, beginning at PA = 1.00 atm, VA = 0.500 L, and TA = 27.0C, is compressed from point A on the PV diagram in Figure P19.31 (page 530) to point B. This represents the compression stroke in a fourcycle gasoline engine. At that point, 132 J of energy is delivered to the gas at constant volume, taking the gas to point C. This represents the transformation of potential energy in the gasoline to internal energy when the spark plug fires. Your supervisor tells you that the internal energy of a gas is proportional to temperature (as we shall find in Chapter 20), the internal energy of the gas at point A is 200 J, and she wants to know what the temperature of the gas is at point C. Figure P19.31 32AP 33AP 34AP Review. Following a collision between a large spacecraft and an asteroid, a copper disk of radius 28.0 m and thickness 1.20 m at a temperature of 850C is floating in space, rotating about its symmetry axis with an angular speed of 25.0 rad/s. As the disk radiates infrared light, its temperature falls to 20.0C. No external torque acts on the disk. (a) Find the change in kinetic energy of the disk. (b) Find the change in internal energy of the disk. (c) Find the amount of energy it radiates.36AP An ice-cube tray is filled with 75.0 g of water. After the filled tray reaches an equilibrium temperature of 20.0C, it is placed in a freezer set at 8.00C to make ice cubes. (a) Describe the processes that occur as energy is being removed from the water to make ice. (b) Calculate the energy that must be removed from the water to make ice cubes at 8.00C.38AP An iron plate is held against an iron wheel so that a kinetic friction force of 50.0 N acts between the two pieces of metal. The relative speed at which the two surfaces slide over each other is 40.0 m/s. (a) Calculate the rate at which mechanical energy is converted to internal energy. (b) The plate and the wheel each have a mass of 5.00 kg, and each receives 50.0% of the internal energy. If the system is run as described for 10.0 s and each object is then allowed to reach a uniform internal temperature, what is the resultant temperature increase?One mole of an ideal gas is contained in a cylinder with a movable piston. The initial pressure, volume, and temperature are Pi, Vi, and Ti, respectively. Find the work done on the gas in the following processes. In operational terms, describe how to carry out each process and show each process on a PV diagram. (a) an isobaric compression in which the final volume is one-half the initial volume (b) an isothermal compression in which the final pressure is four times the initial pressure (c) an isovolumetric process in which the final pressure is three times the initial pressure 41AP 42AP 43AP A student measures the following data in a calorimetry experiment designed to determine the specific heat of aluminum: Initial temperature of water and calorimeter: 70.0C Mass of water: 0.400 kg Mass of calorimeter: 0.040 kg Specific heat of calorimeter: 0.63 kJ/kg C Initial temperature of aluminum: 27.0C Mass of aluminum: 0.200 kg Final temperature of mixture: 66.3C (a) Use these data to determine the specific heat of aluminum. (b) Explain whether your result is within 15% of the value listed in Table 19.1.(a) The inside of a hollow cylinder is maintained at a temperature Ta, and the outside is at a lower temperature, Tb (Fig. P19.45). The wall of the cylinder has a thermal conductivity k. Ignoring end effects, show that the rate of energy conduction from the inner surface to the outer surface in the radial direction is dQdt=2Lk[TaTbln(b/a)] Suggestions: The temperature gradient is dT/dr. A radial energy current passes through a concentric cylinder of area 2rL. (b) The passenger section of a jet airliner is in the shape of a cylindrical tube with a length of 35.0 m and an inner radius of 2.50 m. Its walls are lined with an insulating material 6.00 cm in thickness and having a thermal conductivity of 4.00 105 cal/s cm C. A heater must maintain the interior temperature at 25.0C while the outside temperature is 35.0C. What power must be supplied to the heater? Figure P19.45 46CP 47CP Two containers hold an ideal gas at the same temperature and pressure. Both containers hold the same type of gas, but container B has twice the volume of container A. (i) What is the average translational kinetic energy per molecule in container B? (a) twice that of container A (b) the same as that of container A (c) half that of container A (d) impossible to determine (ii) From the same choices, describe the internal energy of the gas in container B.(i) How does the internal energy of an ideal gas change as it follows path i f in Figure 20.4? (a) Eint increases. (b) Eint decreases. (c) Eint stays the same. (d) There is not enough information to determine how Eint changes. (ii) From the same choices, how does the internal energy of an ideal gas change as it follows path f f along the isotherm labeled T + T in Figure 20.4? Figure 20.4 Energy is transferred by heat to an ideal gas in two ways.20.3QQ 20.4QQ A spherical balloon of volume 4.00 103 cm3 contains helium at a pressure of 1.20 105 Pa. How many moles of helium are in the balloon if the average kinetic energy of the helium atoms is 3.60 1022 J?A spherical balloon of volume V contains helium at a pressure P. How many moles of helium are in the balloon if the average kinetic energy of the helium atoms is Kavg?A 2.00-mol sample of oxygen gas is confined to a 5.00-L vessel at a pressure of 8.00 atm. Find the average translational kinetic energy of the oxygen molecules under these conditions.4P A 5.00-L vessel contains nitrogen gas at 27.0C and 3.00 atm. Find (a) the total translational kinetic energy of the gas molecules and (b) the average kinetic energy per molecule.6P In a period of 1.00 s, 5.00 1023 nitrogen molecules strike a wall with an area of 8.00 