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All Textbook Solutions for Physics for Scientists and Engineers

A meterstick of uniform density is hung from a string tied at the 25-cm mark. A 0.50-kg object is hung from the zero end of the meterstick, and the meterstick is balanced horizontally. What is the mass of the meterstick? (a) 0.25 kg (b) 0.50 kg (c) 0.75 kg (d) 1.0 kg (e) 2.0 kg (f) impossible to determine For the three parts of this Quick Quiz, choose from the following choices the correct answer for the elastic modulus that describes the relationship between stress and strain for the system of interest, which is in italics: (a) Youngs modulus (b) shear modulus (c) bulk modulus (d) none of those choices (i) A block of iron is sliding across a horizontal floor. The friction force between the sliding block and the floor causes the block to deform. (ii) A trapeze artist swings through a circular arc. At the bottom of the swing, the wires supporting the trapeze are longer than when the trapeze artist simply hangs from the trapeze due to the increased tension in them. (iii) A spacecraft carries a steel sphere to a planet on which atmospheric pressure is much higher than on the Earth. The higher pressure causes the radius of the sphere to decrease.You are building additional storage space in your garage. You decide to suspend a 10.0-kg sheet of plywood of dimensions 0.600 m wide by 2.25 m long from the ceiling. The plywood will be held in a horizontal orientation by four light vertical chains attached to the plywood at its corners and mounted to the ceiling. After you complete the job of suspending the plywood from the ceiling, you choose three cubic boxes to place on the shelf. Each box is 0.750 m on a side. Box 1 has a mass of 50.0 kg, box 2 has a mass of 100 kg, and box 3 has a mass of 125 kg. The mass of each box is uniformly distributed within the box and each box is centered on the front-to-back width of the shelf. Unbeknownst to you, one of the chains on the right-hand end of your shelf is defective and will break if subjected to a force of more than 700 N. There are six possible arrangements of the three boxes on the shelf, for example, from left to right, Box 1, Box 2, Box 3, and Box 1, Box 3, Box, 2, and four more. Which arrangements are safe (that is, the defective chain will not break if the boxes are arranged in this way), and which arrangements are dangerous?Why is the following situation impossible? A uniform beam of mass mk = 3.00 kg and length = 1.00 m supports blocks with masses m1 = 5.00 kg and m2 = 15.0 kg at two positions as shown in Figure P12.2. The beam rests on two triangular blocks, with point P a distance d = 0.300 m to the right of the center of gravity of the beam. The position of the object of mass m2 is adjusted along the length of the beam until the normal force on the beam at O is zero. Figure P12.2 3P A circular pizza of radius R has a circular piece of radius R/2 removed from one side as shown in Figure P12.4. The center of gravity has moved from C to C along the x axis. Show that the distance from C to C is R/6. Assume the thickness and density of the pizza are uniform throughout. Figure P12.4 Your brother is opening a skateboard shop. He has created a sign for his shop made from a uniform material and in the shape shown in Figure P12.5. The shape of the sign represents one of the hills in the skateboard park he plans on building on land adjacent to the shop. The curve on the top of the sign is described by the function y = (x 3)2/9. When the sign arrives in his shop, your brother wants to hang it from a single wire outside the shop. But he doesnt know where on the sign to attach the wire so that the bottom edge of the sign will hang in a horizontal orientation. He asks for your help. Figure P12.5 A uniform beam of length 7.60 m and weight 4.50 102 N is carried by two workers, Sam and Joe, as shown in Figure P12.6. Determine the force that each person exerts on the beam. Figure P12.6 7P A uniform beam of length L and mass m shown in Figure P12.8 is inclined at an angle to the horizontal. Its upper end is connected to a wall by a rope, and its lower end rests on a rough, horizontal surface. The coefficient of static friction between the beam and surface is s. Assume the angle is such that the static friction force is at its maximum value. (a) Draw a force diagram for the beam. (b) Using the condition of rotational equilibrium, find an expression for the tension T in the rope in terms of m, g, and . (c) Using the condition of translational equilibrium, find a second expression for T in terms of s, m, and g. (d) Using the results from parts (a) through (c), obtain an expression for s involving only the angle . (e) What happens if the ladder is lifted upward and its base is placed back on the ground slightly to the left of its position in Figure P12.8? Explain. Figure P12.8 A flexible chain weighing 40.0 N hangs between two hooks located at the same height (Fig. P12.9). At each hook, the tangent to the chain makes an angle = 42.0 with the horizontal. Find (a) the magnitude of the force each hook exerts on the chain and (b) the tension in the chain at its midpoint. Suggestion: For part (b), make a force diagram for half of the chain. Figure P12.9 A 20.0-kg floodlight in a park is supported at the end of a horizontal beam of negligible mass that is hinged to a pole as shown in Figure P12.10. A cable at an angle of = 30.0 with the beam helps support the light. (a) Draw a force diagram for the beam. By computing torques about an axis at the hinge at the left-hand end of the beam, find (b) the tension in the cable, (c) the horizontal component of the force exerted by the pole on the beam, and (d) the vertical component of this force. Now solve the same problem from the force diagram from part (a) by computing torques around the junction between the cable and the beam at the right-hand end of the beam. Find (e) the vertical component of the force exerted by the pole on the beam, (f) the tension in the cable, and (g) the horizontal component of the force exerted by the pole on the beam. (h) Compare the solution to parts (b) through (d) with the solution to parts (c) through (g). Is either solution more accurate? Figure P12.10 11P Review. While Lost-a-Lot ponders his next move in the situation described in Problem 11 and illustrated in Figure P12.11, the enemy attacks! An incoming projectile breaks off the stone ledge so that the end of the drawbridge can be lowered past the wall where it usually rests. In addition, a fragment of the projectile bounces up and cuts the drawbridge cable! The hinge between the castle wall and the bridge is frictionless, and the bridge swings down freely until it is vertical and smacks into the vertical castle wall below the castle entrance. (a) How long does Lost-a-Lot stay in contact with the bridge while it swings downward? (b) Find the angular acceleration of the bridge just as it starts to move. (c) Find the angular speed of the bridge when it strikes the wall below the hinge. Find the force exerted by the hinge on the bridge (d) immediately after the cable breaks and (e) immediately before it strikes the castle wall.Figure P12.13 shows a claw hammer being used to pull a nail out of a horizontal board. The mass of the hammer is 1.00 kg. A force of 150 N is exerted horizontally as shown, and the nail does not yet move relative to the board. Find (a) the force exerted by the hammer claws on the nail and (b) the force exerted by the surface on the point of contact with the hammer head. Assume the force the hammer exerts on the nail is parallel to the nail. Figure P12.13 A 10.0-kg monkey climbs a uniform ladder with weight 1.20 102 N and length L = 3.00 m as shown in Figure P12.14. The ladder rests against the wall and makes an angle of = 60.0 with the ground. The upper and lower ends of the ladder rest on frictionless surfaces. The lower end is connected to the wall by a horizontal rope that is frayed and can support a maximum tension of only 80.0 N. (a) Draw a force diagram for the ladder. (b) Find the normal force exerted on the bottom of the ladder. (c) Find the tension in the rope when the monkey is two-thirds of the way up the ladder. (d) Find the maximum distance d that the monkey can climb up the ladder before the rope breaks. (e) If the horizontal surface were rough and the rope were removed, how would your analysis of the problem change? What other information would you need to answer parts (c) and (d)? Figure P12.14 John is pushing his daughter Rachel in a wheelbarrow when it is stopped by a brick 8.00 cm high (Fig. P12.15). The handles make an angle of = 15.0 with the ground. Due to the weight of Rachel and the wheelbarrow, a downward force of 400 N is exerted at the center of the wheel, which has a radius of 20.0 cm. (a) What force must John apply along the handles to just start the wheel over the brick? (b) What is the force (magnitude and direction) that the brick exerts on the wheel just as the wheel begins to lift over the brick? In both parts, assume the brick remains fixed and does not slide along the ground. Also assume the force applied by John is directed exactly toward the center of the wheel.16P The deepest point in the ocean is in the Mariana Trench, about 11 km deep, in the Pacific. The pressure at this depth is huge, about 1.13 108 N/m2. (a) Calculate the change in volume of 1.00 m3 of seawater carried from the surface to this deepest point. (b) The density of seawater at the surface is 1.03 105 kg/m3. Find its density at the bottom. (c) Explain whether or when it is a good approximation to think of water as incompressible.A steel wire of diameter 1 mm can support a tension of 0.2 kN. A steel cable to support a tension of 20 kN should have diameter of what order of magnitude?A child slides across a floor in a pair of rubber-soled shoes. The friction force acting on each foot is 20.0 N. The footprint area of each shoe sole is 14.0 cm2, and the thickness of each sole is 5.00 mm. Find the horizontal distance by which the upper and lower surfaces of each sole are offset. The shear modulus of the rubber is 3.00 MN/m2.Evaluate Youngs modulus for the material whose stressstrain curve is shown in Figure 12.12. Figure 12.12 Stress-versus-strain curve for an elastic solid.Assume if the shear stress in steel exceeds about 4.00 108 N/m2, the steel ruptures. Determine the shearing force necessary to (a) shear a steel bolt 1.00 cm in diameter and (b) punch a 1.00-cm-diameter hole in a steel plate 0.500 cm thick.When water freezes, it expands by about 9.00%. What pressure increase would occur inside your automobile engine block if the water in it froze? (The bulk modulus of ice is 2.0 109 N/m2.)Review. A 30.0-kg hammer, moving with speed 20.0 m/s, strikes a steel spike 2.30 cm in diameter. The hammer rebounds with speed 10.0 m/s after 0.110 s. What is the average strain in the spike during the impact?A uniform beam resting on two pivots has a length L = 6.00 m and mass M = 90.0 kg. The pivot under the left end exerts a normal force n1 on the beam, and the second pivot located a distance = 4.00 m from the left end exerts a normal force n. A woman of mass m = 55.0 kg steps onto the left end of the beam and begins walking to the right as in Figure P12.24. The goal is to find the womans position when the beam begins to tip. (a) What is the appropriate analysis model for the beam before it begins to tip? (b) Sketch a force diagram for the beam, labeling the gravitational and normal forces acting on the beam and placing the woman a distance x to the right of the first pivot, which is the origin. (c) Where is the woman when the normal force n1 is the greatest? (d) What is n1 when the beam is about to tip? (e) Use Equation 12.1 to find the value of n2 when the beam is about to tip. (f) Using the result of part (d) and Equation 12.2, with torques computed around the second pivot, find the womans position x when the beam is about to tip. (g) Check the answer to part (e) by computing torques around the first pivot point. Figure P12.24 A bridge of length 50.0 m and mass 8.00 104 kg is supported on a smooth pier at each end as shown in Figure P12.25. A truck of mass 3.00 104 kg is located 15.0 m from one end. What are the forces on the bridge at the points of support? Figure P12.25 26AP The lintel of prestressed reinforced concrete in Figure P12.27 is 1.50 m long. The concrete encloses one steel reinforcing rod with cross-sectional area 1.50 cm2. The rod joins two strong end plates. The cross-sectional area of the concrete perpendicular to the rod is 50.0 cm2. Youngs modulus for the concrete is 30.0 109 N/m2. After the concrete cures and the original tension T1 in the rod is released, the concrete is to be under compressive stress 8.00 106 N/m2. (a) By what distance will the rod compress the concrete when the original tension in the rod is released? (b) What is the new tension T2 in the rod? (c) The rod will then be how much longer than its unstressed length? (d) When the concrete was poured, the rod should have been stretched by what extension distance from its unstressed length? (e) Find the required original tension T1 in the rod. Figure P12.27 The following equations are obtained from a force diagram of a rectangular farm gate, supported by two hinges on the left-hand side. A bucket of grain is hanging from the latch. A+C=0+B392N50.0N=0A(0)+B(0)+C(1.80m)392N(1.50m)50.0N(3.00m)=0 (a) Draw the force diagram and complete the statement of the problem, specifying the unknowns. (b) Determine the values of the unknowns and state the physical meaning of each.A hungry bear weighing 700 N walks out on a beam in an attempt to retrieve a basket of goodies hanging at the end of the beam (Fig. P12.29). The beam is uniform, weighs 200 N, and is 6.00 m long, and it is supported by a wire at an angle of = 60.0. The basket weighs 80.0 N. (a) Draw a force diagram for the beam. (b) When the bear is at x = 1.00 m, find the tension in the wire supporting the beam and the components of the force exerted by the wall on the left end of the beam. (c) What If? If the wire can withstand a maximum tension of 900 N, what is the maximum distance the bear can walk before the wire breaks? Figure P12.29 A 1 200-N uniform boom at = 65 to the vertical is supported by a cable at an angle = 25.0 to the horizontal as shown in Figure P12.30. The boom is pivoted at the bottom, and an object of weight m = 2 000 N hangs from its top. Find (a) the tension in the support cable and (b) the components of the reaction force exerted by the floor on the boom. Figure P12.30 A uniform sign of weight Fg and width 2L hangs from a light, horizontal