6.4 Motion in the Presence of Resistive Forces 165 Table 6.1 Terminal Speed for Various Objects Falling Through Air Object Skydiver Baseball (radius 3.7 cm) Golf ball (radius 2.1 cm) Hailstone (radius 0.50 cm) Raindrop (radius 0.20 cm) Mass Cross-Sectional Area vT (kg) (m 2) (m/s) 75 0.70 60 0.145 4.2 1023 43 0.046 1.4 1023 44 24 4.8 10 7.9 1025 14 3.4 1025 1.3 1025 9.0 so vT 2mg Å DrA (6.10) Table 6.1 lists the terminal speeds for several objects falling through air Q uick Quiz 6.4 A baseball and a basketball, 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 m or more Which one strikes the ground first? (a) The baseball strikes the ground first (b) The basketball strikes the ground first (c) Both strike the ground at the same time Conceptual Example 6.9 The Skysurfer Consider a skysurfer (Fig 6.15) who jumps from a plane with his feet attached firmly to his surfboard, does some tricks, and then opens his parachute Describe the forces acting on him during these maneuvers S o l u ti o n Oliver Furrer/Jupiter Images When the surfer first steps out of the plane, he has no vertical velocity The downward gravitational force causes him to accelerate toward the ground As his downward speed increases, so does the upward resistive force exerted by the air on his body and the board This upward force reduces their acceleration, and so their speed increases more slowly Eventually, they are going so fast that the upward resistive force matches the downward gravitational force Now the net force is zero and they no longer accelerate, but instead reach their terminal speed At some point after reaching terminal speed, he opens his parachute, resulting in a drastic increase in the upward resistive force The net force (and therefore the acceleration) is now upward, in the direction opposite the direction of the velocity The downward velocity therefore decreases rapidly, and the Figure 6.15 (Conceptual Example resistive force on the parachute also decreases Eventually, the upward resistive 6.9) A skysurfer force and the downward gravitational force balance each other again and a much smaller terminal speed is reached, permitting a safe landing (Contrary to popular belief, the velocity vector of a skydiver never points upward You may have seen a video in which a skydiver appears to “rocket” upward once the parachute opens In fact, what happens is that the skydiver slows down but the person holding the camera continues falling at high speed.) Example 6.10 Falling Coffee Filters AM The dependence of resistive force on the square of the speed is a simplification model Let’s test the model for a specific situation Imagine an experiment in which we drop a series of bowl-shaped, pleated coffee filters and measure their terminal speeds Table 6.2 on page 166 presents typical terminal speed data from a real experiment using these coffee filters as continued 166 Chapter 6 Circular Motion and Other Applications of Newton’s Laws ▸ 6.10 c o n t i n u e d they fall through the air The time constant t is small, so a dropped filter quickly reaches terminal speed Each filter has a mass of 1.64 g When the filters are nested together, they combine in such a way that the front-facing surface area does not increase Determine the relationship between the resistive force exerted by the air and the speed of the falling filters S o l u ti o n Conceptualize Imagine dropping the coffee filters through the air (If you have some coffee filters, try dropping them.) Because of the relatively small mass of the coffee filter, you probably won’t notice the time interval during which there is an acceleration The filters will appear to fall at constant velocity immediately upon leaving your hand Categorize Because a filter moves at constant velocity, we model it as a particle in equilibrium Analyze At terminal speed, the upward resistive force on the filter balances the downward gravitational force so that R 5 mg R mg 1.64 g a Evaluate the magnitude of the resistive force: Likewise, two filters nested together experience 0.032 N of resistive force, and so forth These values of resistive force are shown in the far right column of Table 6.2 A graph of the resistive force on the filters as a function of terminal speed is shown in Figure 6.16a A straight line is not a good fit, indicating that the resistive force is not proportional to the speed The behavior is more clearly seen in Figure 6.16b, in which the resistive force is plotted as a function of the square of the terminal speed This graph indicates that the resistive force is proportional to the square of the speed as suggested by Equation 6.7 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 The data points not lie along a straight line, but instead suggest a curve Resistive force (N) Resistive force (N) Finalize Here is a good opportunity for you to take some actual data at home on real coffee filters and see if you can reproduce the results shown in Figure 6.16 If you have shampoo and a marble as mentioned in Example 6.8, take data on that system too and see if the resistive force is appropriately modeled as being proportional to the speed 000 g Table 6.2 b 9.80 m/s2 0.016 N Terminal Speed and Resistive Force for Nested Coffee Filters Number of Filters vT (m/s)a R (N) 1.01 1.40 1.63 2.00 2.25 2.40 2.57 2.80 3.05 3.22 0.016 0.032 0.048 0.064 0.080 0.096 0.112 0.128 0.144 0.161 10 a All values of vT are approximate The fit of the straight line to the data points indicates that the resistive force is proportional to the terminal speed squared 10 12 Terminal speed squared (m/s)2 Terminal speed (m/s) a 0.18 0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 kg b Figure 6.16 (Example 6.10) (a) Relationship between the resistive force acting on falling coffee filters and their terminal speed (b) Graph relating the resistive force to the square of the terminal speed Example 6.11 Resistive Force Exerted on a Baseball AM A pitcher hurls a 0.145-kg baseball past a batter at 40.2 m/s (5 90 mi/h) Find the resistive force acting on the ball at this speed Summary 167 ▸ 6.11 c o n t i n u e d S o l u ti o n Conceptualize This example is different from the previous ones in that the object is now moving horizontally through the air instead of moving vertically under the influence of gravity and the resistive force The resistive force causes the ball to slow down, and gravity causes its trajectory to curve downward We simplify the situation by assuming the velocity vector is exactly horizontal at the instant it is traveling at 40.2 m/s Categorize In general, the ball is a particle under a net force Because we are considering only one instant of time, however, we are not concerned about acceleration, so the problem involves only finding the value of one of the forces Analyze To determine the drag coefficient D, imagine that we drop the baseball and allow it to reach terminal speed Solve Equation 6.10 for D: D5 2mg v T2rA 2mg v a brAv mg a b vt v t rA Use this expression for D in Equation 6.7 to find an expression for the magnitude of the resistive force: R 12DrAv Substitute numerical values, using the terminal speed from Table 6.1: R 0.145 kg 9.80 m/s2 a 40.2 m/s b 1.2 N 43 m/s Finalize The magnitude of the resistive force is similar in magnitude to the weight of the baseball, which is about 1.4 N Therefore, air resistance plays a major role in the motion of the ball, as evidenced by the variety of curve balls, floaters, sinkers, and the like thrown by baseball pitchers Summary Concepts and Principles A particle moving in uniform circular motion has a centripetal acceleration; this acceleration must be provided by a net force directed toward the center of the circular path An object moving through a liquid or gas experiences a speed-dependent resistive force This resistive force is in a direction opposite that of the velocity of the object relative to the medium and generally increases with speed The magnitude of the resistive force depends on the object’s size and shape and on the properties of the medium through which the object is moving In the limiting case for a falling object, when the magnitude of the resistive force equals the object’s weight, the object reaches its terminal speed An observer in a noninertial (accelerating) frame of reference introduces fictitious forces when applying Newton’s second law in that frame Analysis Model for Problem-Solving Particle in Uniform Circular Motion (Extension) With our new knowledge of forces, we can extend the model of a particle in uniform circular motion, first introduced in Chapter Newton’s second law applied to a particle moving in uniform circular motion states that the net force causing the particle to undergo a centripetal acceleration (Eq 4.14) is related to the acceleration according to v a F ma c m r (6.1) S ⌺F S ac r S v 168 Chapter 6 Circular Motion and Other Applications of Newton’s Laws Objective Questions 1. denotes answer available in Student Solutions Manual/Study Guide A child is practicing for a BMX race His B speed remains conN stant as he goes counA C terclockwise around W E a level track with two S straight sections and D E two nearly semicircular sections as shown in Figure OQ6.1 the aerial view of Figure OQ6.1 (a) Rank the magnitudes of his acceleration at the points A, B, C, D, and E from largest to smallest If his acceleration is the same size at two points, display that fact in your ranking If his acceleration is zero, display that fact (b) What are the directions of his velocity at points A, B, and C ? For each point, choose one: north, south, east, west, or nonexistent (c) What are the directions of his acceleration at points A, B, and C ? Consider a skydiver who has stepped from a helicopter and is falling through air Before she reaches terminal speed and long before she opens her parachute, does her speed (a) increase, (b) decrease, or (c) stay constant? A door in a hospital has a pneumatic closer that pulls the door shut such that the doorknob moves with constant speed over most of its path In this part of its motion, (a) does the doorknob experience a centripetal acceleration? (b) Does it experience a tangential acceleration? A pendulum