cm2. Assume the molecules move with a speed of 300 m/s and strike the wall head-on in elastic collisions. What is the pressure exerted on the wall? Note: The mass of one N2 molecule is 4.65 1026 kg.A 7.00-L vessel contains 3.50 moles of gas at a pressure of 1.60 106 Pa. Find (a) the temperature of the gas and (b) the average kinetic energy of the gas molecules in the vessel. (c) What additional information would you need if you were asked to find the average speed of the gas molecules?Calculate the change in internal energy of 3.00 mol of helium gas when its temperature is increased by 2.00 K.10P In a constant-volume process, 209 J of energy is transferred by heat to 1.00 mol of an ideal monatomic gas initially at 300 K. Find (a) the work done on the gas, (b) the increase in internal energy of the gas, and (c) its final temperature.A vertical cylinder with a heavy piston contains air at 300 k. The initial pressure is 2.00 105 Pa, and the initial volume is 0.350 m3. Take the molar mass of air as 28.9 g/mol and assume CV=52R. (a) Find the specific heat of air at constant volume in units of J/kg C. (b) Calculate the mass of the air in the cylinder. (c) Suppose the piston is held fixed. Find the energy input required to raise the temperature of the air to 700 K. (d) What If? Assume again the conditions of the initial state and assume the heavy piston is free to move. Find the energy input required to raise the temperature to 700 K.13P A certain molecule has f degrees of freedom. Show that an ideal gas consisting of such molecules has the following properties: (a) its total internal energy is fnRT/2, (b) its molar specific heat at constant volume is fR/2, (c) its molar specific heat at constant pressure is (f + 2)R/2, and (d) its specific heat ratio is = CP/CV = (f + 2)/f.15P Why is the following situation impossible? A team of researchers discovers a new gas, which has a value of = CP/CV of 1.75.You and your younger brother are designing an air rifle that will shoot a lead pellet with mass m = 1.10 g and cross-sectional area A = 0.030 0 cm3. The rifle works by allowing high-pressure air to expand, propelling the pellet down the rifle barrel. Because this process happens very quickly, no appreciable thermal conduction occurs and the expansion is essentially adiabatic. Your design is such that, once the pressure begins pushing on the pellet, it moves a distance of L = 50.0 cm before leaving the open end of the rifle at your desired speed of v = 120 m/s. Your design also includes a chamber of volume V = 12.0 cm3 in which the high-pressure air is stored until it is released. Your brother reminds you that you need to purchase a pump to pressurize the chamber. To determine what kind of pump to buy, you need to find what the pressure of the air must be in the chamber to achieve your desired muzzle speed. Ignore the effects of the air in front of the bullet and friction with the inside walls of the barrel.During the compression stroke of a certain gasoline engine, the pressure increases from 1.00 atm to 20.0 atm. If the process is adiabatic and the air-fuel mixture behaves as a diatomic ideal gas. (a) by what factor does the volume change and (b) by what factor does the temperature change? Assuming the compression stalls with 0.016 0 mol of gas at 27.0C, find the values of (c) Q, (d) Eint, and (e) W that characterize the process.Air in a thundercloud expands as it rises. If its initial temperature is 300 K and no energy is lost by thermal conduction on expansion, what is its temperature when the initial volume has doubled?20P Air (a diatomic ideal gas) at 27.0C and atmospheric pressure is drawn into a bicycle pump that has a cylinder with an inner diameter of 2.50 cm and length 50.0 cm. The downstroke adiabatically compresses the air, which readies a gauge pressure of 8.00 105 Pa before entering the tire. We wish to investigate the temperature increase of the pump. (a) What is the initial volume of the air in the pump? (b) What is the number of moles of air in the pump? (c) What is the absolute pressure of the compressed air? (d) What is the volume of the compressed air? (c) What is the temperature of the compressed air? (f) What is the increase in internal energy of the gas during the compression? What If? The pump is made of steel that is 2.00 mm thick. Assume 4.00 cm of the cylinders length is allowed to come to thermal equilibrium with the air. (g) What is the volume of steel in this 4.00-cm length? (h) What is the mass of steel in this 4.00-cm length? (i) Assume the pump is compressed once. After the adiabatic expansion, conduction results in the energy increase in part (f) being shared between the gas and the 4.00-cm length of steel. What will be the increase in temperature of the steel after one compression?22P 23P 24P 25P 26P 27AP 28AP The dimensions of a classroom are 4.20 m 3.00 m 2.50 m. (a) Find the number of molecules of air in the classroom at atmospheric pressure and 20.0C. (b) Find the mass of this air, assuming the air consists of diatomic molecules with molar mass 28.9 g/mol. (c) Find the average kinetic energy of the molecules. (d) Find the rms molecular speed. (c) What If? Assume the molar specific heat of the air is independent of temperature. Find the change in internal energy of the air in the room as the temperature is raised to 25.0C. (f) Explain how you could convince a fellow student that your answer to part (e) is correct, even though it sounds surprising.30AP The Earths atmosphere consists primarily of oxygen (21%) 30 and nitrogen (78%). The rms speed of oxygen molecules (O2) in the atmosphere at a certain location is 535 m/s. (a) What is the temperature of the atmosphere at this location? (b) Would the rms speed of nitrogen molecules (N2) at this location be higher, equal to, or lower than 535 m/s? Explain. (c) Determine the rms speed of