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All Textbook Solutions for Physics for Scientists and Engineers

In a machine shop, two cams are produced, one of aluminum and one of iron. Both cams have the same mass. Which cam is larger? (a) The aluminum cam is larger. (b) The iron cam is larger. (c) Both cams have the same size.True or False: Dimensional analysis can give you the numerical value of constants of proportionality that may appear in an algebraic expression.The distance between two cities is 100 mi. What is the number of kilometers between the two cities? (a) smaller than 100 (b) larger than 100 (c) equal to 100 1P 2P 3P 4P You have been hired by the defense attorney as an expert witness in a lawsuit. The plaintiff is someone who just returned from being a passenger on the first orbital space tourist flight. Based on a travel brochure offered by the space travel company, the plaintiff expected to be able to see the Great Wall of China from his orbital height of 200 km above the Earths surface. He was unable to do so, and is now demanding that his fare he refunded and to receive additional financial compensation to cover his great disappointment. Construct the basis for an argument for the defense that shows that his expectation of seeing the Great Wall from orbit was unreasonable. The Wall is 7 m wide at its widest point and the normal visual acuity of the human eye is 3 104 rad. (Visual acuity is the smallest subtended angle that an object can make at the eye and still be recognized; the subtended angle in radians is the ratio of the width of an object to the distance of the object from your eyes.)A surveyor measures the distance across a straight river by the following method (Fig. P1.6). Starting directly across from a tree on the opposite bank, she walks d = 100 m along the riverbank to establish a baseline. Then she sights across to the tree. The angle from her baseline to the tree is 0 = 35.0. How wide is the river? Figure P1.6 A crystalline solid consists of atoms stacked up in a repeating lattice structure. Consider a crystal as shown in Figure P1.7a. The atoms reside at the corners of cubes of side L = 0.200 nm. One piece of evidence for the regular arrangement of atoms comes from the flat surfaces along which a crystal separates, or cleaves, when it is broken. Suppose this crystal cleaves along a face diagonal as shown in Figure P1.7b. Calculate the spacing d between two adjacent atomic planes that separate when the crystal cleaves.The position of a particle moving under uniform acceleration is some function of time and the acceleration. Suppose we write this position as x = kat, where k is a dimensionless constant. Show by dimensional analysis that this expression is satisfied if m = 1 and n = 2. Can this analysis give the value of k?9P (a) Assume the equation x = At3 + Bt describes the motion of a particular object, with x having the dimension of length and t having the dimension of time. Determine the dimensions of the constants A and B. (b) Determine the dimensions of the derivative dx/dt = 3At2 + B.A solid piece of lead has a mass of 23.94 g and a volume of 2.10 cm3. From these data, calculate the density of lead in SI units (kilograms per cubic meter).Why is the following situation impossible? A students dormitory room measures 3.8 m by 3.6 m, and its ceiling is 2.5 m high. After the student completes his physics course, he displays his dedication by completely wallpapering the walls of the room with the pages from his copy of volume 1 (Chapters 121) of this textbook. He even covers the door and window.13P Let AI represent the density of aluminum and Fe that of iron. Kind the radius of a solid aluminum sphere that balances a solid iron sphere of radius rFe on an equal-arm balance.One gallon of paint (volume = 3.78 103 m3) covers an area of 25.0 m2. What is the thickness of the fresh paint on the wall?16P (a) Compute the order of magnitude of the mass of a bathtub half full of water. (b) Compute the order of magnitude of the mass of a bathtub half full of copper coins.To an order of magnitude, how many piano tuners reside in New York City? The physicist Enrico Fermi was famous for asking questions like this one on oral Ph.D. qualifying examinations.Your roommate is playing a video game from the latest Star Wars movie while you are studying physics. Distracted by the noise, you go to see what is on the screen. The game involves trying to fly a spacecraft through a crowded field of asteroids in the asteroid belt around the Sun. You say to him. Do you know that the game you are playing is very unrealistic? The asteroid belt is not that crowded and you dont have to maneuver through it like that! Distracted by your statement, he accidentally allows his spacecraft to strike an asteroid, just missing the high score. He turns to you in disgust and says, Yeah, prove it. You say. Okay, Ive learned recently that the highest concentration of asteroids is in a doughnut-shaped region between the Kirkwood gaps at radii of 2.06 AU and 3.27 AU from the Sun. There are an estimated 109 asteroids of radius 100 m or larger, like those in your video game, in this region Finish your argument with a calculation to show that the number of asteroids in the space near a spacecraft is tiny. (An astronomical unitAUis the mean distance of the Earth from the Sun: 1 AU = 1.496 1011 m.)How many significant figures are in the following numbers? (a) 78.9 0.2 (b) 3.788 109 (c) 2.46 106 (d) 0.005 3 The tropical year, the time interval from one vernal equinox to the next vernal equinox, is the basis for our calendar. It contains 365.242 199 days. Find the number of seconds in a tropical year.22P Review. In a community college parking lot, the number of ordinary cars is larger than the number of sport utility vehicles by 94.7%. The difference between the number of cars and the number of SUVs is 18. Find the number of SUVs in the lot.24P Review. The ratio of the number of sparrows visiting a bird feeder to the number of more interesting birds is 2.25. On a morning when altogether 91 birds visit the feeder, what is the number of sparrows?Review. Prove that one solution of the equation 2.00x43.00x3+5.00x=70.0 is x = 2.22.27P 28P 29AP (a) What is the order of magnitude of the number of micro organisms in the human intestinal tract? A typical bacterial length scale is 106 m. Estimate the intestinal volume and assume 1% of it is occupied by bacteria. (b) Does the number of bacteria suggest whether the bacteria are beneficial, dangerous, or neutral for the human body? What functions could they serve?The distance from the Sun to the nearest star is about 4 1016 m. The Milky Way galaxy (Fig. P1.31) is roughly a disk of diameter 1021 in and thickness 1019 m. Find the order of magnitude of the number of stars in the Milky Way. Assume the distance between the Sun and our nearest neighbor is typical. Figure P1.31 The Milky Way galaxy.Why is the following situation impossible? In an effort to boost interest in a television game show, each weekly winner is offered an additional 1 million bonus prize if he or she can personally count out that exact amount from a supply of one-dollar bills. The winner must do this task under supervision by television show executives and within one 40-hour work week. To the dismay of the shows producers, most contestants succeed at the challenge.Bacteria and other prokaryotes are found deep underground, in water, and in the air. One micron (106 m) is a typical length scale associated with these microbes. (a) Estimate the total number of bacteria and other prokaryotes on the Earth. (b) Estimate the total mass of all such microbes.A spherical shell has an outside radius of 2.60 cm and an inside radius of a. The shell wall has uniform thickness and is made of a material with density 4.70 g/cm3. The space inside the shell is filled with a liquid having a density of 1.23 g/cm3. (a) Find the mass m of the sphere, including its contents, as a function of a. (b) For what value of the variable a does m have its maximum possible value? (c) What is this maximum mass? (d) Explain whether the value from part (c) agrees with the result of a direct calculation of the mass of a solid sphere of uniform density made of the same material as the shell. (e) What If? Would the answer to part (a) change if the inner wall were not concentric with the outer wall?Air is blown into a spherical balloon so that, when its radius is 6.50 cm, its radius is increasing at the rate 0.900 cm/s. (a) Find the rate at which the volume of the balloon is increasing. (b) If this volume flow rate of air entering the balloon is constant, at what rate will the radius be increasing when the radius is 13.0 cm? (c) Explain physically why the answer to part (b) is larger or smaller than 0.9 cm/s, if it is different.In physics, it is important to use mathematical approximations. (a) Demonstrate that for small angles ( 20) tansin=180 where is in radians and is in degrees. (b) Use a calculator to find the Largest angle for which tan may be approximated by with an error less than 10.0%.The consumption of natural gas by a company satisfies the empirical equation V = 1.50t + 0.008 00t2, where V is the volume of gas in millions of cubic feet and t is the time in months. Express this equation in units of cubic feet and seconds. Assume a month is 30.0 days.A woman wishing to know the height of a mountain measures the angle of elevation of the mountaintop as 12.0. After walking 1.00 km closer to the mountain on level ground, she finds the angle to be 14.0. (a) Draw a picture of the problem, neglecting the height of the womans eyes above the ground. Hint: Use two triangles. (b) Using the symbol y to represent the mountain height and the symbol x to represent the womans original distance from the mountain, label the picture. (c) Using the labeled picture, write two trigonometric equations relating the two selected variables. (d) Find the height y.39CP Which of the following choices best describes what can be determined exactly from Table 2.1 and Figure 2.1 for the entire 50-s interval? (a) The distance the car moved. (b) The displacement of the car. (c) Both (a) and (b). (d) Neither (a) nor (b).2.2QQ Are officers in the highway patrol more interested in (a) your average speed or (b) your instantaneous speed as you drive?Make a velocitytime graph for the car in Figure 2.1a. Suppose the speed limit for the road on which the car is driving is 30 km/h. True or False? The car exceeds the speed limit at some time within the time interval 050 s.If a car is traveling eastward and slowing down, what is the direction of the force on the car that causes it to slow down? (a) eastward (b) westward (c) neither eastward nor westward Which one of the following statements is true? (a) If a car is traveling eastward, its acceleration must be eastward. (b) If a car is slowing down, its acceleration must be negative. (c) A particle with constant acceleration can never stop and stay stopped.In Figure 2.12, match each vxt graph on the top with the axt graph on the bottom that best describes the motion.Consider the following choices: (a) increases, (b) decreases, (c) increases and then decreases, (d) decreases and then increases, (e) remains the same. From these choices, select what happens to (i) the acceleration and (ii) the speed of a ball after it is thrown upward into the air.The speed of a nerve impulse in the human body is about 100 m/s. If you accidentally stub your toe in the dark, estimate the time it takes the nerve impulse to travel to your brain.A particle moves according to the equation x = 10t2, where x is in meters and t is in seconds. (a) Find the average velocity for the time interval from 2.00 s to 3.00 s. (b) Find the average velocity for the time interval from 2.00 to 2.10 s.The position of a pinewood derby car was observed at various times; the results are summarized in the following table. Find the average velocity of the car for (a) the first second, (b) the last 3 s, and (c) the entire period of observation.An athlete leaves one end of a pool of length L at t = 0 and arrives at the other end at time t1. She swims back and arrives at the starting position at time t2. If she is swimming initially in the positive x direction, determine her average velocities symbolically in (a) the first half of the swim, (b) the second half of the swim, and (c) the round trip. (d) What is her average speed for the round trip?A positiontime graph for a particle moving along the x axis is shown in Figure P2.5. (a) Find the average velocity in the time interval t = 1.50 s to t = 4.00 s. (b) Determine the instantaneous velocity at t = 2.00 s by measuring the slope of the tangent line shown in the graph. (c) At what value of t is the velocity zero? Figure P2.5 A car travels along a straight line at a constant speed of 60.0 mi/h for a distance d and then another distance d in the same direction at another constant speed. The average velocity for the entire trip is 30.0 mi/h. (a) What is the constant speed with which the car moved during the second distance d? (b) What If? Suppose the second distance d were traveled in the opposite direction; you forgot something and had to return home at the same constant speed as found in part (a). What is the average velocity for this trip? (c) What is the average speed for this new trip?A person takes a trip, driving with a constant speed of 89.5 km/h, except for a 22.0-min rest stop. If the persons average speed is 77.8 km/h, (a) how much time is spent on the trip and (b) how far does the person travel?A child rolls a marble on a bent track that is 100 cm long as shown in Figure P2.8. We use x to represent the position of the marble along the track. On the horizontal sections from x = 0 to x = 20 cm and from x = 40 cm to x = 60 cm, the marble rolls with constant speed. On the sloping sections, the marbles speed changes steadily. At the places where the slope changes, the marble stays on the track and does not undergo any sudden changes in speed. The child gives the marble some initial speed at x = 0 and t = 0 and then watches it roll to x = 90 cm, where it turns around, eventually returning to x = 0 with the same speed with which the child released it. Prepare graphs of x versus t, vx versus t, and ax versus t, vertically aligned with their time axes identical, to show the motion of the marble. You will not be able to place numbers other than zero on the horizontal axis or on the velocity or acceleration axes, but show the correct graph shapes. Figure P2.8 Figure P2.9 shows a graph of vx versus t for the motion of a motorcyclist as he starts from rest and moves along the road in a straight line. (a) Find the average acceleration for the time interval t = 0 to t = 6.00 s. (b) Estimate the time at which the acceleration has its greatest positive value and the value of the acceleration at that instant. (c) When is the acceleration zero? (d) Estimate the maximum negative value of the acceleration and the time at which it occurs. Figure P2.9 (a) Use the data in Problem 3 to construct a smooth graph of position versus time. (b) By constructing tangents to the x(t) curve, find