beam hinged at the wall and supported by a cable (Fig. P12.31). Determine (a) the tension in the cable and (b) the components of the reaction force exerted by the wall on the beam in terms of Fg, d, L, and . Figure P12.31 When a person stands on tiptoe on one foot (a strenuous position), the position of the foot is as shown in Figure P12.32a. The total gravitational force Fg on the body is supported by the normal force n exerted by the floor on the toes of one foot. A mechanical model of the situation is shown in Figure P12.32b, where T is the force exerted on the foot by the Achilles tendon and R is the force exerted on the foot by the tibia. Find the values of T, R, and when Fg = 700 N. Figure P12.32 A 10 000-N shark is supported by a rope attached to a 4.00-m rod that can pivot at the base. (a) Calculate the tension in the cable between the rod and the wall, assuming the cable is holding the system in the position shown in Figure P12.33. Find (b) the horizontal force and (c) the vertical force exerted on the base of the rod. Ignore the weight of the rod. Figure P12.33 Assume a person bends forward to lift a load with his back as shown in Figure P12.34a. The spine pivots mainly at the fifth lumbar vertebra, with the principal supporting force provided by the erector spinalis muscle in the back. To see the magnitude of the forces involved, consider the model shown in Figure P12.34b for a person bending forward to lift a 200-N object. The spine and upper body are represented as a uniform horizontal rod of weight 350 N, pivoted at the base of the spine. The erector spinalis muscle, attached at a point two-thirds of the way up the spine, maintains the position of the back. The angle between the spine and this muscle is = 12.0. Find (a) the tension T in the back muscle and (b) the compressional force in the spine. (c) Is this method a good way to lift a load? Explain your answer, using the results of parts (a) and (b). (d) Can you suggest a better method to lift a load? Figure P12.34 A uniform beam of mass m is inclined at an angle to the horizontal. Its upper end (point P) produces a 90 bend in a very rough rope tied to a wall, and its lower end rests on a rough floor (Fig. P12.35). Let s represent the coefficient of static friction between beam and floor. Assume s is less than the cotangent of . (a) Find an expression for the maximum mass M that can be suspended from the top before the beam slips. Determine (b) the magnitude of the reaction force at the floor and (c) the magnitude of the force exerted by the beam on the rope at P in terms of m, M, and s. Figure P12.35 Why is the following situation impossible? A worker in a factory pulls a cabinet across the floor using a rope as shown in Figure P12.36a. The rope make an angle = 37.0 with the floor and is tied h1 = 10.0 cm from the bottom of the cabinet. The uniform rectangular cabinet has height = 100 cm and width w = 60.0 cm, and it weighs 400 N. The cabinet slides with constant speed when a force F = 300 N is applied through the rope. The worker tires of walking backward. He fastens the rope to a point on the cabinet h2 = 65.0 cm off the floor and lays the rope over his shoulder so that he can walk forward and pull as shown in Figure P12.36b. In this way, the rope again makes an angle of = 37.0 with the horizontal and again has a tension of 300 N. Using this technique, the worker is able to slide the cabinet over a long distance on the floor without tiring. Figure P12.36 Problems 36 and 44.When a circus performer performing on the rings executes the iron cross, he maintains the position at rest shown in Figure P12.37a. In this maneuver, the gymnasts feet (not shown) are off the floor. The primary muscles involved in supporting this position are the latissimus dorsi (lats) and the pectoralis major (pecs). One of the rings exerts an upward force Fk on a hand as show n in Figure P12.37b. The force Fs, is exerted by the shoulder joint on the arm. The latissimus dorsi and pectoralis major muscles exert a total force Fm on the arm. (a) Using the information in the figure, find the magnitude of the force Fm for an athlete of weight 750 N. (b) Suppose a performer in training cannot perform the iron cross but can hold a position similar to the figure in which the arms make a 45 angle with the horizontal rather than being horizontal. Why is this position easier for the performer? Figure P12.37 Figure P12.38 shows a light truss formed from three struts lying in a plane and joined by three smooth hinge pins at their ends. The truss supports a downward force of F=1000N applied at the point B. The truss has negligible weight. The piers at A and C are smooth. (a) Given 1 = 30.0 and 2 = 45.0, find nA and nC. (b) One can show that the force any strut exerts on a pin must be directed along the length of the strut as a force of tension or compression. Use that fact to identify the directions of the forces that the struts exert on the pins joining them. Find the force of tension or of compression in each of the three bars. Figure P12.38 39AP A stepladder of negligible weight is constructed as shown in Figure P12.40, with AC = BC = = 4.00 m. A painter of mass m = 70.0 kg stands on the ladder d = 3.00 m from the bottom. Assuming the floor is frictionless, find (a) the tension in the horizontal bar DE connecting the two halves of the ladder, (b) the normal forces at A and B, and (c) the components of the reaction force at the single hinge C that the left half of the ladder exerts on the right half. Suggestion: Treat the ladder as a single object, but also treat each half of the ladder separately. Figure P12.40 Problems 40 and 41.A stepladder of negligible weight is constructed as shown in Figure P12.40, with AC = BC = . A painter of mass m stands on the ladder a distance d from the bottom. Assuming the floor is frictionless, find (a) the tension in the horizontal bar DE connecting the two halves of the ladder, (b) the normal forces at A and B, and (c) the components of the reaction force at the single hinge C that the left half of the ladder exerts on the right half. Suggestion: Treat the ladder as a single object, but also treat each half of the ladder separately. Figure P12.40 Problems 40 and 41.Review. A wire of length L, Youngs modulus Y, and cross-sectional area A is stretched elastically by an amount L. By Hookes law, the restoring force is kL. (a) Show that k = YA/L. (b) Show that the work done in stretching the wire by an amount L is W=12YA(L)2/L.Two racquetballs, each having a mass of 170 g, are placed in a glass jar as shown in Figure P12.43. Their centers lie on a straight line that makes a 45 angle with the horizontal. (a) Assume the walls are frictionless and determine P1, P2, and P3. (b) Determine the magnitude of the force exerted by the left ball on the right ball. Figure P12.43 44AP Review. An aluminum wire is 0.850 m long and has a circular cross section of diameter 0.780 mm. Fixed at the top end, the wire supports a 1.20-kg object that swings in a horizontal circle. Determine the angular speed of the object required to produce a strain of 1.00 103.You have been hired as an expert witness in a case involving an injury in a factory. The attorney who hired you represents the injured worker. The worker was told to lift one end of a long, heavy crate that was lying horizontally on the floor and tilt it up so that it is standing on end. He began lifting the end of the crate, always applying a force that was perpendicular to the top of the crate. As the end of the crate got higher, at a certain angle, the bottom of the crate slipped on the floor, and the worker, in trying to recover, stepped forward and the crate landed on his foot, injuring it badly. As part of your investigation, you go to the factory and measure the coefficient of static friction between a crate and the smooth concrete floor. You find it to be 0.340. Prepare an argument for the attorney showing that it was impossible to lift the crate in the manner described without it slipping on the floor.A 500-N uniform rectangular sign 4.00 m wide and 3.00 m high is suspended from a horizontal, 6.00-m-long, uniform. 100-N rod as indicated in Figure P12.47. The left end of the rod is supported by a hinge, and the right end is supported by a thin cable making a 30.0 angle with the vertical. (a) Find the tension T in the cable. (b) Find the horizontal and vertical components of force exerted on the left end of the rod by the hinge. Figure P12.47 A steel cable 3.00 cm2 in cross-sectional area has a mass of 2.40 kg per meter of length. If 500 m of the cable is hung over a vertical cliff, how much does the cable stretch under its own weight? Take Ysteel = 2.00 1011 N/m2.A uniform rod of weight Fg and length L is supported at its ends by a frictionless trough as shown in Figure P12.49. (a) Show that the center of gravity of the rod must be vertically over point O when the rod is in equilibrium. (b) Determine the equilibrium value of the angle . (c) Is the equilibrium of the rod stable or unstable? Figure P12.49 In the What If? section of Example 12.2, let d represent the distance in meters between the person and the hinge at the left end of the beam. (a) Show that the cable tension is given by T = 93.9d + 125, with T in newtons. (b) Show that the direction angle of the hinge force is described by tan=(323d+41)tan53.0 (c) Show that the magnitude of the hinge force is given by R=8.82103d29.65104d+4.96105 (d) Describe how the changes in T, , and R as d increases differ from one another.A planet has two moons of equal mass. Moon 1 is in a circular orbit of radius r. Moon 2 is in a circular orbit of radius 2r. What is the magnitude of the gravitational force exerted by the planet on Moon 2? (a) four times as large as that on Moon 1 (b) twice as large as that on Moon 1 (c) equal to that on Moon 1 (d) half as large as that on Moon 1 (e) one-fourth as large as that on Moon 1 Superman stands on top of a very tall mountain and throws a baseball horizontally with a speed such that the baseball goes into a circular orbit around the Earth. While the baseball is in orbit, what is the magnitude of the acceleration of the ball? (a) It depends on how fast the baseball is thrown. (b) It is zero because the ball does not fall to the ground. (c) It is slightly less than 9.80 m/s2. (d) It is equal to 9.80 m/s2.An asteroid is in a highly eccentric elliptical orbit around the Sun. The period of the asteroids orbit is 90 days. Which of the following statements is true about the possibility of a collision between this asteroid and the Earth? (a) There is no possible danger of a collision. (b) There is a possibility of a collision. (c) There is not enough information to determine whether there is danger of a collision.13.4QQ In introductory physics laboratories, a typical Cavendish balance for measuring the gravitational constant G uses lead spheres with masses of 1.50 kg and 15.0 g whose centers are separated by about 4.50 cm. Calculate the gravitational force between these spheres, treating each as a particle located at the spheres center.During a solar eclipse, the Moon, the Earth, and the Sun all lie on the same line, with the Moon between the Earth and the Sun. (a) What force is exerted by the Sun on the Moon? (b) What force is exerted by the Earth on the Moon? (c) What force is exerted by the Sun on the Earth? (d) Compare the answers to parts (a) and (b). Why doesnt the Sun capture the Moon away from the Earth?Determine the order of magnitude of the gravitational force that you exert on another person 2 m away. In your solution, state the quantities you measure or estimate and their values.4P Review. Miranda, a satellite of Uranus, is shown in Figure P13.5a. It can be modeled as a sphere of radius 242 km and mass 6.68 1019 kg. (a) Find the free-fall acceleration on its surface. (b) A cliff on Miranda is 5.00 km high. It appears on the limb at the 11 oclock position in Figure P13.5a and is magnified in Figure P13.5b. If a devotee of extreme sports runs horizontally off the top of the cliff at 8.50 m/s, for what time interval is he in flight? (c) How far from the base of the vertical cliff does he strike the icy surface of Miranda? (d) What will be his vector impact velocity? Figure P13.5 (a) Compute the vector gravitational field at a point P on the perpendicular bisector of the line joining two objects of equal mass separated by a distance 2a as shown in Figure P13.6. (b) Explain physically why the field should approach zero as r 0. (c) Prove mathematically that the answer to part (a) behaves in this way. (d) Explain physically why the magnitude of the field should approach 2GM/r2 as r . (e) Prove mathematically that the answer to part (a) behaves correctly in this limit. Figure P13.6 A spacecraft in the shape of a long cylinder has a length of 100 m, and its mass with occupants is 1 000 kg. It has strayed too close to a black hole having a mass 100 times that of the Sun (Fig. P13.7). The nose of the spacecraft points toward the black hole, and the distance between the nose and the center of the black hole is 10.0 km. (a) Determine the total force on the spacecraft. (b) What is the difference in the gravitational fields acting on the occupants in the nose of the ship and on those in the rear of the ship, farthest from the black hole? (This difference in accelerations grows rapidly as the ship approaches the black hole. It puts the body of the ship under extreme tension and eventually tears it apart.) Figure P13.7 An artificial satellite circles the Earth in a circular orbit at a location where the acceleration due to gravity is 9.00 m/s2. Determine the orbital period of the satellite.9P A particle of mass m moves along a straight line with constant velocity v0 in the x direction, a distance b from the x axis (Fig. P13.10). (a) Does the particle possess any angular momentum about the origin? (b) Explain why the amount of its angular momentum should change or should stay constant. (c) Show that Keplers second law is satisfied by showing that the two shaded triangles in the figure have the same area when . Figure P13.10 Use Keplers third law to determine how many days it takes a spacecraft to travel in an elliptical orbit from a point 6 670 km from the Earths center to the Moon, 385 000 km from the Earths center.12P Suppose the Suns gravity were switched off. The planets would leave their orbits and fly away in straight lines as described by Newtons first law. (a) Would Mercury ever be farther from the Sun than Pluto? (b) If so, find how long it would take Mercury to achieve this passage. If not, give a convincing argument that Pluto is always farther from the Sun than is Mercury.