N2 at his location.32AP 33AP In a cylinder, a sample of an ideal gas with number of moles n undergoes an adiabatic process. (a) Starting with the expression W=PdV and using the condition PV = constant, show that the work done on the gas is W=(11)(PfVfPiVi) (b) Starting with the first law of thermodynamics, show that the work done on the gas is equal to nCV(Tf Ti). (c) Are these two results consistent with each other? Explain.As a 1.00-mol sample of a monatomic ideal gas expands adiabatically, the work done on it is 2.50 103 J. The initial temperature and pressure of the gas are 500 K and 3.60 atm. Calculate (a) the final temperature and (b) the final pressure.36AP 37AP 38AP 39AP 40AP 41AP On the PV diagram for an ideal gas, one isothermal curve and one adiabatic curve pass through each point as shown in Figure P20.42. Prove that the slope of the adiabatic curve is steeper than the slope of the isotherm at that point by the factor .43AP 44AP 45CP The energy input to an engine is 4.00 times greater than the work it performs. (i) What is its thermal efficiency? (a) 4.00 (b) 1.00 (c) 0.250 (d) impossible to determine (ii) What fraction of the energy input is expelled to the cold reservoir? (a) 0.250 (b) 0.750 (c) 1.00 (d) impossible to determine The energy entering an electric heater by electrical transmission can be converted to internal energy with an efficiency of 100%. By what factor does the cost of heating your home change when you replace your electric heating system with an electric heat pump that has a COP of 4.00? Assume the motor running the heat pump is 100% efficient. (a) 4.00 (b) 2.00 (c) 0.500 (d) 0.250 Three engines operate between reservoirs separated in temperature by 300 K. The reservoir temperatures are as follows: Engine A: Th = 1 000 K, Tc = 700 K; Engine B: Th = 800 K, Tc = 500 K; Engine C: Th = 600 K, Tc = 300 K. Rank the engines in order of theoretically possible efficiency from highest to lowest.(a) Suppose you select four cards at random from a standard deck of playing cards and end up with a macrostate of four deuces. How many microstates are associated with this macrostate? (b) Suppose you pick up two cards and end up with a macrostate of two aces. How many microstates are associated with this macrostate?An ideal gas is taken from an initial temperature Ti to a higher final temperature Tf along two different reversible paths as shown in Figure 21.15. Path A is at constant pressure, and path B is at constant volume. What is the relation between the entropy changes of the gas for these paths? (a) SA SB (b) SA = SB (c) SA SB Figure 21.15 (Quick Quiz 21.5) An ideal gas is taken from temperature Ti to Tf via two different paths.True or False: The entropy change in an adiabatic process must be zero because Q = 0.A particular heat engine has a mechanical power output of 5.00 kW and an efficiency of 25.0%. The engine expels 8.00 103 J of exhaust energy in each cycle. Find (a) the energy taken in during each cycle and (b) the time interval for each cycle.The work done by an engine equals one-fourth the energy it absorbs from a reservoir. (a) What is its thermal efficiency? (b) What fraction of the energy absorbed is expelled to the cold reservoir?Suppose a heat engine is connected to two energy reservoirs, one a pool of molten aluminum (660C) and the other a block of solid mercury (38.9C). The engine runs by freezing 1.00 g of aluminum and melting 15.0 g of mercury during each cycle. The heat of fusion of aluminum is 3.97 105 J/kg; the heat of fusion of mercury is 1.18 104 J/kg. What is the efficiency of this engine?During each cycle, a refrigerator ejects 625 kJ of energy to a high-temperature reservoir and takes in 550 kJ of energy from a low-temperature reservoir. Determine (a) the work done on the refrigerant in each cycle and (b) the coefficient of performance of the refrigerator.A freezer has a coefficient of performance of 6.30. It is advertised as using electricity at a rate of 457 kWh/yr. (a) On average, how much energy does it use in a single day? (b) On average, how much energy docs it remove from the refrigerator in a single day? (c) What maximum mass of water at 20.0C could the freezer freeze in a single day? Note: One kilowatt-hour (kWh) is an amount of energy equal to running a 1-kW appliance for one hour.6P One of the most efficient heat engines ever built is a coal-fired steam turbine in the Ohio River valley, operating between 1 870C and 430C. (a) What is its maximum theoretical efficiency? (b) The actual efficiency of the engine is 42.0%. How much mechanical power does the engine deliver if it absorbs 1.40 103 J of energy each second from its hot reservoir?8P If a 35.0% -efficient Carnot heat engine (Fig. 21.2) is run in reverse so as to form a refrigerator (Fig. 21.4), what would be this refrigerators coefficient of performance? Figure P21.2 Schematic representation of a heat engine. Figure P21.4 Schematic representation of a heat pump.10P 11P A power plant operates at a 32.0% efficiency during the summer when the seawater used for cooling is at 20.0C. The plant uses 350C steam to drive turbines. If the plants efficiency changes in the same proportion as the ideal efficiency, what would be the plants efficiency in the winter, when the seawater is at 10.0C?You are working on a summer job at a company that designs non-traditional energy systems. The company is working on a proposed electric power plant that would make use of the temperature gradient in the ocean. The system includes a heat engine that would operate between 20.0C (surface-water temperature) and 5.00C (water temperature at a depth of about 1 km). (a) Your supervisor asks you to determine the maximum efficiency of such a system. (b) In addition, if the electric power output of the plant is 75.0 MW and it operates at the maximum theoretically possible