the instantaneous velocity of the car at several instants. (c) Plot the instantaneous velocity versus time and, from this information, determine the average acceleration of the car. (d) What was the initial velocity of the car?A particle starts from rest and accelerates as shown in Figure P2.11. Determine (a) the particles speed at t = 10.0 s and at t = 20.0 s, and (b) the distance traveled in the first 20.0 s. Figure P2.11 Draw motion diagrams for (a) an object moving to the right at constant speed, (b) an object moving to the right and speeding up at a constant rate, (c) an object moving to the right and slowing down at a constant rate, (d) an object moving to the left and speeding up at a constant rate, and (e) an object moving to the left and slowing down at a constant rate. (f) How would your drawings change if the changes in speed were not uniform, that is, if the speed were not changing at a constant rate?Each of the strobe photographs (a), (b), and (c) in Figure P2.13 was taken of a single disk moving toward the right, which we take as the positive direction. Within each photograph the time interval between images is constant. For each photograph, prepare graphs of x versus t, vx versus t, and ax versus t, vertically aligned with their time axes identical, to show the motion of the disk. You will not be able to place numbers other than zero on the axes, but show the correct shapes for the graph lines. Figure P2.13 An electron in a cathode-ray tube accelerates uniformly from 2.00 104 m/s to 6.00 106 m/s over 1.50 cm. (a) In what time interval does the electron travel this 1.50 cm? (b) What is its acceleration?A parcel of air moving in a straight tube with a constant acceleration of 4.00 m/s2 has a velocity of 13.0 m/s at 10:05:00 a.m. (a) What is its velocity at 10:05:01 a.m.? (b) At 10:05:04 a.m.? (c) At 10:04:50 a.m.? (d) Describe the shape of a graph of velocity versus time for this parcel of air. (e) Argue for or against the following statement: Knowing the single value of an objects constant acceleration is like knowing a whole list of values for its velocity.In Example 2.7, we investigated a jet landing on an aircraft carrier. In a later maneuver, the jet comes in for a landing on solid ground with a speed of 100 m/s, and its acceleration can have a maximum magnitude of 5.00 m/s2 as it comes to rest. (a) From the instant the jet touches the runway, what is the minimum time interval needed before it can come to rest? (b) Can this jet land at a small tropical island airport where the runway is 0.800 km long? (c) Explain your answer.An object moving with uniform acceleration has a velocity of 12.0 cm/s in the positive x direction when its x coordinate is 3.00 cm. If its x coordinate 2.00 s later is 5.00 cm, what is its acceleration?Solve Example 2.8 by a graphical method. On the same graph, plot position versus time for the car and the trooper. From the intersection of the two curves, read the time at which the trooper overtakes the car. Example 2.8 Watch Out for the Speed Limit! You are driving at a constant speed of 45.0 m/s when you pass a trooper on a motorcycle hidden behind a billboard. One second after your car passes the billboard, the trooper sets out from the billboard to catch you, accelerating at a constant rate of 3.00 m/s2. How long does it take the trooper to overtake your car?A glider of length moves through a stationary photogate on an air track. A photogate (Fig. P2.19) is a device that measures the time interval td during which the glider blocks a beam of infrared light passing across the photogate. The ratio vd = /td is the average velocity of the glider over this part of its motion. Suppose the glider moves with constant acceleration. (a) Argue for or against the idea that vd is equal to the instantaneous velocity of the glider when it is halfway through the photogate in space. (b) Argue for or against the idea that vd is equal to the instantaneous velocity of the glider when it is halfway through the photogate in time. Figure P2.19 Why is the following situation impossible? Starting from rest, a charging rhinoceros moves 50.0 m in a straight line in 10.0 s. Her acceleration is constant during the entire motion, and her final speed is 8.00 m/s.A glider of length 12.4 cm moves on an air track with constant acceleration (Fig P2.19). A time interval of 0.628 s elapses between the moment when its front end passes a fixed point along the track and the moment when its back end passes this point. Next, a time interval of 1.39 s elapses between the moment when the back end of the glider passes the point and the moment when the front end of the glider passes a second point farther down the track. After that, an additional 0.431 s elapses until the back end of the glider passes point . (a) Find the average speed of the glider as it passes point . (b) Find the acceleration of the glider. (c) Explain how you can compute the acceleration without knowing the distance between points and .In the particle under constant acceleration model, we identify the variables and parameters vxi, vxf, ax, t, and xf xi. Of the equations in the model. Equations 2.132.17, the first does not involve xf xi, the second and third do not contain ax, the fourth omits vxf, and the last leaves out t. So, to complete the set, there should be an equation not involving vxi. Derive it from the others.At t = 0, one toy car is set rolling on a straight track with initial position 15.0 cm, initial velocity 3.50 cm/s, and constant acceleration 2.40 cm/s2. At the same moment, another toy car is set rolling on an adjacent track with initial position 10.0 cm, initial velocity +5.50 cm/s, and constant acceleration zero. (a) At what time, if any, do the two cars have equal speeds? (b) What are their speeds at that time? (c) At what time(s), if any, do the cars pass each other? (d) What are their locations at that time? (e) Explain the difference between question (a) and question (c) as clearly as possible.You are observing the poles along the side of the road as described in the opening storyline of the chapter. You have already stopped and measured the distance between adjacent poles as 40.0 m. You are now driving again and have activated your smartphone stopwatch. You start the stopwatch at t = 0 as you pass pole #1. At pole #2, the stopwatch reads 10.0 s. At pole #3, the stopwatch reads 25.0 s. Your friend tells you that he was pressing the brake and slowing down the car uniformly during the entire time interval from pole #1 to pole #3. (a) What was the acceleration of the car between poles #1 and #3? (b) What was the velocity of the car at pole #1? (c) If the motion of the car continues as described, what is the number of the last pole passed before the car conics to rest?Why is the following situation impossible? Emily challenges David to catch a 1 bill as follows. She holds the bill vertically as shown in Figure P2.25, with the center of the bill between but not touching Davids index finger and thumb. Without warning, Emily releases the bill. David catches the bill without moving his hand downward. Davids reaction time is equal to the average human reaction time. Figure P2.25 An attacker at the base of a castle wall 3.65 m high throws a rock straight up with speed 7.40 m/s from a height of 1.55 m above the ground. (a) Will the rock reach the top of the wall? (b) If so, what is its speed at the top? If not, what initial speed must it have to reach the top? (c) Find the change in speed of a rock thrown straight down from the top of the wall at an initial speed of 7.40 m/s and moving between the same two points. (d) Does the change in speed of the downward-moving rock agree with the magnitude of the speed change of the rock moving upward between the same elevations? (e) Explain physically why it does or does not agree.The height of a helicopter above the ground is given by h = 3.00t5, where h is in meters and t is in seconds. At t = 2.00 s, the helicopter releases a small mailbag. How long after its release does the mailbag reach the ground?A ball is thrown upward from the ground with an initial speed of 25 m/s; at the same instant, another ball is dropped from a building 15 m high. After how long will the balls be at the same height above the ground?A student throws a set of keys vertically upward to her sorority sister, who is in a window 4.00 m above. The second student catches the keys 1.50 s later. (a) With what initial velocity were the keys thrown? (b) What was the velocity of the keys just before they were caught?At time t = 0, a student throws a set of keys vertically upward to her sorority sister, who is in a window at distance h above. The second student catches the keys at time t. (a) With what initial velocity were the keys thrown? (b) What was the velocity of the keys just before they were caught?You have been hired by the prosecuting attorney as an expert witness in a robbery case. The defendant is accused of stealing an expensive and massive diamond ring in its box from a jewelry store. A witness to the alleged crime testified that she saw the defendant run from the store, stop next to an apartment building, and throw the box straight upward to an accomplice leaning out a fourth-floor window. When captured, the defendant did not have the stolen box with him and claimed innocence. When the witness testified in court about the defendants throwing of the box to an accomplice, the defending attorney argued that it would he impossible to throw the box upward that high to reach the window in question. The bottom of the window is 19.0 in above the sidewalk. You have set up a demonstration in which the defendant was asked by the judge to throw a baseball horizontally as fast as he could and a radar device was used to determine that he can throw the ball at 20 m/s. (a) What testimony can you provide about the ability of the defendant to throw the box to the window in question? (b) What argument might the defense attorney make about the process used to develop your expert testimony? What might be your counter argument? Ignore any effects of air resistance on the box.A student drives a moped along a straight road as described by the velocitytime graph in Figure P2.32. Sketch this graph in the middle of a sheet of graph paper. (a) Directly above your graph, sketch a graph of the position versus time, aligning the time coordinates of the two graphs. (b) Sketch a graph of the acceleration versus time directly below the velocitytime graph, again aligning the time coordinates. On each graph, show the numerical values of x and ax for all points of inflection. (c) What is the acceleration at t = 6.00 s? (d) Find the position (relative to the starting point) at t = 6.00 s. (e) What is the mopeds final position at t = 9.00 s? Figure P2.32 Automotive engineers refer to the time rate of change of acceleration as the jerk. Assume an object moves in one dimension such that its jerk J is constant. (a) Determine expressions for its acceleration ax(t), velocity vx(t), and position x(t), given that its initial acceleration, velocity, and position are axi, vxi, and xi, respectively. (b) Show that ax2 = axi2 + 2J(vx vxi).In Figure 2.11b, the area under the velocitytime graph and between the vertical axis and time t (vertical dashed line) represents the displacement. As shown, this area consists of a rectangle and a triangle. (a) Compute their areas. (b) Explain how the sum of the two areas compares with the expression on the right-hand side of Equation 2.16.The froghopper Philaenus spumarius is supposedly the best jumper in the animal kingdom. To start a jump, this insect can accelerate at 4.00 km/s2 over a distance of 2.00 mm as it straightens its specially adapted jumping legs. Assume the acceleration is constant. (a) Find the upward velocity with which the insect takes off. (b) In what time interval does it reach this velocity? (c) How high would the insect jump if air resistance were negligible? The actual height it reaches is about 70 cm, so air resistance must be a noticeable force on the leaping froghopper.A woman is reported to have fallen 144 ft from the 17th floor of a building, landing on a metal ventilator box that she crushed to a depth of 18.0 in. She suffered only minor injuries. Ignoring air resistance, calculate (a) the speed of the woman just before she collided with the ventilator and (b) her average acceleration while in contact with the box. (c) Modeling her acceleration as constant, calculate the time interval it took to crush the box.At t = 0, one athlete in a race running on a long, straight track with a constant speed v1 is a distance d1 behind a second athlete running with a constant speed v2. (a) Under what circumstances is the first athlete able to overtake the second athlete? (b) Find the time t at which the first athlete overtakes the second athlete, in terms of d1, v1, and v2. (c) At what minimum distance d2 from the leading athlete must the finish line be located so that the trailing athlete can at least tie for first place? Express d2 in terms of d1, v1, and v2 by using the result of part (b).Why is the following situation impossible? A freight train is lumbering along at a constant speed of 16.0 m/s. Behind the freight train on the same track is a passenger train traveling in the same direction at 40.0 m/s. When the front of the passenger train is 58.5 m from the back of the freight train, the engineer on the passenger train recognizes the danger and hits the brakes of his train, causing the train to move with acceleration 3.00 m/s2. Because of the engineers action, the trains do not collide.Hannah tests her new sports car by racing with Sam, an experienced racer. Both start from rest, but Hannah leaves the starting line 1.00 s after Sam does. Sam moves with a constant acceleration of 3.50 m/s2, while Hannah maintains an acceleration of 4.90 m/s2. Find (a) the time at which Hannah overtakes Sam, (b) the distance she travels before she catches him, and (c) the speeds of both cars at the instant Hannah overtakes Sam.Two objects, A and B, are connected by hinges to a rigid rod that has a length L. The objects slide along perpendicular guide rails as shown in Figure P2.40. Assume object A slides to the left with a constant speed v. (a) Find the velocity vB of object B as a function of the angle . (b) Describe vB relative to v. Is vB always smaller than v, larger than v, or the same as v, or does it have some other relationship? Figure P2.40 Lisa rushes down onto a subway platform to find her train already departing. She stops and watches the cars go by. Each car is 8.60 m long. The first moves past her in 1.50 s and the second in 1.10 s. Find the constant acceleration of the train.Two thin rods are fastened to the inside of a circular ring as shown in Figure P2.42. One rod of length D is vertical, and the other of length L makes an angle with the horizontal. The two rods and the ring lie in a vertical plane. Two small beads are free to slide without friction along the rods. (a) If the two beads are released from rest simultaneously from the positions shown, use your intuition and guess which bead reaches the bottom first. (b) Find an expression for the time interval required for the red head to fall from point to point in terms of g and D. (c) Find an expression for the time interval required for the blue bead to slide from point to point in terms of g, L, and . (d) Show that the two time intervals found in parts (b) and (c) are equal. Hint: What is the angle between the chords of the circle and ? (e) Do these results surprise you? Was your intuitive guess in part (a) correct? This problem was inspired by an article by Thomas B. Greenslade, Jr., Galileos Paradox, Phys. Teach. 