consists of a small object called a bob hanging from a light cord of fixed length, with the top end of the cord fixed, as represented in Figure OQ6.4 The bob moves without friction, swinging equally high on both sides It moves from its turning point A through point B and reaches its maximum speed at point C (a) Of these points, is there a point where the bob has nonzero radial acceleration and zero tangential acceleration? If so, which point? What is the Conceptual Questions direction of its total acceleration at this point? (b) Of these points, is there a point where the bob has nonzero tangential acceleration and zero radial acceleration? If so, which point? What is the direction of its total acceleraA tion at this point? (c) Is there a B C point where the bob has no acceleration? If so, which point? (d) Is Figure OQ6.4 there a point where the bob has both nonzero tangential and radial acceleration? If so, which point? What is the direction of its total acceleration at this point? As a raindrop falls through the atmosphere, its speed initially changes as it falls toward the Earth Before the raindrop reaches its terminal speed, does the magnitude of its acceleration (a) increase, (b) decrease, (c) stay constant at zero, (d) stay constant at 9.80 m/s2, or (e) stay constant at some other value? An office door is given a sharp push and swings open against a pneumatic device that slows the door down and then reverses its motion At the moment the door is open the widest, (a) does the doorknob have a centripetal acceleration? (b) Does it have a tangential acceleration? Before takeoff on an airplane, an inquisitive student on the plane dangles an iPod by its earphone wire It hangs straight down as the plane is at rest waiting to take off The plane then gains speed rapidly as it moves down the runway (i) Relative to the student’s hand, does the iPod (a) shift toward the front of the plane, (b) continue to hang straight down, or (c) shift toward the back of the plane? (ii) The speed of the plane increases at a constant rate over a time interval of several seconds During this interval, does the angle the earphone wire makes with the vertical (a) increase, (b) stay constant, or (c) decrease? 1. denotes answer available in Student Solutions Manual/Study Guide What forces cause (a) an automobile, (b) a propellerdriven airplane, and (c) a rowboat to move? tion is constant in magnitude at all times and parallel to the velocity A falling skydiver reaches terminal speed with her parachute closed After the parachute is opened, what parameters change to decrease this terminal speed? The observer in the accelerating elevator of Example 5.8 would claim that the “weight” of the fish is T, the scale reading, but this answer is obviously wrong Why does this observation differ from that of a person outside the elevator, at rest with respect to the Earth? If someone told you that astronauts are weightless in orbit because they are beyond the pull of gravity, would you accept the statement? Explain It has been suggested that rotating cylinders about 20 km in length and km in diameter be placed in An object executes circular motion with constant speed whenever a net force of constant magnitude acts perpendicular to the velocity What happens to the speed if the force is not perpendicular to the velocity? Describe the path of a moving body in the event that (a) its acceleration is constant in magnitude at all times and perpendicular to the velocity, and (b) its accelera- Problems space and used as colonies The purpose of the rotation is to simulate gravity for the inhabitants Explain this concept for producing an effective imitation of gravity Consider a small raindrop and a large raindrop falling through the atmosphere (a) Compare their terminal speeds (b) What are their accelerations when they reach terminal speed? Why does a pilot tend to black out when pulling out of a steep dive? 169 10 A pail of water can be whirled in a vertical path such that no water is spilled Why does the water stay in the pail, even when the pail is above your head? 11 “If the current position and velocity of every particle in the Universe were known, together with the laws describing the forces that particles exert on one another, the whole future of the Universe could be calculated The future is determinate and preordained Free will is an illusion.” Do you agree with this thesis? Argue for or against it Problems The problems found in this chapter may be assigned online in Enhanced WebAssign straightforward; intermediate; challenging full solution available in the Student Solutions Manual/Study Guide AMT Analysis Model tutorial available in Enhanced WebAssign GP Guided Problem M Master It tutorial available in Enhanced WebAssign W Watch It video solution available in Enhanced WebAssign BIO Q/C S Section 6.1 Extending the Particle in Uniform Circular Motion Model A light string can AMT support a stationM ary hanging load r m of 25.0 kg before breaking An object of mass m 3.00 kg attached to the string rotates on a frictionless, horizontal table in a circle of radius r 0.800 m, and Figure P6.1 the other end of the string is held fixed as in Figure P6.1 What range of speeds can the object have before the string breaks? 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 astronaut’s orbital speed and (b) the period of the orbit 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 10 m/s Find (a) the force acting on the electron as it revolves in a circular orbit of radius 0.529 10210 m and (b) the centripetal acceleration of the electron 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? y A car initially traveling eastward turns north by W traveling in a circular path at uniform speed x as shown in Figure P6.6 35.0Њ C O The length of the arc ABC is 235 m, and the B car completes the turn in 36.0 s (a) What is the A acceleration when the car is at B located at an Figure P6.6 angle of 35.08? Express your answer in terms of the unit vectors i^ and j^ Determine (b) the car’s average speed and (c) its average acceleration during the 36.0-s interval 7 A space station, in the form of a wheel 120 m in diameter, rotates to provide an “artificial gravity” of 3.00 m/s2 for persons who walk around on the inner wall of the outer rim Find the rate of the wheel’s rotation in revolutions per minute that will produce this effect Consider a conical pendulum (Fig P6.8) with a bob W of mass m 80.0 kg on a string of length L 10.0 m that makes an angle of u 5 5.008 with the vertical Determine (a) the horizontal and vertical components of the 170 Chapter 6 Circular Motion and Other Applications of Newton’s Laws 16 A roller-coaster car (Fig P6.16) has a mass of 500 kg L A coin placed 30.0 cm from the center u M of a rotating, horizontal turntable slips m when its speed is 50.0 cm/s (a) What force causes the centripetal acceleration when the coin is stationary relative to Figure P6.8 the turntable? (b) What is the coefficient of static friction between coin and turntable? 10 Why is the following situation impossible? The object of mass m 4.00 kg in Figure P6.10 is , attached to a vertical rod by two m strings of length , 2.00 m The d 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 Figure P6.10 rotates along with the object so that the strings not wrap onto the rod What If? Could this situation be possible on another planet? 11 A crate of eggs is located in the middle of the flatbed W of a pickup truck as the truck negotiates a curve in the flat road The curve may be regarded as an arc of a circle of radius 35.0 m If the coefficient of static friction between crate and truck is 0.600, how fast can the truck be moving without the crate sliding? Section 6.2 Nonuniform Circular Motion 12 A pail of water is rotated in a vertical circle of radius W 1.00 m (a) What two external forces act on the water in Q/C the pail? (b) Which of the two forces is most important in causing the water to move in a circle? (c) What is the pail’s minimum speed at the top of the circle if no water is to spill out? (d) Assume the pail with the speed found in part (c) were to suddenly disappear at the top of the circle Describe the subsequent motion of the water Would it differ from the motion of a projectile? 13 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 hawk’s speed is 4.00 m/s 14 A 40.0-kg child swings in a swing supported by two M chains, each 3.00 m long The tension in each chain at the lowest point is 350 N Find (a) the child’s 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.) 15 A child of mass m swings in a swing supported by two S chains, each of length R If the tension in each chain at the lowest point is T, find (a) the child’s 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.) AMT when fully loaded with passengers The path of the W coaster from its initial point shown in the figure to point B involves only up-and-down motion (as seen by the riders), with no motion to the left or right (a) If the vehicle has a speed of 20.0 m/s at point A, what is the force exerted by the track on the car at this point? (b) What is the maximum speed the vehicle can have at point B and still remain on the track? Assume the roller-coaster tracks at points A and B are parts of vertical circles of radius r 10.0 m and r 15.0 m, respectively B r2 r1 A Figure P6.16 Problems 16 and 38 17 A roller coaster at the Six Q/C Flags Great America amuse- ment 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 P6.17) The cars ride on the inside of the loop at the top, and the speeds are fast enough to ensure the Figure P6.17 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 Frank Cezus/Getty Images force exerted by the string on the pendulum and (b) the radial acceleration of the bob 18 One end of a cord is fixed and a small Q/C 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 u S shown in Figure P6.18 When u 20.08, v the speed of the object is 8.00 m/s At this instant, find (a) the tension in the string, (b) the tangential and Figure P6.18 radial components of acceleration, and (c) the total acceleration (d) Is your answer changed if the object is swinging down toward its Problems 171 lowest point