(a) Given that the period of the Moons orbit about the Earth is 27.32 days and the nearly constant distance between the center of the Earth and the center of the Moon is 3.84 108 m, use Equation 13.11 to calculate the mass of the Earth. (b) Why is the value you calculate a bit too large?How much energy is required to move a 1 000-kg object from the Earths surface to an altitude twice the Earths radius?An object is released from rest at an altitude h above the surface of the Earth. (a) Show that its speed at a distance r from the Earths center, where RE r RE + h, is v=2GME(1r1RE+h) (b) Assume the release altitude is 500 km. Perform the integral t=ifdt=ifdrv to find the time of fall as the object moves from the release point to the Earths surface. The negative sign appears because the object is moving opposite to the radial direction, so its speed is v = dr/dt. Perform the integral numerically.A system consists of three particles, each of mass 5.00 g, located at the corners of an equilateral triangle with sides of 30.0 cm. (a) Calculate the gravitational potential energy of the system. (b) Assume the particles are released simultaneously. Describe the subsequent motion of each. Will any collisions take place? Explain.18P A 500-kg satellite is in a circular orbit at an altitude of 500 km above the Earths surface. Because of air friction, the satellite eventually falls to the Earths surface, where it hits the ground with a speed of 2.00 km/s. How much energy was transformed into internal energy by means of air friction?20P 21P 22P Ganymede is the largest of Jupiters moons. Consider a rocket on the surface of Ganymede, at the point farthest from the planet (Fig. P13.23). Model the rocket as a particle. (a) Does the presence of Ganymede make Jupiter exert a larger, smaller, or same size force on the rocket compared with the force it would exert if Ganymede were not interposed? (b) Determine the escape speed for the rocket from the planetsatellite system. The radius of Ganymede is 2.64 106 m, and its mass is 1.495 1023 kg. The distance between Jupiter and Ganymede is 1.071 109 m, and the mass of Jupiter is 1.90 1027 kg. Ignore the motion of Jupiter and Ganymede as they revolve about their center of mass. Figure P13.23 24AP Voyager 1 and Voyager 2 surveyed the surface of Jupiters moon Io and photographed active volcanoes spewing liquid sulfur to heights of 70 km above the surface of this moon. Find the speed with which the liquid sulfur left the volcano. Ios mass is 8.9 1022 kg, and its radius is 1 820 km.26AP 27AP Why is the following situation impossible? A spacecraft is launched into a circular orbit around the Earth and circles the Earth once an hour.Let gM represent the difference in the gravitational fields produced by the Moon at the points on the Earths surface nearest to and farthest from the Moon. Find the fraction gM/g, where g is the Earths gravitational field. (This difference is responsible for the occurrence of the lunar tides on the Earth.)30AP 31AP 32AP 33AP Two spheres having masses M and 2M and radii R and 3R, respectively, are simultaneously released from rest when the distance between their centers is 12R. Assume the two spheres interact only with each other and we wish to find the speeds with which they collide. (a) What two isolated system models are appropriate for this system? (b) Write an equation from one of the models and solve it for v1, the velocity of the sphere of mass M at any time after release in terms of v2, the velocity of 2M. (c) Write an equation from the other model and solve it for speed v1 in terms of speed v2 when the spheres collide. (d) Combine the two equations to find the two speeds v1 and v2 when the spheres collide.(a) Show that the rate of change of the free-fall acceleration with vertical position near the Earths surface is dgdr=2GMERE3 This rate of change with position is called a gradient. (b) Assuming h is small in comparison to the radius of the Earth, show that the difference in free-fall acceleration between two points separated by vertical distance h is g=2GMEhRE3 (c) Evaluate this difference for h = 6.00 m, a typical height for a two-story building.36AP Studies of the relationship of the Sun to our galaxythe Milky Wayhave revealed that the Sun is located near the outer edge of the galactic disc, about 30 000 ly (1 ly = 9.46 1015 m) from the center. The Sun has an orbital speed of approximately 250 km/s around the galactic center. (a) What is the period of the Suns galactic motion? (b) What is the order of magnitude of the mass of the Milky Way galaxy? (c) Suppose the galaxy is made mostly of stars of which the Sun is typical. What is the order of magnitude of the number of stars in the Milky Way?Review. Two identical hard spheres, each of mass m and radius r, are released from rest in otherwise empty space with their centers separated by the distance R. They are allowed to collide under the influence of their gravitational attraction. (a) Show that the magnitude of the impulse received by each sphere before they make contact is given by [Gm3(1/2r 1/R)1/2. (b) What If? Find the magnitude of the impulse each receives during their contact if they collide elastically.39AP 40AP 41AP 42AP As thermonuclear fusion proceeds in its core, the Sun loses mass at a rate of 3.64 109 kg/s. During the 5 000-yr period of recorded history, by how much has the length of the year changed due to the loss of mass from the Sun? Suggestions: Assume the Earths orbit is circular. No external torque acts on the EarthSun system, so the angular momentum of the Earth is constant.Two stars of masses M and m, separated by a distance d, revolve in circular orbits about their center of mass (Fig. P13.44). Show that each star has a period given by T2=42d3G(M+m) Figure P13.44 The Solar and Heliospheric Observatory (SOHO) spacecraft has a special orbit, located between the Earth and the Sun along the line joining them, and it is always close enough to the Earth to transmit data easily. Both objects exert gravitational forces on the observatory. It moves around the Sun in a near-circular orbit that is smaller than the Earths circular orbit. Its period, however, is not less than 1 yr but just equal to 1 yr. Show that its distance from the Earth must be 1.48 109 m. In 1772, Joseph Louis Lagrange determined theoretically the special location allowing this orbit. Suggestions: Use data that are precise to four digits. The mass of the Earth is 5.974 1024 kg. You will not be able to easily solve the equation you generate; instead, use a computer to verify that 1.48 109 m is the correct value.Suppose you are standing directly behind someone who steps back and accidentally stomps on your foot with the heel of one shoe. Would you be better off if that person were (a) a large, male professional basketball player wearing sneakers or (b) a petite woman wearing spike-heeled shoes?The pressure at the bottom of a filled glass of water ( = 1 000 kg/m3) is P. The water is poured out, and the glass is filled with ethyl alcohol ( = 806 kg/m3). What is the pressure at the bottom of the glass? (a) smaller than P (b) equal to P (c) larger than P (d) indeterminate Several common barometers are built, with a variety of fluids. For which of the following fluids will the column of fluid in the barometer he the highest? (a) mercury (b) water (c) ethyl alcohol (d) benzene You are shipwrecked and floating in the middle of the ocean on a raft. Your cargo on the raft includes a treasure chest full of gold that you found before your ship sank, and the raft is just barely afloat. To keep you floating as high as possible in the water, should you (a) leave the treasure chest on top of the raft, (b) secure the treasure chest to the underside of the raft, or (c) hang the treasure chest in the water with a rope attached to the raft? (Assume throwing the treasure chest overboard is not an option you wish to consider.)You observe two helium balloons floating next to each other at the ends of strings secured to a table. The facing surfaces of the balloons are separated by 1-2 cm. You blow through the small space between the balloons. What happens to the balloons? (a) They move toward each other. (b) They move away from each other. (c) They are unaffected.A large man sits on a four-legged chair with his feet off the floor. The combined mass of the man and chair is 95.0 kg. If the chair legs are circular and have a radius of 0.500 cm at the bottom, what pressure does each leg exert on the floor?2P Estimate the total mass of the Earths atmosphere. (The radius of the Earth is 6.37 106 m, and atmospheric pressure at the surface is 1.013 105 Pa.)4P What must be the contact area between a suction cup (completely evacuated) and a ceiling if the cup is to support the weight of an 80.0-kg student?6P Review. A solid sphere of brass (bulk modulus of 14.0 1010 N/m2) with a diameter of 3.00 m is thrown into the ocean. By how much does the diameter of the sphere decrease as it sinks to a depth of 1.00 km?The human brain and spinal cord are immersed in the cerebrospinal fluid. The fluid is normally continuous between the cranial and spinal cavities and exerts a pressure of 100 to 200 mm of H2O above the prevailing atmospheric pressure. In medical work, pressures are often measured in units of millimeters of H2O because body fluids, including the cerebrospinal fluid, typically have the same density as water. The pressure of the cerebrospinal fluid can be measured by means of a spinal tap as illustrated in Figure P14.8. A hollow tube is inserted into the spinal column, and the height to which the fluid rises is observed. If the fluid rises to a height of 160 mm, we write its gauge pressure as 160 mm H2O. (a) Express this pressure in pascals, in atmospheres, and in millimeters of mercury. (b) Some conditions that block or inhibit the flow of cerebrospinal fluid can be investigated by means of Queckenstedts test. In this procedure, the veins in the patients neck are compressed to make the blood pressure rise in the brain, which in turn should be transmitted to the cerebrospinal fluid. Explain how the level of fluid in the spinal tap can be used as a diagnostic tool for the condition of the patients spine. Figure P14.8 Blaise Pascal duplicated Torricellis barometer using a red Bordeaux wine, of density 984 kg/m3, as the working liquid (Fig. P14.9). (a) What was the height h of the wine column for normal atmospheric pressure? (b) Would you expect the vacuum above the column to be as good as for mercury? Figure P14.9 A tank with a flat bottom of area A and vertical sides is filled to a depth h with water. The pressure is P0 at the top surface. (a) What is the absolute pressure at the bottom of the tank? (b) Suppose an object of mass M and density less than the density of water is placed into the tank and floats. No water overflows. What is the resulting increase in pressure at the bottom of the tank?11P A 10.0-kg block of metal measuring 12.0 cm by 10.0 cm by 10.0 cm is suspended from a scale and immersed in water as shown in Figure P14.11b. The 12.0-cm dimension is vertical, and the top of the block is 5.00 cm below the surface of the water. (a) What are the magnitudes of the forces acting on the top and on the bottom of the block due to the surrounding water? (b) What is the reading of the spring scale? (c) Show that the buoyant force equals the difference between the forces at the top and bottom of the block. Figure P14.11 Problems 11 and 12.A plastic sphere floats in water with 50.0% of its volume submerged. This same sphere floats in glycerin with 40.0% of its volume submerged. Determine the densities of (a) the glycerin and (b) the sphere.The weight of a rectangular block of low-density material is 15.0 N. With a thin string, the center of the horizontal bottom face of the block is tied to the bottom of a beaker partly filled with water. When 25.0% of the blocks volume is submerged, the tension in the string is 10.0 N. (a) Find the buoyant force on the block. (b) Oil of density 800 kg/m3 is now steadily added to the beaker, forming a layer above the water and surrounding the block. The oil exerts forces on each of the four sidewalls of the block that the oil touches. What are the directions of these forces? (c) What happens to the string tension as the oil is added? Explain how the oil has this effect on the string tension. (d) The string breaks when its tension reaches 60.0 N. At this moment, 25.0% of the blocks volume is still below the water line. What additional fraction of the blocks volume is below the top surface of the oil?A wooden block of volume 5.24 104 m3 floats in water, and a small steel object of mass m is placed on top of the block. When m = 0.310 kg, the system is in equilibrium and the top of the wooden block is at the level of the water. (a) What is the density of the wood? (b) What happens to the block when the steel object is replaced by an object whose mass is less than 0.310 kg? (c) What happens to the block when the steel object is replaced by an object whose mass is greater than 0.310 kg?A hydrometer is an instrument used to determine liquid density. A simple one is sketched in Figure P14.16. The bulb of a syringe is squeezed and released to let the atmosphere lift a sample of the liquid of interest into a tube containing a calibrated rod of known density. The rod, of length L and average density 0, floats partially immersed in the liquid of density . A length h of the rod protrudes above the surface of the liquid. Show that the density of the liquid is given by =0LLh Figure P14.16 Problems 16 and 17 Refer to Problem 16 and Figure P14.16. A hydrometer is to be constructed with a cylindrical floating rod. Nine fiduciary marks are to be placed along the rod to indicate densities of 0.98 g/cm3, 1.00 g/cm3, 1.02 g/cm3, 1.01 g/cm3, 1.14 g/cm3. The row of marks is to start 0.200 cm from the top end of the rod and end 1.80 cm from the top end. (a) What is the required length of the rod? (b) What must be its average density? (c) Should the marks be equally spaced? Explain your answer.On October 21, 2001, Ian Ashpole of the United Kingdom achieved a record altitude of 3.35 km (11 000 ft) powered by 600 toy balloons filled with helium. Each filled balloon had a radius of about 0.50 m and an estimated mass of 0.30 kg. (a) Estimate the total buoyant force on the 600 balloons. (b) Estimate the net upward force on all 600 balloons. (c) Ashpole parachuted to the Earth after the balloons began to burst at the high altitude and the buoyant force decreased. Why did the balloons burst?19P Water flowing through a garden hose of diameter 2.74 cm fills a 25-L bucket in 1.50 min. (a) What is the speed of the water leaving