46, 294 (May 2008). Figure P2.42 In a womens 100-m race, accelerating uniformly, Laura takes 2.00 s and Healan 3.00 s to attain their maximum speeds, which they each maintain for the rest of the race. They cross the finish line simultaneously, both setting a world record of 10.4 s. (a) What is the acceleration of each sprinter? (b) What are their respective maximum speeds? (c) Which sprinter is ahead at the 6.00-s mark, and by how much? (d) What is the maximum distance by which Healan is behind Laura, and at what time does that occur?Which of the following are vector quantities and which are scalar quantities? (a) your age (b) acceleration (c) velocity (d) speed (e) mass The magnitudes of two vectors A and B are A = 12 units and B = 8 units. Which pair of numbers represents the largest and smallest possible values for the magnitude of the resultant vector R=A+B? (a) 14.4 units, 4 units (b) 12 units, 8 units (c) 20 units, 4 units (d) none of these answers If vector B is added to vector A, which two of the following choices must be true for the resultant vector to be equal to zero? (a) A and B are parallel and in the same direction. (b) A and B are parallel and in opposite directions. (c) A and B have the same magnitude. (d) A and B are perpendicular.Choose the correct response to make the sentence true: A component of a vector is (a) always, (b) never, or (c) sometimes larger than the magnitude of the vector.For which of the following vectors is the magnitude of the vector equal to one of the components of the vector? (a) A=2i+5j (b) B=3j (c) C=+5k Two points in the xy plane have Cartesian coordinates (2.00, 4.00) m and (3.00, 3.00) m. Determine (a) the distance between these points and (b) their polar coordinates.2P The polar coordinates of a certain point are (r = 4.30 cm, = 214). (a) Find its Cartesian coordinates x and y. Find the polar coordinates of the points with Cartesian coordinates (b) (x, y), (c) (2x, 2y), and (d) (3x, 3y).Let the polar coordinates of the point (x, y) be (r, ). Determine the polar coordinates for the points (a) (x, y), (b) (2x, 2y), and (c) (3x, 3y).Why is the following situation impossible? A skater glides along a circular path. She defines a certain point on the circle as her origin. Later on, she passes through a point at which the distance she has traveled along the path from the origin is smaller than the magnitude of her displacement vector from the origin.Vector A has a magnitude of 29 units and points in the positive y direction. When vector B is added to A, the resultant vector A+B points in the negative y direction with a magnitude of 14 units. Find the magnitude and direction of B.A force F1 of magnitude 6.00 units acts on an object at the origin in a direction = 30.0 above the positive x axis (Fig. P3.7). A second force F2 of magnitude 5.00 units acts on the object in the direction of the positive y axis. Find graphically the magnitude and direction of the resultant force F1+F2. Figure P3.7 Three displacements are A=200m due south, B=250m due west, and C=150m at 30.0 east of north. (a) Construct a separate diagram for each of the following possible ways of adding these vectors: R1=A+B+C;R2=B+C+A;R3=C+B+A. (b) Explain what you can conclude from comparing the diagrams.The displacement vectors A and B shown in Figure P3.9 both have magnitudes of 3.00 m. The direction of vector A is = 30.0. Find graphically (a) A+B, (b) AB, (c) BA, and (d) A2B. (Report all angles counterclockwise from the positive x axis.)A roller-coaster car moves 200 ft horizontally and then rises 135 ft at an angle of 30.0 above the horizontal. It next travels 135 ft at an angle of 40.0 downward. What is its displacement from its starting point? Use graphical techniques.A minivan travels straight north in the right lane of a divided highway at 28.0 m/s. A camper passes the minivan and then changes from the left Lane into the right lane. As it does so. the campers path on the road is a straight displacement at 8.50 east of north. To avoid cutting off the minivan, the northsouth distance between the campers back bumper and the minivans front bumper should not decrease. (a) Can the camper be driven to satisfy this requirement? (b) Explain your answer.A person walks 25.0 north of east for 3.10 km. How far would she have to walk due north and due east to arrive at the same location?Your dog is running around the grass in your back yard. He undergoes successive displacements 3.50 m south, 8.20 m northeast, and 15.0 m west. What is the resultant displacement?Given the vectors A=2.00i+6.00j and B=3.00i2.00j, (a) draw the vector sum C=A+B and the vector difference D=AB. (b) Calculate C and D, in terms of unit vectors, (c) Calculate C and D in terms of polar coordinates, with angles measured with respect to the positive x axis.The helicopter view in Fig. P3.15 shows two people pulling on a stubborn mule. The person on the right pulls with a force F1 of magnitude 120 X and direction of 1 = 60.0. The person on the left pulls with a force F2 of magnitude 80.0 N and direction of 2 = 75.0. Find (a) the single force that is equivalent to the two forces shown and (b) the force that a third person would have to exert on the mule to make the resultant force equal to zero. The forces are measured in units of newtons (symbolized N). Figure P3.15 A snow-covered ski slope makes an angle of 35.0 with the horizontal. When a ski jumper plummets onto the hill, a parcel of splashed snow is thrown up to a maximum displacement of 1.50 m at 16.0 from the vertical in the uphill direction as shown in Figure P3.16. Find the components of its maximum displacement (a) parallel to the surface and (b) perpendicular to the surface. Figure P3.16 Consider the three displacement vectors m, m, and m. Use the component method to determine (a) the magnitude and direction of and (b) the magnitude and direction of . Vector A has x and y components of 8.70 cm and 15.0 cm, respectively; vector R has x and y components of 13.2 cm and 6.60 cm, respectively. If AB+3C=0, what are the components of C?19P Given the displacement vectors A=(3i4j+4k)m and B=(2i+3j7k)m, find the magnitudes of the following vectors and express each in terms of its rectangular components. (a) C=A+B (b) D=2AB Vector A has a negative x component 3.00 units in length and a positive y component 2.00 units in length. (a) Determine an expression for A in unit-vector notation. (b) Determine the magnitude and direction of A. (c) What vector B when added to A gives a resultant vector with no x component and a negative y component 4.00 units in length?Three displacement vectors of a croquet ball are shown in Figure P3.22, where A=20.0 units, B=40.0 units, and C=30.0 units. Find (a) the resultant in unit-vector notation and (b) the magnitude and direction of the resultant displacement. Figure P3.22 23P 24P Use the component method to add the vectors A and B shown in Figure P3.9. Both vectors have magnitudes of 3.00 m and vector A makes an angle of = 30.0 with the x axis. Express the resultant A+B in unit-vector notation.A girl delivering newspapers covers her route by traveling 3.00 blocks west, 4.00 blocks north, and then 6.00 blocks east. (a) What is her resultant displacement? (b) What is the total distance she travels?A man pushing a mop across a floor causes it to undergo two displacements. The first has a magnitude of 150 cm and makes an angle of 120 with the positive x axis. The resultant displacement has a magnitude of 140 cm acid is directed at an angle of 35.0 to the positive x axis. Find the magnitude and direction of the second displacement.Figure P3.28 illustrates typical proportions of male (m) and female (f) anatomies. The displacements d1m and d1f from the soles of the feet to the navel have magnitudes of 104 cm and 64.0 cm. respectively. The displacements d2m and d2f from the navel to outstretched fingertips have magnitudes of 100 cm and 86.0 cm, respectively. Find the vector sum of these displacements d3=d1+d2 for both people. Figure P3.28 Review. As it passes over Grand Bahama Island, the eye of a hurricane is moving in a direction 60.0 north of west with a speed of 41.0 km/h. (a) What is the unit-vector expression for the velocity of the hurricane? It maintains this velocity for 3.00 h. at which time the course of the hurricane suddenly shifts due north, and its speed slows to a constant 25.0 km/h. This new velocity is maintained for 1.50 h. (b) What is the unit-vector expression for the new velocity of the hurricane? (c) What is the unit-vector expression for the displacement of the hurricane during the first 3.00 h? (d) What is the unit-vector expression for the displacement of the hurricane during the latter 1.50 h? (e) How far from Grand Bahama is the eye 4.50 h after it passes over the island?In an assembly operation illustrated in Figure P3.30. a robot moves an object first straight upward and then also to the east, around an arc forming one-quarter of a circle of radius 4.80 cm that lies in an eastwest vertical plane. The robot then moves the object upward and to the north, through one-quarter of a circle of radius 3.70 cm that lies in a northsouth vertical plane. Find (a) the magnitude of the total displacement of the object and (b) the angle the total displacement makes with the vertical. Figure P3.30 Review. You are standing on the ground at the origin of a coordinate system. An airplane flies over you with constant velocity parallel to the x axis and at a fixed height of 7.60 103 m. At time t = 0, the airplane is directly above you so that the vector leading from you to it is P0=7.60103m. At t = 30.0 s, the position vector leading from you to the airplane is P30=(8.04103i+7.60103j)m as suggested in Figure P3.31. Determine the magnitude and orientation of the airplanes position vector at t = 45.0 s. Figure P3.31 Why is the following situation impossible? A shopper pushing a cart through a market follows directions to the canned goods and moves through a displacement 8.00im down one aisle. He then makes a 90.0 turn and moves 3.00 m along the y axis. He then makes another 90.0 turn and moves 4.00 m along the x axis. Every shopper who follows these directions correctly ends up 5.00 m from the starting point.In Figure P3.33, the line segment represents a path from the point with position vector (5i+3j)m to the point with location (16i+12j)m. Point is along this path, a fraction f of the way to the destination. (a) Find the position vector of point in terms of f. (b) Evaluate the expression from part (a) for f = 0. (c) Explain whether the result in part (b) is reasonable. (d) Evaluate the expression for f = 1. (c) Explain whether the result in part (d) is reasonable. Figure P3.33 Point is a fraction f of the distance from the initial point (5,3) to the final point (16, 12).You are spending the summer as an assistant learning how to navigate on a large ship carrying freight across Lake Erie. One day, you and your ship are to travel across the lake a distance of 200 km traveling due north from your origin port to your destination port. Just as you leave your origin port, the navigation electronics go down. The captain continues sailing, claiming he can depend on his years of experience on the water as a guide. The engineers work on the navigation system while the ship continues to sail, and winds and waves push it off course. Eventually, enough of the navigation system comes back up to tell you your location. The system tells you that your current position is 50.0 km north of the origin port and 25.0 km east of the port. The captain is a little embarrassed that his ship is so far off course and barks an order to you to tell him immediately what heading he should set from your current position to the destination port. Give him an appropriate heading angle.A person going for a walk follows the path shown in Figure P3.35. The total trip consists of four straight-line paths. At the end of the walk, what is the persons resultant displacement measured from the starting point?A ferry transports tourists between three islands. It sails from the first island to the second island, 4.76 km away, in a direction 37.0 north of east. It then sails from the second island to the third island in a direction 69.0 west of north. Finally it returns to the first island, sailing in a direction 28.0 east of south. Calculate the distance between (a) the second and third islands and (b) the first and third islands.Two vectors A and B have precisely equal magnitudes. For the magnitude of A+B to be 100 times larger than the magnitude of AB, what must be the angle between them?Two vectors A and B have precisely equal magnitudes. For the magnitude of A+B to be larger than the magnitude of AB by the factor n, what must be the angle between them?Review. The biggest stuffed animal in the world is a snake 420 m long, constructed by Norwegian children. Suppose the snake is laid out in a park as shown in Figure P3.39, forming two straight sides of a 105 angle, with one side 240 m long. Olaf and Inge run a race they invent. Inge runs directly from the tail of the snake to its head, and Olaf starts from the same place at the same moment but runs along the snake. (a) If both children run steadily at 12.0 km/h. Inge reaches the head of the snake how much earlier than Olaf? (b) If Inge runs the race again at a constant speed of 12.0 km/h. at what constant speed must Olaf run to reach the end of the snake at the same time as Inge? Figure P3.39 Ecotourists use their global positioning system indicator to determine their location inside a botanical garden as latitude 0.002 43 degree south of the equator, longitude 75.642 38 degrees west. They wish to visit a tree at latitude 0.001 62 degree north, longitude 75.644 26 degrees west. (a) Determine the straight-line distance and the direction in which they can walk to reach the tree as follows. First model the Earth as a sphere of radius 6.37 106 m to determine the westward and northward displacement components required, in meters. Then model the Earth as a flat surface to complete the calculation. (b) Explain why it is possible to use these two geometrical models together to solve the problem.A vector is given by R=2i+j+3k. Find (a) the magnitudes of the x, y, and z components; (b) the magnitude of R; and (c) the angles between R and the x, y, and z axes.You are working as an assistant to an air-traffic controller at the local airport, from which small airplanes take off and land. Your job is to make sure that airplanes are not closer to each other than a minimum safe separation distance of 2.00 km. You observe two small aircraft on your