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All Textbook Solutions for University Physics Volume 1
Water is shot nearly vertically upward in a decorative fountain and the steam observed to broaden as it rises. Conversely, a stream of water falling straight down from a faucet narrows. Explain why.Look back to figure 14.29. Answer the following two question. Why is poless than atmospheric? Why is pogreater than pi?A tube with a narrow segment designed to enhance entrainment is called a Venturi, such as below. Venturis are very commonly used in carburetors and aspirators. How does this structure bolster entrainment?Some chimney pipes have a T-shape, with a crosspiece on top that helps draw up gases whenever there is even a slight breeze. Explain how this works in terms of Bernoulli’s principle.Is there a limit to the height to which an entrainment device can raise a fluid? Explain your answer.Why is it preferable for airplanes to take off into the wind rather than with the wind?Roofs are sometimes pushed off vertically a tropical cyclone, and buildings sometimes explode outward when hit by a tornado. Use Bernoulli’s principle to explain these phenomena.It is dangerous to stud close to railroad tracks when a rapidly moving commuter train passes. Explain why atmospheric pressure would push you toward the moving train.Water pressure inside a hose nozzle can be less atmospheric pressure due to the Bernoulli effect. Explain in terms of energy how the water can emerge from the nozzle against the opposing atmospheric pressure.David rolled down the window on his car while driving on freeway. An empty plastic bag on the floor promptly flew out the window. Explain why.Based on Bernoulli’s equation, what are three forms of energy a fluid? (Note that these forms are conservative, unlike heat transfer and other dissipative forms not included Bernoulli’s equation.)The old rubber boot below has leaks. To what maximum height can water squirt from Leak 1? How does the velocity of water emerging from Leak 2 differ from that of Leak 1? Explain your responses in terms of energy.Water pressure inside a hose nozzle can be less than atmospheric pressure due to the Bernoulli effect. Explain in terms of energy how the water can emerge from the nozzle against the opposing atmospheric pressure.Explain why the viscosity of a liquid decreases with temperature, that is, how might an increase in temperature reduce the effects of cohesive forces in a liquid? Also explain why the viscosity of a gas increases with temperature, that is, how does increased gas temperature create more collisions between atoms and molecules?When paddling a canoe upstream, it is wisest to travel as near to the shore as possible. When canoeing downstream, it is genrally better to stay near the middle. Explain why.Plumbing usually includes air-filled tubes tear water faucets (see the following figure). Explain why they are needed and how the work.Doppler ultrasound can be used to measure the speed of blood in the body. If there is a constriction of an artery, where would you blood speed to be greatest: at or after the constriction? What are the two distinct causes of higher resistance the constriction?Sink drains often have a device such as that shown below to help speed the flow of water. How does this work?Gold is sold by the troy ounce (31.103 g). What is the volume of 1 troy ounce of pure sold?Mercury is commonly supplied in flasks containing 34.5 kg (about 76 Ib.). What is the volume in liters of this much mercury?What is the mass deep breath of air having a volume of 2.00 L? Discuss this effect taking such a breath has on your body's volume and density.A straightforward method of finding the density of an object is to measure its mass and then measure its volume by submerging it in a graduated cylinder. What is the density of a 240-g rock 89.0 cm3 of water? (Note that the accuracy and practical applications of this technique are more limited than a variety of others that are based on Archimedes' principle.)Suppose you have a coffee with a circular cross-section and vertical sides (uniform radius). What is its inside radius if it holds 375 g of coffee filled to a depth of 7.50 cm? Assume coffee has the same density as water.A rectangular gasoline tank bold 30.0 kg of gasoline when full. What is the depth of the tank if is 0.500-m wide by 0.900-m long? (b) Discuss whether gas tank has a reasonable volume for a passenger car.A trash compactor can compress its contents to 0.350 times their original volume. Neglecting mass of air expelled, by what factor is the density of be rubbish increased?A 2.50-kg steel gasoline can holds 20.0 L of gasoline when full. What is the average density of gas can, taking into account be volume occupied by steel as well as by gasoline?What is the density of 18.0-karat gold that is a mixture of 18 parts gold, 5 parts silver, and 1 part copper? (These value are parts by mass, not volume.) Assume that this is a simple mixture having an average density equal to the weighted densities of its constituents.The tip of a nail exerts tremendous pressure when hit by a hammer because it exerts a large force over a small area. What force must be exerted a nail a circular tip of 1.00-mm diameter to create a pressure of 3.00109 N/m2? (This pressure is possible because the hammer striking the nail is brought to rest in such a short distance.)A glass tube mercury. What would be the height of the column of mercury which would create pressure equal to 1.00 atm?The greatest ocean depths on Earth are found in the Marianas Trench near the Philippines. Calculate the pressure due to ocean at bottom of this trench, given its depth is 11.0 km and assuming density of seawater is constant all way down.Verigy that the SI of hpg is N/m2.What pressure is exerted the bottom of a gas tank that is 0.500-m wide and 0.900-m long and can hold 50.0 kg of gasoline full?A dam is used to hold back a river. The dam has a height H=12m and a width W=10m . Assume that density of the water is p=1000kg/m3 . (a) Determine be net force on tie dam. (b) Why does the thickness of be dam increase with?Find ae gauge and absolute pressures in be balloon and peanut jar shown Figure 14.12, assuming be manometer connected the balloon uses water and be manometer connected to the jar contains mercury. Express in units of centimeters of water for the balloon and millimeters of mercury for the jar, taking h=0.0500m for each.How tall must be to measure blood pressure as high as 300 mm Hg?Assuming bicycle tires are perfectly flexible and support the weight of bicycle and rider by pressure alone, calculate the total area of the tires in contact with be ground if a bicycle and rider have a total mass of 80.0 kg, and gauge pressure in the tires is 3.50105 Pa.Pascal’s Principle and Hydraulics 59. How much pressure is transmitted in the hydraulic system considered in Example 14.3? Express your answer in atmospheres.What force must exerted on the master cylinder of a hydraulic lift to support the weight of a 2000-kg car (a large car) resting on a second cylinder? The master cylinder has a 2.00-cm diameter and second cylinder has 24.0-cm diameter.A host pours the remnants of several of wine into a jug a party. The host then inserts a cork with a 2.00-cm diameter into the bottle, placing it in direct contact with the wine. The host is amazed when the host pounds the cork into place and the bottom of the jug (with a 14.0-cm diameter) breaks away. Calculate the extra force exerted against the bottom if he pounded the cork with a 120-N force.A certain hydraulic system is designed to exert a force 100 times as large as the put into it. (a) What must be the ratio of the area of the cylinder that is being controlled to area of the master cylinder? (b) What must be the ratio of their diameters? (c) By what factor is the distance through which the output force moves reduced relative to the distance through which input force moves? Assume no losses due to friction.Verify that work input equals work output for a hydraulic system assuming no losses due to fiction. Do this by showing that distance output force moves is reduced by the same factor output force is increased. Assume the volume of the fluid is constant. What effect would friction within the fluid and between components in the system have output force? How would this depend on whether or not fluid is moving?What fraction of ice is submerged when it floats in freshwater, given the density of water 0°C is very close to 1000 kg/m3?If a person's body has a density of 995 kg/m3, what fraction of the body will be submerged when floating gently in (a) freshwater? (b) In salt water with a density of 1027 kg/m3?A rock with a mass of 540 g in air is found to have an apparent mass of 342 g when submerged in water. (a) What mass of water is displaced? (b) What is the volume of the rock? (c) What is its average density? Is this consistent with the value for granite?Archimedes' principle can be used to calculate the density of a fluid as well as that of a solid. Suppose a chunk of iron with a mass of 390.0 g in air is found to have an apparent mass of 350.5 g when completely submerged in an unknown liquid. (a) What mass of fluid does the iron displace? (b) What is the volume of iron, using its density as given Table 14.1? (c) Calculate the fluid's density and identify it.Calculate the buoyant force a 200-L helium balloon. (b) Given mass of rubber balloon is 1.50 g, what net vertical force the balloon if it is let go? Neglect the volume of be rubber.What is density of a woman floats in fresh water with 4.00% of her volume above the surface? (This could be measured by placing her in a Wii marks on the side to measure how much water she displaces when floating and when held under water.) (b) What percent of her volume is above surface when she floats in seawater?A man has a mass of 80 kg and a density of 955kg/m3 (excluding the air in his lungs). (a) Calculate his volume. (b) Find the buoyant force air exerts on him. (c) What the ratio of the buoyant