efficiency, you must determine the rate at which energy is taken in from the warm reservoir. (c) From this information, if an electric bill for a typical home shows a use of 950 kWh per month, your supervisor wants to know how many homes can be provided with power from this energy system operating at its maximum efficiency. (d) As energy is drawn from the warm surface water to operate the engine, it is replaced by energy absorbed from sunlight on the surface. If the average intensity absorbed from sunlight is 650 W/m2 for 12 daylight hours on a clear day, you need to find the area of the ocean surface that is necessary for sunlight to replace the energy absorbed into the engine. (e) From this information, you need to determine if there is enough ocean surface on the Earth to use such engines to supply the electrical needs for all the homes associated with the Earths population. Assume the energy use for a home in part (c) is an average over the entire planet. (f) In view of your results in this problem, your supervisor has asked for your conclusion as to whether such a system is worthwhile to pursue. Note that the fuel (sunlight) is free.14P 15P Suppose you build a two-engine device with the exhaust energy output from one heat engine supplying the input energy for a second heat engine. We say that the two engines arc running in series. Let e1 and e2 represent the efficiencies of the two engines. (a) The overall efficiency of the two-engine device is defined as the total work output divided by the energy put into the first engine by heat. Show that the overall efficiency e is given by e=e1+e2e1e2 What If? For parts (b) through (e) that follow, assume the two engines are Carnot engines. Engine 1 operates between temperatures Th and Ti. The gas in engine 2 varies in temperature between Ti and Tc. In terms of the temperatures, (b) what is the efficiency of the combination engine? (c) Does an improvement in net efficiency result from the use of two engines instead of one? (d) What value of the intermediate temperature Ti results in equal work being done by each of the two engines in series? (e) What value of Ti results in each of the two engines in series having the same efficiency?A heat pump used for heating shown in Figure P21.17 is essentially an air conditioner installed backward. It extracts energy from colder air outside and deposits it in a warmer room. Suppose the ratio of the actual energy entering the room to the work done by the devices motor is 10.0% of the theoretical maximum ratio. Determine the energy entering the room per joule of work done by the motor given that the inside temperature is 20.0C, and the outside temperature is 5.00C. Figure P21.17 Note: For problems in this section, assume the gas in the engine is diatomic with = 1.40.18P An idealized diesel engine operates in a cycle known as the air-standard diesel cycle shown in Figure P21.19. Fuel is sprayed into the cylinder at the point of maximum compression, B. Combustion occurs during the expansion B C, which is modeled as an isobaric process. Show that the efficiency of an engine operating in this idealized diesel cycle is e=11(TDTATCTB) Figure P21.19 20P 21P A Styrofoam cup holding 125 g of hot water at 100C cools to room temperature. 20.0C. What is the change in entropy of the room? Neglect the specific heat of the cup and any change in temperature of the room.A 1 500-kg car is moving at 20.0 m/s. The driver brakes to a stop. The brakes cool off to the temperature of the surrounding air, which is nearly constant at 20.0C. What is the total entropy change?A 2.00-L container has a center partition that divides it into two equal parts as shown in Figure P21.24. The left side contains 0.044 0 mol of H2 gas, and the right side contains 0.044 0 mol of O2 gas. Both gases are at room temperature and at atmospheric pressure. The partition is removed, and the gases are allowed to mix. What is the entropy increase of the system? Figure P21.24 Calculate the change in entropy of 250 g of water warmed slowly from 20.0C to 80.0C.What change in entropy occurs when a 27.9-g ice cube at 12C is transformed into steam at 115C?27P 28P 29P 30AP 31AP In 1993, the U.S. government instituted a requirement that all room air conditioners sold in the United States must have an energy efficiency ratio (EER) of 10 or higher. The EER is defined as the ratio of the cooling capacity of the air conditioner, measured in British thermal units per hour, or Btu/h, to its electrical power requirement in watts. (a) Convert the EER of 10.0 to dimensionless form, using the conversion 1 Btu = 1 055 J. (b) What is the appropriate name for this dimensionless quantity? (c) In the 1970s, it was common to find room air conditioners with EERs of 5 or lower. State how the operating costs compare for 10 000-Btu/h air conditioners with EERs of 5.00 and 10.0. Assume each air conditioner operates for 1 500 h during the summer in a city where electricity costs 17.0 per kWh.In 1816, Robert Stirling, a Scottish clergyman, patented the Stirling engine, which has found a wide variety of applications ever since, including current use in solar energy collectors to transform sunlight into electricity. Fuel is burned externally to warm one of the engines two cylinders. A fixed quantity of inert gas moves cyclically between the cylinders, expanding in the hot one and contracting in the cold one. Figure P21.33 represents a model for its thermodynamic cycle. Consider n moles of an ideal monatomic gas being taken once through the cycle, consisting of two isothermal processes at temperatures 3Ti and Ti and two constant-volume processes. Let us find the efficiency of this engine. (a) Find the energy transferred by heat into the gas during the isovolumetric process AB. (b) Find the energy transferred by heat into the gas during the isothermal process