the end of the hose? (b) A nozzle is now attached to the end of the hose. If the nozzle diameter is one-third the diameter of the hose, what is the speed of the water leaving the nozzle?Water falls over a dam of height h with a mass flow rate of Iv, in units of kilograms per second. (a) Show that the power available from the water is P = Ivgh where g is the free-fall acceleration. (b) Each hydroelectric unit at the Grand Coulee Dam takes in water at a rate of 8.50 105 kg/s from a height of 87.0 m. The power developed by the falling water is converted to electric power with an efficiency of 85.0%. How much electric power does each hydroelectric unit produce?A legendary Dutch boy saved Holland by plugging a hole of diameter 1.20 cm in a dike with his finger. If the hole was 2.00 m below the surface of the North Sea (density 1 030 kg/m3). (a) what was the force on his finger? (b) If he pulled his finger out of the hole, during what time interval would the released water fill 1 acre of land to a depth of 1 ft? Assume the hole remained constant in size.Water is pumped up from the Colorado River to supply Grand Canyon Village, located on the rim of the canyon. The river is at an elevation of 564 m, and the village is at an elevation of 2 096 m. Imagine that the water is pumped through a single long pipe 15.0 cm in diameter, driven by a single pump at the bottom end. (a) What is the minimum pressure at which the water must be pumped if it is to arrive at the village? (b) If 4 500 m3 of water is pumped per day, what is the speed of the water in the pipe? Note: Assume the free-fall acceleration and the density of air are constant over this range of elevations. The pressures you calculate are too high for an ordinary pipe. The water is actually lifted in stages by several pumps through shorter pipes.In ideal flow, a liquid of density 850 kg/m3 moves from a horizontal tube of radius 1.00 cm into a second horizontal tube of radius 0.500 cm at the same elevation as the first tube. The pressure differs by P between the liquid in one tube and the liquid in the second tube. (a) Find the volume flow rate as a function of P. Evaluate the volume flow rate for (b) P = 6.00 kPa and (c) P = 12.0 kPa.Review. Old Faithful Geyser in Yellowstone National Park erupts at approximately one-hour intervals, and the height of the water column reaches 40.0 m (Fig. P14.25). (a) Model the rising stream as a series of separate droplets. Analyze the free-fall motion of one of the droplets to determine the speed at which the water leaves the ground. (b) What If? Model the rising stream as an ideal fluid in streamline flow. Use Bernoullis equation to determine the speed of the water as it leaves ground level. (c) How does the answer from part (a) compare with the answer from part (b)? (d) What is the pressure (above atmospheric) in the heated underground chamber if its depth is 175 m? Assume the chamber is large compared with the geysers vent. Figure P14.25 You are working as an expert witness for the owner of a skyscraper complex in a downtown area. The owner is being sued by pedestrians on the streets below his buildings who were injured by falling glass when windows popped outward from the sides of the building. The Bernoulli effect can have important consequences for windows in such buildings. For example, wind can blow around a skyscraper at remarkably high speed, creating low pressure on the outside surface of the windows. The higher atmospheric pressure in the still air inside the buildings can cause windows to pop out. (a) In your research into the case, you find some overhead views of your clients project, as shown below. The project includes two tall skyscrapers and some park area on a square plot. Plan (i) (Fig. P14.26(i), page 382) was submitted by the original architects and planners. At the last minute, the owner decided he didnt want the park grounds to be divided into two areas and submitted Plan (ii) (Fig. P14.26(ii), which is the way the project was built. Explain to your client why Plan (ii) is a much more dangerous situation in terms of windows popping out than Plan (i). (b) Your client is not convinced by your conceptual argument in part (a), so you provide a numerical argument. Suppose a horizontal wind blows with a speed of 11.2 m/s outside a large pane of plate glass with dimensions 4.00 m 1.50 m. Assume the density of the air to be constant at 1.20 kg/ m3. The air inside the building is at atmospheric pressure. Calculate the total force exerted by air on the windowpane for your client. (c) What If? To further convince your client of the problems with the building design, calculate the total force exerted by air on the windowpane if the wind speed between the buildings is 22.4 m/s, twice as high as in part (b). Figure P14.26 A thin 1.50-mm coating of glycerin has been placed between two microscope slides of width 1.00 cm and length 4.00 cm. Find the force required to pull one of the microscope slides at a constant speed of 0.300 m/s relative to the other slide.A hypodermic needle is 3.00 cm in length and 0.300 mm in diameter. What pressure difference between the input and output of the needle is required so that the flow rate of water through it will be 1.00 g/s? (Use 1.00 103 Pa s as the viscosity of water.)What radius needle should be used to inject a volume of 500 cm3 of a solution into a patient in 30.0 min? Assume the length of the needle is 2.50 cm and the solution is elevated 1.00 m above the point of injection. Further, assume the viscosity and density of the solution are those of pure water, and that the pressure inside the vein is atmospheric.An airplane has a mass of 1.60 104 kg, and each wing has an area of 40.0 m2. During level flight, the pressure on the lower wing surface is 7.00 104 Pa. (a) Suppose the lift on the airplane were due to a pressure difference alone. Determine the pressure on the upper wing surface. (b) More realistically, a significant part of the lift is due to deflection of air downward by the wing. Does the inclusion of this force mean that the pressure in part (a) is higher or lower? Explain.31P Decades ago, it was thought that huge herbivorous dinosaurs such as Apatosaurus and Brachiosaurus habitually walked on the bottom of lakes, extending their long necks up to the surface to breathe. Brarhiosaurus had its nostrils on the top of its head. In 1977, Knut Schmidt-Nielsen pointed out that breathing would be too much work for such a creature. For a simple model, consider a sample consisting of 10.0 L of air at absolute pressure 2.00 atm, with density 2.40 kg/m3, located at the surface of a freshwater lake. Find the work required to transport it to a depth of 10.3 m, with its temperature, volume, and pressure remaining constant. This energy investment is greater than the energy that can be obtained by metabolism of food with the oxygen in that quantity of air.33AP The true weight of an object can be measured in a vacuum, where buoyant forces are absent. A measurement in air, however, is disturbed by buoyant forces. An object of volume V is weighed in air on an equal-arm balance with the use of counterweights of density . Representing the density of air as air and the balance reading as Fg, show that the true weight Fg is Fg=Fg+(VFgg)airg 35AP Review. Assume a certain liquid, with density 1 230 kg/m3, exerts no friction force on spherical objects. A ball of mass 2.10 kg and radius 9.00 cm is dropped from rest into a deep tank of this liquid from a height of 3.30 m above the surface. (a) Find the speed at which the hall enters the liquid. (b) Evaluate the magnitudes of the two forces that are exerted on the ball as it moves through the liquid. (c) Explain why the ball moves down only a limited distance into the liquid and calculate this distance. (d) With what speed will the ball pop up out of the liquid? (c) How does the time interval tdown, during which the ball moves from the surface down to its lowest point, compare with the lime interval tup for the return trip between the same two points? (f) What If? Now modify the model to suppose the liquid exerts a small friction force on the ball, opposite in direction to its motion. In this case, how do the time intervals tdown and tup compare? Explain your answer with a conceptual argument rather than a numerical calculation.Evangelista Torricelli was the first person to realize that we live at the bottom of an ocean of air. He correctly surmised that the pressure of our atmosphere is attributable to the weight of the air. The density of air at 0C at the Earths surface is 1.29 kg/m3. The density decreases with increasing altitude (as the atmosphere thins). On the other hand, if we assume the density is constant at 1.29 kg/m3 up to some altitude h and is zero above that altitude, then h would represent the depth of the ocean of air. (a) Use this model to determine the value of h that gives a pressure of 1.00 atm at the surface of the Earth. (b) Would the peak of Mount Everest rise above the surface of such an atmosphere?A common parameter that can be used to predict turbulence in fluid flow is called the Reynolds number. The Reynolds number for fluid flow in a pipe is a dimensionless quantity defined as Re=vd where is the density of the fluid, v is its speed, d is the inner diameter of the pipe, and is the viscosity of the fluid. The criteria for the type of flow are as follows: It Re 2 300, the flow is laminar. If 2 300 Re 4 000, the flow is in a transition region between laminar and turbulent. If Re 4 000, the flow is turbulent. (a) Lets model Mood of density 1.06 103 kg/m3 and viscosity 3.00 103 Pa s as a pure liquid, that is, ignore the fact that it contains red blood cells. Suppose it is flowing in a large artery of radius 1.50 cm with a speed of 0.067 0 m/s. Show that the flow is laminar. (b) Imagine that the artery ends in a single capillary so that the radius of the artery reduces to a much smaller value. What is the radius of the capillary that would cause the flow to become turbulent? (c) Actual capillaries have radii of about 510 micrometers, much smaller than the value in part (b). Why doesnt the flow in actual capillaries become turbulent?In 1983, the United States began coining the one-cent piece out of copper-clad zinc rather than pure copper. The mass of the old copper penny is 3.083 g and that of the new cent is 2.517 g. The density of copper is 8.920 g/cm3 and that of zinc is 7.133 g/cm3. The new and old coins have the same volume. Calculate the percent of zinc (by volume) in the new cent.Review. With reference to the dam studied in Example 14.4 and shown in Figure 14.5, (a) show that the total torque exerted by the water behind the dam about a horizontal axis through O is 16gwH3. (b) Show that the effective line of action of the total force exerted by the water Ls at a distance 13Habove O.The spirit-in-glass thermometer, invented in Florence, Italy, around 1654, consists of a tube of liquid (the spirit) containing a number of submerged glass spheres with slightly different masses (Fig. P14.41). At sufficiently low temperatures, all the spheres float, but as the temperature rises, the spheres sink one after another. The device is a crude but interesting tool for measuring temperature. Suppose the tube is filled with ethyl alcohol, whose density is 0.789 45 g/cm3 at 20.0C and decreases to 0.780 97 g/cm3 at 30.0C. (a) Assuming that one of the spheres has a radius of 1.000 cm and is in equilibrium halfway up the tube at 20.0C, determine its mass. (b) When the temperature increases to 30.0C, what mass must a second sphere of the same radius have to be in equilibrium at the halfway point? (c) At 30.0C, the first sphere has fallen to the bottom of the tube. What upward force does the bottom of the tube exert on this sphere? Figure P14.41 A woman is draining her fish tank by siphoning the water into an outdoor drain as shown in Figure P14.42. The rectangular tank has footprint area A and depth h. The drain is located a distance d below the surface of the water in the tank, where d h. The crosssectional area of the siphon tube is A. Model the water as flowing without friction. Show that the time interval required to empty the tank is given by t=AhA2gd Figure P14.42 43AP 44AP 45AP Review. In a water pistol, a piston drives water through a large tube of area A1 into a smaller tube of area A2 as shown in Figure P14.46. The radius of the large tube is 1.00 cm and that of the small tube is 1.00 mm. The smaller tube is 3.00 cm above the larger tube. (a) If the pistol is fired horizontally at a height of 1.50 m, determine the time interval required for the water to travel from the nozzle to the ground. Neglect air resistance and assume atmospheric pressure is 1.00 atm. (b) If the desired range of the stream is 8.00 m, with what speed v2 must the stream leave the nozzle? (c) At what speed v1 must the plunger be moved to achieve the desired range? (d) What is the pressure at the nozzle? (e) Find the pressure needed in the larger tube. (f) Calculate the force that must be exerted on the trigger to achieve the desired range. (The force that must be exerted is due to pressure over and above atmospheric pressure.) Figure P14.46 47AP The hull of an experimental boat is to be lifted above the water by a hydrofoil mounted below its keel as shown in Figure P14.48. The hydrofoil has a shape like that of an airplane wing. Its area projected onto a horizontal surface is A. When the boat is towed at sufficiently high speed, water of density moves in streamline flow so that its average speed at the top of the hydrofoil is n times larger than its speed vb below the hydrofoil. (a) Ignoring the buoyant force, show that the upward lift force exerted by the water on the hydrofoil has a magnitude F=12(n21)vb2A (b) The boat has mass M. Show that the liftoff speed is given by v=2Mg(n21)A Figure P14.4 8 Show that the variation of atmospheric pressure with altitude is given by P = P0ey where = 0g/P0, P0 is atmospheric pressure at some reference level y = 0, and 0 is the atmospheric density at this level. Assume the decrease in atmospheric pressure over an infinitesimal change in altitude (so that the density is approximately uniform over the infinitesimal change) can be expressed from Equation 14.4 as dP = pg dy. Also assume the density of air is proportional to the pressure, which, as we will see in Chapter 18, is equivalent to assuming the temperature of the air is the same at all altitudes.Why is the following situation impossible? A barge is carrying a load of small pieces of iron along a river. The iron pile is in the shape of a cone for which the radius r of the base of the cone is equal to the central height h of