force to his weight?A simple compass cute made by placing a small bar magnet on a cork floating in water. (a) What fraction of a plain cork will be submerged when floating in water? (b) If the cork has a mass of 10.0 g and a 20.0-g magnet is placed on it, what fraction of the cork will be submerged? (c) Will the bar magnet and cork float in ethyl alcohol?What percentage of an iron anchor’s weight will be supported by buoyant force when submerged in salt water?Referring to Figure 14.20, prove that the buoyant force on be cylinder is equal to the weight of the fluid displaced (Archimedes' principle). You may assume that be buoyant force is F2F1 and that the ends of be cylinder have equal areas A. Note Eat be volume of be cylinder (and that of be fluid displaces) equals (h2h1)A .A 75.0-kg floats in freshwater 3.00% of his volume above water when his are empty, and 5.00% of his volume above water when his lungs are full. Calculate the volume of air inhales—called his lung capacity—in liters. (b) Does lung volume seem reasonable?What is the average flow rate in cm3/s of gasoline to the engine of a car traveling at 100 km/h if it averages 10.0 km/L?The heart of a resting adult pumps blood at a rate of 5.00 L/min. (a) Convert this to cm3/s. (b) What is this rate m3/s?The Huka Falls on the Waikato River is one of New Zealand's most visited natural tourist attractions. On average, the river has a flow rate of about 300,000 L/s. At be gorge, the river narrows to 20-m wide and averages 20-m deep. (a) What be average speed of the river in the gorge? (b) What is the average speed of the water in the river downstream of the falls when it widens to 60 m and its depth increases to an average of 40 m?(a) Estimate the time it would take to a private swimming pool with a capacity of 80,000 L using a garden hose delivering 60 L/min. (b) How long would it take if you could divert a moderate size river, flowing at 5000 m3/s into pool?What is the fluid speed a hose a 9.00-cm diameter 80.0 L of water per second? (b) What is the flow rate in cubic meters per second? (c) Would your answers be different if salt water replaced the fresh water in the fire hose?Water is moving at a velocity of 2.00 m/s through a hose with internal diameter of 1.60 cm. (a) What is the flow rate in liters per second? (b) The fluid velocity in this hose's nozzle is 15.0 m/s. What is the nozzle's inside diameter?Prove the sped of an incompressible fluid through a constriction, such as a Venturi tube, increases by a factor equal to the square of the factor by which the diameter decreases. (The converse applies for flow out of a constriction into a larger-diameter region.)Water emerges straight down from a faucet with a 1.80-cm diameter at a speed of 0.500 m/s. (Because of the construction of the faucet, there is no variation in speed across the stream.) (a) What is flow rate in cm3/s? (b) What is the diameter of the stream 0.200 m below the faucet? Neglect any effects due to surface tension.Verify that pressure has units of enery per unit volume.Suppose you have a wind speed gauge like the pitot tube shown in Figure 14.32. By what factor must wad speed increase to double the value of h in the manometer? Is independent of be moving fluid and be fluid the Figure 14.32 Measurement of fluid speed on Bernoulli’s principle. (a) A manometer is connected to two tubes close together and small enough not to disturb the flow. Tube 1 is open at the end facing the flow. A dead spot having zero speed is created there. Tube 2 has an opening on the side, so the fluid has a speed v across; thus, pressure there drops. The difference in pressure at the manometer is 12v22 , so h is proportional to . 12v22 (b) This type of velocity measuring device is a Prandtl tube, also known as a pitot tube.If be pressure reading of your pitot tube is 15.0 mm Hg at a speed of 200 km/h, what will it be at 700 km/h at the same altitude?Every few years, winds in Boulder, Colorado, attain sustained speeds of 43.0 m/s (about 100 mph) when jet steam descends during early spring. Approximately what is the force due to the Bernoulli equation on a roof having an area of 220m2? Typical air density in Boulder is 1.14kg/m3, and be corresponding atmospheric pressure is 8.89104 N/m2. (Bernoulli’s principal as stated in the text assumes laminar flow. Using the principle here produces only an approximate result, because there is significant turbulence.)What is the pressure drop due to the Bernoulli Effect as water goes into a 3.00-cm-diameter nozzle from a 9.00-cm-diameter fire hose while carrying a flow of 40.0 L/s? (b) To what maximum above be nozzle can this water rise? (The actual height will be significantly smaller due to air resistance.)(a) Using Bernoulli's equation, show that be measured fluid speed v for a pitot tube, like the one in figure 14.32(b), is given by v=( 2gh)1/h , where h is be height of be manometer fluid, p' is the density of the manometer fluid, p is the density of the moving fluid, and g is be acceleration due to gravity. (Note that v is indeed proportional to the square root of h, as stated in text) (b) Calculate v for moving air if a mercury manometer's h is 0.200 m. Figure 14.32 Measurement of fluid speed on Bernoulli’s principle. (a) A manometer is connected to two tubes close together and small enough not to disturb the flow. Tube 1 is open at the end facing the flow. A dead spot having zero speed is created there. Tube 2 has an opening on the side, so the fluid has a speed v across; thus, pressure there drops. The difference in pressure at the manometer is 12v22 , so h is proportional to . 12v22 (b) This type of velocity measuring device is a Prandtl tube, also known as a pitot tube.A container of water has a cross-sectional area of A=0.1m2 . A piston sits top of the water (see be following figure). There is a spout located 0.15 m from the bottom of tank, open to the atmosphere, and a stream of water exits the spout. The cross sectional area of the spat is As=7.0104m2 . (a) What is the velocity of the water as it leaves the spout? (b) If the opening of the spout is located 1.5 m above the grand, how far from be pout does water hit the floor? Ignore friction and dissipative forces.A fluid of a constant density flows through a reduction in a pipe. Find an equation for be change in pressure, in terms of v1, A2, A2, and the density.(a) Calculate the retarding force due to viscosity of the air layer between a cart and a level air track given the following information: air temperature is 20°C, the cart is moving at 0.400 m/s, surface area is 2.50102 m2, ad thickness of air layer is 6.00105 m. (b) What is the ration of this force to the weight of the 0.300-kg cart?The arterioles (small arteries) leading to organ constrict in order to decrease flow to the organ. To down organ, blood is reduced naturally to 1.00% of its original value. By what factor do the radii of the arterioles constrict?A spherical particle falling at a terminal speed in a liquid must have the gravitational force balanced by the drag force the buoyant force. The buoyant force is equal to the weight of the displaced fluid, while drag force assumed to be given by Stokes Law, Fs=6rv . Show that the terminal speed is given by v=2R2g9(s1) , where R is be radius of sphere, s is its ad 1 is of be fluid, and the coefficient of viscosity.Using the equation of the previous problem, find the viscosity of motor oil in which a steel ball of radius 0.8 mm falls a terminal speed of 4.32 cm/s. The densities of the ball and the oil are 7.86 and 0.88 g/mL, respectively.A skydiver will reach a terminal velocity when the air drag equals his or her weight. For a skydiver with a large body, turbulence is a factor at high speeds. The drag force then is approximately proportional to the square of the velocity. Taking the drag force to be FD=12Av2 , and setting this equal to the skydiver's weight, find the terminal speed for a person falling "spread eagle.(a) Verify that a 19.0% decrease in laminar flow through a tube is caused by a 5.00% decrease in radius, assuming that all other factors remain constant. (b) What increase in flow is obtained from a 5.00% increase in radius, again assuming all other factors remain constant?When physicians diagnose arterial blockages, they quote reduction flow rate. If the flow rate an artery has been reduced to 10.0% of its normal value by a blood clot and be average pressure difference has increased by 20.0%, by what factor has the clot reduced the radius of the artery?An oil gusher shoots crude 25.0 m the through a pipe with a 0.100-m diameter. Neglecting resistance but not resistance of the pipe, and assuming laminar flow, calculate pressure at be entrance of be 50.0-m-Iong vertical pipe. Take of the oil to be 900 kg/m3 and its viscosity to be 100(N/m2) s (or 1.00 Pa s). Note that you must take into account the pressure due to 50.0-m column of oil in pipe.Concrete is pumped from a cement mixer to the place it is being lad, instead of being carried wheelbarrows. The flow rate is 200 L/min through a 50.0-m-long, 8.00-cm-diameter hose, and he pressure it the pump is 8.00106 N/m2. (a) Calculate the resistance of the hose. (b) What is the viscosity of the concrete, assuming the flow is laminar? (c) How much power is being supplied, assuming the point of use is at the same level as the pump? You may neglect the power supplied to increase be concrete's velocity.Verify that flow of oil is laminar for an oil gusher that shoots crude oil 25.0 m into the air through a pipe with a 0.100-m diameter. The vertical pipe is 50 m long. Take the density of the oil to be 900 kg/m3 and its viscosity to be 1.00(N/m2)s (or 1.00Pas ).Calculate Reynolds numbers for flow of trough (a) a nozzle a radius of 0.250 cm and (b) a garden hose with a radius of 0.900 cm, when be nozzle is attached to hose. The flow rate through hose and nozzle is 0.500 L/s. Can flow in either possibly be laminar?A fire hose has an inside diameter of 6.40 cm. Suppose such a hose caries a flow of 40.0 L/s starting at a gauge pressure of 1.62106 N/m2. The hose goes 10.0 m up a ladder to a nozzle having an inside diameter of 3.00 cm. Calculate the Reynolds numbers for flow in the fire hose and nozzle to show that flow in each must be turbulent.At what rate might turbulence begin to develop in a water main with a 0.200-m diameter? Assume a 20°C temperature.Before digital storage devices, such as the memory in your cell phone, music was stored vinyl disks with grooves with varying depths cut into disk. A phonograph used a needle, which moved over the grooves, measuring the depth of the grooves. The pressure exerted by a phonograph needle on a record is surprisingly large. If the equivalent of 1.00 g is supported by a needle, the tip of which is a circle with a 0.200-mm radius, what pressure is exerted on be record Pa?Water towers store water above the level of consumers for times of heavy use, eliminating need for high-speed pumps. How high above a user must the water level be to create a gauge pressure of 3.00105 N/m2?The aqueous humor in a person's eye is exerting a force of 0.300 N on the 1.10-cm2 area of the cornea. What pressure is this in mm Hg?