instead of swinging up? (e) Explain your answer to part (d) of kinetic friction mk between the backpack and the elevator floor 19 An adventurous archeologist (m 85.0 kg) tries to cross a river by swinging from a vine The vine is 10.0 m long, and his speed at the bottom of the swing is 8.00 m/s The archeologist doesn’t know that the vine has a breaking strength of 000 N Does he make it across the river without falling in? 25 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? Section 6.4 Motion in the Presence of Resistive Forces Section 6.3 Motion in Accelerated Frames 20 An object of mass m 5.00 kg, attached to a spring scale, rests on a m frictionless, horizontal surface as shown in Figure P6.20 The spring scale, attached to the Figure P6.20 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 21 An object of mass m 5 M 0.500 kg is suspended from the ceiling of an accelerating truck as shown in Figure P6.21 Taking a 3.00 m/s2, find (a) the angle u that the string makes with the vertical and (b) the tension T in the string S a m u Figure P6.21 22 A child lying on her back experiences 55.0 N tension in the muscles on both sides of her neck when she raises her head to look past her toes Later, sliding feet first down a water slide at terminal speed 5.70 m/s and riding high on the outside wall of a horizontal curve of radius 2.40 m, she raises her head again to look forward past her toes Find the tension in the muscles on both sides of her neck while she is sliding 23 A person stands on a scale in an elevator As the elevator M 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 person’s mass, and (c) the acceleration of the elevator 24 Review A student, along with her backpack on the S floor next to her, are 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 26 Review (a) Estimate the terminal speed of a wooden sphere (density 0.830 g/cm3) falling through air, taking its radius as 8.00 cm and its drag coefficient as 0.500 (b) From what height would a freely falling object reach this speed in the absence of air resistance? 27 The mass of a sports car is 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 A skydiver of mass 80.0 kg jumps from a slow-moving aircraft and reaches a terminal speed of 50.0 m/s (a) What is her acceleration when her speed is 30.0 m/s? What is the drag force on the skydiver when her speed is (b) 50.0 m/s and (c) 30.0 m/s? 29 Calculate the force required to pull a copper ball of radius 2.00 cm upward through a fluid at the constant speed 9.00 cm/s Take the drag force to be proportional to the speed, with proportionality constant 0.950 kg/s Ignore the buoyant force 30 A small piece of Styrofoam packing material is dropped W 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? 31 A small, spherical bead of mass 3.00 g is released from M rest at t from a point under the surface of a viscous liquid The terminal speed is observed to be vT 2.00 cm/s Find (a) the value of the constant b that appears in Equation 6.2, (b) the time t at which the bead reaches 0.632vT , and (c) the value of the resistive force when the bead reaches terminal speed 32 At major league baseball games, it is commonplace to flash on the scoreboard a speed for each pitch This speed is determined with a radar gun aimed by an operator positioned behind home plate The gun uses the Doppler shift of microwaves reflected from the baseball, an effect we will study in Chapter 39 The gun determines the speed at some particular point on the baseball’s path, depending on when the operator pulls the trigger Because the ball is subject to a drag force due to air proportional to the square of its speed given by R kmv 2, it slows as it travels 18.3 m toward the 172 Chapter 6 Circular Motion and Other Applications of Newton’s Laws plate according to the formula v vie2kx Suppose the ball leaves the pitcher’s hand at 90.0 mi/h 40.2 m/s Ignore its vertical motion Use the calculation of R for baseballs from Example 6.11 to determine the speed of the pitch when the ball crosses the plate 33 Assume the resistive force acting on a speed skater is S proportional to the square of the skater’s speed v and is given by f 2kmv 2, where k is a constant and m is the skater’s 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 skater’s speed at any time t after crossing the finish line is v(t) vi /(1 ktvi ) Review A window washer pulls a rubber squeegee AMT 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 squeegee’s 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 220.0v, 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) 35 A motorboat cuts its engine when its speed is 10.0 m/s and then coasts to rest The equation describing the motion of the motorboat during this period is v vie2ct , where v is the speed at time t, vi is the initial speed at t 0, and c is a constant At t 20.0 s, the speed is 5.00 m/s (a) Find the constant c (b) What is the speed at t 40.0 s? (c) Differentiate the expression for v(t) and thus show that the acceleration of the boat is proportional to the speed at any time 36 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 Additional Problems 37 A car travels clockwise at constant speed around a circular section of a horizontal road as shown in the aerial view of Figure P6.37 Find the directions of its velocity and acceleration at (a) position A and (b) position B A N W E S B Figure P6.37 38 The mass of a roller-coaster car, including its passengers, is 500 kg Its speed at the bottom of the track in Figure P6.16 is 19 m/s The radius of this section of the track is r 1 5 25 m Find the force that a seat in the roller-coaster car exerts on a 50-kg passenger at the lowest point 39 A string under a tension of 50.0 N is used m to whirl a rock in a R horizontal circle of radius 2.50 m at a speed of 20.4 m/s on a frictionless surface as shown in Figure P6.39 As the string is pulled in, the speed of the rock Figure P6.39 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? 40 Disturbed by speeding cars outside his workplace, Nobel laureate Arthur Holly Compton designed a speed bump (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 P6.40 (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? S v (b) What If? What is the maximum speed the car can have without losing contact with Figure P6.40 the road as it passes this Problems 40 and 41 highest point? 41 A car of mass m passes over a hump in a road that folS lows the arc of a circle of radius R as shown in Figure P6.40 (a) If the car travels at a speed v, 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? m 42 A child’s toy consists of a small S wedge that has an acute angle u L (Fig P6.42) The sloping side of the wedge is frictionless, and an u 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 Figure P6.42 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 u)1/2 43 A seaplane of total mass m lands on a lake with initial S speed v i i^ 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: S R 2bS v Newton’s second law applied to the plane is 2bv i^ m dv/dt i^ From the fundamental theorem Problems of calculus, this differential equation implies that the speed changes according to dv b dt m vi v v t (a) Carry out 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? 4 An object of mass m1 5 m1 W 4.00 kg is tied to an String , object of mass m S 3.00 kg with String of m2 v length , 0.500 m , The combination is String swung in a vertical circular path on a second Figure P6.44 string, String 2, of length , 0.500 m During the motion, the two strings are collinear at all times as shown in Figure P6.44 At the top of its motion, m is traveling at v 4.00 m/s (a) What is the tension in String at this instant? (b) What is the tension in String at this instant? (c) Which string will break first if the combination is rotated faster and faster? 45 A ball of mass m 0.275 kg swings in a vertical circular path on a string L 0.850 m long as in Figure P6.45 (a) What are the forces L acting on the ball at any point on the path? (b) Draw force diagrams m for the ball when it is at the bottom of the circle and when it is at the Figure P6.45 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? 46 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 ride’s structure, which is shaped like a cone with its axis vertical and its sloped sides making an angle of u 20.08 with the horizontal as shown in Figure P6.46 This part d u Figure P6.46 173 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 47 (a) A luggage carousel at an airport has the form of a section of a large cone, steadily rotating about its vertical axis Its metallic surface slopes downward toward the outside, making an angle of 20.08 with the horizontal A piece of luggage having mass 30.0 kg is placed on the carousel at a position 7.46 m measured horizontally from the axis of rotation The travel bag goes around once in 38.0 s Calculate the force of static friction exerted by the carousel on the bag (b) The drive motor is shifted to turn the carousel at a higher constant rate of rotation, and the piece of luggage is bumped to another position, 7.94 m from the axis of rotation Now going around once in every 34.0 s, the bag is on the verge of slipping down the sloped surface Calculate the coefficient of static friction between the bag and the carousel 48 In a home laundry dryer, a cylindrical tub containing wet clothes is rotated steadily about a horizontal axis as shown in Figure P6.48 So that the clothes will dry uniformly, they are made to tumble The rate of rotation of the smooth-walled tub is chosen so that a small piece of cloth will lose contact with the tub when the cloth is at an angle of u 68.08 above the horizontal If the radius of the tub is r 0.330 m, what rate of revolution is needed? r u Figure P6.48 49 Interpret the graph in Figure 6.16(b), which describes the results for falling coffee filters discussed in Example 6.10 Proceed as follows (a) Find the slope of the straight line, including its units (b) From Equation 6.6, R 12 DrAv 2, identify the theoretical slope of a graph of resistive force versus squared speed (c) Set the experimental and theoretical slopes equal to each other and proceed to