radar screen, out over the ocean surface. The first is at altitude 800 m above the surface, horizontal distance 19.2 km. and 25.0 south of west. The second aircraft is at altitude 1 100 m, horizontal distance 17.6 km, and 20.0 south of west. Your supervisor is concerned that the two aircraft are too close together and asks for a separatism distance for the two airplanes. (Place the x axis west, the y axis south, and the z axis vertical.)Review. The instantaneous position of an object is specified by its position vector leading from a fixed origin to the location of the object, modeled as a particle. Suppose for a certain object the position vector is a function of time given by r=4i+3j2tk, where r is in meters and t is in seconds. (a) Evaluate dr/dt. (b) What physical quantity does dr/dt represent about the object?Vectors A and B have equal magnitudes of 5.00. The sum of A and B is the vector 6.00j. Determine the angle between A and B.A rectangular parallelepiped has dimensions a, b, and c as shown in Figure P3.45. (a) Obtain a vector expression for the face diagonal vector R1. (b) What is the magnitude of this vector? (c) Notice that R1,ck, and R2 make a right triangle. Obtain a vector expression for the body diagonal vector R2.A pirate has buried his treasure on an island with five trees located at the points (30.0 m, 20.0 m), (60.0 m, 80.0 m). (10.0 m, 10.0 m), (40.0 m, 30.0 m), and (70.0 m, 60.0 m), all measured relative to some origin, as shown in Figure P3.46. His ships log instructs you to start at tree A and move toward tree B, but to cover only one-half the distance between A and B. Then move toward tree C, covering one-third the distance between your current location and C. Next move toward tree D, covering one-fourth the distance between where you are and D. Finally move toward tree E, covering one-fifth the distance between you and E, stop, and dig. (a) Assume you have correctly determined the order in which the pirate labeled the trees as A, B, C, D, and E as shown in the figure. What are the coordinates of the point where his treasure is buried? (b) What If? What if you do not really know the way the pirate labeled the trees? What would happen to the answer if you rearranged the order of the trees, for instance, to B (30 m, 20 m), A (60 m, 80 m), E (10 m, 10 m), C (40 m, 30 m), and D (70 m, 60 m)? State reasoning to show that the answer does not depend on the order in which the trees are labeled. Figure P3.46 Consider the following controls in an automobile in motion: gas pedal, brake, steering wheel. What are the controls in this list that cause an acceleration of the car? (a) all three controls (b) the gas pedal and the brake (c) only the brake (d) only the gas pedal (e) only the steering wheel (i) As a projectile thrown at an upward angle moves in its parabolic path (such as in Fig. 4.9), at what point along its path are the velocity and acceleration vectors for the projectile perpendicular to each other? (a) nowhere (b) the highest point (c) the launch point (ii) From the same choices, at what point are the velocity and acceleration vectors for the projectile parallel to each other?Rank the launch angles for the five paths in Figure 4.11 with respect to time of flight from the shortest time of flight to the longest.A particle moves in a circular path of radius r with speed v. It then increases its speed to 2v while traveling along the same circular path. (i) The centripetal acceleration of the particle has changed by what factor? Choose one: (a) 0.25 (b) 0.5 (c) 2 (d) 4 (e) impossible to determine (ii) From the same choices, by what factor has the period of the particle changed?A particle moves along a path, and its speed increases with time. (i) In which of the following cases are its acceleration and velocity vectors parallel? (a) when the path is circular (h) when the path is straight (c) when the path is a parabola (d) never (ii) From the same choices, in which case are its acceleration and velocity vectors perpendicular everywhere along the path?Suppose the position vector for a particle is given as a function of time by r(t)=x(t)i+y(t)j, with x(t) = at + b and y(t) = ct2 + d, where a = 1.00 m/s, b = 1.00 m, c = 0.125 m/s2, and d = 1.00 m. (a) Calculate the average velocity during the time interval from t = 2.00 s to t = 4.00 s. (b) Determine the velocity and the speed at t = 2.00 s.The coordinates of an object moving in the xy plane vary with time according to the equations x = 5.00 sin t and y = 4.00 5.00 cos t, where is a constant, x and y are in meters, and t is in seconds. (a) Determine the components of velocity of the object at t = 0. (b) Determine the components of acceleration of the object at t = 0. (c) Write expressions for the position vector, the velocity vector, and the acceleration vector of the object at any time t 0. (d) Describe the path of the object in an xy plot.The vector position of a particle varies in time according to the expression r=3.00i6.00t2j, where r is in meters and t is in seconds. (a) Find an expression for the velocity of the particle as a function of time. (b) Determine the acceleration of the particle as a function of time. (c) Calculate the particles position and velocity at t = 1.00 s.It is not possible to see very small objects, such as viruses, using an ordinary light microscope. An electron microscope, however, can view such objects using an electron beam instead of a light beam. Electron microscopy has proved invaluable for investigations of viruses, cell membranes and subcellular structures, bacterial surfaces, visual receptors, chloroplasts, and the contractile properties of muscles. The lenses of an electron microscope consist of electric and magnetic fields that control the electron beam. As an example of the manipulation of an electron beam, consider an electron traveling away from the origin along the x axis in the xy plane with initial velocity vi=vii. As it passes through the region x = 0 to x = d, the electron experiences acceleration a(t)=axi+ayj, where ax and ay are constants. For the case vi = 1.80 107 m/s, ax = 8.00 1014 m/s2, and ay = 1.60 1015 m/s2 determine at x = d = 0.010 0 m (a) the position of the electron, (b) the velocity of the electron, (c) the speed of the electron, and (d) the direction of travel of the electron (i.e., the angle between its velocity and the x axis).Review. A snowmobile is originally at the point with position vector 29.0 m at 95.0 counterclockwise from the x axis, moving with velocity 4.50 m/s at 40.0. It moves with constant acceleration 1.90 m/s2 at 200. After 5.00 s have elapsed, find (a) its velocity and (b) its position vector.In a local bar, a customer slides an empty beer mug down the counter for a refill. The height of the counter is h. The mug slides off the counter and strikes the floor at distance d from the base of the counter. (a) With what velocity did the mug leave the counter? (b) What was the direction of the mugs velocity just before it hit the floor?Mayan kings and many school sports teams are named for the puma, cougar, or mountain lionFelis concolorthe best jumper among animals. It can jump to a height of 12.0 ft when leaving the ground at an angle of 45.0. With what speed, in SI units, does it leave the ground to make this leap?A projectile is fired in such a way that its horizontal range is equal to three times its maximum height. What is the angle of projection?The speed of a projectile when it reaches its maximum height is one-half its speed when it is at half its maximum height. What is the initial projection angle of the projectile?A rock is thrown upward from level ground in such a way that the maximum height of its flight is equal to its horizontal range R. (a) At what angle is the rock thrown? (b) In terms of its original range R, what is the range Rmax the rock can attain if it is launched at the same speed but at the optimal angle for maximum range? (c) What If? Would your answer to part (a) be different if the rock is thrown with the same speed on a different planet? Explain.A firefighter, a distance d from a burning building, directs a stream of water from a fire hose at angle i above the horizontal as shown in Figure P4.11. If the initial speed of the stream is vi, at what height h does the water strike the building? Figure P4.11 A basketball star covers 2.80 m horizontally in a jump to dunk the ball (Fig. P4.12a). His motion through space can be modeled precisely as that of a particle at his center of mass, which we will define in Chapter 9. His center of mass is at elevation 1.02 m when he leaves the floor. It reaches a maximum height of 1.85 m above the floor and is at elevation 0.900 m when he touches down again. Determine (a) his time of flight (his hang time), (b) his horizontal and (c) vertical velocity components at the instant of takeoff, and (d) his takeoff angle. (e) For comparison, determine the hang time of a whitetail deer making a jump (Fig. P4.12b) with center of mass elevations yi = 1.20 m, ymax = 2.50 m, and yf = 0.700 m. Figure P4.12 A student stands at the edge of a cliff and throws a stone horizontally over the edge with a speed of vi = 18.0 m/s. The cliff is h = 50.0 m above a body of water as shown in Figure P4.13. (a) What are the coordinates of the initial position of the stone? (b) What are the components of the initial velocity of the stone? (c) What is the appropriate analysis model for the vertical motion of the stone? (d) What is the appropriate analysis model for the horizontal motion of the stone? (e) Write symbolic equations for the x and y components of the velocity of the stone as a function of time. (f) Write symbolic equations for the position of the stone as a function of time. (g) How long after being released does the stone strike the water below the cliff? (h) With what speed and angle of impact does the stone land? Figure P4.13 The record distance in the sport of throwing cowpats is 81.1 m. This record toss was set by Steve Urner of the United States in 1981. Assuming the initial launch angle was 45 and neglecting air resistance, determine (a) the initial speed of the projectile and (b) the total time interval the projectile was in flight. (c) How would the answers change it the range were the same but the launch angle were greater than 45? Explain.A home run is hit in such a way that the baseball just clears a wall 21.0 m high, located 130 m from home plate. The ball is hit at an angle of 35.0 to the horizontal, and air resistance is negligible. Find (a) the initial speed of the ball, (b) the time it takes the ball to reach the wall, and (c) the velocity components and the speed of the ball when it reaches the wall. (Assume the ball is hit at a height of 1.00 m above the ground.)A projectile is fired from the top of a cliff of height h above the ocean below. The projectile is fired at an angle above the horizontal and with an initial speed vi. (a) Find a symbolic expression in terms of the variables vi, g, and for the time at which the projectile reaches its maximum height. (b) Using the result of part (a), find an expression for the maximum height hmax above the ocean attained by the projectile in terms of h, vi, g, and .A boy stands on a diving board and tosses a stone into a swimming pool. The stone is thrown from a height of 2.50 m above the water surface with a velocity of 4.00 m/s at an angle of 60.0 above the horizontal. As the stone strikes the water surface, it immediately slows down to exactly half the speed it had when it struck the water and maintains that speed while in the water. After the stone enters the water, it moves in a straight line in the direction of the velocity it had when it struck the water. If the pool is 3.00 m deep, how much tune elapses between when the stone is thrown and when it strikes the bottom of the pool?In Example 4.6, we found the centripetal acceleration of the Earth as it revolves around the Sun. From information on the endpapers of this book, compute the centripetal acceleration of a point on the surface of the Earth at the equator caused by the rotation of the Earth about its axis.The astronaut orbiting the Earth in Figure P4.19 is preparing to dock with a Westar VI satellite. The satellite is in a circular orbit 600 km above the Earths surface, where the free-fall acceleration is 8.21 m/s2. Take the radius of the Earth as 6.400 km. Determine the speed of the satellite and the time interval required to complete one orbit around the Earth, which is the period of the satellite. Figure P4.19 An athlete swings a ball, connected to the end of a chain, in a horizontal circle. The athlete is able to rotate the ball at the rate of 8.00 rev/s when the length of the chain is 6.600 m. When be increases the length to 0.900 m, he is able to rotate the ball only 6.00 rev/s. (a) Which rate of rotation gives the greater speed for the ball? (b) What is the centripetal acceleration of the ball at 8.00 rev/s? (c) What is the centripetal acceleration at 6.00 rev/s?The athlete shown in Figure P4.21 rotates a 1.00-kg discus along a circular path of radius 1.06 m. The maximum speed of the discus is 20.0 m/s. Determine the magnitude of the maximum radial acceleration of the discus. Figure P4.21 A tire 0.500 m in radius rotates at a constant rate of 200 rev/min. Find the speed and acceleration of a small stone lodged in the tread of the tire (on its outer edge).