(a) Convert normal blood pressure readings of 120 over 80 mm Hg to newtons per meter squared using be relationship for pressure due to the weight of a fluid (p=hg) rater a conversion factor. (b) Explain why be blood pressure of an infant would likely be smaller than that of an adult. Specifically, consider the smaller height to which blood mast be pumped.Pressure cookers have been around for more than 300 years, although their use has greatly declined in recent years (early models had a nasty of exploding). How much force must the latches holding the lid onto a pressure cooker be able to if the is 25.0 cm in diameter and he gauge pressure inside is 300 atm? Neglect the weight of be lid.Bird bones have air pockets to reduce their weight—this also gives them an average density significantly less than that of the bones of other animals. Suppose an ornithologist weighs a bird bone air and in water and finds its mass is 45.0 g ad its apparent mass when submerged is 3.60 g (assume the bone is watertight.)(a) What mass of is displaced? (b) What is the volume of the bone? (c) What is its average density?In an immersion measurement of a woman's density, she is found to have a mass of 62.0 kg in air an apparent mass of 0.0850 kg completely submerged with lungs empty. (a) What of water does she displace? (b) What is her volume? (c) Calculate her density. (d) If her lung capacity is 1.7S L, is she able to that without treading water with her lungs filled air?Some have a density slightly less than that of water and must exert a force (swim) to stay submerged. What force must an 85.0-kg grouper exert to stay submerged salt water if its body density is 1015 kg/m3?The human circulation system has approximately 1109 vessels. Each vessel has a diameter of about 8m . Assuming cardiac output is 5 L/min, determine the average velocity of blood flow through each capillary vessel.The flow of blood through a 2.00106 m -radius capillary is 3.80109cm3/s . (a) What is the speed of be blood now? (b) Assuming all the blood in the body passes through capillaries, how many of them must there be to carry a total flow of 90.0 cm3/s?The left ventricle of a resting adult's heart pumps blood at a flow rate of 83.0 cm3/s , increasing its pressure by 110 mm Hg, its speed from zero to 30.0 cm/s, and its height by 5.00 cm. (All cumbers are averaged over the entire heartbeat) Calculate the total power output of left ventricle. Note that most of the power is used to increase blood pressure.A sump pump (used to drain water from be basement of houses built below the water table) is draining a flooded basement at rate of 0.750 L/S, with an output pressure of 3.00105N/m2 . (a) The water enters a hose with a 3.00-cm inside diameter and rises 2.50 m above the pump. What is its pressure at this point? (b) The hose goes over the foundation wall, losing 0.500 m in height and widens to 4.00 cm in diameter. What is the pressure now? You may neglect frictional losses both parts of the problem.A glucose solution being administered with an IV has a flow rate of 4.00 cm3/min. What will the new flow rate be if the glucose is replaced by whole blood having the same density but a viscosity 2.50 times that of glucose? All other factors remain constant.A small artery has a length of 1.1103m and a radius of 2.55105m . If the pressure drop across the artery is 1.3 kPa, what is the flow rate through artery? (Assume Eat the temperature 37°C)Angioplasty is a technique in which arteries partially blocked with plaque are dilated to increase blood flow. By what actor must the radius of an artery be increased in order to increase blood by a factor of 10?Suppose a blood vessel's radius is decreased to 90.0% of its original value by plague deposits and the body compensates by increasing pressure difference along the vessel to keep the flow rate constant. By what factor must pressure difference increase? (b) If turbulence is created by the obstruction, what additional effect would it have on the flow rate?The pressure dam early in problems section increases with depth. Therefore, there is net torque on the dam. Find be net torque.The temperature of atmosphere is not always constant and can increase or decrease with height. In a neutral atmosphere, where there is a significant amount of vertical mixing, the temperature decreases at a of approximately 6.5 K per km. The magnitude of be decrease in temperature as height increases is known as be lapse rate (I’). (The symbol is upper case Greek letter gamma.) Assume that be surface pressure is p0=1.013105Pa where T=293K and the lapse rate is (I=6.5Kkm) . Estimate the pressure 3.0 km above the surface of Earth.A submarine is stranded on the bottom of the ocean with its hatch 25.0 m below surface. Calculate force needed to open the hatch from the inside, given it is circular and 0.450 m in diameter. Air pressure inside the submarine is 1.00 atm.Logs sometimes float vertically a lake because one end has become water-logged and denser than the other. What is the average density of a uniform-diameter log that floats with 20.0% of its length above water?Scurrilous con artists have been known to represent gold-plated tungsten ingots as pure gold and sell them at prices much below gold value but high above the cost of tungsten. With what accuracy must you be able to measure the mass of such an ingot in and out of water to tell that it is almost pure tungsten rather than pure gold?The inside volume of a house is equivalent to that of a rectangular solid 13.0 m wide by 200 m long by 2.75 m high. The house is heated by a forced air gas heater. The main uptake air duct of heater is 0.300 m in diameter. What is the average speed of the duct if it carries a volume equal to that of the house’s interior every 15 minutes?A garden hose with a diameter of 2.0 cm is used to fill a bucket, which has a volume of 0.10 cubic meters. It takes 1.2 minutes to fill. An adjustable nozzle is attached to the hose to decrease the diameter of the opening, which increases the speed of the water. The hose is held level to the ground at a height of 1.0 meters and the diameter is decreased until a flower bed 3.0 meters away is reached. (a) What is the volume flow rate of the through the nozzle when the diameter 2.0 cm? (b) What does is the speed of coming out of the hose? (c) What does the speed of the water coming out of the hose need to be to reach the flower bed 3.0 meters away? (d) What is be diameter of nozzle needed to reach be flower bed?A frequency quoted rule of thumb aircraft design is that wings should produce about 1000 N of lift per square meter of wing. (The fact that a wing has a top and bottom surface does not double its area.) (a) At takeoff, an aircraft travels at 60.0 m/s, so that the air speed relative to the bottom of the wing is 60.0 m/s. Given be sea level density of air as 1.29kg/m3, how fast must it move over be upper surface to create the ideal lift? (b) How fast must air move over the upper surface at a cruising speed of 245 m/s and at an altitude where air density is one-fourth that at sea level? (Note that his not all of be aircraft's lift—some comes from be body of the plane, some from engine thrust, and so on. Furthermore, Bernoulli's principle gives approximate answer because flow over wing creates turbulence.)Two pipes of equal and constant diameter leave a water pumping and dump water cut of open end that is open to the atmosphere (see the following figure). The water enters at pressure of atmosphere and a speed of (v1=1.0m/s) . One pipe drops a height of 10 m. What is the velocity of the water as the water leaves each pipe?Fluid originally flows through a tube at a rate of 100 cm3/s. To illustrate the sensitivity of flow rate to various factors, calculate be new flow rate for following changes with all other factors remaining the same as in original conditions. (a) Pressure difference increases by a factor of 1.50. (b) A new fluid wit 3.00 times greater viscosity is substituted. (c) The tube is replaced by one having 4.00 times the length. (d) Another tube used with a 0.100 times the original. (e) Yet another tube is substituted with a radius 0.100 times the original and half length, and pressure difference is increased by factor of 1.50.During a marathon race, a runner's blood flow increases to 10.0 times her resting rate. Her blood's viscosity dropped to 95.0% of its normal value, and the blood pressure difference across the circulatory system has increased by 50.0%. By what factor bas be average radii of her blood vessels increased?Water supplied to a house by a water main has a pressure of 3.00105N/m2 early on a summer day when neighborhood use is low. This pressure produces a flow of 20.0 L/min through a garden hose. Later in the day, pressure at the exit of the water main and entrance to the house drops, and a flow of only 8.00 L/min is obtained through the same hose. (a) What pressure is now being supplied to the