calculate the drag coefficient of the filters Model the cross-sectional area of the filters as that of a circle of radius 10.5 cm and take the density of air to be 1.20 kg/m3 (d) Arbitrarily choose the eighth data point on the graph and find its vertical 174 Chapter 6 Circular Motion and Other Applications of Newton’s Laws 50 A basin surrounding a drain has the shape of a circular Q/C cone opening upward, having everywhere an angle of 35.0° with the horizontal A 25.0-g ice cube is set sliding around the cone without friction in a horizontal circle of radius R (a) Find the speed the ice cube must have as a function of R (b) Is any piece of data unnecessary for the solution? Suppose R is made two times larger (c) Will the required speed increase, decrease, or stay constant? If it changes, by what factor? (d) Will the time required for each revolution increase, decrease, or stay constant? If it changes, by what factor? (e) Do the answers to parts (c) and (d) seem contradictory? Explain S 51 A truck is moving with a S constant acceleration a up a hill that makes an angle f with the u m horizontal as in Figure P6.51 A small sphere of mass m is suspended f from the ceiling of the truck by a light cord If Figure P6.51 the pendulum makes a constant angle u with the perpendicular to the ceiling, what is a? 52 The pilot of an airplane executes a loop-the-loop maneuver in a vertical circle The speed of the airplane is 300 mi/h at the top of the loop and 450 mi/h at the bottom, and the radius of the circle is 200 ft (a) What is the pilot’s apparent weight at the lowest point if his true weight is 160 lb? (b) What is his apparent weight at the highest point? (c) What If? Describe how the pilot could experience weightlessness if both the radius and the speed can be varied Note: His apparent weight is equal to the magnitude of the force exerted by the seat on his body 53 Review While learning to drive, you are in a 1 200-kg Q/C car moving at 20.0 m/s across a large, vacant, level parking lot Suddenly you realize you are heading straight toward the brick sidewall of a large supermarket and are in danger of running into it The pavement can exert a maximum horizontal force of 7 000 N on the car (a) Explain why you should expect the force to have a well-defined maximum value (b) Suppose you apply the brakes and not turn the steering wheel Find the minimum distance you must be from the wall to avoid a collision (c) If you not brake but instead maintain constant speed and turn the steering wheel, what is the minimum distance you must be from the wall to avoid a collision? (d) Of the two methods in parts (b) and (c), which is better for avoiding a collision? Or should you use both the brakes and the steering wheel, or neither? Explain (e) Does the conclusion in part (d) depend on the numerical values given in this problem, or is it true in general? Explain 54 A puck of mass m1 is tied Q/C to a string and allowed S to revolve in a circle of m1 R radius R on a frictionless, horizontal table The other end of the string passes through a m2 small hole in the center of the table, and an object of mass m2 is Figure P6.54 tied to it (Fig P6.54) The suspended object remains in equilibrium while the puck on the tabletop revolves Find symbolic expressions for (a) the tension in the string, (b) the radial force acting on the puck, and (c) the speed of the puck (d) Qualitatively describe what will happen in the motion of the puck if the value of m2 is increased by placing a small additional load on the puck (e) Qualitatively describe what will happen in the motion of the puck if the value of m2 is instead decreased by removing a part of the hanging load 55 Because the Earth rotates about its axis, a point on M the equator experiences a centripetal acceleration of 0.033 7 m/s2, whereas a point at the poles experiences no centripetal acceleration If a person at the equator has a mass of 75.0 kg, calculate (a) the gravitational force (true weight) on the person and (b) the normal force (apparent weight) on the person (c) Which force is greater? Assume the Earth is a uniform sphere and take g 9.800 m/s2 56 Galileo thought about whether acceleration should be Q/C defined as the rate of change of velocity over time or as S the rate of change in velocity over distance He chose the former, so let’s use the name “vroomosity” for the rate of change of velocity over distance For motion of a particle on a straight line with constant acceleration, the equation v vi at gives its velocity v as a function of time Similarly, for a particle’s linear motion with constant vroomosity k, the equation v vi kx gives the velocity as a function of the position x if the particle’s speed is vi at x (a) Find the law describing the total force acting on this object of mass m (b) Describe an example of such a motion or explain why it is unrealistic Consider (c) the possibility of k positive and (d) the possibility of k negative 57 Figure P6.57 shows AMT a photo of a swing W ride at an amusement park The structure consists of a horizontal, rotating, circular platform of diameter D from which seats of mass m are suspended at the end of massless chains of length d When the system rotates at Stuart Gregory/Getty Images separation from the line of best fit Express this scatter as a percentage (e) In a short paragraph, state what the graph demonstrates and compare it with the theoretical prediction You will need to make reference to the quantities plotted on the axes, to the shape of the graph line, to the data points, and to the results of parts (c) and (d) Figure P6.57 200 Chapter 7 Us ϭ Ϫ kx Energy of a System energy function for a block–spring system, given by Us 12kx This function is plotted versus x in Figure 7.20a, where x is the position of the block The force Fs exerted by the spring on the block is related to Us through Equation 7.29: Us E dUs 2kx dx As we saw in Quick Quiz 7.8, the x component of the force is equal to the negative of the slope of the U-versus-x curve When the block is placed at rest at the equilibrium position of the spring (x 0), where Fs 0, it will remain there unless some external force Fext acts on it If this external force stretches the spring from equilibrium, x is positive and the slope dU/dx is positive; therefore, the force Fs exerted by the spring is negative and the block accelerates back toward x when released If the external force compresses the spring, x is negative and the slope is negative; therefore, Fs is positive and again the mass accelerates toward x upon release From this analysis, we conclude that the x position for a block–spring system is one of stable equilibrium That is, any movement away from this position results in a force directed back toward x In general, configurations of a system in stable equilibrium correspond to those for which U(x) for the system is a minimum If the block in Figure 7.20 is moved to an initial position x max and then released from rest, its total energy initially is the potential energy 12kx 2max stored in the spring As the block starts to move, the system acquires kinetic energy and loses potential energy The block oscillates (moves back and forth) between the two points x 2x max and x 1x max, called the turning points In fact, because no energy is transformed to internal energy due to friction, the block oscillates between 2x max and 1x max forever (We will discuss these oscillations further in Chapter 15.) Another simple mechanical system with a configuration of stable equilibrium is a ball rolling about in the bottom of a bowl Anytime the ball is displaced from its lowest position, it tends to return to that position when released Now consider a particle moving along the x axis under the influence of a conservative force Fx , where the U-versus-x curve is as shown in Figure 7.21 Once again, Fx at x 0, and so the particle is in equilibrium at this point This position, however, is one of unstable equilibrium for the following reason Suppose the particle is displaced to the right (x 0) Because the slope is negative for x 0, Fx 2dU/dx is positive and the particle accelerates away from x If instead the particle is at x and is displaced to the left (x , 0), the force is negative because the slope is positive for x , and the particle again accelerates away from the equilibrium position The position x in this situation is one of unstable equilibrium because for any displacement from this point, the force pushes the particle farther away from equilibrium and toward a position of lower potential energy A pencil balanced on its point is in a position of unstable equilibrium If the pencil is displaced slightly from its absolutely vertical position and is then released, it will surely fall over In general, configurations of a system in unstable equilibrium correspond to those for which U(x) for the system is a maximum Finally, a configuration called neutral equilibrium arises when U is constant over some region Small displacements of an object from a position in this region produce neither restoring nor disrupting forces A ball lying on a flat, horizontal surface is an example of an object in neutral equilibrium Fs Ϫx max x max x a The restoring force exerted by the spring always acts toward x ϭ 0, the position of stable equilibrium S Fs m xϭ0 b x max Figure 7.20 (a) Potential energy as a function of x for the frictionless block–spring system shown in (b) For a given energy E of the system, the block oscillates between the turning points, which have the coordinates x 6x max Pitfall Prevention 7.10 Energy Diagrams A common mistake is to think that potential energy on the graph in an energy diagram represents the height of some object For example, that is not the case in Figure 7.20, where the block is only moving horizontally U Figure 7.21 A plot of U versus x for a particle that has a position of unstable equilibrium located at x For any finite displacement of the particle, the force on the particle is directed away from x Positive slope xϽ0 Negative slope xϾ0 x Summary 201 Example 7.9 Force and Energy on an Atomic Scale The potential energy associated with the force between two neutral atoms in a molecule can be