(a) Can a particle moving with instantaneous speed 3.00 m/s on a path with radius of curvature 2.00 m have an acceleration of magnitude 6.00 m/s2? (b) Can it have an acceleration of magnitude 4.00 m/s2? In each case, if the answer is yes, explain how it can happen; if the answer is no, explain why not.A ball swings counterclockwise in a vertical circle at the end of a rope 1.50 m long. When the ball is 36.9 past the lowest point on its way up, its total acceleration is (2.25i+20.2j)m/s2. For that instant, (a) sketch a vector diagram showing the components of its acceleration, (b) determine the magnitude of its radial acceleration, and (c) determine the speed and velocity of the ball.A bolt drops from the ceiling of a moving train car that is accelerating northward at a rate of 2.50 m/s2. (a) What is the acceleration of the bolt relative to the train car? (b) What is the acceleration of the bolt relative to the Earth? (c) Describe the trajectory of the bolt as seen by an observer inside the train car. (d) Describe the trajectory of the bolt as seen by an observer fixed on the Earth.The pilot of an airplane notes that the compass indicates a heading due west. The airplanes speed relative to the air is 150 km/h. The air is moving in a wind at 30.0 km/h toward the north. Find the velocity of the airplane relative to the ground.You are taking flying lessons from an experienced pilot. You and the pilot are up in the plane, with you in the pilot seat. The control tower radios the plane, saying that, while you have been airborne, a 25-mi/h crosswind has arisen, with the direction of the wind perpendicular to the runway on which you plan to land. The pilot tells you that your normal airspeed as you land will be 80 mi/h relative to the ground. This speed is relative to the air, in the direction in which the nose of the airplane points. He asks you to determine the angle at which the aircraft must be crabbed. that is, the angle between the centerline of the aircraft and the centerline of the runway that will allow the airplanes velocity relative to the ground to be parallel to the runway.A car travels due east with a speed of 50.0 km/h. Raindrops are falling at a constant speed vertically with respect to the Earth. The traces of the rain on the side windows of the car make an angle of 60.0 with the vertical. Find the velocity of the rain with respect to (a) the car and (b) the Earth.A science student is riding on a flatcar of a train traveling along a straight, horizontal track at a constant speed of 10.0 m/s. The student throws a ball into the air along a path that he judges to make an initial angle of 60.0 with the horizontal and to be in line with the track. The students professor, who is standing on the ground nearby, observes the ball to rise vertically. How high does she see the ball rise?A river has a steady speed of 0.500 m/s. A student swims upstream a distance of 1.00 km and swims back to the starting point. (a) If the student can swim at a speed of 1.20 m/s in still water, how long does the trip take? (b) How much time is required in still water for the same length swim? (c) Intuitively, why does the swim take longer when there is a current?A river flows with a steady speed v. A student swims upstream a distance d and then back to the starling point. The student can swim at speed c in still water. (a) In terms of d, v, and c, what time interval is required for the round trip? (b) What time interval would be required if the water were still? (c) Which time interval is larger? Explain whether it is always larger.You are participating in a summer internship with the Coast Guard. You have been assigned the duty of determining the direction in which a Coast Guard speedboat should travel to intercept unidentified vessels. One day, the radar operator detects an unidentified vessel at a distance of 20.0 km from the radar installation in the direction 15.0 east of north. The vessel is traveling at 26.0 km/h on a course at 40.0 east of north. The Coast Guard wishes to send a speedboat, which travels at 50.0 km/h, to travel in a straight line from the radar installation to intercept and investigate the vessel, and asks you for the heading for the speedboat to take. Express the direction as a compass bearing with respect to due north.A farm truck moves due east with a constant velocity of 9.50 m/s on a limitless, horizontal stretch of road. A boy riding on the back of the truck throws a can of soda upward (Fig. P4.33) and catches the projectile at the same location on the truck bed, hut 16.0 m farther down the road. (a) In the frame of reference of the truck, at what angle to the vertical does the boy throw the can? (b) What is the initial speed of the can relative to the truck? (c) What is the shape of the cans trajectory as seen by the boy? An observer on the ground watches the boy throw the can and catch it. In this observers frame of reference, (d) describe the shape of the cans path and (e) determine the initial velocity of the can. Figure P4.33 A ball on the end of a string is whirled around in a horizontal circle of radius 0.300 m. The plane of the circle is 1.20 m above the ground. The string breaks and the ball lands 2.00 m (horizontally) away from the point on the ground directly beneath the balls location when the string breaks. Find the radial acceleration of the ball during its circular motion.Why is the following situation impassible? A normally proportioned adult walks briskly along a straight line in the +x direction, standing straight up and holding his right arm vertical and next to his body so that the arm does not swing. His right hand holds a ball at his side a distance h above the floor. When the ball passes above a point marked as x = 0 on the horizontal floor, he opens his fingers to release the ball from rest relative to his hand. The ball strikes the ground for the first time at position x = 7.00h.A particle starts from the origin with velocity 5im/s at t = 0 and moves in the xy plane with a varying acceleration given by a=(6tj), where a is in meters per second squared and t is in seconds. (a) Determine the velocity of the particle as a function of time. (b) Determine the position of the particle as a function of time.Lisa in her Lamborghini accelerates at (3.00i2.00j)m/s2, while Jill in her Jaguar accelerates at (1.00i3.00j)m/s2. They both start from rest at the origin. After 5.00 s, (a) what is Lisas speed with respect to Jill. (b) how far apart are they, and (c) what is Lisas acceleration relative to Jill?A boy throws a stone horizontally from the top of a cliff of height h toward the ocean below. The stone strikes the ocean at distance d from the base of the cliff. In terms of h, d, and g, find expressions for (a) the time t at which the stone lands in the ocean, (b) the initial speed of the stone, (c) the speed of the stone immediately before it reaches the ocean, and (d) the direction of the stones velocity immediately before it reaches the ocean.Why is the following situation impossible? Albert Pujols hits a home run so that the baseball just clears the top row of bleachers, 24.0 m high, located 130 m from home plate. The ball is hit at 41.7 m/s at an angle of 35.0 to the horizontal, and air resistance is negligible.As some molten metal splashes, one droplet flies off to the east with initial velocity vi at angle i above the horizontal, and another droplet flies off to the west with the same speed at the same angle above the horizontal as shown in Figure P4.40. In terms of vi and i, find the distance between the two droplets as a function of time. Figure P4.40 An astronaut on the surface of the Moon fires a cannon to launch an experiment package, which leaves the barrel moving horizontally. Assume the free-fall acceleration on the Moon is one-sixth of that on the Earth. (a) What must the muzzle speed of the package be so that it travels completely around the Moon and returns to its original location? (b) What time interval does this trip around the Moon require?A pendulum with a cord of length r = 1.00 m swings in a vertical plane (Fig. P4.42). When the pendulum is in the two horizontal positions = 90.0 and = 270, its speed is 5.00 m/s. Find the magnitude of (a) the radial acceleration and (b) the tangential acceleration for these positions. (c) Draw vector diagrams to determine the direction of the total acceleration for these two positions. (d) Calculate the magnitude and direction of the total acceleration at these two positions. Figure P4.42 A spring cannon is located at the edge of a table that is 1.20 m above the floor. A steel ball is launched from the cannon with speed vi at 35.0 above the horizontal. (a) Find the horizontal position of the ball as a function of vi at the instant it lands on the floor. We write this function as x(vi). Evaluate x for (b) vi = 0.100 m/s and for (c) vi = 100 m/s. (d) Assume vi is close to but not equal to zero. Show that one term in the answer to part (a) dominates so that the function x(vi) reduces to a simpler form. (c) If vi is very large, what is the approximate form of x(v)? (f) Describe the overall shape of the graph of the function x(vi).A projectile is launched from the point (x = 0, y = 0), with velocity (12.0i19.0j)m/s2, at t = 0. (a) Make a table listing the projectiles distance r from the origin at the end of each second thereafter, for 0 t 10 s. Tabulating the x and y coordinates and the components of velocity vx and vy will also be useful. (b) Notice that the projectiles distance from its starting point increases with time, goes through a maximum, and starts to decrease. Prove that the distance is a maximum when the position vector is perpendicular to the velocity. Suggestion: Argue that if v is not perpendicular to r, then r must lie increasing or decreasing. (c) Determine the magnitude of the maximum displacement. (d) Explain your method for solving part (c).A fisherman sets out upstream on a river. His small boat, powered by an outboard motor, travels at a constant speed v in still water. The water (Urns at a lower constant speed vw. The fisherman has traveled upstream for 2.00 km when his ice chest falls out of the boat. He notices that the chest is missing only after he has gone upstream for another 15.0 min. At that point, he turns around and heads back downstream, all the time traveling at the same speed relative to the water. He catches up with the floating ice chest just as he returns to his starting point. How last is the river flowing? Solve this problem in two ways. (a) First, use the Earth as a reference frame. With respect to the Earth, the boat travels upstream at speed v vw, and downstream at v + vw. (b) A second much simpler and more elegant solution is obtained by using the water as the reference frame. This approach has important applications in many more complicated problems; examples are calculating the motion of rockets and satellites and analyzing the scattering of subatomic particles from massive targets.An outfielder throws a baseball to his catcher in an attempt to throw out a runner at home plate. The ball bounces once before reaching the catcher. Assume the angle at which the bounced ball leaves the ground is the same as the angle at which the outfielder threw it as shown in Figure P4.46, but that the balls speed after the bounce is one-half of what it was before the bounce. (a) Assume the ball is always thrown with the same initial speed and ignore air resistance. At what angle should the fielder throw the ball to make it go the same distance D with one bounce (blue path) as a ball thrown upward at 45.0 with no bounce (green path)? (b) Determine the ratio of the time interval for the one-bounce throw to the flight time for the no-bounce throw. Figure P4.46 Do not hurt yourself; do not strike your hand against anything. Within these limitations, describe what you do to give your hand a large acceleration. Compute an order-of-magnitude estimate of this acceleration, stating the quantities you measure or estimate and their values.You are on the Pirates of the Caribbean attraction in the Magic kingdom at Disney World. Your boat rules through a pirate battle, in which cannons on a ship and in a fort are firing at each other. While you are aware that the splashes in the water do not represent actual cannonballs, you begin to wonder about such battles in the days of the pirates. Suppose the fort and the ship are separated by 75.0 m. You sec that the cannons in the fort are aimed so that their cannonballs would be fired horizontally from a height of 7.00 m above the water. (a) You wonder at what speed they must be fired in order to hit the ship before falling in the water. (b) Then, you think about the sludge that must build up inside the barrel of a cannon. This sludge should slow down the cannonballs. A question occurs in your mind: if the cannonballs can be fired at only 50.0% of the speed found earlier, is it possible to fire them upward at some angle to the horizontal so that they would reach the ship?A skier leaves the ramp of a ski jump with a velocity of v = 10.0 m/s at = 15.0 above the horizontal as shown in Figure P4.49 (page 94). The slope where she will land is inclined downward at = 50.0, and air resistance is negligible. Find (a) the distance from the end of the ramp to where the jumper lands and (b) her velocity components just before the landing. (c) Explain how you think the results might be affected if air resistance were included. Figure P4.49 A projectile is fired up an incline (incline angle ) with an initial speed vi at an angle i with respect to the horizontal (i ) as shown in Figure P4.50. (a) Show that the projectile travels a distance d up the incline, where d=2vi2cosisin(i)gcos2 (b) For what value of 0i is d a maximum, and what is that maximum value? Figure P4.50 Two swimmers, Chris and Sarah, start together at the same point on the hank, of a wide stream that flows with a speed v. Both move at the same speed c (where c v) relative to the water. Chris swims downstream a distance L and then upstream the same distance. Sarah swims so that her motion relative to the Earth is perpendicular to the hanks of the stream. She swims the distance L and then back the same distance, with both swimmers returning to the starting point. In terms of L, c, and v, find the time intervals required (a) for Chriss round trip and (b) for Sarahs round trip. (c) Explain which swimmer returns first.In the What If? section of Example 4.5, it was claimed that the maximum range of a ski jumper occurs for a launch angle given by =452 where is the angle the hill makes with the horizontal in Figure 4.15. Prove this claim by deriving the equation above. Figure P4.15 A fireworks rocket explodes at height h, the peak of its vertical trajectory. It throws out burning fragments in all directions, but all at the same speed v. Pellets of solidified metal fall to the ground without air resistance. Find the smallest angle that the final velocity of an impacting fragment makes with the horizontal.Which of the following statements is correct? (a) It is possible for an object to have motion in the absence of forces on the object. (b) It is possible to have forces on an object in the absence of motion of the object. (c) Neither statement (a) nor statement (b) is correct. (d) Both statements (a) and (b) are correct.An object experiences no acceleration. Which of the following cannot be true for the object? (a) A single force acts on the object. (b) No forces act on the object. (c) Forces act on the object, but the forces cancel.You push an object, initially at rest, across a frictionless floor with a constant force for a time interval t, resulting in a final speed of v for the object. You then repeat the experiment, but with a force that is twice as large. What time interval is now required to reach the same final speed v? (a) 4 t (b) 2 t (c) t (d) t/2 (e) t/4 Suppose you are talking by interplanetary telephone to a friend who lives on the Moon. He tells you that he has just won a newton of gold in a contest. Excitedly, you tell him that you entered the Earth version of the same contest and also won a newton of gold! Who is richer? (a) You are. (b) Your friend is. (c) You are equally rich.