house, assuming resistance is constant? (b) By what factor did the flow rate be water main increase in order to cause this decrease in delivered pressure? The pressure at the entrance of the water main is 5.00105N/m2 , and the original rate was 200 L/min. (c) How many more users are there, assuming each would consume 20.0 L/min in be morning?Gasoline is piped underground from refineries to major users. The flow rate is 3.00102m3/s (about 500 gal/min), viscosity of gasoline is 3.00102m3/s , and its density is 680 kg/m3. (a) What minimum diameter must be pipe have if the Reynolds number is to be less 2000? (b) What pressure difference must be along each kilometer of be pipe to maintain flow rate?Check Your Understanding Why would it hurt more if you snapped your hand with a ruler than with a loose spring, even if the displacement of each system is equal?Check Your Understanding Identify one way you could decrease the maximum velocity of a simple harmonic oscillator.Check Your Understanding Identify an object that undergoes uniform circular motion. Describe how you could trace the SHM of this object.Check Your Understanding An engineer builds two simple pendulums. Both are suspended from small wires secured to the ceiling of a room. Each pendulum hovers 2 cm above the floor. Pendulum 1 has a bob with a mass of 10 kg. Pendulum 2 has a bob with a mass of 100 kg. Describe how the motion of the pendulums will differ if the bobs are both displaced by 12°.Check Your Understanding Why are completely undamped harmonic oscillators so rare?Check Your Understanding A famous magic trick involves a performer singing a note toward a crystal glass until the glass shatters. Explain why the trick works in terms of resonance and natural frequency.What conditions must be met to produce SHM?(a) If frequency is not constant for some oscillation, can the oscillation be SHM? (b) Can you think of any examples of harmonic motion where the frequency may depend on the amplitude?Give an example of a simple harmonic oscillator, specifically noting how its frequency is independent of amplitude.Explain why you expect an object made of a stiff material to vibrate at a higher frequency than a similar object made of a more pliable material.As you pass a freight truck with a trailer on a highway, you notice that its trailer is bouncing up and down slowly. Is it more likely that the trailer is heavily loaded or nearly empty? Explain your answer.Some people modify cars to be much closer to the ground than when manufactured. Should they install stiffer springs? Explain your answer.Describe a system in which elastic potential energy is stored.Explain in terms of energy how dissipative forces such as friction reduce the amplitude of a harmonic oscillator. Also explain how a driving mechanism can compensate. (A pendulum clock is such a system.)The temperature of the atmosphere oscillates from a maximum near noontime and a minimum near sunrise. Would you consider the atmosphere to be in stable or unstable equilibrium?Can this analogy of SHM to circular motion be carried out with an object oscillating on a spring vertically hung from the ceiling? Why or why not? If given the choice, would you prefer to use a sine function or a cosine function to model the motion?Can this analogy of SHM to circular motion be carried out with an object oscillating on a spring vertically hung from the ceiling? Why or why not? If given the choice, would you prefer to use a sine function or a cosine function to model the motion?Can this analogy of SHM to circular motion be carried out with an object oscillating on a spring vertically hung from the ceiling? Why or why not? If given the choice, would you prefer to use a sine function or a cosine function to model the motion?A pendulum clock works by measuring the period of a pendulum. In the springtime the clock runs with perfect time, but in the summer and winter the length of the pendulum changes. When most materials are heated, they expand. Does the clock run too fast or too slow in the summer? What about the winter?With the use of a phase shift, the position of an object may be modeled as a cosine or sine function. If given the option, which function would you choose? Assuming that the phase shift is zero, what are the initial conditions of function; that is, the initial position, velocity, and acceleration, when using a sine function? How about when a cosine function is used?Give an example of a damped harmonic oscillator. (They are more common than undamped or simple harmonic oscillators.)How would a car bounce after a bump under each of these conditions? (a) overdamping (b) underdamping (c) critical dampingMost harmonic oscillators are damped and, if undriven, eventually come to a stop. Why?Why are soldiers in general ordered to “route step” (walk out of step) across a bridge?Do you think there is any harmonic motion in the physical world that is not damped harmonic motion? Try to make a list of five examples of undamped harmonic motion and damped harmonic motion. Which list was easier to make?Some engineers use sound to diagnose performance problems with car engines. Occasionally, a part of the engine is designed that resonates at the frequency of the engine. The unwanted oscillations can cause noise that irritates the driver or could lead to the part failing prematurely. In one case, a part was located that had a length L made of a material with a mass M. What can be done to correct this problem?Prove that using x(t)=Asin(t+) will produce the same results for the period for the oscillations of a mass and a spring. Why do you think the cosine function was chosen?What is the period of 60.0 Hz of electrical power?If your heart rate is 150 beats per minute during strenuous exercise, what is the time per beat in units of seconds?Find the frequency of a tuning fork that takes 2.50103 s to complete one oscillation.A stroboscope is set to flash every 8.00105 s. What is the frequency of the flashes?A tire has a tread pattern with a crevice every 2.00 cm. Each crevice makes a single vibration as the tire moves. What is the frequency of these vibrations if the car moves at 30.0 m/s?Each piston of an engine makes a sharp sound every other revolution of the engine. (a) How fast is a race car going if its eight-cylinder engine emits a sound of frequency 750 Hz, given that the engine makes 2000 revolutions per kilometer? (b) At how many revolutions per minute is the engine rotating?A type of cuckoo clock keeps time by having a mass bouncing on a spring, usually something cute like a cherub in a chair. What force constant is needed to produce a period of 0.500 s for a 0.0150-kg mass?A mass m0is attached to a spring and hung vertically. The mass is raised a short distance in the vertical direction and released. The mass oscillates with a frequency f0. If the mass is replaced with a mass nine times as large, and the experiment was repeated, what would be the frequency of the oscillations in terms of f0?A 0.500-kg mass suspended from a spring oscillates with a period of 1.50 s. How much mass must be added to the object to change the period to 2.00 s?By how much leeway (both percentage and mass) would you have in the selection of the mass of the object in the previous problem if you did not wish the new period to be greater than 2.01 s or less than 1.99 s?Fish are hung on a spring scale to determine their mass. (a) What is the force constant of the spring in such a scale if it the spring stretches 8.00 cm for a 10.0 kg load? (b) What is the mass of a fish that stretches the spring 5.50 cm? (c) How far apart are the half-kilogram marks on the scale?It is weigh-in time for the local under-85-kg rugby team. The bathroom scale used to assess eligibility can be described by Hooke’s law and is depressed 0.75 cm by its maximum load of 120 kg. (a) What is the spring’s effective force constant? (b) A player stands on the scales and depresses it by 0.48 cm. Is he eligible to play on this under-85-kg team?One type of BB gun uses a spring-driven plunger to blow the BB from its barrel. (a) Calculate the force constant of its plunger’s spring if you must compress it 0.150 m to drive the 0.0500-kg plunger to a top speed of 20.0 m/s. (b) What force must be exerted to compress the spring?When an 80.0-kg man stands on a pogo stick, the spring is compressed 0.120 m. (a) What is the force constant of the spring? (b) Will the spring be compressed more when he hops down the road?A spring has a length of 0.200 m when a 0.300-kg mass hangs from it, and a length of 0.750 m when a 1.95-kg mass hangs from it. (a) What is the force constant of the spring? (b) What is the unloaded length of the spring?The length of nylon rope from which a mountain climber is suspended has an effective force constant of 1.40104 N/m . (a) What is the frequency at which he bounces, given his mass plus and the mass of his equipment are 90.0 kg? (b) How much would this rope stretch to break the climber’s fall if he free-falls 2.00 m before the rope runs out of slack? (Hint: Use conservation of energy.) (c) Repeat both parts of this problem in the situation where twice this length of nylon rope is used.The motion of a mass on a spring hung vertically, where the mass oscillates up and down, can also be modeled using the rotating disk. Instead of the lights being placed horizontally along the top and pointing down, place the lights vertically and have the lights shine on the side of the rotating disk. A shadow will be produced on a nearby wall, and will move up and down. Write the equations of motion for the shadow taking the position at t=0.0 s to be t=0.0 m with the mass moving in the positive y-direction.