modeled by the Lennard–Jones potential energy function: s 12 s U x 4P c a b a b d x x where x is the separation of the atoms The function U(x) contains two parameters s and P that are determined from experiments Sample values for the interaction between two atoms in a molecule are s 0.263 nm and P 1.51 10222 J Using a spreadsheet or similar tool, graph this function and find the most likely distance between the two atoms S o l u ti o n Conceptualize We identify the two atoms in the molecule as a system Based on our understanding that stable molecules exist, we expect to find stable equilibrium when the two atoms are separated by some equilibrium distance Categorize Because a potential energy function exists, we categorize the force between the atoms as conservative For a conservative force, Equation 7.29 describes the relationship between the force and the potential energy function Analyze Stable equilibrium exists for a separation distance at which the potential energy of the system of two atoms (the molecule) is a minimum dU x d s 12 s 212s12 6s6 d 4P c a b a b d 4P c 13 x dx dx x x x Take the derivative of the function U(x): 212s12 6s6 d 50 13 x eq x eq S x eq 2 1/6s Minimize the function U(x) by setting its derivative equal to zero: 4P c Evaluate x eq, the equilibrium separation of the two atoms in the molecule: x eq 2 1/6 0.263 nm 2.95 10210 m We graph the Lennard–Jones function on both sides of this critical value to create our energy diagram as shown in Figure 7.22 U (10Ϫ23 J ) x (10Ϫ10 m) Finalize Notice that U(x) is extremely large when the atoms are very close together, is a minimum when the atoms are at their critical separation, and then increases again as the atoms move apart When U(x) is a minimum, the atoms are in stable equilibrium, indicating that the most likely separation between them occurs at this point –10 x eq –20 Figure 7.22 (Example 7.9) Potential energy curve associated with a molecule The distance x is the separation between the two atoms making up the molecule Summary Definitions A system is most often a single particle, a collection of particles, or a region of space, and may vary in size and shape A system boundary separates the system from the environment The work W done on a system by an agent exerting a constant S force F on the system is the product of the magnitude Dr of the displacement of the point of application of the force and the component F cos u of the force along the direction of the displacement DS r: W ; F Dr cos u continued (7.1) continued 202 Chapter 7 Energy of a System If a varying force does work on a particle as the particle moves along the x axis from xi to xf , the work done by the force on the particle is given by W Fx dx xf (7.7) xi where Fx is the component of force in the x direction The kinetic energy of a particle of mass m moving with a speed v is K ; 12mv (7.16) The scalar product (dot product) of two S S vectors A and B is defined by the relationship S S A ? B ; AB cos u (7.2) where the result is a scalar quantity and u is the angle between the two vectors The scalar product obeys the commutative and distributive laws If a particle of mass m is at a distance y above the Earth’s surface, the gravitational potential energy of the particle–Earth system is (7.19) Ug ; mgy The elastic potential energy stored in a spring of force constant k is Us ; 12kx A force is conservative if the work it does on a particle that is a member of the system as the particle moves between two points is independent of the path the particle takes between the two points Furthermore, a force is conservative if the work it does on a particle is zero when the particle moves through an arbitrary closed path and returns to its initial position A force that does not meet these criteria is said to be nonconservative (7.22) The total mechanical energy of a system is defined as the sum of the kinetic energy and the potential energy: E mech ; K U (7.25) Concepts and Principles The work–kinetic energy theorem states that if work is done on a system by external forces and the only change in the system is in its speed, Wext K f K i DK 12mv f 2 12mv i (7.15, 7.17) A potential energy function U can be associated only with S a conservative force If a conservative force F acts between members of a system while one member moves along the x axis from xi to xf , the change in the potential energy of the system equals the negative of the work done by that force: Uf Ui 23 Fx dx xf (7.27) xi Systems can be in three types of equilibrium configurations when the net force on a member of the system is zero Configurations of stable equilibrium correspond to those for which U(x) is a minimum Objective Questions Configurations of unstable equilibrium correspond to those for which U(x) is a maximum Neutral equilibrium arises when U is constant as a member of the system moves over some region 1. denotes answer available in Student Solutions Manual/Study Guide Alex and John are loading identical cabinets onto a truck Alex lifts his cabinet straight up from the ground to the bed of the truck, whereas John slides his cabinet up a rough ramp to the truck Which statement is correct about the work done on the cabinet– Earth system? (a) Alex and John the same amount of work (b) Alex does more work than John (c) John does more work than Alex (d) None of those state- ments is necessarily true because the force of friction is unknown (e) None of those statements is necessarily true because the angle of the incline is unknown If the net work done by external forces on a particle is zero, which of the following statements about the particle must be true? (a) Its velocity is zero (b) Its velocity is decreased (c) Its velocity is unchanged (d) Its speed is unchanged (e) More information is needed 203 Objective Questions A worker pushes a wheelbarrow with a horizontal force of 50 N on level ground over a distance of 5.0 m If a friction force of 43 N acts on the wheelbarrow in a direction opposite that of the worker, what work is done on the wheelbarrow by the worker? (a) 250 J (b) 215 J (c) 35 J (d) 10 J (e) None of those answers is correct A cart is set rolling across a level table, at the same speed on every trial If it runs into a patch of sand, the cart exerts on the sand an average horizontal force of N and travels a distance of cm through the sand as it comes to a stop If instead the cart runs into a patch of gravel on which the cart exerts an average horizontal force of N, how far into the gravel will the cart roll before stopping? (a) cm (b) 6 cm (c) cm (d) cm (e) none of those answers ^ represent the direction horizontally north, Let N NE represent northeast (halfway between north and east), and so on Each direction specification can be thought of as a unit vector Rank from the largest to the smallest the following dot products Note that zero is larger than a negative number If two quantities ^ ?N ^ are equal, display that fact in your ranking (a) N ^ ? NE (c) N ^ ? S^ (d) N ^ ? E^ (e) SE ? S^ (b) N 7 Is the work required to be done by an external force on an object on a frictionless, horizontal surface to accelerate it from a speed v to a speed 2v (a) equal to the work required to accelerate the object from v to v, (b) twice the work required to accelerate the object from v to v, (c) three times the work required to accelerate the object from v 5 to v, (d) four times the work required to accelerate the object from to v, or (e) not known without knowledge of the acceleration? A block of mass m is dropped from the fourth floor of an office building and hits the sidewalk below at speed v From what floor should the block be dropped to double that impact speed? (a) the sixth floor (b) the eighth floor (c) the tenth floor (d) the twelfth floor (e) the sixteenth floor As a simple pendulum swings back and forth, the forces acting on the suspended object are (a) the gravitational force, (b) the tension in the supporting cord, and (c) air resistance (i) Which of these forces, if any, does no work on the pendulum at any time? (ii) Which of these forces does negative work on the pendulum at all times during its motion? Bullet has twice the mass of bullet Both are fired so that they have the same speed If the kinetic energy of bullet is K, is the kinetic energy of bullet (a) 0.25K, (b) 0.5K, (c) 0.71K, (d) K, or (e) 2K? 10 Figure OQ7.10 shows a light extended spring exerting a force Fs to the left on a block (i) Does the block exert a force on the spring? Choose every correct answer (a) No, it doesn’t (b) Yes, it does, to the left (c) Yes, it does, to the right (d) Yes, it does, and its magnitude is larger than Fs (e) Yes, it does, and its magnitude is equal to Fs (ii) Does the spring exert a force S xϭ0 Fs x x Figure OQ7.10 on the wall? Choose your answers from the same list (a) through (e) 11 If the speed of a particle is doubled, what happens to its kinetic energy? (a) It becomes four times larger (b) It becomes two times larger (c) It becomes !2 times larger (d) It is unchanged (e) It becomes half as large 12 Mark and David are loading identical cement blocks onto David’s pickup truck Mark lifts his block straight up from the ground to the truck, whereas David slides his block up a ramp containing frictionless rollers Which statement is true about the work done on the block–Earth system? (a) Mark does more work than David (b) Mark and David the same amount of work (c) David does more work than Mark (d) None of those statements is necessarily true because the angle of the incline is unknown (e) None of those statements is necessarily true because the mass of one block is not given 13 (i) Rank the gravitational accelerations you would measure for the following falling objects: (a) a 2-kg object cm above the floor, (b) a 2-kg object 120 cm above the floor, (c) a 3-kg object 120 cm above the floor, and (d) a 3-kg object 80 cm above the floor List the one with the largest magnitude of acceleration first If any are equal, show their equality in your list (ii) Rank the gravitational forces on the same four objects, listing the one with the largest magnitude first (iii) Rank the gravitational potential energies (of the object–Earth