(i) If a fly collides with the windshield of a fast-moving bus, which experiences an impact force with a larger magnitude? (a) The fly. (b) The bus. (c) The same force is experienced by both. (ii) Which experiences the greater acceleration? (a) The fly. (b) The bus. (c) The same acceleration is experienced by both.You press your physics textbook flat against a vertical wall with your hand. What is the direction of the friction force exerted by the wall on the book? (a) downward (b) upward (c) out from the wall (d) into the wall Charlie is playing with his daughter Toney in the snow. She sits on a sled and asks him to slide her across a flat, horizontal field. Charlie has a choice of (a) pushing her from behind by applying a force downward on her shoulders at 30 below the horizontal (Fig. 5.18a) or (b) attaching a rope to the front of the sled and pulling with a force at 30 above the horizontal (Fig. 5.18b). Which would be easier for him and why?A certain orthodontist uses a wire brace to align a patients crooked tooth as in Figure P5.1. The tension in the wire is adjusted to have a magnitude of 18.0 N. Find the magnitude of the net force exerted by the wire on the crooked tooth. Figure P5.1 One or more external forces, large enough to be easily measured, are exerted on each object enclosed in a dashed box shown in Figure 5.1. Identify the reaction to each of these forces.A 3.00-kg object undergoes an acceleration given by a=(2.00i+5.00j)m/s2. Find (a) the resultant force acting on the object and (b) the magnitude of the resultant force.The average speed of a nitrogen molecule in air is about 6.70 102 m/s, and its mass is 4.68 1026 kg. (a) If it takes 3.00 1015 s for a nitrogen molecule to hit a wall and rebound with the same speed but moving in the opposite direction, what is the average acceleration of the molecule during this time interval? (b) What average force does the molecule exert on the wall?Two forces, F1=(6.00i4.00j)N and F2=(3.00i+7.00j)N, act on a particle of mass 2.00 kg that is initially at rest at coordinates (2.00 m, + 4.00 m). (a) What are the components of the particles velocity at t = 10.0 s? (b) In what direction is the particle moving at t = 10.0 s? (c) What displacement does the particle undergo during the first 10.0 s? (d) What are the coordinates of the particle at t = 10.0 s?The force exerted by the wind on the sails of a sailboat is 390 N north. The water exerts a force of 180 N east. If the boat (including its crew) has a mass of 270 kg, what are the magnitude and direction of its acceleration?Review. Three forces acting on an object are given by F1=(2.00i2.00j)N, and F1=(5.00i3.00j)N, and F1=(45.0i)N. The object experiences an acceleration of magnitude 3.75 m/s2. (a) What is the direction of the acceleration? (b) What is the mass of the object? (c) If the object is initially at rest, what is its speed after 10.0 s? (d) What are the velocity components of the object after 10.0 s?If a single constant force acts on an object that moves on a straight line, the objects velocity is a linear function of time. The equation v = vi + at gives its velocity v as a function of time, where a is its constant acceleration. What if velocity is instead a linear function of position? Assume that as a particular object moves through a resistive medium, its speed decreases as described by the equation v = vi kx, where k is a constant coefficient and x is the position of the object. Find the law describing the total force acting on this object.Review. The gravitational force exerted on a baseball is 2.21 N down. A pitcher throws the ball horizontally with velocity 18.0 m/s by uniformly accelerating it along a straight horizontal line for a time interval of 170 ms. The ball starts from rest, (a) Through what distance does it move before its release? (b) What are the magnitude and direction of the force the pitcher exerts on the hall?Review. The gravitational force exerted on a baseball is Fgj. A pitcher throws the ball with velocity vi by uniformly accelerating it along a straight horizontal line for a time interval of t = t 0 = t. (a) Starting from rest, through what distance does the hall move before its release? (b) What force does the pitcher exert on the hall?Review. An electron of mass 9. 11 1031 kg has an initial speed of 3.00 105 m/s. It travels in a straight line, and its speed increases to 7.00 105 m/s in a distance of 5.00 cm. Assuming its acceleration is constant, (a) determine the magnitude of the force exerted on the electron and (b) compare this force with the weight of the electron, which we ignored.If a man weighs 900 N on the Earth, what would he weigh on Jupiter, where the free-fall acceleration is 25.9 m/s2?You stand on the seat of a chair and then hop off. (a) During the time interval you are in flight down to the floor, the Earth moves toward you with an acceleration of what order of magnitude? In your solution, explain your logic. Model the Earth as a perfectly solid object, (b) The Earth moves toward you through a distance of what order of magnitude?A brick of mass M has been placed on a rubber cushion of mass m. Together they are sliding to the right at constant velocity on an ice-covered parking lot. (a) Draw a free-body diagram of the brick and identify each force acting on it. (b) Draw a free-body diagram of the cushion and identify each force acting on it. (c) Identify all of the action-reaction pairs of forces in the brick-cushion-planet system.Review. Figure P5.15 shows a worker poling a boata very efficient mode of transportation across a shallow lake. He pushes parallel to the length of the light pole, exerting a force of magnitude 240 N on the bottom of the lake. Assume the pole lies in the vertical plane containing the keel of the boat. At one moment, the pole makes an angle of 35.0 with the vertical and the water exerts a horizontal drag force of 47.5 N on the boat, opposite to its forward velocity of magnitude 0.857 m/s. The mass of the boat including its cargo and the worker is 370 kg. (a) The water exerts a buoyant force vertically upward on the boat. Find the magnitude of this force. (b) Model the forces as constant over a short interval of time to find live velocity of live boat 0.450 s after the moment described. Figure P5.15 An iron bolt of mass 65.0 g hangs from a string 35.7 cm long. The top end of the string is fixed. Without touching it, a magnet attracts the bolt so that it remains stationary, but is displaced horizontally 28.0 cm to the tight from the previously vertical line of the string. The magnet is located to the right of the bolt and on the same vertical level as the bolt in the final configuration. (a) Draw a free-body diagram of the bolt. (b) Find the tension in the string, (c) Find the magnetic force on the bolt.A block slides down a frictionless plane having an inclination of 15.0. The block starts from rest at the top. and the length of the incline is 2.00 m. (a) Draw a free-body diagram of the block. Find (b) the acceleration of the block and (c) its speed when it reaches the bottom of the incline.A bag of cement whose weight is Fg hangs in equilibrium from three wires as shown in Figure P5.18. Two of the wires make angles 1 and 2 with the horizontal. Assuming the system is in equilibrium, show that the tension in the left-hand wire is T1=Fgcos2sin(1+2) Figure P5.18 The distance between two telephone poles is 50.0 m. When a 1.00-kg bird lands on the telephone wire midway between the poles, the wire sags 0.200 m. (a) Draw a free-body diagram of the bird. (b) How much tension does the bird produce in the wire? Ignore the weight of the wire.An object of mass m = 1.00 kg is observed to have an acceleration awith a magnitude of 10.0 m/s2 in a direction 60.0 east of north. Figure P5.20 shows a view of the object from above. The force F2 acting on the object has a magnitude of 5.0 N and is directed north. Determine the magnitude and direction of the one other horizontal force F1 acting on the object. Figure P5.20 A simple accelerometer is constructed inside a car by suspending an object of mass m from a string of length L that is tied to the cars ceiling. As the car accelerates the string-object system makes a constant angle of with the vertical. (a) Assuming that the string mass is negligible compared with m, derive an expression for the cars acceleration in terms of and show that it is independent of the mass m and the length L. (b) Determine the acceleration of the car when = 23.0.An object of mass m1 = 5.00 kg placed on a frictionless, horizontal table is connected to a string that passes over a pulley and then is fastened to a hanging object of mass m2 = 9.00 kg as shown in Figure P5.22. (a) Draw free-body diagrams of both objects. Find (b) the magnitude of the acceleration of the objects and (c) the tension in the string. Figure P5.22 Problems 22 and 29.In the system shown in Figure P5.23, a horizontal force Facts on an object of mass m2 = 8.00 kg. The horizontal surface is frictionless. Consider the acceleration of the sliding object as a function of Fr. (a) For what values of Fr does the object of mass m1 = 2.00 kg accelerate upward? (b) For what values of Fr is the tension in the cord zero? (c) Plot the acceleration of the m2 object versus F1. Include values of Fr from 100 N to +100 N. Figure P5.23 A car is stuck in the mud. A tow truck pulls on the car with the arrangement shown in Figure P5.24. The tow cable is under a tension of 2 500 N and pulls downward and to the left on the pin at its upper end. The light pin is held in equilibrium by forces exerted by the two bars A and B. Each bar is a strut; that is, each is a bar whose weight is small compared to the forces it exerts and which exerts forces only through hinge pins at its ends. Each strut exerts a force directed parallel to its length. Determine the force of tension or compression in each strut. Proceed as follows. Make a guess as to which way (pushing or pulling) each force acts on the top pin. Draw a free-body diagram of the pin. Use the condition for equilibrium of the pin to translate the free-body diagram into equations. From the equations calculate the forces exerted by struts A and B. If you obtain a positive answer, you correctly guessed the direction of the force. A negative answer means that the direction should be reversed, but the absolute value correctly gives the magnitude of the force. If a strut pulls on a pin. it is in tension. If it pushes, the strut is in compression. Identify whether each strut is in tension or in compression.An object of mass m1 hangs from a string that passes over a very light fixed pulley P1 as shown in Figure P5.25. The string connects to a second very light pulley P2. A second string passes around this pulley with one end attached to a wall and the other to an object of mass m2 on a frictionless, horizontal table. (a) If a1 and a2 are the accelerations of m1 and m2, respectively, what is the relation between these accelerations? Find expressions for (b) the tensions in the strings and (c) the accelerations a1 and a2 in terms of the masses m1 and m2 and g. Figure P5.25 Why is the following situation impassible? Your 3.80-kg physics book is placed next to you on the horizontal seat of your car. The coefficient of static friction between the book and the seat is 0.650, and the coefficient of kinetic friction is 0.550. You are traveling forward at 72.0 km/h and brake to a stop with constant acceleration over a distance of 30.0 m. Your physics book remains on the seat rather than sliding forward onto the floor.Consider a large truck carrying a heavy load, such as steel beams. A significant hazard for the driver is that the load may slide forward, crushing the cab, if the truck stops suddenly in an accident or even in braking. Assume, for example, that a 10 000-kg load sits on the flatbed of a 20 000-kg truck moving at 12.0 m/s. Assume that the load is not tied down to the truck, but has a coefficient of friction of 0.500 with the flatbed of the truck. (a) Calculate the minimum stopping distance for which the load will not slide forward relative to the truck. (b) Is any piece of data unnecessary for the solution?Before 1960m people believed that the maximum attainable coefficient of static friction for an automobile tire on a roadway was s = 1. Around 1962, three companies independently developed racing tires with coefficients of 1.6. This problem shows that tires have improved further since then. The shortest time interval in which a piston-engine car initially at rest has covered a distance of one-quarter mile is about 4.43 s. (a) Assume the cars rear wheels lift the front wheels off the pavement as shown in Figure P5.28. What minimum value of s is necessary to achieve the record time? (b) Suppose the driver were able to increase his or her engine power, keeping other things equal. How would this change affect the elapsed time? Figure P5.28 A 9.00-kg hanging object is connected by a light, in extensible cord over a light, frictionless pulley to a 5.00-kg block that is sliding on a flat table (Fig. P5.22). Taking the coefficient of kinetic friction as 0.200, find the tension in the string.The person in Figure P5.30 weighs 170 lb. As seen from the front, each light crutch makes an angle of 22.0 with the vertical. Half of the persons weight is supported by the crutches. The other half is supported by the vertical forces of the ground on the persons feet. Assuming that the person is moving with constant velocity and the force exerted by the ground on the crutches acts along the crutches, determine (a) the smallest possible coefficient of friction between crutches and ground and (b) the magnitude of the compression force in each crutch. Figure P5.30 Three objects are connected on a table as shown in Figure P5.31. The coefficient of kinetic friction between the block of mass m2 and the table is 0.350. The objects have masses of m1 = 4.00 kg, m2 = 1.00 kg, and m3 = 2.00 kg, and the pulleys are frictionless. (a) Draw a free-body diagram of each object. (b) Determine the acceleration of each object, including its direction. (c) Determine the tensions in the two cords. What If? (d) If the tabletop were smooth, would the tensions increase, decrease, or remain the same? Explain. Figure P5.31 You are working as a letter sorter in a U.S Post Office. Postal regulations require that employees footwear must have a minimum coefficient of static friction of 0.5 on a specified tile surface. You are wearing athletic shoes for which you do not know the coefficient of static friction. In order to determine the coefficient, you imagine that there is an emergency and start running across the room. You have a coworker time you, and find that you can begin at rest and move 4.23 m in 1.20 s. If you try to move faster than this, your feet slip. Assuming your acceleration is constant, does your footwear qualify for the postal regulation?You have been called as an expert witness for a trial in which a driver has been charged with speeding but is claiming innocence. He claims to have slammed on his brakes to avoid rear-ending another car, but tapped the back of the other car just as he came to rest. You have been hired by the prosecution to prove that the driver was indeed speeding. You have received data as follows from the police: Skid marks left by the driver are 56.0 m long and the roadway is level. Tires matching those on the car of the driver have been dragged over the same roadway to determine that the coefficient of kinetic friction between the tires and the roadway is 0.82 at all points along the skid mark. The speed limit on the road is 35 mi/h. Construct an argument to be used in court to show that the driver was indeed speeding.A block of mass 3.00 kg is pushed up against a wall by a force P that makes an angle of = 50.0 with the horizontal as shown in Figure P5.34. The coefficient of static friction between the block and the wall is 0.250. (a) Determine the possible values for the magnitude of P that allow the block to remain stationary. (b) Describe what happens if P has a larger value and what happens if it is smaller. (c) Repeal parts (a) and (b), assuming the force makes an angle of = 13.0 with the horizontal. Figure P5.34 Review. A Chinook salmon can swim underwater at 3.58 m/s, and it can also jump vertically upward, leaving the water with a speed of 6.26 m/s. A record salmon has length 1.50 in and mass 61.0 kg. Consider the fish swimming straight upward in the water below the surface of a lake. The gravitational force exerted on it is very nearly canceled out by a buoyant force exerted by the water as we will study in Chapter 14. The fish experiences an upward force P exerted by the water on its threshing tail fin and a downward fluid friction force that we model as acting on its front end. Assume the fluid friction force disappears as soon as the fishs head breaks the water surface and assume the force on its tail is constant. Model the gravitational force as suddenly switching full on when half the length of the fish is out of the water. Find the value of P.A 5.00-kg block is placed on top of a 10.0-kg block (Fig. P5.36). A horizontal force of 45.0 N is applied to the 10-kg block, and the 5.00-kg block is tied to the wall. The coefficient of kinetic friction between all moving surfaces is 0. 