(a) A novelty clock has a 0.0100-kg-mass object bouncing on a spring that has a force constant of 1.25 N/m. What is the maximum velocity of the object if the object bounces 3.00 cm above and below its equilibrium position? (b) How many joules of kinetic energy does the object have at its maximum velocity?Reciprocating motion uses the rotation of a motor to produce linear motion up and down or back and forth. This is how a reciprocating saw operates, as shown below. If the motor rotates at 60 Hz and has a radius of 3.0 cm,estimate the maximum speed of the saw blade as it moves up and down. This design is known as a scotch yoke.A student stands on the edge of a merry-go-round which rotates five times a minute and has a radius of two meters one evening as the sun is setting. The student produces a shadow on the nearby building. (a) Write an equation for the position of the shadow. (b) Write an equation for the velocity of the shadow.What is the length of a pendulum that has a period of 0.500 s?Some people think a pendulum with a period of 1.00 s can be driven with “mental energy” or psycho kinetically, because its period is the same as an average heartbeat. True or not, what is the length of such a pendulum?What is the period of a 1.00-m-long pendulum?How long does it take a child on a swing to complete one swing if her center of gravity is 4.00 m below the pivot?The pendulum on a cuckoo clock is 5.00-cm long. What is its frequency?Two parakeets sit on a swing with their combined CMs 10.0 cm below the pivot. At what frequency do they swing?(a) A pendulum that has a period of 3.00000 s and that is located where the acceleration due to gravity is 9.79 m/s2 is moved to a location where the acceleration due to gravity is 9.82 m/s2. What is its new period? (b) Explain why so many digits are needed in the value for the period, based on the relation between the period and the acceleration due to gravity.A pendulum with a period of 2.00000 s in one location (g=9.80m/s2) is moved to a new location where the period is now 1.99796 s. What is the acceleration due to gravity at its new location?(a) What is the effect on the period of a pendulum if you double its length? (b) What is the effect on the period of a pendulum if you decrease its length by 5.00%?The amplitude of a lightly damped oscillator decreases by 3.0% during each cycle. What percentage of the mechanical energy of the oscillator is lost in each cycle?How much energy must the shock absorbers of a 1200-kg car dissipate in order to damp a bounce that initially has a velocity of 0.800 m/s at the equilibrium position? Assume the car returns to its original vertical position.If a car has a suspension system with a force constant of 5.00104 N/m , how much energy must the car’s shocks remove to dampen an oscillation starting with a maximum displacement of 0.0750 m?(a) How much will a spring that has a force constant of 40.0 N/m be stretched by an object with a mass of 0.500 kg when hung motionless from the spring? (b) Calculate the decrease in gravitational potential energy of the 0.500-kg object when it descends this distance. (c) Part of this gravitational energy goes into the spring. Calculate the energy stored in the spring by this stretch, and compare it with the gravitational potential energy. Explain where the rest of the energy might go.Suppose you have a 0.750-kg object on a horizontal surface connected to a spring that has a force constant of 150 N/m. There is simple friction between the object and surface with a static coefficient of friction s=0.100 . (a) How far can the spring be stretched without moving the mass? (b) If the object is set into oscillation with an amplitude twice the distance found in part (a), and the kinetic coefficient of friction is k=0.0850 , what total distance does it travel before stopping? Assume it starts at the maximum amplitude.Suppose you attach an object with mass m to a vertical spring originally at rest, and let it bounce up and down. You release the object from rest at the spring’s original rest length, the length of the spring in equilibrium, without the mass attached. The amplitude of the motion is the distance between the equilibrium position of the spring without the mass attached and the equilibrium position of the spring with the mass attached. (a) Show that the spring exerts an upward force of 2.00mg on the object at its lowest point. (b) If the spring has a force constant of 10.0 N/m, is hung horizontally, and the position of the free end of the spring is marked as y = 0.00 m , where is the new equilibrium position if a 0.25-kg-mass object is hung from the spring? (c) If the spring has a force constant of 10.0 M/m and a 0.25-kg-mass object is set in motion as described, find the amplitude of the oscillations. (d) Find the maximum velocity.A diver on a diving board is undergoing SHM. Her mass is 55.0 kg and the period of her motion is 0.800 s. The next diver is a male whose period of simple harmonic oscillation is 1.05 s. What is his mass if the mass of the board is negligible?Suppose a diving board with no one on it bounces up and down in a SHM with a frequency of 4.00 Hz. The board has an effective mass of 10.0 kg. What is the frequency of the SHM of a 75.0-kg diver on the board?The device pictured in the following figure entertains infants while keeping them from wandering. The child bounces in a harness suspended from a door frame by a spring. (a) If the spring stretches 0.250 m while supporting an 8.0-kg child, what is its force constant? (b) What is the time for one complete bounce of this child? (c) What is the child’s maximum velocity if the amplitude of her bounce is 0.200 m?A mass is placed on a frictionless, horizontal table. A spring (k=100N/m) , which can be stretched or compressed, is placed on the table. A 5.00-kg mass is attached to one end of the spring, the other end is anchored to the wall. The equilibrium position is marked at zero. A student moves the mass out to x=4.0 cm and releases it from rest. The mass oscillates in SHM. (a) Determine the equations of motion. (b) Find the position, velocity, and acceleration of the mass at time t=3.00 s.Find the ratio of the new/old periods of a pendulum if the pendulum were transported from Earth to the Moon, where the acceleration due to gravity is 1.63 m/s2.At what rate will a pendulum clock run on the Moon, where the acceleration due to gravity is 1.63 m/s2, if it keeps time accurately on Earth? That is, find the time (in hours) it takes the clock’s hour hand to make one revolution on the Moon.If a pendulum-driven clock gains 5.00 s/day, what fractional change in pendulum length must be made for it to keep perfect time?A 2.00-kg object hangs, at rest, on a 1.00-m-long string attached to the ceiling. A 100-g mass is fired with a speed of 20 m/s at the 2.00-kg mass, and the 100.00-g mass collides perfectly elastically with the 2.00-kg mass. Write an equation for the motion of the hanging mass after the collision. Assume air resistance is negligible.A 2.00-kg object hangs, at rest, on a 1.00-m-long string attached to the ceiling. A 100-g object is fired with a speed of 20 m/s at the 2.00-kg object, and the two objects collide and stick together in a totally inelastic collision. Write an equation for the motion of the system after the collision. Assume air resistance is negligible.Assume that a pendulum used to drive a grandfather clock has a length L0=1.00 m and a mass M at temperature T=20.00 °C. It can be modeled as a physical pendulum as a rod oscillating around one end. By what percentage will the period change if the temperature increases by 10°C? Assume the length of the rod changes linearly with temperature, where L=L0(1+T) and the rod is made of (=18106C1) .A 2.00-kg block lies at rest on a frictionless table. A spring, with a spring constant of 100 N/m is attached to the wall and to the block. A second block of 0.50 kg is placed on top of the first block. The 2.00-kg block is gently pulled to a position x=+A and released from rest. There is a coefficient of friction of 0.45 between the two blocks. (a) What is the period of the oscillations? (b) What is the largest amplitude of motion that will allow the blocks to oscillate without the 0.50-kg block sliding off?A suspension bridge oscillates with an effective force constant of 1.00108 N/m . (a) How much energy is needed to make it oscillate with an amplitude of 0.100 m? (b) If soldiers march across the bridge with a cadence equal to the bridge’s natural frequency and impart 1.00104 J of energy each second, how long does it take for the bridge’s oscillations to go from 0.100 m to 0.500 m amplitude.Near the top of the Citigroup Center building in New York City, there is an object with mass of 4.00105 kg on springs that have adjustable force constants. Its function is to dampen wind-driven oscillations of the building by oscillating at the same frequency as the building is being driven—the driving force is transferred to the object, which oscillates instead of the entire building. (a) What effective force constant should the springs have to make the object oscillate with a period of 2.00 s? (b) What energy is stored in the springs for a 2.00-m displacement from equilibrium?Parcels of air (small volumes of air) in a stable atmosphere (where the temperature increases with height) can oscillate up and down, due to the restoring force provided by the buoyancy of the air parcel. The frequency of the oscillations are a measure of the stability of the atmosphere. Assuming that the acceleration of an air parcel can be modeled as 2zt2=g(z)0zz , prove that z=z0etN2 is a solution, where N is known as the Brunt-Väisälä frequency. Note that in a stable atmosphere, the density decreases with height and parcel oscillates up and down.Consider the van der Waals potential U(r)=U0[( R 0 r)122( R 0 r)6] , used to model the potential energy function of two molecules, where the minimum potential is at r=R0 . Find the force as a function of r. Consider a small displacement R=R0+r and use the binomial theorem: (1+x)n=1+nx+n( n1)2!x2+n( n1)( n2)3!x3+ , to show that the force does approximate a Hooke’s law force.Suppose the length of a clock’s pendulum is changed by 1.000%, exactly at noon one day. What time will the clock read 24.00 hours later, assuming it the pendulum has kept perfect time before the change? Note that there are two answers, and perform the calculation to four-digit precision.