system) for the same four objects, largest first, taking y at the floor 14 A certain spring that obeys Hooke’s law is stretched by an external agent The work done in stretching the spring by 10 cm is J How much additional work is required to stretch the spring an additional 10 cm? (a) J (b) J (c) J (d) 12 J (e) 16 J 15 A cart is set rolling across a level table, at the same speed on every trial If it runs into a patch of sand, the cart exerts on the sand an average horizontal force of N and travels a distance of cm through the sand as it comes to a stop If instead the cart runs into a patch of flour, it rolls an average of 18 cm before stopping What is the average magnitude of the horizontal force the cart exerts on the flour? (a) N (b) N (c) N (d) 18 N (e) none of those answers 16 An ice cube has been given a push and slides without friction on a level table Which is correct? (a) It is in stable equilibrium (b) It is in unstable equilibrium (c) It is in neutral equilibrium (d) It is not in equilibrium 204 Chapter 7 Energy of a System Conceptual Questions 1. denotes answer available in Student Solutions Manual/Study Guide Can a normal force work? If not, why not? If so, give an example Object pushes on object as the objects move together, like a bulldozer pushing a stone Assume object does 15.0 J of work on object Does object work on object 1? Explain your answer If possible, determine how much work and explain your reasoning A student has the idea that the total work done on an object is equal to its final kinetic energy Is this idea true always, sometimes, or never? If it is sometimes true, under what circumstances? If it is always or never true, explain why (a) For what values of the angle u between two vectors is their scalar product positive? (b) For what values of u is their scalar product negative? Can kinetic energy be negative? Explain Discuss the work done by a pitcher throwing a baseball What is the approximate distance through which the force acts as the ball is thrown? Discuss whether any work is being done by each of the following agents and, if so, whether the work is positive or negative (a) a chicken scratching the ground (b) a person studying (c) a crane lifting a bucket of concrete (d) the gravitational force on the bucket in part (c) (e) the leg muscles of a person in the act of sitting down If only one external force acts on a particle, does it necessarily change the particle’s (a) kinetic energy? (b) Its velocity? Preparing to clean them, you pop all the removable keys off a computer keyboard Each key has the shape of a tiny box with one side open By accident, you spill the keys onto the floor Explain why many more keys land letter-side down than land open-side down 10 You are reshelving books in a library You lift a book from the floor to the top shelf The kinetic energy of the book on the floor was zero and the kinetic energy of the book on the top shelf is zero, so no change occurs in the kinetic energy, yet you did some work in lifting the book Is the work–kinetic energy theorem violated? Explain 11 A certain uniform spring has spring constant k Now the spring is cut in half What is the relationship between k and the spring constant k9 of each resulting smaller spring? Explain your reasoning 12 What shape would the graph of U versus x have if a particle were in a region of neutral equilibrium? 13 Does the kinetic energy of an object depend on the frame of reference in which its motion is measured? Provide an example to prove this point 14 Cite two examples in which a force is exerted on an object without doing any work on the object Problems The problems found in this chapter may be assigned online in Enhanced WebAssign straightforward; intermediate; challenging full solution available in the Student Solutions Manual/Study Guide AMT Analysis Model tutorial available in Enhanced WebAssign GP Guided Problem M Master It tutorial available in Enhanced WebAssign W Watch It video solution available in Enhanced WebAssign BIO Q/C S Section 7.2 Work Done by a Constant Force A shopper in a supermarket pushes a cart with a Q/C force of 35.0 N directed at an angle of 25.08 below the horizontal The force is just sufficient to balance various friction forces, so the cart moves at constant speed (a) Find the work done by the shopper on the cart as she moves down a 50.0-m-long aisle (b) The shopper goes down the next aisle, pushing horizontally and maintaining the same speed as before If the friction force doesn’t change, would the shopper’s applied force be larger, smaller, or the same? (c) What about the work done on the cart by the shopper? A raindrop of mass 3.35 1025 kg falls vertically at W constant speed under the influence of gravity and air resistance Model the drop as a particle As it falls 100 m, what is the work done on the raindrop (a) by the gravitational force and (b) by air resistance? In 1990, Walter Arfeuille of Belgium lifted a 281.5-kg object through a distance of 17.1 cm using only his teeth (a) How much work was done on the object by Arfeuille in this lift, assuming the object was lifted at constant speed? (b) What total force was exerted on Arfeuille’s teeth during the lift? The record number of boat lifts, including the boat and its ten crew members, was achieved by Sami Heinonen and Juha Räsänen of Sweden in 2000 They lifted a total mass of 653.2 kg approximately in off the ground a total of 24 times Estimate the total work done by the two men on the boat in this record lift, ignoring the negative work done by the men when they lowered the boat back to the ground 205 Problems S A block of mass m F 2.50 kg is pushed a disu tance d 5 2.20 m along m a frictionless, horizontal table by a constant applied d force of magnitude F 16.0 N directed at an angle Figure P7.5 u 25.08 below the horizontal as shown in Figure P7.5 Determine the work done on the block by (a) the applied force, (b) the normal force exerted by the table, (c) the gravitational force, and (d) the net force on the block Section 7.4 Work Done by a Varying Force Spiderman, whose mass is 80.0 kg, is dangling on the M free end of a 12.0-m-long rope, the other end of which is fixed to a tree limb above By repeatedly bending at the waist, he is able to get the rope in motion, eventually getting it to swing enough that he can reach a ledge when the rope makes a 60.08 angle with the vertical How much work was done by the gravitational force on Spiderman in this maneuver? Section 7.3 The Scalar Product of Two Vectors S S S S that SA ? B A x Bx For any two vectors A and B , show S S Ay By A z Bz Suggestions: Write A and B in unit-vector form and use Equations 7.4 and 7.5 S S Vector A has a magnitude of 5.00 units, and vector B has a magnitude of 9.00 units The twoS vectors make S an angle of 50.08 with each other Find A ? B Note: In Problems through 12, calculate numerical answers to three significant figures as usual S S S For A i^ j^ 2Sk^ , BS i^ j^ 5k^ , and C S W j^ 3k^ , find C ? A B 10 Find the scalar product of the vectors in Figure P7.10 y 118Њ x 32.8 N Fx (N) 4 10 x (m) Ϫ2 Figure P7.14 15 A particle is subject to a force Fx that varies with posiW tion as shown in Figure P7.15 Find the work done by the force on the particle as it moves (a) from x to x 5.00 m, (b) from x 5.00 m to x 10.0 m, and (c) from x 10.0 m to x 15.0 m (d) What is the total work done by the force over the distance x to x 15.0 m? Fx (N) 2 10 12 14 16 x (m) Figure P7.15 Problems 15 and 34 16 In a control system, an accelerometer consists of a 4.70-g object sliding on a calibrated horizontal rail A low-mass spring attaches the object to a flange at one end of the rail Grease on the rail makes static friction negligible, but rapidly damps out vibrations of the sliding object When subject to a steady acceleration of 0.800g, the object should be at a location 0.500 cm away from its equilibrium position Find the force constant of the spring required for the calibration to be correct 17 When a 4.00-kg object is vertically on a cer- 132Њ AMT tain light spring that obeys Hooke’s law, the spring M stretches 2.50 cm If the 4.00-kg object is removed, 17.3 cm Figure P7.10 S 11 A force F i^ 2 j^ N acts on a particle that underS M goes a displacement D r i^ j^ m Find (a) the work done by the force on the particle and (b) the angle S r between F and DS 12 Using the definition Sof the scalar product, find the ^ 2 ^j and S i^ 2S4 j^ , i B angles between (a) A S S (b) A 22i^ ^j Sand B 3i^ ^j 2k^ , and (c) A i^ 2 j^ 2k^ and B j^ 4k^ S 14 The force acting on a particle varies as shown in Figure M P7.14 Find the work done by the force on the particle W as it moves (a) from x to x 8.00 m, (b) from x 8.00 m to x 10.0 m, and (c) from x to x 10.0 m S 13 Let B 5.00 mSat 60.0° Let the vector C have the same angle greater than magnitude as A and aSdirection S S S S B?C that of A by 25.0° Let A ? B 30.0 m2 and S 35.0 m2 Find the magnitude and direction of A (a) how far will the spring stretch if a 1.50-kg block is on it? (b) How much work must an external agent to stretch the same spring 4.00 cm from its unstretched position? 18 Hooke’s law describes a certain light spring of unstretched length 35.0 cm When one end is attached to the top of a doorframe and a 7.50-kg object is from the other end, the length of the spring is 41.5 cm (a) Find its spring constant (b) The load and the spring are taken down Two people pull in opposite directions on the ends of the spring, each with a force of 190 N Find the length of the spring in this situation 19 An archer pulls her bowstring back 0.400 m by exerting a force that increases uniformly from zero to 230 N (a) What is the equivalent spring constant of the bow? 206 Chapter 7 Energy of a System (b) How much work does the archer on the string in drawing the bow? 20 A light spring with spring constant 200 N/m is from an elevated support From its lower end hangs a second light spring, which has spring constant 800 N/m An object of mass 1.50 kg is at rest from the lower end of the second spring (a) Find the total extension distance of the pair of springs (b) Find the effective spring constant of the pair of springs as a system We describe these springs as in series 21 A light spring with spring constant k1 is from an S elevated support From its lower end a second light spring is hung, which has spring constant k An object of mass m is at rest from the lower end of the second spring (a) Find the total extension distance of the pair of springs (b) Find the effective spring constant of the pair of springs as a system 22 Express the units of the force constant of a spring in SI S fundamental units 23 A cafeteria tray dispenser supports a stack of trays on a shelf that hangs from four identical spiral springs under tension, one near each corner of the shelf Each tray is rectangular, 45.3 cm by 35.6 cm, 0.450 cm thick, and with mass 580 g (a) Demonstrate that the top tray in the stack can always be at the same height above the floor, however many trays are in the dispenser (b) Find the spring constant each spring should have for the dispenser to function in this convenient way (c) Is any piece of data unnecessary for this determination? 