200. (a) Draw a free-body diagram for each block and identify the actionreaction forces between the blocks. (b) Determine the tension in the string and the magnitude of the acceleration of the 10.0-kg block. Figure P5.36 A black aluminum glider floats on a film of air above a level aluminum air track. Aluminum feels essentially no force in a magnetic field, and air resistance is negligible. A strong magnet is attached to the top of the glider, forming a total mass of 240 g. A piece of scrap iron attached to one end stop on the track attracts the magnet with a force of 0.823 N when the iron and the magnet are separated by 2.50 cm. (a) Find the acceleration of the glider at this instant. (b) The scrap iron is now attached to another green glider, forming total mass 120 g. Find the acceleration of each glider when the gliders are simultaneously released at 2.50-cm separation.Why is the following situation impossible? A book sits on an inclined plane on the surface of the Earth. The angle of the plane with the horizontal is 60.0. The coefficient of kinetic friction between the book and the plane is 0.300. At time t = 0, the book is released from rest. The book then slides through a distance of 1.00 m, measured along the plane, in a time interval of 0.483 s.Two blocks of masses m1 and m2, are placed on a table in contact with each other as discussed in Example 5.7 and shown in Figure 5.13a. The coefficient of kinetic friction between the block of mass m1 and the table is 1, and that between the block of mass m2 and the table is 2. A horizontal force of magnitude F is applied to the block of mass m1. We wish to find P, the magnitude of the contact force between the blocks. (a) Draw diagrams showing the forces for each block. (b) What is the net force on the system of two blocks? (c) What is the net force acting on m1? (d) What is the net force acting on m2? (e) Write Newtons second law in the x direction for each block. (f) Solve the two equations in two unknowns for the acceleration a of the blocks in terms of the masses, the applied force F, the coefficients of friction, and g. (g) Find the magnitude P of the contact force between the blocks in terms of the same quantities.A 1.00-kg glider on a horizontal air track is pulled by a string at an angle . The taut string runs over a pulley and is attached to a hanging object of mass 0.500 kg as shown in Figure P5.40. (a) Show that the speed vx of the glider and the speed vy of the hanging object are related by vx = uvy, where u = z(z2 h02)1/2. (b) The glider is released from rest. Show that at that instant the acceleration ax of the glider and the acceleration ay of the hanging object are related by ax = uay. (c) Find the tension in the string at the instant the glider is released for h0 = 80.0 cm and = 30.0. Figure P5.40 An inventive child named Nick wants to reach an apple in a tree without climbing the tree. Sitting in a chair connected to a rope that passes over a frictionless pulley (Fig. P5.41), Nick pulls on the loose end of the rope with such a force that the spring scale reads 250 N. Nicks true weight is 320 N, and the chair weighs 160 N. Nicks feet are not touching the ground. (a) Draw one pair of diagrams showing the forces for Nick and the chair considered as separate systems and another diagram for Nick and the chair considered as one system. (b) Show that the acceleration of the system is upward and find its magnitude. (c) Find the force Nick exerts on the chair. Figure P5.41 Problems 41 and 44.A rope with mass mr is attached to a block with mass mb as in Figure P5.42. The block rests on a frictionless, horizontal surface. The rope does not stretch. The free end of the rope is pulled to the right with a horizontal force F. (a) Draw force diagrams for the rope and the block, noting that the tension in the rope is not uniform. (b) Find the acceleration of the system in terms of mb, mr, and F. (c) Find the magnitude of the force the rope exerts on the block. (d) What happens to the force on the block as the ropes mass approaches zero? What can you state about the tension in a light cord joining a pair of moving objects? Figure P5.42 In Example 5.7, we pushed on two blocks on a table. Suppose three blocks are in contact with one another on a frictionless, horizontal surface as shown in Figure P5.43. A horizontal force F is applied to m1. Take m1 = 2.00 kg, m2 = 3.00 kg, m3 = 4.00 kg, and F = 18.0 N. (a) Draw a separate free-body diagram for each block. (b) Determine the acceleration of the blocks. (c) Find the resultant force on each block. (d) Find the magnitudes of the contact forces between the blocks. (e) You are working on a construction project. A coworker is nailing up plasterboard on one side of a light partition, and you are on the opposite side, providing backing by leaning against the wall with your back pushing on it. Every hammer blow makes your back sting. The supervisor helps you put a heavy block of wood between the wall and your back. Using the situation analyzed in parts (a) through (d) as a model, explain how this change works to make your job more comfortable.In the situation described in Problem 41 and Figure P5.41, the masses of the rope, spring balance, and pulley are negligible. Nicks feet are not touching the ground. (a) Assume Nick is momentarily at rest when he stops pulling down on the rope and passes the end of the rope to another child, of weight 440 N, who is standing on the ground next to him. The rope docs not break. Describe the ensuing motion. (b) Instead, assume Nick is momentarily at rest when he ties the end of the rope to a strong hook projecting from the tree trunk. Explain why this action can make the rope break.A crate of weight Fg is pushed by a force P on a horizontal floor as shown in Figure P5.45. The coefficient of static friction is s, and P is directed at angle below the horizontal. (a) Show that the minimum value of P that will move the crate is given by P=sFgsec1stan Figure P5.45 (b) Find the condition on in terms of s, for which motion of the crate is impossible for any value of P.In Figure P5.46, the pulleys and pulleys the cord are light, all surfaces are frictionless, and the cord does not stretch. (a) How does the acceleration of block 1 compare with the acceleration of block 2? Explain your reasoning. (b) The mass of block 2 is 1.30 kg. Find its acceleration as it depends on the mass m1 of block 1. (c) What If? What does the result of part (b) predict if m1 is very much less than 1.30 kg? (d) What docs the result of part (b) predict if m2 approaches infinity? (e) In this last case, what is the tension in the cord? (f) Could you anticipate the answers to parts (c), (d), and (e) without first doing part (b)? Explain. Figure P5.46 You are working as an expert witness for the defense of a container ship captain whose ship ran into a reef surrounding a Caribbean island. The captain is being charged with intentionally running the ship into the reef. In discovery, the following information has been presented, and attorneys on both sides have stipulated that the information is correct: The ship was traveling at 2.50 m/s toward the reef when a mechanical failure caused the rudder to jam in the straight-ahead position. At that point in time, the ship was 900 m from the reef. The wind was blowing directly toward the reef, and exerting a constant force of 9.00 103 N on the boat in a direction toward the reef. The mass of the ship and its cargo was 5.50 107 kg. During the preparation for the trial, the captain claims that without control of the direction of travel, the only choice he had was to put the engines in reverse at maximum power, such that the total force exerted by the frictional drag force of the water and the force of the water on the propellers was 1.25 105 N in a direction away from the reef. From this information, construct a convincing argument that nothing the captain could do in this situation could have prevented the ship from striking the reef.A flat cushion of mass m is released from rest at the corner of the roof of a building, at height h. A wind blowing along the side of the building exerts a constant horizontal force of magnitude F on the cushion as it drops as shown in Figure P5.48. The air exerts no vertical force. (a) Show that the path of the cushion is a straight line. (b) Does die cushion fall with constant velocity? Explain. (c) If m = 1.20 kg, h = 8.00 m, and F = 2.40 N, how far from the building will the cushion hit the level ground? What If? (d) If the cushion is thrown downward with a nonzero speed at the top of the building, what will be the shape of its trajectory? Explain. Figure P5.48 What horizontal force must be applied to a large block of mass M shown in Figure P5.49 so that the tan blocks remain stationary relative to M? Assume all surfaces and the pulley are frictionless. Notice that the force exerted by the string accelerates m2. Figure P5.49 Problems 49 and 53 An 8.40-kg object slides down a fixed, frictionless, inclined plane. Use a computer to determine and tabulate (a) the normal force exerted on the object and (b) its acceleration for a series of incline angles (measured from the horizontal) ranging from 0 to 90 in 5 increments. (c) Plot a graph of the normal force and the acceleration as functions of the incline angle. (d) In the limiting cases of 0 and 90, are your results consistent with the known behavior?A block of mass 2.20 kg is accelerated across a rough surface by a light cord passing over a small pulley as shown in Figure P5.51. The tension T in the cord is maintained at 10.0 N, and the pulley is 0.100 m above the top of the block. The coefficient of kinetic friction is 0.400. (a) Determine the acceleration of the block when x = 0.400 m. (b) Describe the general behavior of the acceleration as the block slides from a location where x is large to x = 0. (c) Find the maximum value of the acceleration and the position x for which it occurs. (d) Find the value of x for which the acceleration is zero. Figure P5.51 Why is the following situation impossible? A 1.30-kg toaster is not plugged in. The coefficient of static friction between the toaster and a horizontal countertop is 0.350. To make the toaster start moving, you carelessly pull on its electric cord. Unfortunately, the cord has become frayed from your previous similar actions and will break if the tension in the cord exceeds 4.00 N. By pulling on the cord at a particular angle, you successfully start the toaster moving without breaking the cord.Initially, the system of objects shown in Figure P5.49 is held motionless. The pulley and all surfaces and wheels are frictionless. Let the force F be zero and assume that m1 can move only vertically. At the instant after the system of objects is released, Find (a) the tension T in the string, (b) the acceleration of m2, (c) the acceleration of M, and (d) the acceleration of m1. (Note: The pulley accelerates along with the cart.) Figure P5.49 Problems 49 and 53 A mobile is formed by supporting four metal butterflies of equal mass m from a string of length L. The points of support are evenly spaced a distance apart as shown in Figure P5.54. The string forms an angle 1 with the ceiling at each endpoint. The center section of string is horizontal. (a) Find the tension in each section of string in terms of 1, m, and g. (b) In terms of 1, find the angle 2 that the sections of string between the outside butterflies and the inside butterflies form with the horizontal. (c) Show that the distance D between the endpoints of the string is D=L5{2cos1+2cos[tan1(12tan1)]+1} Figure P5.54 In Figure P5.55, the incline has mass M and is fastened to the stationary horizontal tabletop. The block of mass m is placed near the bottom of the incline and is released with a quick push that sets it sliding upward. The block stops near the top of the incline as shown in the figure and then slides down again, always without friction. Find the force that the tabletop exerts on the incline throughout this motion in terms of m, M, g, and . Figure P5.55 You are riding on a Ferris wheel that is rotating with constant speed. The car in which you are riding always maintains its correct upward orientation; it does not invert. (i) What is the direction of the normal force on you from the seat when you are at the top of the wheel? (a) upward (b) downward (c) impossible to determine (ii) From the same choices, what is the direction of the net force on you when you are at the top of the wheel?A bead slides at constant speed along a curved wire lying on a horizontal surface as shown in Figure 6.8. (a) Draw the vectors representing the force exerted by the wire on the bead at points , ,and . (b) Suppose the bead in Figure 6.8 speeds up with constant tangential acceleration as it moves toward the right. Draw the vectors representing the forces on the bead at points , ,and .Consider the passenger in the car making a left turn in Figure 6.10. Which of the following is correct about forces in the horizontal direction if she is making contact with the right-hand door? (a) The passenger is in equilibrium between real forces acting to the right and real forces acting to the left. (b) The passenger is subject only to real forces acting to the right. (c) The passenger is subject only to real forces acting to the left. (d) None of those statements is true.A basketball and a 2-inch-diameter steel ball, having the same mass, are dropped through air from rest such that their bottoms are initially at the same height above the ground, on the order of 1 m or more. Which one strikes the ground first? (a) The steel ball strikes the ground first. (b) The basketball strikes the ground first. (c) Both strike the ground at the same time.In the Bohr model of the