the cone. The barge is square in shape, with vertical sides of length 2r, so that the pile of iron comes just up to the edges of the barge. The barge approaches a low bridge, and the captain realizes that the top of the pile of iron is not going to make it under the bridge. The captain orders the crew to shovel iron pieces from the pile into the water to reduce the height of the pile. As iron is shoveled from the pile, the pile always has the shape of a cone whose diameter is equal to the side length of the barge. After a certain volume of iron is removed from the barge, it makes it under the bridge without the top of the pile striking the bridge.A block on the end of a spring is pulled to position x = A and released from rest. In one full cycle of its motion, through what total distance does it travel? (a) A/2 (b) A (c) 2A (d) 4A Consider a graphical representation (Fig. 15.3) of simple harmonic motion as described mathematically in Equation 15.6. When the particle is at point on the graph, what can you say about its position and velocity? (a) The position and velocity are both positive. (b) The position and velocity are both negative. (c) The position is positive, and the velocity is zero. (d) The position is negative, and the velocity is zero. (e) The position is positive, and the velocity is negative. (f) The position is negative, and the velocity is positive.Figure 15.4 shows two curves representing particles undergoing simple harmonic motion. The correct description of these two motions is that the simple harmonic motion of particle B is (a) of larger angular frequency and larger amplitude than that of particle A, (b) of larger angular frequency and smaller amplitude than that of particle A, (c) of smaller angular frequency and larger amplitude than that of particle A, or (d) of smaller angular frequency and smaller amplitude than that of particle A.An object of mass m is hung from a spring and set into oscillation. The period of the oscillation is measured and recorded as T. The object of mass m is removed and replaced with an object of mass 2m. When this object is set into oscillation, what is the period of the motion? (a) 2T (b) 2T (c) T (d) T/2 (e) T/2 The ball in Figure 15.13 moves in a circle of radius 0.50 m. At t = 0, the ball is located on the left side of the turntable, exactly opposite its position in Figure 15.13. What are the correct values for the amplitude and phase constant (relative to an x axis to the right) of the simple harmonic motion of the shadow? (a) 0.50 m and 0 (b) 1.00 m and 0 (c) 0.50 m and (d) 1.00 m and Figure 15.13 An experimental setup for demonstrating the connection between a particle in simple harmonic motion and a corresponding particle in uniform circular motion.The grandfather clock in the opening storyline depends on the period of a pendulum to keep correct time. (i) Suppose the clock is calibrated correctly and then a mischievous child slides the bob of the pendulum downward on the oscillating rod. Does the grandfather clock run (a) slow, (b) fast, or (c) correctly? (ii) Suppose a grandfather clock is calibrated correctly at sea level and is then taken to the top of a very tall mountain. Does the grandfather clock now run (a) slow, (b) fast, or (c) correctly?A 0.60-kg block attached to a spring with force constant 130 N/m is free to move on a frictionless, horizontal surface as in Figure 15.1. The block is released from rest when the spring is stretched 0.13 m. At the instant the block is released, find (a) the force on the block and (b) its acceleration. Figure 15.1 A block attached to a spring and moving on a frictionless surface.A piston in a gasoline engine is in simple harmonic motion. The engine is running at the rate of 3 600 rev min. Taking the extremes of its position relative to its center point as 5.00 cm, find the magnitudes of the (a) maximum velocity and (b) maximum acceleration of the piston.The position of a particle is given by the expression x = 4.00 cos {3.00 t + }, where x is in meters and t is in seconds. Determine (a) the frequency and (b) period of the motion, (c) the amplitude of the motion, (d) the phase constant, and (e) the position of the particle at t = 0.250 s.A 7.00-kg object is hung from the bottom end of a vertical spring fastened to an overhead beam. The object is set into vertical oscillations having a period of 2.60 s. Find the force constant of the spring.Review. A particle moves along the x axis. It is initially at the position 0.270 m, moving with velocity 0.140 m/s and acceleration 0.320 m/s2. Suppose it moves as a particle under constant acceleration for 4.50 s. Find (a) its position and (b) its velocity at the end of this time interval. Next, assume it moves as a particle in simple harmonic motion for 4.50 s and x = 0 is its equilibrium position. Find (c) its position and (d) its velocity at the end of this time interval.A ball dropped from a height of 4.00 m makes an elastic collision with the ground. Assuming no decrease in mechanical energy due to air resistance, (a) show that the ensuing motion is periodic and (b) determine the period of the motion. (c) Is the motion simple harmonic? Explain.A particle moving along the x axis in simple harmonic motion starts from its equilibrium position, the origin, at t = 0 and moves to the right. The amplitude of its motion is 2.00 cm, and the frequency is 1.50 Hz. (a) Find an expression for the position of the particle as a function of time. Determine (b) the maximum speed of the particle and (c) the earliest time (t 0) at which the panicle has this speed. Find (d) the maximum positive acceleration of the particle and (e) the earliest time (t 0) at which the particle has this acceleration. (f) Find the total distance traveled by the particle between t = 0 and t = 1.00 s.The initial position, velocity, and acceleration of an object moving in simple harmonic motion are xi vi and ai; the angular frequency of oscillation is . (a) Show that the position and velocity of the object for all time can be written as x(t)=xicost+(vi)sintv(t)=xisint+vicost (b) Using A to represent the amplitude of the motion, show that v2ax=vi2aixi=2A2 You attach an object to the bottom end of a hanging vertical spring. It hangs at rest alter extending the spring 18.3 cm. You then set the object vibrating. (a) Do you have enough information to find its period? (b) Explain your answer and state whatever you can about its period.To test the resiliency of its bumper during low-speed collisions, a 1 000-kg automobile is driven into a brick wall. The cars bumper behaves like a spring with a force constant 5.00 106 N/m and compresses 3.16 cm as the car is brought to rest. What was the speed of the car before impact, assuming no mechanical energy is transformed or transferred away during impact with the wall?A particle executes simple harmonic motion with an amplitude of 3.00 cm. At what position does its speed equal half of its maximum speed?The amplitude of a system moving in simple harmonic motion is doubled. Determine the change in (a) the total energy, (b) the maximum speed, (c) the maximum acceleration, and (d) the period.A simple harmonic oscillator of amplitude A has a total energy Determine E, (a) the kinetic energy and (b) the potential energy when the position is one-third the amplitude. (c) For what values of the position does the kinetic energy equal one-half the potential energy? (d) Are there any values of the position where the kinetic energy is greater than the maximum potential energy? Explain.Review. A 65.0-kg bungee jumper steps off a bridge with a light bungee cord tied to her body and to the bridge. The outstretched length of the cord is 11.0 m. The jumper reaches the bottom of her motion 36.0 m below the bridge before bouncing back. We wish to find the time interval between her leaving the bridge and her arriving at the bottom of her motion. Her overall motion can be separated into an 11.0-m tree fall and a 25.0-m section of simple harmonic oscillation. (a) For the free-fall part, what is the appropriate analysis model to describe her motion? (b) For what time interval is she in free fall? (c) For the simple harmonic oscillation part of the plunge, is the system of the bungee jumper, the spring, and the Earth isolated or nonisolated? (d) From your response in part (c) find the spring constant of the bungee cord. (c) What is the location of the equilibrium point where the spring force balances the gravitational force exerted on the jumper? (f) What is the angular frequency of the oscillation? (g) What time interval is required for the cord to stretch by 25.0 m? (h) What is the total time interval for the entire 36.0-m drop?Review. A 0.250-kg block resting on a frictionless, horizontal surface is attached to a spring whose force constant is 83.8 N/m as in Figure P15.15. A horizontal force F causes the spring to stretch a distance of 5.46 cm from its equilibrium position. (a) Find the magnitude of F. (b) What is the total energy stored in the system when the spring is stretched? (c) Find the magnitude of the acceleration of the block just after the applied force is removed. (d) Find the speed of the block when it first reaches the equilibrium position. (e) If the surface is not frictionless but the block still reaches the equilibrium position, would your answer to part (d) be larger or smaller? (f) What other information would you need to know to find the actual answer to part (d) in this case? (g) What is the largest value of the coefficient of friction that would allow the block to reach the equilibrium position? Figure P15.15 While driving behind a car traveling at 3.00 m/s, you notice that one of the cars tires has a small hemispherical bump on its rim as shown in Figure P15.16. (a) Explain why the bump, from your viewpoint behind the car, executes simple harmonic motion. (b) If the radii of the cars tires are 0.300 m, what is the bumps period of oscillation? (c) What If? You hang a spring with spring constant k = 100 N/m from the rear view mirror of your car. What is the mass that needs to be hung from this spring to produce simple harmonic motion with the same period as the bump on the tire? (d) What would be the maximum speed of the hanging mass in your car if you initially pulled the mass down 8.00 cm beyond equilibrium before releasing it? Figure P15.16 A simple pendulum makes 120 complete oscillations in 3.00 min at a location where g = 9.80 m/s2. Find (a) the period of the pendulum and (b) its length.A particle of mass m slides without friction inside a hemispherical bowl of radius R. Show that if the particle starts from rest with a small displacement from equilibrium, it moves in simple harmonic motion with an angular frequency equal to that of a simple pendulum of length R. That is, =g/R.A physical pendulum in the form of a planar object moves in simple harmonic motion with a frequency of 0.450 Hz. The pendulum has a mass of 2.20 kg, and the pivot is located 0.350 m from the center of mass. Determine the moment of inertia of the pendulum about the pivot point.A physical pendulum in the form of a planar object moves in simple harmonic motion with a frequency f. The pendulum has a mass m, and the pivot is located a distance d from the center of mass. Determine the moment of inertia of the pendulum about the pivot point.21P Consider the physical pendulum of Figure 15.16. (a) Represent its moment of inertia about an axis passing through its center of mass and parallel to the axis passing through its pivot point as ICM. Show that its period is T=2ICM+md2mgd where d is the distance between the pivot point and the center of mass. (b) Show that the period has a minimum value when d satisfies Figure 15.16 A physical pendulum pivoted at O.A watch balance wheel (Fig. P15.25) has a period of oscillation of 0.250 s. The wheel is constructed so that its mass of 20.0 g is concentrated around a rim of radius 0.500 cm. What are (a) the wheels moment of inertia and (b) the torsion constant of the attached spring? Figure P15.23 Show that the time rate of change of mechanical energy for a damped, undriven oscillator is given by dE/dt = bv2 and hence is always negative. To do so, differentiate the expression for the mechanical energy of an oscillator, E=12mv2+12kx2, and use Equation 15.51.Show that Equation 15.32 is a solution of Equation 15.31 provided that b2 4 mk.As you enter a fine restaurant, you realize that you have accidentally brought a small electronic timer from home instead of your cell phone. In frustration, you drop the timer into a side pocket of your suit coat, not realizing that the timer is operating. The arm of your chair presses the light cloth of your coat against your body at one spot. Fabric with a length L hangs freely below that spot, with the timer at the bottom. At one point during your dinner, the timer goes off and a buzzer and a vibrator turn on and off with a frequency of 1.50 Hz. It makes the hanging part of your coat swing hack and forth with remarkably large amplitude, drawing everyones attention. Find the value of L.A 2.00-kg object attached to a spring moves without friction (b = 0) and is driven by an external force given by the expression F = 3.00 sin (2t), where F is in newtons and t is in seconds. The force constant of the spring is 20.0 N/m. Find (a) the resonance angular frequency of the system, (b) the angular frequency of the driven system, and (c) the amplitude of the motion.Considering an undamped, forced oscillator (b = 0), show that Equation 15.35 is a solution of Equation 15.34, with an amplitude given by Equation 15.36.29P 30P An object of mass m moves in simple harmonic motion with amplitude 12.0 cm on a light spring. Its maximum acceleration is 108 cm/s2. Regard m as a variable. (a) Find the period T of the object. (b) Find its frequency f. (c) Find the maximum speed vmax of the object. (d) Find the total energy E of the objectspring system. (e) Find the force constant k of the spring. (f) Describe the pattern of dependence of each of the quantities T, f, vmax, E, and k on m.Review. This problem extends the reasoning of Problem 41 in Chapter 9. Two gliders are set in motion on an air track. Glider 1 has mass m1 = 0.240 kg and moves to the right with speed 0.740 m/s. It will have a rear-end collision with glider 2, of mass m2 = 0.360 kg, which initially moves to the right with speed 0.120 m/s. A light spring of force constant 45.0 N/m is attached to the back end of glider 2 as shown in Figure P9.41. When glider 1 touches the spring, superglue instantly and permanently makes it stick to its end of the spring. (a) Find the common speed the two gliders have when the spring is at maximum compression. (b) Find the