(a) The springs of a pickup truck act like a single spring with a force constant of 1.30105 N/m . By how much will the truck be depressed by its maximum load of 1000 kg? (b) If the pickup truck has four identical springs, what is the force constant of each?Check Your Understanding When a guitar string is plucked, the guitar string oscillates as a result of waves moving through the string. The vibrations of the string cause the air molecules to oscillate, forming sound waves. The frequency of the sound waves is equal to the frequency of the vibrating string. Is the wavelength of the sound wave always equal to the wavelength of the waves on the string?Check Your Understanding The propagation velocity of a transverse or longitudinal mechanical wave may be constant as the wave disturbance moves through the medium. Consider a transverse mechanical wave: Is the velocity of the medium also constant?Check Your Understanding The wave function above is derived using a sine function. Can a cosine function be used instead?Check Your Understanding The wave equation 2y(x,t)x2=1v22y(x,t)t2 works for any wave of the form y(x,t)=f(xvt) . In the previous section, we stated that a cosine function could also be used to model a simple harmonic mechanical wave. Check if the wave y(x,t)=0.50mcos(0.20m1x4.00s1t+10) is a solution to the wave equation. Any disturbance that complies with the wave equation can propagate as a wave moving along the x-axis with a wave speed v. It works equally well for waves on a string, sound waves, and electromagnetic waves. This equation is extremely useful. For example, it can be used to show that electromagnetic waves move at the speed of light.Check Your Understanding The wave speed of a wave on a string depends on the tension and the linear mass density. If the tension is doubled, what happens to the speed of the waves on the string?Check Your Understanding Is the time-averaged power of a sinusoidal wave on a string proportional to the linear density of the string?Check Your Understanding The equations for the wavelengths and the frequencies of the modes of a wave produced on a string: n=2nLn=1,2,3,4,5...andn=nv2L=nf1n=1,2,3,4,5... were derived by considering a wave on a string where there were symmetric boundary conditions of a node at each end. These modes resulted from two sinusoidal waves with identical characteristics except they were moving in opposite directions, confined to a region L with nodes required at both ends.Will the same equations work if there were symmetric boundary conditions with antinodes at each end? What would the normal modes look like for a medium that was free to oscillate on each end? Don’t worry for now if you cannot imagine such a medium, just consider two sinusoidal wave functions in a region of length L, with antinodes on each end.Give one example of a transverse wave and one example of a longitudinal wave, being careful to note the relative directions of the disturbance and wave propagation in each.A sinusoidal transverse wave has a wavelength of 2.80 m. It takes 0.10 s for a portion of the string at a position x to move from a maximum position of y=0.03 m to the equilibrium position y=0 . What are the period, frequency, and wave speed of the wave?What is the difference between propagation speed and the frequency of a mechanical wave? Does one or both affect wavelength? If so, how?Consider a stretched spring, such as a slinky. The stretched spring can support longitudinal waves and transverse waves. How can you produce transverse waves on the spring? How can you produce longitudinal waves on the spring?Consider a wave produced on a stretched spring by holding one end and shaking it up and down. Does the wavelength depend on the distance you move your hand up and down?A sinusoidal, transverse wave is produced on a stretched spring, having a period T. Each section of the spring moves perpendicular to the direction of propagation of the wave, in simple harmonic motion with an amplitude A. Does each section oscillate with the same period as the wave or a different period? If the amplitude of the transverse wave were doubled but the period stays the same, would your answer be the same?An electromagnetic wave, such as light, does not require a medium. Can you think of an example that would support this claim?If you were to shake the end of a taut spring up and down 10 times a second, what would be the frequency and the period of the sinusoidal wave produced on the spring?If you shake the end of a stretched spring up and down with a frequency f, you can produce a sinusoidal, transverse wave propagating down the spring. Does the wave number depend on the frequency you are shaking the spring?Does the vertical speed of a segment of a horizontal taut string through which a sinusoidal, transverse wave is propagating depend on the wave speed of the transverse wave?In this section, we have considered waves that move at a constant wave speed. Does the medium accelerate?If you drop a pebble in a pond you may notice that several concentric ripples are produced, not just a single ripple. Why do you think that is?If the tension in a string were increased by a factor of four, by what factor would the wave speed of a wave on the string increase?Does a sound wave move faster in seawater or fresh water, if both the sea water and fresh water are at the same temperature and the sound wave moves near the surface? (w1000kgm3,s1030kgm3,Bw=2.15109Pa,Bs=2.34109Pa)Guitars have strings of different linear mass density. If the lowest density string and the highest density string are under the same tension, which string would support waves with the higher wave speed?Shown below are three waves that were sent down a string at different times. The tension in the string remains constant. (a) Rank the waves from the smallest wavelength to the largest wavelength. (b) Rank the waves from the lowest frequency to the highest frequency.Electrical power lines connected by two utility poles are sometimes heard to hum when driven into oscillation by the wind. The speed of the waves on the power lines depend on the tension. What provides the tension in the power lines?Two strings, one with a low mass density and one with a high linear density are spliced together. The higher density end is tied to a lab post and a student holds the free end of the low-mass density string. The student gives the string a flip and sends a pulse down the strings. If the tension is the same in both strings, does the pulse travel at the same wave velocity in both strings? If not, where does it travel faster, in the low density string or the high density string?Consider a string with under tension with a constant linear mass density. A sinusoidal wave with an angular frequency and amplitude produced by some external driving force. If the frequency of the driving force is decreased to half of the original frequency, how is the timeaveraged power of the wave affected? If the amplitude of the driving force is decreased by half, how is the timeaveraged power affected? Explain your answer.Circular water waves decrease in amplitude as they move away from where a rock is dropped. Explain why.In a transverse wave on a string, the motion of the string is perpendicular to the motion of the wave. If this is so, how is possible to move energy along the length of the string?The energy from the sun warms the portion of the earth facing the sun during the daylight hours. Why are the North and South Poles cold while the equator is quite warm?The intensity of a spherical waves decreases as the wave moves away from the source. If the intensity of the wave at the source is I0, how far from the source will the intensity decrease by a factor of nine?An incident sinusoidal wave is sent along a string that is fixed to the wall with a wave speed of v. The wave reflects off the end of the string. Describe the reflected wave.A string of a length of 2.00 m with a linear mass density of =0.006 kg/m is attached to the end of a 2.00-m-long string with a linear mass density of =0.012 kg/m. The free end of the higher-density string is fixed to the wall, and a student holds the free end of the low-density string, keeping the tension constant in both strings. The student sends a pulse down the string. Describe what happens at the interface between the two strings.A long, tight spring is held by two students, one student holding each end. Each student gives the end a flip sending one wavelength of a sinusoidal wave down the spring in opposite directions. When the waves meet in the middle, what does the wave look like?Many of the topics discussed in this chapter are useful beyond the topics of mechanical waves. It is hard to conceive of a mechanical wave with sharp corners, but you could encounter such a wave form in your digital electronics class, as shown below. This could be a signal from a device known as an analog to digital converter, in which a continuous voltage signal is converted into a discrete signal or a digital recording of sound. What is the result of the superposition of the two signals?A string of a constant linear mass density is held taut by two students, each holding one end. The tension in the string is constant. The students each send waves down the string by wiggling the string. (a) Is it possible for the waves to have different wave speeds? (b) Is it possible for the waves to have different frequencies? (c) Is it possible for the waves to have different wavelengths?A truck manufacturer finds that a strut in the engine is failing prematurely. A sound engineer determines that the strut resonates at the frequency of the engine and suspects that this could be the problem. What are two possible characteristics of the strut can be modified to correct the problem?Why do roofs of gymnasiums and churches seem to fail more than family homes when an earthquake occurs?Wine glasses can be set into resonance by moistening your finger and rubbing it around the rim of the glass. Why?Air conditioning units are sometimes placed on