hydrogen atom, an electron moves in a circular path around a proton. The speed of the electron is approximately 2.20 106 m/s. Find (a) the force acting on the electron as it revolves in a circular orbit of radius 0.529 1010 m and (b) the centripetal acceleration of the electron.Whenever two Apollo astronauts were on the surface of the Moon, a third astronaut orbited the Moon. Assume the orbit to be circular and 100 km above the surface of the Moon, where the acceleration due to gravity is 1.52 m/s2. The radius of the Moon is 1.70 106 m. Determine (a) the astronauts orbital speed and (b) the period of the orbit.A car initially traveling eastward turns north by traveling in a circular path at uniform speed as shown in Figure P6.3. The length of the arc ABC is 235 m, and the car completes the turn in 36.0 s. (a) What is the acceleration when the car is at B located at an angle of 35.0? Express your answer in terms of the unit vectors i and j. Determine (b) the cars average speed and (c) its average acceleration during the 36.0-s interval. Figure P6.3 A curve in a road forms part of a horizontal circle. As a car goes around it at constant speed 14.0 m/s, the total horizontal force on the driver has magnitude 130 N. What is the total horizontal force on the driver if the speed on the same curve is 18.0 m/s instead?In a cyclotron (one type of particle accelerator), a deuteron (of mass 2.00 u) reaches a final speed of 10.0% of the speed of light while moving in a circular path of radius 0.480 m. What magnitude of magnetic force is required to maintain the deuteron in a circular path?Why is the following situation impossible? The object of mass m = 4.00 kg in Figure P6.6 is attached to a vertical rod by two strings of length = 2.00 m. The strings are attached to the rod at points a distance d = 3.00 m apart. The object rotates in a horizontal circle at a constant speed of v = 3.00 m/s, and the strings remain taut. The rod rotates along with the object so that the strings do not wrap onto the rod. What If? (amid this situation be possible on another planet? Figure P6.6 You are working during your summer break as an amusement park ride operator. The ride you are controlling consists of a large vertical cylinder that spins about its axis fast enough that any person inside is held up against the wall when the floor drops away (Fig. P6.7). The coefficient of static friction between a person of mass m and the wall is s, and the radius of the cylinder is R. You are rotating the ride with an angular speed suggested by your supervisor. (a) Suppose a very heavy person enters the ride. Do you need to increase the angular speed so that this person will not slide down the wall? (b) Suppose someone enters the ride wearing a very slippery satin workout outfit. In this case, do you need to increase the angular speed so that this person will not slide down the wall? Figure P6.7 A driver is suing the state highway department after an accident on a curved freeway. The driver lost control and crashed into a tree located a short distance from the outside edge of the curved roadway. The driver is claiming that the radius of curvature of the unbanked roadway was too small for the speed limit, causing him to slide outward on the curve and hit the tree. You have been hired as an expert witness for the defense, and have been requested to use your knowledge of physics to testify that the radius of curvature of the roadway is appropriate for the speed limit. State regulations show that the radius of curvature of an unbanked roadway on which the speed limit is 65 mi/h must be at least 150 m. You build an accelerometer, which is a plumb bob with a protractor that you attach to the roof of your car. An associate riding in your car with you observes that the plumb bob hangs at an angle of 15.0 from the vertical when the car is driven at a safer speed of 23.0 m/s on the curve in question. What is your testimony regarding the radius of the curve?A hawk flies in a horizontal arc of radius 12.0 m at constant speed 4.00 m/s. (a) Find its centripetal acceleration. (b) It continues to fly along the same horizontal arc, but increases its speed at the rate of 1.20 m/s2. Find the acceleration (magnitude and direction) in this situation at the moment the hawks speed is 4.00 m/s.A 40.0-kg child swings in a swing supported by two chains, each 3.00 m long. The tension in each chain at the lowest point is 350 N. Find (a) the childs speed at the lowest point and (b) the force exerted by the seat on the child at the lowest point. (Ignore the mass of the seat.)A child of mass m swings in a swing supported by two chains, each of length R. If the tension in each chain at the lowest point is T, find (a) the childs speed at the lowest point and (b) the force exerted by the seat on the child at the lowest point. (Ignore the mass of the seat.)One end of a cord is fixed and a small 0.500-kg object is attached to the other end, where it swings in a section of a vertical circle of radius 2.00 m as shown in Figure P6.12. When = 20.0, the speed of the object is 8.00 m/s. At this instant, find (a) the tension in the string, (b) the tangential and radial components of acceleration, and (c) the total acceleration. (d) Is your answer changed if the object is swinging down toward its lowest point instead of swinging up? (e) Explain your answer to part (d). Figure P6.12 A roller coaster at the Six Flags Great America amusement park in Gurnee, Illinois, incorporates some clever design technology and some basic physics. Each vertical loop, instead of being circular, is shaped like a teardrop (Fig. P6.13). The cars ride on the inside of the loop at the top, and the speeds are fast enough to ensure the cars remain on the track. The biggest loop is 40.0 m high. Suppose the speed at the top of the loop is 13.0 m/s and the corresponding centripetal acceleration of the riders is 2g. (a) What is the radius of the arc of the teardrop at the top? (b) If the total mass of a car plus the riders is M, what force does the rail exert on the car at the top? (c) Suppose the roller coaster had a circular loop of radius 20.0 m. If the cars have the same speed, 13.0 m/s at the top, what is the centripetal acceleration of the riders at the top? (d) Comment on the normal force at the top in the situation described in part (c) and on the advantages of having teardrop-shaped loops. Figure P6.13 An object of mass m = 5.00 kg, attached to a spring scale, rests on a frictionless, horizontal surface as shown in Figure P6.14. The spring scale, attached to the front end of a boxcar, reads zero when the car is at rest. (a) Determine the acceleration of the car if the spring scale has a constant reading of 18.0 N when the car is in motion. (b) What constant reading will the spring scale show if the car moves with constant velocity? Describe the forces on the object as observed (c) by someone in the car and (d) by someone at rest outside the car. Figure P6.14 A person stands on a scale in an elevator. As the elevator starts, the scale has a constant reading of 591 N. As the elevator later stops, the scale reading is 391 N. Assuming the magnitude of the acceleration is the same during starting and stopping, determine (a) the weight of the person, (b) the persons mass, and (c) the acceleration of the elevator.Review. A student, along with her backpack on the floor next to her, is in an elevator that is accelerating upward with acceleration a. The student gives her backpack a quick kick at t = 0, imparting to it speed v and causing it to slide across the elevator floor. At time t, the backpack hits the opposite wall a distance L away from the student. Find the coefficient of kinetic friction k between the backpack and the elevator floor.A small container of water is placed on a turntable inside a microwave oven, at a radius of 12.0 cm from the center. The turntable rotates steadily, turning one revolution in each 7.25 s. What angle does the water surface make with the horizontal?The mass of a sports car is 1 200 kg. The shape of the body is such that the aerodynamic drag coefficient is 0.250 and the frontal area is 2.20 m2. Ignoring all other sources of friction, calculate the initial acceleration the car has if it has been traveling at 100 km/h and is now shifted into neutral and allowed to coast.Review. A window washer pulls a rubber squeegee down a very tall vertical window. The squeegee has mass 160 g and is mounted on the end of a light rod. The coefficient of kinetic friction between the squeegee and the dry glass is 0.900. The window washer presses it against the window with a force having a horizontal component of 4.00 N. (a) If she pulls the squeegee down the window at constant velocity, what vertical force component must she exert? (b) The window washer increases the downward force component by 25.0%, while all other forces remain the same. Find the squeegees acceleration in this situation. (c) The squeegee is moved into a wet portion of the window, where its motion is resisted by a fluid drag force R proportional to its velocity according to R = 20.0v, where R is in newtons and v is in meters per second. Find the terminal velocity that the squeegee approaches, assuming the window washer exerts the same force described in part (b).A small piece of Styrofoam packing material is dropped from a height of 2.00 m above the ground. Until it reaches terminal speed, the magnitude of its acceleration is given by a = g Bv. After falling 0.500 m, the Styrofoam effectively reaches terminal speed and then takes 5.00 s more to reach the ground. (a) What is the value of the constant B? (b) What is the acceleration at t = 0? (c) What is the acceleration when the speed is 0.150 m/s?21P Assume the resistive force acting on a speed skater is proportional to the square of the skaters speed v and is given by f = kmv2, where k is a constant and m is the skaters mass. The skater crosses the finish line of a straight-line race with speed vi and then slows down by coasting on his skates. Show that the skaters speed at any time t after crossing the finish line is v(t) = vi/(1 + ktvi).You can feel a force of air drag on your hand if you stretch your arm out of the open window of a speeding car. Note: Do not endanger yourself. What is the order of magnitude of this force? In your solution, state the quantities you measure or estimate and their values.A car travels clockwise at constant speed around a circular section of a horizontal road as shown in the aerial view of Figure P6.24. Find the directions of its velocity and acceleration at (a) position and (b) position . Figure P6.24 A string under a tension of 50.0 N is used to whirl a rock in a horizontal circle of radius 2.50 m at a speed of 20.4 m/s on a fricitonless surface as shown in Figure P6.25. As the string is pulled in, the speed of the rock increases. When the string on the table is 1.00 m long and the speed of the rock is 51.0 m/s, the string breaks. What is the breaking strength, in newtons, of the string? Figure P6.25 Disturbed by speeding cars outside his workplace, Nobel laureate Arthur Holly Compton designed a speed hump (called the Holly hump) and had it installed. Suppose a 1 800-kg car passes over a hump in a roadway that follows the arc of a circle of radius 20.4 m as shown in Figure P6.26. (a) If the car travels at 30.0 km/h, what force does the road exert on the car as the car passes the highest point of the hump? (b) What If? What is the maximum speed the car can have without losing contact with the road as it passes this highest point? Figure P6.26 Problems 26 and 27.A car of mass m passes over a hump in a road that follows the arc of a circle of radius R as shown in Figure P6.26. (a) If the car travels at a speed v, what force docs the road exert on the car as the car passes the highest point of the hump? (b) What If? What is the maximum speed the car can have without losing contact with the road as it passes this highest point?A childs toy consists of a small wedge that has an acute angle (Fig. P6.28). The sloping side of the wedge is frictionless, and an object of mass m on it remains at constant height if the wedge is spun at a certain constant speed. The wedge is spun by rotating, as an axis, a vertical rod that is firmly attached to the wedge at the bottom end. Show that, when the object sits at rest at a point at distance L up along the wedge, the speed of the object must be v = (gL sin )1/2. Figure P6.28 A seaplane of total mass m lands on a lake with initial speed vii. The only horizontal force on it is a resistive force on its pontoons from the water. The resistive force is proportional to the velocity of the seaplane: R=hv. Newtons second law applied to the plane is bvi=m(dv/dt)i. From the fundamental theorem of calculus, this differential equation implies that the speed changes according to vivdvv=bm0tdt (a) Carry nut the integration to determine the speed of the seaplane as a function of time. (b) Sketch a graph of the speed as a function of time. (c) Does the seaplane come to a complete stop after a finite interval of time? (d) Does the seaplane travel a finite distance in stopping?An object of mass m1 = 4.00 kg is tied to an object of mass m2 = 3.00 kg with String 1 of length ( = 0.500 m. The combination is swung in a vertical circular path on a second string, String 2, of length = 0.500 m. During the motion, the two strings are collinear at all times as shown in Figure P6.30. At the top of its motion, m2 is traveling at v = 4.00 m/s. (a) What is the tension in String 1 at this instant? (b) What is the tension in String 2 at this instant? (c) Which string will break first if the combination is rotated faster and faster? Figure P6.30 A ball of mass m = 0.275 kg swings in a vertical circular path on a string L = 0.850 in long as in Figure P6.31. (a) What are the forces acting on the ball at any point on the path? (b) Draw force diagrams for the ball when it is at the bottom of the circle and when it is at the top. (c) If its speed is 5.20 m/s at the top of the circle, what is the tension in the string there? (d) If the string breaks when its tension exceeds 22.5 N, what is the maximum speed the ball can have at the bottom before that happens? Figure P6.31 Why is the following situation impossible? A mischievous child goes to an amusement park with his family. On one ride, after a severe scolding from his mother, he slips out of his seat and climbs to the top of the rides structure, which is shaped like a cone with its axis vertical and its sloped sides making an angle of = 20.0 with the horizontal as shown in Figure P6.32. This part of the structure rotates about the vertical central axis when the ride operates. The child sits on the sloped surface at a point d = 5.32 m down the sloped side from the center of the cone and pouts. The coefficient of static friction between the boy and the cone is 0.700. The ride operator does not notice that the child has slipped away from his seat and so continues to operate the ride. As a result, the sitting, pouting boy rotates in a circular path at a speed of 3.75 m/s. Figure P6.32

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