24 A light spring with force constant 3.85 N/m is compressed by 8.00 cm as it is held between a 0.250-kg block on the left and a 0.500-kg block on the right, both resting on a horizontal surface The spring exerts a force on each block, tending to push the blocks apart The blocks are simultaneously released from rest Find the acceleration with which each block starts to move, given that the coefficient of kinetic friction between each block and the surface is (a) 0, (b) 0.100, and (c) 0.462 S 25 A small particle of mass F S m is pulled to the top of a frictionless halfm cylinder (of radius R) by R a light cord that passes u over the top of the cylinder as illustrated in Figure P7.25 (a) AssumFigure P7.25 ing the particle moves at a constant speed, show that F mg cos u Note: If the particle moves at constant speed, the component of its acceleration tangent to the cylinder must be zero at all S r , find times (b) By directly integrating W e F ? d S the work done in moving the particle at constant speed from the bottom to the top of the half-cylinder 26 The force acting on a particle is Fx (8x 16), where F is in newtons and x is in meters (a) Make a plot of this force versus x from x to x 3.00 m (b) From your graph, find the net work done by this force on the particle as it moves from x to x 3.00 m 27 When different loads hang on a spring, the spring Q/C stretches to different lengths as shown in the follow- ing table (a) Make a graph of the applied force versus the extension of the spring (b) By least-squares fitting, determine the straight line that best fits the data (c) To complete part (b), you want to use all the data points, or should you ignore some of them? Explain (d) From the slope of the best-fit line, find the spring constant k (e) If the spring is extended to 105 mm, what force does it exert on the suspended object? F (N) 2.0 4.0 6.0 8.0 10 12 14 16 18 20 22 L (mm) 15 32 49 64 79 98 112 126 149 175 190 A 100-g bullet is fired from a rifle having a barrel 0.600 m long Choose the origin to be at the location where the bullet begins to move Then the force (in newtons) exerted by the expanding gas on the bullet is 15 000 10 000x 25 000x 2, where x is in meters (a) Determine the work done by the gas on the bullet as the bullet travels the length of the barrel (b) What If? If the barrel is 1.00 m long, how much work is done, and (c) how does this value compare with the work calculated in part (a)? S S 29 A force F 4x i^ 3y j^ , where F is in newtons and W x and y are in meters, acts on an object as the object moves in the x direction from the origin to x 5.00 m.SFind the work u (N) r done by W e F ?d S the force on the object b 30 Review The graph in Figure P7.30 specifies a functional relationship between the two variables u and v (a) Find b a ea u dv (b)b Find eb u dv (c) Find ea v du –4 a 10 20 v (cm) 30 Figure P7.30 Section 7.5 Kinetic Energy and the Work–Kinetic Energy Theorem 31 A 3.00-kg object has a velocity 6.00 i^ 2.00 j^ m/s W (a) What is its kinetic energy at this moment? (b) What is the net work done on the object if its velocity changes to 8.00 i^ 4.00 j^ m/s? (Note: From the definition of v ?S v ) the dot product, v S 32 A worker pushing a 35.0-kg wooden crate at a constant AMT speed for 12.0 m along a wood floor does 350 J of work Q/C by applying a constant horizontal force of magnitude F on the crate (a) Determine the value of F (b) If the worker now applies a force greater than F, describe the subsequent motion of the crate (c) Describe what would happen to the crate if the applied force is less than F 33 A 0.600-kg particle has a speed of 2.00 m/s at point A W and kinetic energy of 7.50 J at point B What is (a) its kinetic energy at A, (b) its speed at B, and (c) the net work done on the particle by external forces as it moves from A to B? Problems 34 A 4.00-kg particle is subject to a net force that varies W with position as shown in Figure P7.15 The particle starts from rest at x What is its speed at (a) x 5.00 m, (b) x 10.0 m, and (c) x 15.0 m? 35 A 100-kg pile driver is used to drive a steel I-beam into M the ground The pile driver falls 5.00 m before coming into contact with the top of the beam, and it drives the beam 12.0 cm farther into the ground before coming to rest Using energy considerations, calculate the average force the beam exerts on the pile driver while the pile driver is brought to rest 36 Review In an electron microscope, there is an electron AMT gun that contains two charged metallic plates 2.80 cm apart An electric force accelerates each electron in the beam from rest to 9.60% of the speed of light over this distance (a) Determine the kinetic energy of the electron as it leaves the electron gun Electrons carry this energy to a phosphorescent viewing screen where the microscope’s image is formed, making it glow For an electron passing between the plates in the electron gun, determine (b) the magnitude of the constant electric force acting on the electron, (c) the acceleration of the electron, and (d) the time interval the electron spends between the plates 37 Review You can think of the work–kinetic energy theGP orem as a second theory of motion, parallel to NewQ/C ton’s laws in describing how outside influences affect the motion of an object In this problem, solve parts (a), (b), and (c) separately from parts (d) and (e) so you can compare the predictions of the two theories A 15.0-g bullet is accelerated from rest to a speed of 780 m/s in a rifle barrel of length 72.0 cm (a) Find the kinetic energy of the bullet as it leaves the barrel (b) Use the work–kinetic energy theorem to find the net work that is done on the bullet (c) Use your result to part (b) to find the magnitude of the average net force that acted on the bullet while it was in the barrel (d) Now model the bullet as a particle under constant acceleration Find the constant acceleration of a bullet that starts from rest and gains a speed of 780 m/s over a distance of 72.0 cm (e) Modeling the bullet as a particle under a net force, find the net force that acted on it during its acceleration (f) What conclusion can you draw from comparing your results of parts (c) and (e)? 38 Review A 7.80-g bullet moving at 575 m/s strikes the hand of a superhero, causing the hand to move 5.50 cm in the direction of the bullet’s velocity before stopping (a) Use work and energy considerations to find the average force that stops the bullet (b) Assuming the force is constant, determine how much time elapses between the moment the bullet strikes the hand and the moment it stops moving 39 Review A 5.75-kg object passes through the origin at time t such that its x component of velocity is 5.00 m/s and its y component of velocity is 23.00 m/s (a) What is the kinetic energy of the object at this time? (b) At a later time t 2.00 s, the particle is located at x 8.50 m and y 5.00 m What constant force acted 207 on the object during this time interval? (c) What is the speed of the particle at t 5 2.00 s? Section 7.6 Potential Energy of a System 40 A 000-kg roller coaster car is initially at the top of a rise, at point A It then moves 135 ft, at an angle of 40.08 below the horizontal, to a lower point B (a) Choose the car at point B to be the zero configuration for gravitational potential energy of the roller coaster– Earth system Find the potential energy of the system when the car is at points A and B, and the change in potential energy as the car moves between these points (b) Repeat part (a), setting the zero configuration with the car at point A 41 A 0.20-kg stone is held 1.3 m above the top edge of a water well and then dropped into it The well has a depth of 5.0 m Relative to the configuration with the stone at the top edge of the well, what is the gravitational potential energy of the stone–Earth system (a) before the stone is released and (b) when it reaches the bottom of the well? (c) What is the change in gravitational potential energy of the system from release to reaching the bottom of the well? 42 A 400-N child is in a swing that is attached to a pair W of ropes 2.00 m long Find the gravitational potential energy of the child–Earth system relative to the child’s lowest position when (a) the ropes are horizontal, (b) the ropes make a 30.08 angle with the vertical, and (c) the child is at the bottom of the circular arc Section 7.7 Conservative and Nonconservative Forces y (m) 43 A 4.00-kg particle moves M from the origin to posiC Q/C tion C, having coordi(5.00, 5.00) B nates x 5.00 m and y 5.00 m (Fig. P7.43) One force on the particle is the gravitational force acting in the negative y x (m) O A direction Using Equation 7.3, calculate the Figure P7.43 work done by the graviProblems 43 through 46 tational force on the particle as it goes from O to C along (a) the purple path, (b) the red path, and (c) the blue path (d) Your results should all be identical Why? 4 (a) Suppose a constant force acts on an object The force does not vary with time or with the position or the velocity of the object Start with the general definition for work done by a force W F ?d S r f S i and show that the force is conservative (b) As a speS cial case, suppose the