BC. (c) Find the energy transferred by heat into the gas during the isovolumetric process CD. (d) Find the energy transferred by heat into the gas during the isothermal process DA. (e) Identify which of the results from parts (a) through (d) are positive and evaluate the energy input to the engine by heat. (f) From the first law of thermodynamics, find the work done by the engine. (g) From the results of parts (e) and (f), evaluate the efficiency of the engine. A Stirling engine is easier to manufacture than an internal combustion engine or a turbine. It can run on burning garbage. It can run on the energy transferred by sunlight and produce no material exhaust. Stirling engines are not currently used in automobiles due to long startup times and poor acceleration response.34AP 35AP 36AP A 1.00-mol sample of an ideal monatomic gas is taken through the cycle shown in Figure P21.37. The process A B is a reversible isothermal expansion. Calculate (a) the net work done by the gas, (b) the energy added to the gas by heat, (c) the energy exhausted from the gas by heat, and (d) the efficiency of the cycle. (e) Explain how the efficiency compares with that of a Carnot engine operating between the same temperature extremes. Figure P21.37 38AP A heat engine operates between two reservoirs at T2 = 600 K and T1 = 350 K. It takes in 1.00 103 J of energy from the higher-temperature reservoir and performs 250 J of work. Find (a) the entropy change of the Universe SU for this process and (b) the work W that could have been done by an ideal Carnot engine operating between these two reservoirs. (c) Show that the difference between the amounts of work done in parts (a) and (b) is T1, SU.You are working as an assistant to a physics professor. She has seen some presentations you have made to your classes and is aware of your expertise in preparing presentation slides. Her laptop has crashed and she cannot access the presentation slides she needs for her lecture coming up in one hour. Her lecture is on entropy in engine cycles. She asks you to quickly generate two slides on your laptop, both showing TS diagrams, (a) one for the Carnot cycle and (b) one for the Otto cycle. As she leaves, you think, Uh-oh. Whats a TS diagram? Quick, you have no time to waste! Get to work!41AP You are working as an expert witness for an environmental agency. A utility in a neighboring town has proposed a new power plant that produces electrical power P from turbines. The utility claims that the plant will take in steam at temperature Th and reject water at temperature Tc into a flowing cold-water river. The flow rate of the river is m/t. The agency supervisor is concerned about the effect of dumping warm water on the fish in the river. (a) The utility claims that the power plant operates with Carnot efficiency. With that assumption, you need to determine for a trial presentation by how much the temperature of the water downstream from the power plant will rise due to the rejected energy from the power plant. (b) If you abandon the utilitys claim that the power plant operates at Carnot efficiency and assume a more realistic efficiency e, you need to determine the increase in water temperature in the stream. (c) Finally, you need to testify whether the increase in water temperature in part (b) will be higher or lower than that found in part (a).43AP 44AP A sample of an ideal gas expands isothermally, doubling in volume. (a) Show that the work done on the gas in expanding is W = nRT ln 2. (b) Because the internal energy Eint of an ideal gas depends solely on its temperature, the change in internal energy is zero during the expansion. It follows from the first law that the energy input to the gas by heat during the expansion is equal to the energy output by work. Does this process have 100% efficiency in converting energy input by heat into work output? (c) Does this conversion violate the second law? Explain.46AP The compression ratio of an Otto cycle as shown in Figure 21.12 is VA/VB = 8.00. At the beginning A of the compression process, 500 cm3 of gas is at 100 kPa and 20.0C. At the beginning of the adiabatic expansion, the temperature is TC = 750C. Model the working fluid as an ideal gas with = 1.40. (a) Fill in this table to follow the states of the gas: (b) Fill in this table to follow the processes: (c) Identify the energy input |Qh|, (d) the energy exhaust |Qc|, and (e) the net output work Weng. (f) Calculate the efficiency. (g) Find the number of crankshaft revolutions per minute required for a one-cylinder engine to have an output power of 1.00 kW = 1.34 hp. Note: The thermodynamic cycle involves four piston strokes.Three objects are brought close to each other, two at a time. When objects A and B are brought together, they repel. When objects B and C are brought together, they also repel. Which of the following are true? (a) Objects A and C possess charges of the same sign. (b) Objects A and C possess charges of opposite sign. (c) All three objects possess charges of the same sign. (d) One object is neutral. (e) Additional experiments must be performed to determine the signs of the charges.Three objects are brought close to one another, two at a time. When objects A and B are brought together, they attract. When objects B and C are brought together, they repel. Which of the following are necessarily true? (a) Objects A and C possess charges of the same sign. (b) Objects A and C possess charges of opposite sign. (c) All three objects possess charges of the same sign. (d) One object is neutral. (e) Additional experiments must be performed to determine information about the charges on the objects.Object A has a charge of +2 C, and object B has a charge of +6 C. Which statement is true about the electric forces on the objects? (a) FAB=3FBA (b) FAB=FBA (c) 3FAB=FBA (d) FAB=3FBA (e) FAB=FBA (f) 3FAB=FBA