the roof of homes in the city. Occasionally, the air conditioners cause an undesirable hum throughout the upper floors of the homes. Why does this happen? What can be done to reduce the hum?Consider a standing wave modeled as y(x,t)=4.00cmsin(3m1x)cos(4s1t) . Is there a node or an antinode at x=0.00m ? What about a standing wave modeled as y(x,t)=4.00cmsin(3m1x+2)cos(4s1t) ? Is there a node or an antinode at the x=0.00m position?Storms in the South Pacific can create waves that travel all the way to the California coast, 12,000 km away. How long does it take them to travel this distance if they travel at 15.0 m/s?Waves on a swimming pool propagate at 0.75 m/s. You splash the water at one end of the pool and observe the wave go to the opposite end, reflect, and return in 30.00 s. How far away is the other end of the pool?Wind gusts create ripples on the ocean that have a wavelength of 5.00 cm and propagate at 2.00 m/s. What is their frequency?How many times a minute does a boat bob up and down on ocean waves that have a wavelength of 40.0 m and a propagation speed of 5.00 m/s?Scouts at a camp shake the rope bridge they have just crossed and observe the wave crests to be 8.00 m apart. If they shake the bridge twice per second, what is the propagation speed of the waves?What is the wavelength of the waves you create in a swimming pool if you splash your hand at a rate of 2.00 Hz and the waves propagate at a wave speed of 0.800 m/s?What is the wavelength of an earthquake that shakes you with a frequency of 10.0 Hz and gets to another city 84.0 km away in 12.0 s?Radio waves transmitted through empty space at the speed of light (v=c=3.00108m/s) by the Voyager spacecraft have a wavelength of 0.120 m. What is their frequency?Your ear is capable of differentiating sounds that arrive at each ear just 0.34 ms apart, which is useful in determining where low frequency sound is originating from. (a) Suppose a low-frequency sound source is placed to the right of a person, whose ears are approximately 18 cm apart, and the speed of sound generated is 340 m/s. How long is the interval between when the sound arrives at the right ear and the sound arrives at the left ear? (b) Assume the same person was scuba diving and a low-frequency sound source was to the right of the scuba diver. How long is the interval between when the sound arrives at the right ear and the sound arrives at the left ear, if the speed of sound in water is 1500 m/s? (c) What is significant about the time interval of the two situations?(a) Seismographs measure the arrival times of earthquakes with a precision of 0.100 s. To get the distance to the epicenter of the quake, geologists compare the arrival times of S- and P-waves, which travel at different speeds. If S- and P-waves travel at 4.00 and 7.20 km/s, respectively, in the region considered, how precisely can the distance to the source of the earthquake be determined? (b) Seismic waves from underground detonations of nuclear bombs can be used to locate the test site and detect violations of test bans. Discuss whether your answer to (a) implies a serious limit to such detection. (Note also that the uncertainty is greater if there is an uncertainty in the propagation speeds of the S- and P-waves.)A Girl Scout is taking a 10.00-km hike to earn a merit badge. While on the hike, she sees a cliff some distance away. She wishes to estimate the time required to walk to the cliff. She knows that the speed of sound is approximately 343 meters per second. She yells and finds that the echo returns after approximately 2.00 seconds. If she can hike 1.00 km in 10 minutes, how long would it take her to reach the cliff?A quality assurance engineer at a frying pan company is asked to qualify a new line of nonstick-coated frying pans. The coating needs to be 1.00 mm thick. One method to test the thickness is for the engineer to pick a percentage of the pans manufactured, strip off the coating, and measure the thickness using a micrometer. This method is a destructive testing method. Instead, the engineer decides that every frying pan will be tested using a nondestructive method. An ultrasonic transducer is used that produces sound waves with a frequency of f=25 kHz. The sound waves are sent through the coating and are reflected by the interface between the coating and the metal pan, and the time is recorded. The wavelength of the ultrasonic waves in the coating is 0.076 m. What should be the time recorded if the coating is the correct thickness (1.00 mm)?A pulse can be described as a single wave disturbance that moves through a medium. Consider a pulse that is defined at time t=0.00s by the equation y(s)=6.00m3x2+2.00m2 centered around x=0.00m . The pulse moves with a velocity of v=3.00m/s in the positive x-direction. (a) What is the amplitude of the pulse? (b) What is the equation of the pulse as a function of position and time? (c) Where is the pulse centered at time t=5.00s ?A transverse wave on a string is modeled with the wave function y(x,t)=(0.20cm)sin(2.00m1x3.00s1t+16) . What is the height of the string with respect to the equilibrium position at a position x=4.00m and a time t=10.00s ?Consider the wave function y(x,t)=(3.00cm)sin(0.4m1x+2.00s1t+10) . What are the period, wavelength, speed, and initial phase shift of the wave modeled by the wave function?A pulse is defined as y(x,t)=e2.77( 2.00( x2.00m/s ( t ) ) 5.00m)2 . Use a spreadsheet, or other computer program, to plot the pulse as the height of medium y as a function of position x. Plot the pulse at times t=0.00s and t=3.00s on the same graph. Where is the pulse centered at time t=0.00s ? Use your spreadsheet to check your answer.A wave is modeled at time t=0.00s with a wave function that depends on position. The equation is y(x)=(0.30m)sin(6.28m1x) . The wave travels a distance of 4.00 meters in 0.50 s in the positive x-direction. Write an equation for the wave as a function of position and time.A wave is modeled with the function y(x,t)=(0.25m)cos(0.30m1x0.90s1t+3) . Find the (a) amplitude, (b) wave number, (c) angular frequency, (d) wave speed, (e) phase shift, (f) wavelength, and (g) period of the wave.A surface ocean wave has an amplitude of 0.60 m and the distance from trough to trough is 8.00 m. It moves at a constant wave speed of 1.50 m/s propagating in the positive x-direction. At t=0 , the water displacement at x=0 is zero, and vyis positive. (a) Assuming the wave can be modeled as a sine wave, write a wave function to model the wave. (b) Use a spreadsheet to plot the wave function at times t=0.00s and t=2.00s on the same graph. Verify that the wave moves 3.00 m in those 2.00 s.A wave is modeled by the wave function y(x,t)=(0.30m)sin[24.50m(x18.00mst)] . What are the amplitude, wavelength, wave speed, period, and frequency of the wave?A transverse wave on a string is described with the wave function y(x,t)=(0.50cm)sin(1.57m1x6.28s1t) . (a) What is the wave velocity of the wave? (b) What is the magnitude of the maximum velocity of the string perpendicular to the direction of the motion?A swimmer in the ocean observes one day that the ocean surface waves are periodic and resemble a sine wave. The swimmer estimates that the vertical distance between the crest and the trough of each wave is approximately 0.45 m, and the distance between each crest is approximately 1.8 m. The swimmer counts that 12 waves pass every two minutes. Determine the simple harmonic wave function that would describes these waves.Consider a wave described by the wave function y(x,t)=0.3msin(2.00m1x628.00s1t) . (a) How many crests pass by an observer at a fixed location in 2.00 minutes? (b) How far has the wave traveled in that time?Consider two waves defined by the wave functions y1(x,t)=0.50msin(23.00mx+24.00st) and y2(x,t)=0.50msin(26.00mx24.00st) . What are the similarities and differences between the two waves?Consider two waves defined by the wave functions y1(x,t)=0.20msin(26.00mx24.00st) and y2(x,t)=0.20mcos(26.00mx24.00st) . What are the similarities and differences between the two waves? frequency, (d) wave speed, (e) phase shift, (f) wavelength, and (g) period of the wave.The speed of a transverse wave on a string is 300.00 m/s, its wavelength is 0.50 m, and the amplitude is 20.00 cm. How much time is required for a particle on the string to move through a distance of 5.00 km?Transverse waves are sent along a 5.00-m-long string with a speed of 30.00 m/s. The string is under a tension of 10.00 N. What is the mass of the string?A copper wire has a density of =8920 kg/m3, a radius of 1.20 mm, and a length L. The wire is held under a tension of 10.00 N. Transverse waves are sent down the wire. (a) What is the linear mass density of the wire? (b) What is the speed of the waves through the wire?A piano wire has a linear mass density of =4.95103 kg/m. Under what tension must the string be kept to produce waves with a wave speed of 500.00 m/s?A string with a linear mass density of =0.0060 kg/m is tied to the ceiling. A 20-kg mass is tied to the free end of the string. The string is plucked, sending a pulse down the string. Estimate the speed of the pulse as it moves down the string.A cord has a linear mass density of =0.0075 kg/m and a length of three meters. The cord is plucked and it takes 0.20 s for the pulse to reach the end of the string. What is the tension of the string?A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?Two strings are attached to poles, however the first string is twice as long as the second. If both strings have the same tension and mu, what is the ratio of the speed of the pulse of the wave from the first string to the second string?Two strings are attached to poles, however the first string is twice the linear mass density mu of the second. If both strings have the same tension, what is the ratio of the speed of the pulse of the wave from the first string to the second string?Transverse waves travel through a string where the tension equals 7.00 N with a speed of 20.00 m/s. What tension would be required for a wave speed of 25.00 