maximum spring compression distance. The motion after the gliders become attached consists of a combination of (1) the constant-velocity motion of the center of mass of the two-glider system found in part (a) and (2) simple harmonic motion of the gliders relative to the center of mass. (c) Find the energy of the center-of-mass motion. (d) Find the energy of the oscillation.An object attached to a spring vibrates with simple harmonic motion as described by Figure P15.33. For this motion, find (a) the amplitude, (b) the period, (c) the angular frequency, (d) the maximum speed, (e) the maximum acceleration, and (f) an equation for its position x as a function of time. Figure P15.33 Review. A rock rests on a concrete sidewalk. An earthquake strikes, making the ground move vertically in simple harmonic motion with a constant frequency of 2.40 Hz and with gradually increasing amplitude. (a) With what amplitude does the ground vibrate when the rock begins to lose contact with the sidewalk? Another rock is sitting on the concrete bottom of a swimming pool full of water. The earthquake produces only vertical motion, so the water does not slosh from side to side. (b) Present a convincing argument that when the ground vibrates with the amplitude found in part (a), the submerged rock also barely loses contact with the floor of the swimming pool.A pendulum of length L and mass M has a spring of force constant k connected to it at a distance h below its point of suspension (Fig. P15.55). Find the frequency of vibration of the system for small values of the amplitude (small ). Assume the vertical suspension rod of length L is rigid, but ignore its mass. Figure P15.35 To account for the walking speed of a bipedal or quadrupedal animal, model a leg that is not contacting the ground as a uniform rod of length , swinging as a physical pendulum through one-half of a cycle, in resonance. Let max represent its amplitude. (a) Show that the animals speed is given by the expression v=6glsinmax if max is sufficiently small that the motion is nearly simple harmonic. An empirical relationship that is based on the same model and applies over a wider range of angles is v=6glcos(max/2)sinmax (b) Evaluate the walking speed of a human with leg length 0.850 m and leg-swing amplitude 28.0. (c) What leg length would give twice the speed for the same angular amplitude?Review. A particle of mass 4.00 kg is attached to a spring with a force constant of 100 N/m. It is oscillating on a frictionless, horizontal surface with an amplitude of 2.00 m. A 6.00-kg object is dropped vertically on top of the 4.00-kg object as it passes through its equilibrium point. The two objects stick together. (a) What is the new amplitude of the vibrating system after the collision? (b) By what factor has the period of the system changed? (c) By how much does the energy of the system change as a result of the collision? (d) Account for the change in energy.People who ride motorcycles and bicycles learn to look out for bumps in the road and especially for washboarding, a condition in which many equally spaced ridges are worn into the road. What is so bad about washboarding? A motorcycle has several springs and shock absorbers in its suspension, but you can model it as a single spring supporting a block. You can estimate the force constant by thinking about how far the spring compresses when a heavy rider sits on the scat. A motorcyclist traveling at highway speed must be particularly careful of washboard bumps that are a certain distance apart. What is the order of magnitude of their separation distance?A ball of mass m is connected to two rubber bands of length L, each under tension T as shown in Figure P15.39. The ball is displaced by a small distance y perpendicular to the length of the rubber bands. Assuming the tension does not change, show that (a) the restoring force is (2T/L)y and (b) the system exhibits simple harmonic motion with an angular frequency =2T/mL. Figure P15.39 Consider the damped oscillator illustrated in Figure 15.19. The mass of the object is 375 g, the spring constant is 100 N/m, and b = 0.100 N s/m. (a) Over what time interval does the amplitude drop to half its initial value? (b) What If? Over what time interval does the mechanical energy drop to half its initial value? (c) Show that, in general, the fractional rate at which the amplitude decreases in a damped harmonic oscillator is one-half the fractional rate at which the mechanical energy decreases.Review. A lobstermans buoy is a solid wooden cylinder of radius r and mass M. It is weighted at one end so that it floats upright in calm seawater, having density . A passing shark tugs on the slack rope mooring the buoy to a lobster trap, pulling the buoy down a distance x from its equilibrium position and releasing it. (a) Show that the buoy will execute simple harmonic motion if the resistive effects of the water are ignored. (b) Determine the period of the oscillations.Your thumb squeaks on a plate you have just washed. Your sneakers squeak on the gym floor. Car tires squeal when you start or stop abruptly. You can make a goblet sing by wiping your moistened finger around its rim. When chalk squeaks on a blackboard, you can see that it makes a row of regularly spaced dashes. As these examples suggest, vibration commonly results when friction acts on a moving elastic object. The oscillation is not simple harmonic motion, but is called stick-and-slip. This problem models stick-and-slip motion. A block of mass m is attached to a fixed support by a horizontal spring with force constant k and negligible mass (Fig. P15.42). Hookes law describes the spring both in extension and in compression. The block sits on a long horizontal board, with which it has coefficient of static friction k and a smaller coefficient of kinetic friction k. The board moves to the right at constant speed v. Assume the block spends most of its time sticking to the board and moving to the right with it, so the speed v is small in comparison to the average speed the block has as it slips back toward the left. (a) Show that the maximum extension of the spring from its unstressed position is very nearly given by s mg/k. (b) Show that the block oscillates around an equilibrium position at which the spring is stretched by k mg/k. (c) Graph the blocks position versus time. (d) Show that the amplitude of the blocks motion is A=(sk)mgk Figure P15.42 (e) Show that the period of the blocks motion is T=2(sk)mgvk+mk It is the excess of static over kinetic friction that is important for the vibration. The squeaky wheel gets the grease because even a viscous fluid cannot exert a force of static friction.43AP 44AP A block of mass m is connected to two springs of force constants k1 and k2 in two ways as shown in Figure P15.45. In both cases, the block moves on a frictionless table after it is displaced from equilibrium and released. Show that in the two cases the block exhibits simple harmonic motion with periods (a)T=2m(k1+k2)k1k2and(b)T=2mk1+k2 Figure P15.45 Review. A light balloon filled with helium of density 0.179 kg/m3 is tied to a light string of length L = 3.00 m. The string is tied to the ground forming an inverted simple pendulum (Fig. P15.46a). If the balloon is displaced slightly from equilibrium as in Figure P15.46b and released, (a) show that the motion is simple harmonic and (b) determine the period of the motion. Take the density of air to be 1.20 kg/m3. Hint: Use an analogy with the simple pendulum and see Chapter 14. Assume the air applies a buoyant force on the balloon but does not otherwise affect its motion. Figure P15.46 A particle with a mass of 0.500 kg is attached to a horizontal spring with a force constant of 50.0 N/m. At the moment t = 0, the particle has its maximum speed of 20.0 m/s and is moving to the left. (a) Determine the particles equation of motion, specifying its position as a function of time. (b) Where in the motion is the potential energy three times the kinetic energy? (c) Find the minimum time interval required for the particle to move from x = 0 to x = 1.0 m. (d) Find the length of a simple pendulum with the same period.A smaller disk of radius r and mass m is attached rigidly to the face of a second larger disk of radius R and mass M as shown in Figure P15.48. The center of the small disk is located at the edge of the large disk. The large disk is mounted at its center on a frictionless axle. The assembly is rotated through a small angle from its equilibrium position and released. (a) Show that the speed of the center of the small disk as it passes through the equilibrium position is v=2[Rg(1cos)(M/m)+(r/R)2+2]1/2 (b) Show that the period of the motion is v=2[(M/2m)+R2+mr22mgR]1/2 Figure P15.48 Review. A system consists of a spring with force constant k = 1 250 N/m, length L = 1.50 m, and an object of mass m = 5.00 kg attached to the end (Fig. P15.49). The object is placed at the level of the point of attachment with the spring unstretched, at position yi = L, and then it is released so that it swings like a pendulum. (a) Find the y position of the object at the lowest point. (b) Will the pendulums period be greater or less than the period of a simple pendulum with the same mass m and length L? Explain. Figure PI 5.49 Review. Why is the following situation impassible? You are in the high-speed package delivers business. Your competitor in the next building gains the right-of-way to build an evacuated tunnel just above the ground all the way around the Earth. By firing packages into this tunnel at just the right speed, your competitor is able to send the packages into orbit around the Earth in this tunnel so that they arrive on the exact opposite side of the Earth in a very short time interval. You come up with a competing idea. Figuring that the distance through the Earth is shorter than the distance around the Earth, you obtain permits to build an evacuated tunnel through the center of the Earth (Fig. P15.50). By simply dropping packages into this tunnel, they fall downward and arrive at the other end of your tunnel, which is in a building right next to the other end of your competitors tunnel. Because your packages arrive on the other side of the Earth in a shorter time interval, you win the competition and your business flourishes. Note: An object at a distance r from the center of the Earth is pulled toward the center of the Earth only by the mass within the sphere of radius r (the reddish region in Fig. P15.50). Assume the Earth has uniform density. Figure P15.50 A light, cubical container of volume a3 is initially filled with a liquid of mass density as shown in Figure P15.5la. The cube is initially supported by a light string to form a simple pendulum of length Li, measured from the center of mass of the filled container, where Li a. The liquid is allowed to flow from the bottom of the container at a constant rate (dM/dt). At any time t, the level of the liquid in the container is h and the length of the pendulum is L. (measured relative to the instantaneous center of mass) as shown in Figure P15.51b. (a) Find the period of the pendulum as a function of time. (b) What is the period of the pendulum after the liquid completely runs out of the container? Figure P15.51 16.1QQ A sinusoidal wave of frequency f is traveling along a stretched string. The string is brought to rest, and a second traveling wave of frequency 2f is established on the string. (i) What is the wave speed of the second wave? (a) twice that of the first wave (b) half that of the first wave (c) the same as that of the first wave (d) impossible to determine (ii) From the same choices, describe the wavelength of the second wave. (iii) From the same choices, describe the amplitude of the second wave.The amplitude of a wave is doubled, with no other changes made to the wave. As a result of this doubling, which of the following statements is correct? (a) The speed of the wave changes. (b) The frequency of the wave changes. (c) The maximum transverse speed of an element of the medium changes. (d) Statements (a) through (c) are all true. (e) None of statements (a) through (c) is true.Suppose you create a pulse by moving the free end of a taut string up and down once with your hand beginning at t = 0. The string is attached at its other end to a distant wall. The pulse reaches the wall at time t. Which of the following actions, taken by itself, decreases the time interval required for the pulse to reach the wall? More than one choice may be correct. (a) moving your hand more quickly, but still only up and down once by the same amount (b) moving your hand more slowly, but still only up and down once by the same amount (c) moving your hand a greater distance up and down in the same amount of time (d) moving your hand a lesser distance up and down in the same amount of time (e) using a heavier string of the same length and under the same tension (f) using a lighter string of the same length and under the same tension (g) using a string of the same linear mass density but under decreased tension (h) using a string of the same linear mass density but under increased tension Which of the following, taken by itself, would be most effective in increasing the rate at which energy is transferred by a wave traveling along a string? (a) reducing the linear mass density of the string by one-half (b) doubling the wavelength of the wave (c) doubling the tension in the string (d) doubling the amplitude of the wave If you blow across the top of an empty soft-drink bottle, a pulse of sound travels down through the air in the bottle. At the moment the pulse reaches the bottom of the bottle, what is the correct description of the displacement of elements of air from their equilibrium positions and the pressure of the air at this point? (a) The displacement and pressure are both at a maximum. (b) The displacement and pressure are both at a minimum. (c) The displacement is zero, and the pressure is a maximum. (d) The displacement is zero, and the pressure is a minimum.A vibrating guitar string makes very little sound if it is not mounted on the guitar body. Why does the sound have greater intensity if the string is attached to the guitar body? (a) The string vibrates with more energy. (b) The energy leaves the guitar more rapidly. (c) The sound power is spread over a larger area at the listeners position. (d) The sound power is concentrated over a smaller area at the listeners position. (e) The speed of sound is higher in the material of the guitar body. (f) None of these answers is correct.Increasing the intensity of a sound by a factor of 100 causes the sound level to increase by what amount? (a) 100 dB (b) 