A test charge of +3 C is at a point P where an external electric field is directed to the right and has a magnitude of 4 106 N/C. If the test charge is replaced with another test charge of 3 C, what happens to the external electric field at P? (a) It is unaffected. (b) It reverses direction. (c) It changes in a way that cannot be determined.Rank the magnitudes of the electric field at points A, B, and C shown in Figure 22.18 (greatest magnitude first).Find to three significant digits the charge and the mass of the following particles. Suggestion: Begin by looking up the mass of a neutral atom on the periodic table of the elements in Appendix C. (a) an ionized hydrogen atom, represented as H+ (b) a singly ionized sodium atom, Na+ (c) a chloride ion Cl (d) a doubly ionized calcium atom, Ca++ = Ca2+ (e) the center of an ammonia molecule, modeled as an N3 ion (f) quadruply ionized nitrogen atoms, N4+, found in plasma in a hot star (g) the nucleus of a nitrogen atom (h) the molecular ion H2O (a) Find the magnitude of the electric force between a Na+ ion and a Cl ion separated by 0.50 nm. (b) Would the answer change if the sodium ion were replaced by Li+ and the chloride ion by Br ? Explain.In a thundercloud, there may be electric charges of +40.0 C near the top of the cloud and 40.0 C near the bottom of the cloud. These charges are separated by 2.00 km. What is the electric force on the top charge?Nobel laureate Richard Feynman (19181088) once said that if two persons stood at arms length from each other and each person had 1% more electrons than protons, the force of repulsion between them would be enough to lift a weight equal to that of the entire Earth. Carry out an order-of-magnitude calculation to substantiate this assertion.A 7.50-nC point charge is located 1.80 m from a 4.20-nC point charge. (a) Find the magnitude of the electric force that one particle exerts on the other. (b) Is the force attract he or repulsive?This afternoon, you have a physics symposium class, and you are the presenter. You will be presenting a topic to physics majors and faculty. You have been so busy that you have not had time to prepare and you dont even have an idea for a topic. You are frantically reading your physics textbook looking for an idea. In your reading, you have learned that the Earth carries a charge on its surface of about 105 C, which results in electric fields in the atmosphere. This gets you very excited about a new theory. Suppose the Moon also carries a charge on the order of 105 C, with the opposite sign! Maybe the orbit of the Moon around the Earth is due to electrical attraction between the Moon and the Earth! Theres an idea for your symposium presentation! You quickly jot down a few notes and run off to your symposium. While you are speaking, you notice one of the professors doing some calculations on a scrap of paper. Uh-oh! He has just raised his hand with a question. Why are you embarrassed?Two small beads having positive charges q1 = 3q and q2 = q are fixed at the opposite ends of a horizontal insulating rod of length d = 1.50 m. The bead with charge q1 is at the origin. As shown in Figure P22.7, a third small, charged bead is free to slide on the rod. (a) At what position x is the third bead in equilibrium? (b) Can the equilibrium be stable? Figure P22.7 Problems 7 and 8.Two small beads having charges q1 and q2 of the same sign are fixed at the opposite ends of a horizontal insulating rod of length d. The bead with charge q1 is at the origin. As shown in Figure P22.8, a third small, charged bead is free to slide on the rod. (a) At what position x is the third bead in equilibrium? (b) Can the equilibrium be stable? Figure P22.7 Problems 7 and 8.Review. In the Bohr theory of the hydrogen atom, an electron moves in a circular orbit about a proton, where the radius of the orbit is 5.29 1011 m. (a) Find the magnitude of the electric force exerted on each particle. (b) If this force causes the centripetal acceleration of the electron, what is the speed of the electron?Three point charges lie along a straight line as shown in Figure P22.10, where q1 = 6.00 C, q2 =1.50 C, and q3 = 2.00 C. The separation distances are d1 = 3.00 cm and d2 = 2.00 cm. Calculate the magnitude and direction of the net electric force on (a) q1, (b) q2, and (c) q3. Figure P22.10 A point charge +2Q is at the origin and a point charge Q is located along the x axis at x = d as in Figure P22.11. Find a symbolic expression for the net force on a third point charge +Q located along the y axis at y = d. Figure P22.11 12P Review. Two identical particles, each having charge +q, are fixed in space and separated by a distance d. A third particle with charge Q is free to move and lies initially at rest on the perpendicular bisector of the two fixed charges a distance x from the midpoint between those charges (Fig. P22.13). (a) Show that if x is small compared with d, the motion of Q is simple harmonic along the perpendicular bisector. (b) Determine the period of that motion. (c) How fast will the charge Q be moving when it is at the midpoint between the two fixed charges if initially it is released at a distance a d from the midpoint? Figure P22.13 Why is the following situation impossible? Two identical dust particles of mass 1.00 g are floating in empty space, far from any external sources of large gravitational or electric fields, and at rest with respect to each other. Both particles carry electric charges that are identical in magnitude and sign. The gravitational and electric forces between the particles happen to have the same magnitude, so each particle experiences zero net force and the distance between the particles remains constant.15P Consider n equal positively charged particles each of magnitude Q/n placed symmetrically around a circle of radius a. Calculate the magnitude of the