m/s?Two strings are attached between two poles separated by a distance of 2.00 m as shown below, both under the same tension of 600.00 N. String 1 has a linear density of 1=0.0025 kg/m and string 2 has a linear mass density of 2=0.0035 kg/m. Transverse wave pulses are generated simultaneously at opposite ends of the strings. How much time passes before the pulses pass one another?Two strings are attached between two poles separated by a distance of 2.00 meters as shown in the preceding figure, both strings have a linear density of 1=0.0025 kg/m, the tension in string 1 is 600.00 N and the tension in string 2 is 700.00 N. Transverse wave pulses are generated simultaneously at opposite ends of the strings. How much time passes before the pulses pass one another?The note E4 is played on a piano and has a frequency of f=393.88 . If the linear mass density of this string of the piano is =0.012 kg/m and the string is under a tension of 1000.00 N, what is the speed of the wave on the string and the wavelength of the wave?Two transverse waves travel through a taut string. The speed of each wave is v=30.00 m/s. A plot of the vertical position as a function of the horizontal position is shown below for the time t=0.00 s. (a) What is the wavelength of each wave? (b) What is the frequency of each wave? (c) What is the maximum vertical speed of each string?A sinusoidal wave travels down a taut, horizontal string with a linear mass density of =0.060 kg/m . The maximum vertical speed of the wave is vymax=0.30 cm/s. The wave is modeled with the wave equation y(x,t)=Asin(6.00m1x24.00s1t) . (a) What is the amplitude of the wave? (b) What is the tension in the string?The speed of a transverse wave on a string is v=60.00 m/s and the tension in the string is FT=100.00 N . What must the tension be to increase the speed of the wave to v=120.00 m/s?A string of length 5 m and a mass of 90 g is held under a tension of 100 N. A wave travels down the string that is modeled as y(x,t)=0.01msin(0.40m1x1170.12s1) . What is the power over one wavelength?Ultrasound of intensity 1.50102W/m2 is produced by the rectangular head of a medical imaging device measuring 3.00 cm by 5.00 cm. What is its power output?The low-frequency speaker of a stereo set has asurface area of A=0.05 m2 and produces 1 W of acoustical power. (a) What is the intensity at the speaker? (b) If the speaker projects sound uniformly in all directions, at what distance from the speaker is the intensity 0.1 W/m2?To increase the intensity of a wave by a factor of 50, by what factor should the amplitude be increased?A device called an insolation meter is used to measure the intensity of sunlight. It has an area of 100 cm2 and registers 6.50 W. What is the intensity in W/m2?Energy from the Sun arrives at the top of Earth’s atmosphere with an intensity of 1400 W/m2. How long does it take for 1.80109 J to arrive on an area of 1.00 m2?Suppose you have a device that extracts energy from ocean breakers in direct proportion to their intensity. If the device produces 10.0 kW of power on a day when the breakers are 1.20 m high, how much will it produce when they are 0.600 m high?A photovoltaic array of (solar cells) is 10.0% efficient in gathering solar energy and converting it to electricity. If the average intensity of sunlight on one day is 70.00 W/m2, what area should your array have to gather energy at the rate of 100 W? (b) What is the maximum cost of the array if it must pay for itself in two years of operation averaging 10.0 hours per day? Assume that it earns money at the rate of 9.00 cents per kilowatt-hour.A microphone receiving a pure sound tone feeds an oscilloscope, producing a wave on its screen. If the sound intensity is originally 2.00105W/m2 , but is turned up until the amplitude increases by 30.0% , what is the new intensity?A string with a mass of 0.30 kg has a length of 4.00 m. If the tension in the string is 50.00 N, and a sinusoidal wave with an amplitude of 2.00 cm is induced on the string, what must the frequency be for an average power of 100.00 W?The power versus time for a point on a string (=0.05kg/m) in which a sinusoidal traveling wave is induced is shown in the preceding figure. The wave is modeled with the wave equation y(x,t)=Asin(20.93m1x1t) . What is the frequency and amplitude of the wave?A string is under tension FT1. Energy is transmitted by a wave on the string at rate P1by a wave of frequency f1. What is the ratio of the new energy transmission rate P2to P1if the tension is doubled?A 250-Hz tuning fork is struck and the intensity at the source is I1at a distance of one meter from the source. (a) What is the intensity at a distance of 4.00 m from the source? (b) How far from the tuning fork is the intensity a tenth of the intensity at the source?A sound speaker is rated at a voltage of P=120.00 V and a current of I=10.00 A. Electrical power consumption is P=IV . To test the speaker, a signal of a sine wave is applied to the speaker. Assuming that the sound wave moves as a spherical wave and that all of the energy applied to the speaker is converted to sound energy, how far from the speaker is the intensity equal to 3.82 W/m2?The energy of a ripple on a pond is proportional to the amplitude squared. If the amplitude of the ripple is 0.1 cm at a distance from the source of 6.00 meters, what was the amplitude at a distance of 2.00 meters from the source?Consider two sinusoidal waves traveling along a string, modeled as y1(x,t)=0.3msin(4m1x+3s1t) and y2(x,t)=0.6msin(8m1x6s1t) . What is the height of the resultant wave formed by the interference of the two waves at the position x=0.5 m at time t=0.2 s?Consider two sinusoidal sine waves traveling along a string, modeled as y1(x,t)=0.3msin(4m1x+3s1t+3) and y2(x,t)=0.6msin(8m1x6s1t) . What is the height of the resultant wave formed by the interference of the two waves at the position x=1.0 m at time t=3.0 s?Consider two sinusoidal sine waves traveling along a string, modeled as y1(x,t)=0.3msin(4m1x3s1t) and y2(x,t)=0.3msin(4m1x+3s1t) . What is the wave function of the resulting wave? [Hint: Use the trig identity sin(uv)=sinucosvcosusinvTwo sinusoidal waves are moving through a medium in the same direction, both having amplitudes of 3.00 cm, a wavelength of 5.20 m, and a period of 6.52 s, but one has a phase shift of an angle . What is the phase shift if the resultant wave has an amplitude of 5.00 cm? [Hint: Use the trig identity sinu+sinv=2sin(u+v2)cos(uv2)Two sinusoidal waves are moving through a medium in the positive x-direction, both having amplitudes of 6.00 cm, a wavelength of 4.3 m, and a period of 6.00 s, but one has a phase shift of an angle =0.50 rad. What is the height of the resultant wave at a time t=3.15 s and a position x=0.45 m ?Two sinusoidal waves are moving through a medium in the positive x-direction, both having amplitudes of 7.00 cm, a wave number of k=3.00 m-1, an angular frequency of =2.50 s-1, and a period of 6.00 s, but one has a phase shift of an angle =12 rad. What is the height of the resultant wave at a time t=2.00 s and a position x=0.53 m?Consider two waves y1(x,t) and y2(x,t) that are identical except for a phase shift propagating in the same medium. (a)What is the phase shift, in radians, if the amplitude of the resulting wave is 1.75 times the amplitude of the individual waves? (b) What is the phase shift in degrees? (c) What is the phase shift as a percentage of the individual wavelength?Two sinusoidal waves, which are identical except for a phase shift, travel along in the same direction. The wave equation of the resultant wave is yR(x,t)=0.70msin(3.00m1x6.28s1t+/16rad) . What are the angular frequency, wave number, amplitude, and phase shift of the individual waves? .Two sinusoidal waves, which are identical except for a phase shift, travel along in the same direction. The wave equation of the resultant wave is yR(x,t)=0.35cmsin(6.28m1x1.57s1t+4) . What are the period, wavelength, amplitude, and phase shift of the individual waves?Consider two wave functions, y1(x,t)=0.40msin(m1xs1t) and y2(x,t)=0.40msin(m1xs1t+3) . (a) Using a spreadsheet, plot the two wave functions and the wave that results from the superposition of the two wave functions as a function of position (0.00x6.00m) for the time t=0.00 s. (b) What are the wavelength and amplitude of the two original waves? (c) What are the wavelength and amplitude of the resulting wave?Consider two wave functions, y2(x,t)=2.00msin(2m1x3s1t) and y2(x,t)=2.00msin(2m1x3s1t+6) . (a) Verifythat yR=2Acos(2)sin(kxt+2) is the solution forthe wave that results from a superposition of the two waves.Make a column for x,y1,y2,y1+y2 , and yR=2Acos(2)sin(kxt+2) . Plot four waves as a function of position where the range of x is from 0 to 12 m.Consider two wave functions that differ only by a phase shift, y1(x,t)=Acos(kxt) and y2(x,t)=Acos(kxt+) . Use the trigonometric cosu+cosv=2cos(uv2)cos(u+v2) and cos()=cos() to find a wave equation for the wave resulting from the superposition of the two waves. Does the resulting wave function come as a surprise to you?A wave traveling on a Slinky® that is stretched to 4 m takes 2.4 s to travel the length of the Slinky and back again. (a) What is the speed of the wave? (b) Using the same Slinky stretched to the same length, a standing wave is created which consists of three antinodes and four nodes. At what frequency must the Slinky be oscillating?A 2-m long string is stretched between two supports with a tension that produces a wave speed equal to vw=50.00 m/s. What are the wavelength and frequency of the first three modes that resonate on the string?Consider the experimental setup shown below. The length of the string between the string vibrator and the pulley is L=1.00 m. The linear density of the string is =0.006 kg/m. The string vibrator can oscillate at any frequency. The hanging mass is 2.00 kg. (a)What are the wavelength and frequency of n=6 mode? (b) The string oscillates the air around the string. What is the wavelength of the sound if the speed of the sound is vs=343.00 m/s?