20 dB (c) 10 dB (d) 2dB Consider detectors of water waves at three locations A, B, and C in Figure 16.25b. Which of the following statements is true? (a) The wave speed is highest at location A. (b) The wave speed is highest at location C. (c) The detected wavelength is largest at location B. (d) The detected wavelength is largest at location C. (e) The detected frequency is highest at location C. (f) The detected frequency is highest at location A.You stand on a platform at a train station and listen to a train approaching the station at a constant velocity. While the train approaches, but before it arrives, what do you hear? (a) the intensity and the frequency of the sound both increasing (b) the intensity and the frequency of the sound both decreasing (c) the intensity increasing and the frequency decreasing (d) the intensity decreasing and the frequency increasing (e) the intensity increasing and the frequency remaining the same (f) the intensity decreasing and the frequency remaining the same An airplane flying with a constant velocity moves from a cold air mass into a warm air mass. Does the Mach number (a) increase, (b) decrease, or (c) stay the same?A seismographic station receives S and P waves from an earthquake, separated in time by 17.3 s. Assume the waves have traveled over the same path at speeds of 4.30 km/s and 7.80 km/s. Find the distance from the seismograph to the focus of the quake.Two points A and B on the surface of the Earth are at the same longitude and 60.0 apart in latitude as shown in Figure P16.2. Suppose an earthquake at point A creates a P wave that reaches point B by traveling straight through the body of the Earth at a constant speed of 7.80 km/s. The earthquake also radiates a Rayleigh wave that travels at 4.50 km/s. In addition to P and S waves. Rayleigh waves are a third type of seismic wave that travels along the surface of the Earth rather than through the bulk of the Earth. (a) Which of these two seismic waves arrives at B first? (b) What is the time difference between the arrivals of these two waves at B? Figure P16.2 You are working for a plumber who is laying very long sections of copper pipe for a large building project. He spends a lot of time measuring the lengths of the sections with a measuring tape. You suggest a faster way to measure the length. You know that the speed of a one-dimensional compressional wave traveling along a copper pipe is 3.56 km/s. You suggest that a worker give a sharp hammer blow at one end of the pipe. Using an oscilloscope app on your smartphone, you will measure the time interval t between the arrival of the two sound waves due to the blow: one through the 20.0C air and the other through the pipe. (a) To measure the length, you must derive an equation that relates the length L of the pipe numerically to the time interval t. (b) You measure a time interval of t = 127 ms between the arrivals of the pulses and, from this value, determine the length of the pipe. (c) Your smartphone app claims an accuracy of 1.0% in measuring time intervals. So you calculate by how many centimeters your calculation of the length might be in error.You are working on a senior project and are analyzing a human wave at a sports stadium such as that shown in Figure P16.4 (page 446). You are trying to determine the effect of the wave on concession sales because people are standing up and sitting down while they participate in the wave, instead of buying food or drinks. You have made observations at a local stadium and have taken data on one particularly stable wave. This wave took 47.4 s to travel around a specific stadium row consisting of a circular ring of 974 seats. You also find that a typical time interval for spectators to stand and sit back down is 0.95 s. In this wave, how many people in the specific row were out of their seats at any given instant? Figure P16.4 Problems 4 and 44.When a particular wire is vibrating with a frequency of 4.00 Hz, a transverse wave of wavelength 60.0 cm is produced. Determine the speed of waves along the wire.(a) Plot y versus t at x = 0 for a sinusoidal wave of the form y = 0.150 cos (15.7x 50.3t), where x and y are in meters and t is in seconds. (b) Determine the period of vibration. (c) State how your result compares with the value found in Example 16.2.Consider the sinusoidal wave of Example 16.2 with the wave function y=0.150cos(15.7x50.3t) where x and y are in meters and t is in seconds. At a certain instant, let point A be at the origin and point B be the closest point to A along the x axis where the wave is 60.0 out of phase with A. What is the coordinate of B?A sinusoidal wave traveling in the negative x direction (to the left) has an amplitude of 20.0 cm, a wavelength of 35.0 cm, and a frequency of 12.0 Hz. The transverse position of an element of the medium at t = 0, x = 0 is y = 3.00 cm, and the element has a positive velocity here. We wish to find an expression for the wave function describing this wave. (a) Sketch the wave at t = 0. (b) Find the angular wave number k from the wavelength. (c) Find the period T from the frequency. Find (d) the angular frequency and (e) the wave speed v. (f) From the information about t = 0, find the phase constant . (g) Write an expression for the wave function y(x, t).(a) Write the expression for y as a function of x and t in SI units for a sinusoidal wave traveling along a rope in the negative x direction with the following characteristics: A = 8.0 cm, = 80.0 cm, f = 3.00 Hz, and y(0, t) = 0 at t = 0. (b) What If? Write the expression for y as a function of x and t for the wave in part (a) assuming y(x, 0) = 0 at the point x = 10.0 cm.Review. The elastic limit of a steel wire is 2.70 108 Pa. What is the maximum speed at which transverse wave pulses can propagate along this wire without exceeding this stress? (The density of steel is 7.86 103 kg/m3.)Transverse waves travel with a speed of 20.0 m/s on a string under a tension of 6.00 N. What tension is required for a wave speed of 30.0 m/s on the same string?Why is the following situation impossible? An astronaut on the Moon is studying wave motion using the apparatus discussed in Example 16.3 and shown in Figure 16.12. He measures the time interval for pulses to travel along the horizontal wire. Assume the horizontal wire has a mass of 4.00 g and a length of 1.60 m and assume a 3.00-kg object is suspended from its extension around the pulley. The astronaut finds that a pulse requires 26.1 ms to traverse the length of the wire.Tension is maintained in a string as in Figure P16.13. The observed wave speed is v = 24.0 m/s when the suspended mass is m = 3.00 kg. (a) What is the mass per unit length of the string? (b) What is the wave speed when the suspended mass is m = 2.00 kg? Figure P16.13 Problems 13 and 43.14P Transverse waves are being generated on a rope under constant tension. By what factor is the required power increased or decreased if (a) the length of the rope is doubled and the angular frequency remains constant, (b) the amplitude is doubled and the angular frequency is halved, (c) both the wavelength and the amplitude are doubled, and (d) both the length of the rope and the wavelength are halved?In a region far from the epicenter of an earthquake, a seismic wave can be modeled as transporting energy in a single direction without absorption, just as a string wave does. Suppose the seismic wave moves from granite into mudfill with similar density but with a much smaller bulk modulus. Assume the speed of the wave gradually drops by a factor of 25.0, with negligible reflection of the wave. (a) Explain whether the amplitude of the ground shaking will increase or decrease. (b) Does it change by a predictable factor? (This phenomenon led to the collapse of part of the Nimitz Freeway in Oakland, California, during the Loma Prieta earthquake of 1989.)A long string carries a wave; a 6.00-m segment of the string contains four complete wavelengths and has a mass of 180 g. The string vibrates sinusoidally with a frequency of 50.0 Hz and a peak-to-valley displacement of 15.0 cm. (The peak-to-valley distance is the vertical distance from the farthest positive position to the farthest negative position.) (a) Write the function that describes this wave traveling in the positive x direction. (b) Determine the power being supplied to the string.A two-dimensional water wave spreads in circular ripples. Show that the amplitude A at a distance r from the initial disturbance is proportional to 1/r. Suggestion: Consider the energy carried by one outward-moving ripple.A horizontal string can transmit a maximum power P0 (without breaking) if a wave with amplitude A and angular frequency is traveling along it. To increase this maximum power, a student folds the string and uses this double string as a medium. Assuming the tension in the two strands together is the same as the original tension in the single string and the angular frequency of the wave remains the same, determine the maximum power that can be transmitted along the double string.20P Show that the wave function y = eb(x vt) is a solution of the linear wave equation (Eq. 16.27), where b is a constant.22P A sinusoidal sound wave moves through a medium and is described by the displacement wave function s(x,t)=2.00cos(15.7x858t) where s is in micrometers, x is in meters, and t is in seconds. Find (a) the amplitude, (b) the wavelength, and (c) the speed of this wave. (d) Determine the instantaneous displacement from equilibrium of the elements of the medium at the position x = 0.050 0 m at t = 3.00 ms. (c) Determine the maximum speed of the elements oscillatory motion.Earthquakes at fault lines in the Earths crust create seismic waves, which are longitudinal (P waves) or transverse (S waves). The P waves have a speed of about 7 km/s. Estimate the average bulk modulus of the Earths crust given that the density of rock is about 2 500 kg/m3.An experimenter wishes to generate in air a sound wave that has a displacement amplitude of 5.50 106 m. The pressure amplitude is to be limited to 0.840 Pa. What is the minimum wavelength the sound wave can have?A sound wave propagates in air at 27C with frequency 4.00 kHz. It passes through a region where the temperature gradually changes and then moves through air at 0C. Give numerical answers to the following questions to the extent possible and state your reasoning about what happens to the wave physically. (a) What happens to the speed of the wave? (b) What happens to its frequency? (c) What happens to its wavelength?27P A rescue plane flies horizontally at a constant speed searching for a disabled boat. When the plane is directly above the boat, the boats crew blows a loud horn. By the time the planes sound detector receives the horns sound, the plane has traveled a distance equal to half its altitude above the ocean. Assuming it takes the sound 2.00 s to reach the plane, determine (a) the speed of the plane and (b) its altitude.The speed of sound in air (in meters per second) depends on temperature according to the approximate expression v=331.5+0.607TC where TC is the Celsius temperature. In dry air, the temperature decreases about 1C for every 150-m rise in altitude. (a) Assume this change is constant up to an altitude of 9 000 m. What time interval is required for the sound from an airplane flying at 9 000 m to reach the ground on a day when the ground temperature is 30C? (b) What If? Compare your answer with the time interval required if the air were uniformly at 30C. Which time interval is longer?A sound wave moves down a cylinder as in Figure 16.17. Show that the pressure variation of the wave is described by P=smax2s2, where s = s(x, t) is given by Equation 16.28.The intensity of a sound wave at a fixed distance from a speaker vibrating at 1.00 kHz is 0.600 W/m2. (a) Determine the intensity that results if the frequency is increased to 2.50 kHz while a constant displacement amplitude is maintained. (b) Calculate the intensity if the frequency is reduced to 0.500 kHz and the displacement amplitude is doubled.The intensity of a sound wave at a fixed distance from a speaker vibrating at a frequency f is I. (a) Determine the intensity that results if the frequency is increased to f while a constant displacement amplitude is maintained. (b) Calculate the intensity if the frequency is reduced to f/2 and the displacement amplitude is doubled.The power output of a certain public-address speaker is 6.00 W. Suppose it broadcasts equally in all directions. (a) Within what distance from the speaker would the sound be painful to the ear? (b) At what distance from the speaker would the sound be barely audible?A fireworks rocket explodes at a height of 100 m above the ground. An observer on the ground directly under the explosion experiences an average sound intensity of 7.00 102 W/m2 for 0.200 s. (a) What is the total amount of energy transferred away from the explosion by sound? (b) What is the sound level (in decibels) heard by the observer?You are working at an open-air amphitheater, where rock concerts occur regularly. The venue has powerful loudspeakers mounted on 10.6-m-tall columns at various locations surrounding the audience. The loudspeakers emit sound uniformly in all directions. There are ladder steps sticking out from the columns, to help workers service the loudspeakers. Many times, audience members break through the protective fencing around the columns and climb upward on the columns to get a better view of the performers. The upcoming concert is by a group that states that several very-high-volume pulses of sound occur in their concerts, and these sounds are part of their artistic expression. The amphitheater owners are worried about people climbing the columns and being too close to the loudspeakers when these peak sounds are emitted. They do not want to be held responsible for injuries to audience members ears. Based on past performances of the group, you determine that the peak sound level is 150 dB measured 20.0 cm from the speakers on the columns. The owners ask you to determine the heights on the columns at which to mount impassable barricades to keep people from getting too close to the speakers and hearing sound above the threshold of pain.Why is the following situation impossible? It is early on a Saturday morning, and much to your displeasure your next-door neighbor starts mowing his lawn. As you try to get back to sleep, your next-door neighbor on the other side of your house also begins to mow the lawn with an identical mower the same distance away. This situation annoys you greatly because the total sound now has twice the loudness it had when only one neighbor was mowing.Show