electric field at a point a distance x from the center of the circle and on the line passing through the center and perpendicular to the plane of the circle.Two equal positively charged particles are at opposite corners of a trapezoid as shown in Figure P22.17. Find symbolic expressions for the total electric field at (a) the point P and (b) the point P. Figure P22.17 Two charged particles are located on the x axis. The first is a charge +Q at x = a. The second is an unknown charge located at x = +3a. The net electric field these charges produce at the origin has a magnitude of 2kQ/a2. Explain how many values are possible for the unknown charge and find the possible values.Three point charges are located on a circular arc as shown in Figure P22.19. (a) What is the total electric field at P, the center of the arc? (b) Find the electric force that would be exerted on a 5.00-nC point charge placed at P. Figure P22.19 Two 2.00-C point charges are located on the x axis. One is at x = 1.00 m, and the other is at x = 1.00 m. (a) Determine the electric field on the y axis at y = 0.500 m.(b) Calculate the electric force on a 3.00-C charge placed on the y axis at y = 0.500 m.Three point charges are arranged as shown in Figure P22.21. (a) Find the vector electric field that the 6.00-nC and 3.00-nC charges together create at the origin. (b) Find the sector force on the 5.00-nC charge. Figure P22.21 Consider the electric dipole shown in Figure P22.22. Show that the electric field at a distant point on the +x axis is Ex= 4kaqa/x3. Figure P22.22 Three equal positive charges q are at the corners of an equilateral triangle of side a as shown in Figure P22.23. Assume the three charges together create an electric field. (a) Sketch the field lines in the plane of the charges. (b) Find the location of one point (other than ) where the electric field is zero. What are (c) the magnitude and (d) the direction of the electric field at P due to the two charges at the base? Figure P22.23 A proton accelerates from rest in a uniform electric field of 640 N/C. At one later moment, its speed is 1.20 Mm/s (non-relativistic because v is much less than the speed of light). (a) Find the acceleration of the proton. (b) Over what time interval does the proton reach this speed? (c) How far does it move in this time interval? (d) What is its kinetic energy at the end of this interval?A proton moves at 4.50 105 m/s in the horizontal direction. It enters a uniform vertical electric field with a magnitude of 9.60 103 N/C. Ignoring any gravitational effects, find (a) the time interval required for the proton to travels 5.00 cm horizontally, (b) its vertical displacement during the time interval in which it travels 5.00 cm horizontally, and (c) the horizontal and vertical components of its velocity after it has traveled 5.00 cm horizontally.Protons are projected with an initial speed vi = 9.55 km/s from a field-free region through a plane and into a region where a uniform electric field E=720jN/C is present above the plane as shown in Figure P22.26. The initial velocity vector of the protons makes an angle with the plane. The protons are to hit a target that lies at a horizontal distance of R = 1.27 mm from the point where the protons cross the plane and enter the electric field. We wish to find the angle at which the protons must pass through the plane to strike the target. (a) What analysis model describes the horizontal motion of the protons above the plane? (b) What analysis model describes the vertical motion of the protons above the plane? (c) Argue that Equation 4.20 would be applicable to the protons in this situation. (d) Use Equation 4.20 to write an expression for R in terms of vi, E, the charge and mass of the proton, and the angle . (e) Find the two possible values of the angle . (f) Find the time interval during which the proton is above the plane in Figure P22.26 for each of the two possible values of . Figure P22.26 You are still fascinated by the process of inkjet printing, as described in the opening storyline for this chapter. You convince your father to take you to his manufacturing facility to see the machines that print expiration dates on eggs. You strike up a conversation with the technician operating the machine. He tells you that the ink drops are created using a piezoelectric crystal, acoustic waves, and the PlateauRayleigh instability, which creates uniform drops of mass m = 1.25 108 g. While you dont understand the fancy words, you do recognize mass! The technician also tells you that the drops are charged to a controllable value of q and then projected vertically downward between parallel deflecting plates at a constant terminal speed of 18.5 m/s. The plates are = 2.25 cm long and have a uniform electric field of magnitude E = 6.35 104 N/C between them. Noting your interest in the process, the technician asks you, If the position on the egg at which the drop is to be deposited requires that its deflection at the bottom end of the plates be 0.17 mm, what is the required charge on the drop? You quickly get to work to find the answer.You are working on a research project in which you must control the direction of travel of electrons using deflection plates. You have devised the apparatus shown in Figure P22.28. The plates are of length = 0.500 m and are separated by a distance d = 3.00 cm. Electrons are fired at vi = 5.00 106 m/s into a uniform electric field from the left edge of the lower, positive plate, aimed directly at the right edge of the upper, negative plate. Therefore, if there is no electric field between the plates, the electrons will follow the broken line in the figure. With an electric field existing between the plates, the electrons will follow a curved path, bending downward. You need to determine (a) the range of angles over which the electron can leave