that the difference between decibel levels 1 and of 2 a sound is related to the ratio of the distances r1 and r2 from the sound source by 21=20log(r1r2)Submarine A travels horizontally at 11.0 m/s through ocean water. It emits a sonar signal of frequency f = 5.27 103 Hz in the forward direction. Submarine B is in front of submarine A and traveling at 3.00 m/s relative to the water in the same direction as submarine A. A crewman in submarine B uses his equipment to detect the sound waves (pings) from submarine A. We wish to determine what is heard by the crewman in submarine B. (a) An observer on which submarine detects a frequency f as described by Equation 16.46? (b) In Equation 16.46, should the sign of vs be positive or negative? (c) In Equation 16.46, should the sign of vo be positive or negative? (d) In Equation 16.46, what speed of sound should be used? (e) Find the frequency of the sound detected by the crewman on submarine B.39P Why is the following situation impossible? At the Summer Olympics, an athlete runs at a constant speed down a straight track while a spectator near the edge of the track blows a note on a horn with a fixed frequency. When the athlete passes the horn, she hears the frequency of the horn fall by the musical interval called a minor third. That is, the frequency she hears drops to five-sixths its original value.Review. A block with a speaker bolted to it is connected to a spring having spring constant k = 20.0 N/m and oscillates as shown in Figure P16.41. The total mass of the block and speaker is 5.00 kg, and the amplitude of this units motion is 0.500 m. The speaker emits sound waves of frequency 440 Hz. Determine (a) the highest and (b) the lowest frequencies heard by the person to the right of the speaker. (c) If the maximum sound level heard by the person is 60.0 dB when the speaker is at its closest distance d = 1.0 m from him, what is the minimum sound level heard by the observer? Figure P16.41 Problems 41 and 42 Review. A block with a speaker bolted to it is connected to a spring having spring constant k and oscillates as shown in Figure P16.41. The total mass of the block and speaker is m, and the amplitude of this units motion is A. The speaker emits sound waves of frequency f. Determine (a) the highest and (b) the lowest frequencies heard by the person to the right of the speaker. (c) If the maximum sound level heard by the person is when the speaker is at its closest distance d from him, what is the minimum sound level heard by the observer?A sinusoidal wave in a rope is described by the wave function y=0.20sin(0.75x+18t) where x and y are in meters and t is in seconds. The rope has a linear mass density of 0.250 kg/m. The tension in the rope is provided by an arrangement like the one illustrated in Figure P16.13. What is the mass of the suspended object?The wave is a particular type of pulse that can propagate through a large crowd gathered at a sports arena (Fig. P16.4). The elements of the medium are the spectators, with zero position corresponding to their being seated and maximum position corresponding to their standing and raising their arms. When a large fraction of the spectators participates in the wave motion, a somewhat stable pulse shape can develop. The wave speed depends on peoples reaction time, which is typically on the order of 0.1 s. Estimate the order of magnitude, in minutes, of the time interval required for such a pulse to make one circuit around a large sports stadium. State the quantities you measure or estimate and their values.Some studies suggest that the upper frequency limit of hearing is determined by the diameter of the eardrum. The diameter of the eardrum is approximately equal to half the wavelength of the sound wave at this upper limit. If the relationship holds exactly, what is the diameter of the eardrum of a person capable of hearing 20 000 Hz? (Assume a body temperature of 37.0C.)An undersea earthquake or a landslide can produce an ocean wave of short duration carrying great energy, called a tsunami. When its wavelength is large compared to the ocean depth d, the speed of a water wave is given approximately by v=gd. Assume an earthquake occurs all along a tectonic plate boundary running north to south and produces a straight tsunami wave crest moving everywhere to the west. (a) What physical quantity can you consider to be constant in the motion of any one wave crest? (b) Explain why the amplitude of the wave increases as the wave approaches shore. (c) If the wave has amplitude 1.80 m when its speed is 200 m/s, what will be its amplitude where the water is 9.00 m deep? (d) Explain why the amplitude at the shore should be expected to be still greater, but cannot be meaningfully predicted by your model.A sinusoidal wave in a string is described by the wave function y=0.150sin(0.800x50.0t) where x and y are in meters and t is in seconds. The mass per length of the string is 12.0 g/m. (a) Find the maximum transverse acceleration of an element of this string. (b) Determine the maximum transverse force on a 1.00-cm segment of the string. (c) State how the force found in part (b) compares with the tension in the string.A rope of total mass m and length L is suspended vertically. Analysis shows that for short transverse pulses, the waves above a short distance from the free end of the rope can be represented to a good approximation by the linear wave equation discussed in Section 16.5. Show that a transverse pulse travels the length of the rope in a time interval that is given approximately by t=2L/g. Suggestion: First find an expression for the wave speed at any point a distance x from the lower end by considering the ropes tension as resulting from the weight of the segment below that point.A wire of density is tapered so that its cross-sectional area varies with x according to A=1.00105x+1.00106 where A is in meters squared and x is in meters. The tension in the wire is T. (a) Derive a relationship for the speed of a wave as a function of position. (b) What If? Assume the wire is aluminum and is under a tension T = 24.0 N. Determine the waver speed at the origin and at x = 10.0 m.50AP 51AP A train whistle (f = 400 Hz) sounds higher or lower in frequency depending on whether it approaches or recedes. (a) Prove that the difference in frequency between the approaching and receding train whistle is f=2u/v1u2/v2f where u is the speed of the train and v is the speed of sound. (b) Calculate this difference for a train moving at a speed of 130 km/h. Take the speed of sound in air to be 340 m/s.Review. A 150-g glider moves at v1 = 2.30 m/s on an air track toward an originally stationary 200-g glider as shown in Figure P16.53. The gliders undergo a completely inelastic collision and latch together over a time interval of 7.00 ms. A student suggests roughly half the decrease in mechanical energy of the two-glider system is transferred to the environment by sound. Is this suggestion reasonable? To evaluate the idea, find the implied sound level at a position 0.800 m from the gliders. If the students idea is unreasonable, suggest a better idea. Figure P16 53 Consider the following wave function in SI units: P(r,t)=(25.0r)sin(1.36r2030t) Explain how this wave function can apply to a wave radiating from a small source, with r being the radial distance from the center of the source to any point outside the source. Give the most detailed description of the wave that you can. Include answers to such questions as the following and give representative values for any quantities that can be evaluated. (a) Does the wave move more toward the right or the left? (b) As it moves away from the source, what happens to its amplitude? (c) Its speed? (d) Its frequency? (e) Its wavelength? (f) Its power? (g) Its intensity?55AP 56AP A string on a musical instrument is held under tension T and extends from the point x = 0 to the point x = L. The string is overwound with wire in such a way that its mass per unit length (x) increases uniformly from o at x = 0 to L at x = L. (a) Find an expression for (x) as a function of x over the range 0 x L. (b) Find an expression for the time interval required for a transverse pulse to travel the length of the string.Assume an object of mass M is suspended from the bottom of the rope of mass m and length L in Problem 48. (a) Show that the time interval for a transverse pulse to travel the length of the rope is t=2Lmg(M+mM) (b) What If? Show that the expression in part (a) reduces to the result of Problem 48 when M = 0. (c) Show that for m M, the expression in part (a) reduces to t=mLMg Equation 16.40 states that at distance r away from a point source with power (Power)avg, the wave intensity is I=(Power)avg4r2 Study Figure 16.25 and prove that at distance r straight in front of a point source with power (Power)avg moving with constant speed vS the wave intensity is I=(Power)avg4r2(vvSv)In Section 16.7, we derived the speed of sound in a gas using the impulsemomentum theorem applied to the cylinder of gas in Figure 16.20. Let us find the speed of sound in a gas using a different approach based on the element of gas in Figure 16.18. Proceed as follows. (a) Draw a force diagram for this element showing the forces exerted on the left and right surfaces due to the pressure of the gas on either side of the element. (b) By applying Newtons second law to the element, show that (P)xAx=Ax2st2 (c) By substituting P = (B s/x) (Eq. 16.30), derive the following wave equation for sound: B2sx2=2st2 (d) To a mathematical physicist, this equation demonstrates the existence of sound waves and determines their speed. As a physics student, you must take another step or two. Substitute into the wave equation the trial solution s(x, t) = smax cos (kx t). Show that this function satisfies the wave equation, provided /k=v=B/.17.1QQ Consider the waves in Figure 17.8 to be waves on a stretched string. Define the velocity of elements of the string as positive if they are moving upward in the figure. (i) At the moment the string has the shape shown by the red-brown curve in Figure 17.8a, what is the instantaneous velocity of elements along the string? (a) zero for all elements (b) positive for all elements (c) negative for all elements (d) varies with the position of the element (ii) From the same choices, at the moment the string has the shape shown by the red-brown curve in Figure 17.8b, what is the instantaneous velocity of elements along the string?When a standing wave is set up on a string fixed at both ends, which of the following statements is true? (a) The number of nodes is equal to the number of antinodes. (b) The wavelength is equal to the length of the string divided by an integer. (c) The frequency is equal to the number of nodes times the fundamental frequency. (d) The shape of the string at any instant shows a symmetry about the midpoint of the string.17.4QQ Balboa Park in San Diego has an outdoor organ. When the air temperature increases, the fundamental frequency of one of the organ pipes (a) stays the same, (b) goes down, (c) goes up, or (d) is impossible to determine.Two waves on one string are described by the wave functions y1=3.0cos(4.0x1.6t)y2=4.0sin(5.0x2.0)t where x and y are in centimeters and t is in seconds. Find the values of y1 + y2 at the points (a) x = 1.00, t = 1.00; (b) x = 1.00, t = 0.500; and (c) x = 0.500, t = 0. Note: Remember that the arguments of the trigonometric functions are in radians.Two pulses of different amplitudes approach each other, each having a speed of v = 1.00 m/s. Figure P17.2 shows the positions of the pulses at time t = 0. (a) Sketch the resultant wave at t = 2.00 s, 4.00 s, 5.00 s, and 6.00 s. (b) What If? If the pulse on the right is inverted so that it is upright, how would your sketches of the resultant wave change? Figure P17.2 Two wave pulses A and B are moving in opposite directions, each with a speed v = 2.00 cm/s. The amplitude of A is twice the amplitude of B. The pulses are shown in Figure P17.3 at t = 0. Sketch the resultant wave at t = 1.00 s, 1.50 s, 2.00 s, 2.50 s, and 3.00 s. Figure P17.3 Why is the following situation impossible? Two identical loudspeakers are driven by the same oscillator at frequency 200 Hz. They are located on the ground a distance d = 4.00 m from each other. Starting far from the speakers, a man walks straight toward the right-hand speaker as shown in Figure P17.4. After passing through three minima in sound intensity, he walks to the next maximum and stops. Ignore any sound reflection from the ground. Figure P17.4 Two pulses traveling on the same string are described by y1=5(3x4t)2+2y2=5(3x+4t6)2+2 (a) In which direction does each pulse travel? (b) At what instant do the two pulses cancel for all x? (c) At what point do the two pulses cancel at all times t?Two identical loudspeakers 10.0 m apart are driven by the same oscillator with a frequency of f = 21.5 Hz (Fig. P17.6) in an area where the speed of sound is 344 m/s. (a) Show that a receiver at point A records a minimum in sound intensity from the two speakers. (b) If the receiver is moved in the plane of the speakers, show that the path it should take so that the intensity remains at a minimum is along the hyperbola 9x2 16y2 = 144 (shown in red-brown in Fig. P17.6). (c) Can the receiver remain at a minimum and move very far away from the two sources? If so, determine the limiting form of the path it must take. If not, explain how far it can go. Figure P17.6 Two sinusoidal waves on a string are defined by the wave functions y1=2.00sin(20.0x32.0t)y2=2.00sin(25.0x40.0t) where x, y1, and y2 are in centimeters and t is in seconds. (a) What is the phase difference between these two waves at the point x = 5.00 cm at t = 2.00 s? (b) What is the positive x value closest to the origin for which the two phases differ by at t = 2.00 s? (At that location, the two waves add to zero.)Verify by direct substitution that the wave function for a standing wave given in Equation 17.1, y=(2Asinkx)coswt is a solution of the general linear wave equation, Equation 16.27: 2yx2=1v22yt2 9P A standing wave is described by the wave function y=6sin(2x)cos(100t) where x and y are in meters and t is in seconds. (a) Prepare graphs showing y as a function of x for five instants: t = 0, 5 ms, 10 ms, 15 ms, and 20 ms. (b) From the graph, identify the wavelength of the wave and explain how to do so. (c) From the graph, identify the frequency of the wave and explain how to do so. (d) From the equation, directly identify the wavelength of the wave and explain bow to do so. (e) From the equation, directly identify the frequency and explain how to do so.11P

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