the apparatus and (b) the electric field required to give the maximum possible deviation angle. Figure P22.28 Consider an infinite number of identical particles, each with charge q, placed along the x axis at distances a, 2a, 3a, 4a, from the origin. What is the electric field at the origin due to this distribution? Suggestion: Use 1+122+132+142+=26 A particle with charge 3.00 nC is at the origin, and a particle with negative charge of magnitude Q is at x = 50.0 cm. A third particle with a positive charge is in equilibrium at x = 20.9 cm. What is Q?A small block of mass m and charge Q is placed on an insulated, frictionless, inclined plane of angle as in Figure P22.31. An electric field is applied parallel to the incline. (a) Find an expression for the magnitude of the electric field that enables the block to remain at rest. (b) If m = 5.40 g, Q = 7.00 C, and = 25.0, determine the magnitude and the direction of the electric field that enables the block to remain at rest on the incline. Figure P22.31 A small sphere of charge q1 = 0.800 C hangs from the end of a spring as in Figure P22.32a. When another small sphere of charge q2 = 0.600 C is held beneath the first sphere as in Figure P22.32b, the spring stretches by d = 3.50 cm from its original length and reaches a new equilibrium position with a separation between the charges of r = 5.00 cm. What is the force constant of the spring? Figure P22.32 A charged cork ball of mass 1.00 g is suspended on a light string in the presence of a uniform electric field as shown in Figure P22.33. When E=(3.00i+5.00j)105N/C, the ball is in equilibrium at = 37.0. Find (a) the charge on the ball and (b) the tension in the string. Figure P22.33 Problems 33 and 34 A charged cork ball of mass m is suspended on a light string in the presence of a uniform electric field as shown in Figure P22.33. When E=Ai+Bj, where A and B are positive quantities, the ball is in equilibrium at the angle . Find (a) the charge on the ball and (b) the tension in the string. Figure P22.33 Problems 33 and 34 Three charged particles are aligned along the x axis as shown in Figure P22.35. Find the electric field at (a) the position (2.00 m, 0) and (b) the position (0, 2.00 m). Figure P22.35 Two point charges qA = 12.0 C and qB = 45.0 C and a third particle with unknown charge qC are located on the x axis. The particle qA is at the origin, and qB is at x = 15.0 cm. The third particle is to be placed so that each particle is in equilibrium under the action of the electric forces exerted by the other two particles. (a) Is this situation possible? If so, is it possible in more than one way? Explain. Find (b) the required location and (c) the magnitude and the sign of the charge of the third particle.Two small spheres hang in equilibrium at the bottom ends of threads, 40.0 cm long, that have their top ends tied to the same fixed point. One sphere has mass 2.40 g and charge +300 nC. The other sphere has the same mass and charge +200 nC. Find the distance between the centers of the spheres.Four identical charged particles (q = +10.0 C) are located on the corners of a rectangle as shown in Figure P22.38. The dimensions of the rectangle arc L = 60.0 cm and W = 15.0 cm. Calculate (a) the magnitude and (b) the direction of the total electric force exerted on the charge at the lower left corner by the other three charges.39AP 40AP Three identical point charges, each of mass m = 0.100 kg, hang from three strings as shown in Figure P22.41. If the lengths of the left and right strings are each L = 30.0 cm and the angle is 45.0, determine the value of q. Figure P22.41 42AP Two hard rubber spheres, each of mass m = 15.0 g, are rubbed with fur on a dry day and are then suspended with two insulating strings of length L = 5.00 cm whose support points are a distance d = 3.00 cm from each other as shown in Figure P22.43. During the rubbing process, one sphere receives exactly twice the charge of the other. They are observed to hang at equilibrium, each at an angle of = 10.0 with the vertical. Find the amount of charge on each sphere. Figure P22.43 Two identical beads each have a mass m and charge q. When placed in a hemispherical bowl of radius R with frictionless, nonconducting walls, the beads move, and at equilibrium, they are a distance d apart (Fig. P22.44). (a) Determine the charge q on each bead. (b) Determine the charge required for d to become equal to 2R. Figure P22.44 45AP 46AP 47AP Eight charged panicles, each of magnitude q, are located on the corners of a cube of edge s as shown in Figure P22.48. (a) Determine the x, y, and z components of the total force exerted by the other charges on the charge located at point A. What are (b) the magnitude and (c) the direction of this total force? Figure P22.48 Two particles, each with charge 52.0 nC, are located on the y axis at y = 25.0 cm and y = 25.0 cm. (a) Find the vector electric field at a point on the x axis as a function of x. (b) Find the field at x = 36.0 cm. (c) At what location is the field 1.00ikN/C? You may need a computer to solve this equation. (d) At what location is the field 16.0ikN/C?Review. An electric dipole in a uniform horizontal electric field is displaced slightly from its equilibrium position as shown in Figure P22.50, where is small. The separation of the charges is 2a, and each of the two particles has mass m. (a) Assuming the dipole is released from this position, show that its angular orientation exhibits simple harmonic motion with a frequency f=12qEma What If? (b) Suppose the masses of the two charged particles in the dipole are not the same even though each particle continues to have charge q. Let the masses of the particles be m1 and m2. Show that the frequency of the oscillation in this case is f=12qE(m1+m2)2am1m2 Figure P22.50

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