force F i^ 4j^ N acts on a particle that moves from S O to C in Figure P7.43 Calculate the work done by F on the particle as it moves along each one of the three paths shown in the figure 208 Chapter 7 Energy of a System and show that the work done along the three paths is identical 45 A force acting on a particle moving Sin the xy plane is S is in newtons M given by F 2y i^ x j^ , where F Q/C and x and y are in meters The particle moves from the origin to a final position having coordinates x 5.00 m and y 5.00 m as shown in Figure P7.43 S Calculate the work done by F on the particle as it moves along (a) the purple path, (b) the red path, and S (c) the blue path (d) Is F conservative or nonconservative? (e) Explain your answer to part (d) 46 An object moves in the xy plane in Figure P7.43 and Q/C experiences a friction force with constant magnitude 3.00 N, always acting in the direction opposite the object’s velocity Calculate the work that you must to slide the object at constant speed against the friction force as the object moves along (a) the purple path O to A followed by a return purple path to O, (b) the purple path O to C followed by a return blue path to O, and (c) the blue path O to C followed by a return blue path to O (d) Each of your three answers should be nonzero What is the significance of this observation? Section 7.8 Relationship Between Conservative Forces and Potential Energy 47 The potential energy of a system of two particles sepa5 A/r, where A S rated by a distance r is given by U(r) S is a constant Find the radial force F r that each particle exerts on the other 48 Why is the following situation impossible? A librarian lifts a book from the ground to a high shelf, doing 20.0 J of work in the lifting process As he turns his back, the book falls off the shelf back to the ground The gravitational force from the Earth on the book does 20.0 J of work on the book while it falls Because the work done was 20.0 J 20.0 J 40.0 J, the book hits the ground with 40.0 J of kinetic energy 49 A potential energy function for a system in which a two-dimensional force acts is of the form U 3x 3y 7x Find the force that acts at the point (x, y) 50 A single conservative force acting on a particle within a S 2Ax Bx 2 ^i, where A and B are F system varies as S constants, F is in newtons, and x is in meters (a) Calculate the potential energy function U(x) associated with this force for the system, taking U at x Find (b) the change in potential energy and (c) the change in kinetic energy of the system as the particle moves from x 2.00 m to x 3.00 m 51 A single conservative force acts on a 5.00-kg particle M within a system due to its interaction with the rest of the system The equation Fx 2x describes the force, where Fx is in newtons and x is in meters As the particle moves along the x axis from x 1.00 m to x 5.00 m, calculate (a) the work done by this force on the particle, (b) the change in the potential energy of the system, and (c) the kinetic energy the particle has at x 5.00 m if its speed is 3.00 m/s at x 1.00 m Section 7.9 Energy Diagrams and Equilibrium of a System 52 For the potential U ( J) energy curve shown A in Figure P7.52, (a) determine wheB ther the force Fx is positive, negative, or x (m) 10 zero at the five points indicated –2 (b) Indicate points C of stable, unstable, –4 and neutral equilibFigure P7.52 rium (c) Sketch the curve for Fx versus x from x 5 to x 9.5 m 53 A right circular cone can theoretically be balanced on a horizontal surface in three different ways Sketch these three equilibrium configurations and identify them as positions of stable, unstable, or neutral equilibrium Additional Problems The potential energy function for a system of particles is given by U(x) 2x 2x 3x, where x is the position of one particle in the system (a) Determine the force Fx on the particle as a function of x (b) For what values of x is the force equal to zero? (c) Plot U(x) versus x and Fx versus x and indicate points of stable and unstable equilibrium 55 Review A baseball outfielder throws a 0.150-kg baseball at a speed of 40.0 m/s and an initial angle of 30.08 to the horizontal What is the kinetic energy of the baseball at the highest point of its trajectory? 56 A particle moves along the x axis from x 12.8 m to x 23.7 m under the influence of a force F5 375 x 3.75x where F is in newtons and x is in meters Using numerical integration, determine the work done by this force on the particle during this displacement Your result should be accurate to within 2% 57 Two identical steel balls, each of diameter 25.4 mm Q/C and moving in opposite directions at m/s, run into each other head-on and bounce apart Prior to the collision, one of the balls is squeezed in a vise while precise measurements are made of the resulting amount of compression The results show that Hooke’s law is a fair model of the ball’s elastic behavior For one datum, a force of 16 kN exerted by each jaw of the vise results in a 0.2-mm reduction in the diameter The diameter returns to its original value when the force is removed (a) Modeling the ball as a spring, find its spring constant (b) Does the interaction of the balls during the collision last only for an instant or for a nonzero time interval? State your evidence (c) Compute an estimate for the kinetic energy of each of the balls before they collide (d) Compute an estimate for the maximum amount of compression each ball undergoes when the balls collide (e) Compute an order-of-magnitude estimate for the time interval for which the balls are in 209 Problems contact (In Chapter 15, you will learn to calculate the contact time interval precisely.) 58 When an object is displaced by an amount x from staS ble equilibrium, a restoring force acts on it, tending to return the object to its equilibrium position The magnitude of the restoring force can be a complicated function of x In such cases, we can generally imagine the force function F(x) to be expressed as a power series in x as F(x) 2(k1x k 2x 2 k 3x 1 . . .) The first term here is just Hooke’s law, which describes the force exerted by a simple spring for small displacements For small excursions from equilibrium, we generally ignore the higher-order terms, but in some cases it may be desirable to keep the second term as well If we model the restoring force as F 2(k1x k 2x 2), how much work is done on an object in displacing it from x to x x max by an applied force 2F ? 59 A 000-kg freight car rolls along rails with negligible friction The car is brought to rest by a combination of two coiled springs as illustrated in Figure P7.59 Both springs are described by Hooke’s law and have spring constants k1 5 600 N/m and k 400 N/m After the first spring compresses a distance of 30.0 cm, the second spring acts with the first to increase the force as additional compression occurs as shown in the graph The car comes to rest 50.0 cm after first contacting the two-spring system Find the car’s initial speed k1 k2 Total force (N) 000 500 000 500 10 20 30 40 50 Distance (cm) 60 Figure P7.59 60 Why is the following situation impossible? In a new casino, a supersized pinball machine is introduced Casino advertising boasts that a professional basketball player can lie on top of the machine and his head and feet will not hang off the edge! The ball launcher in the machine sends metal balls up one side of the machine and then into play The spring in the launcher (Fig P7.60) has a force constant of 1.20 N/cm The surface on which the ball moves is inclined u 10.08 with respect to the horizontal The spring is initially compressed its maximum distance d 5.00 cm A ball of mass 100 g is projected into play by releasing the plunger Casino visitors find the play of the giant machine quite exciting u Figure P7.60 61 Review Two constant forces act on an object of mass Q/C m 5.00 kg moving in the xy plane as shown in S S Figure P7.61 Force F is 25.0 N at 35.08, and force F is 42.0 N at 1508 At time t 0, the object is at the origin and has velocity 4.00 i^ 2.50 j^ m/s (a) Express the two forces in unit-vector notation Use unit-vector notation for your other answers (b) Find the total force exerted on the object (c) Find the object’s acceleration Now, considering the instant t 3.00 s, find (d) the object’s velocity, (e) its y position, (f) its kinetic energy from 2mv f , and S S F2 (g) its kinetic Senergy F1 S from 2mv i g F ? D r 150Њ (h) What conclusion 35.0Њ x m can you draw by comparing the answers to Figure P7.61 parts (f) and (g)? The spring constant of an automotive suspension spring increases with increasing load due to a spring coil that is widest at the bottom, smoothly tapering to a smaller diameter near the top The result is a softer ride on normal road surfaces from the wider coils, but the car does not bottom out on bumps because when the lower coils collapse, the stiffer coils near the top absorb the load For such springs, the force exerted by the spring can be empirically found to be given by F ax b For a tapered spiral spring that compresses 12.9 cm with a 000-N load and 31.5 cm with a 000-N load, (a) evaluate the constants a and b in the empirical equation for F and (b) find the work needed to compress the spring 25.0 cm 63 An inclined plane of angle u 20.08 has a m d spring of force constant S v k 500 N/m fastened k securely at the bottom u so that the spring is parallel to the surface as shown in Figure P7.63 Figure P7.63 A block of mass m Problems 63 and 64 2.50 kg is placed on the plane at a distance d 5 0.300 m from the spring From this position, the block is projected downward toward the spring with speed v 0.750 m/s By what distance is the spring compressed when the block momentarily comes to rest? An inclined plane of angle u has a spring of force S constant k fastened securely at the bottom so that the 210 Chapter 7 Energy of a System spring is parallel to the surface A block of mass m is placed on the plane at a distance d from the spring From this position, the block is projected downward toward the spring with speed v as shown in Figure P7.63 By what distance is the spring compressed when the block momentarily comes to rest? 65 (a) Take U 5 for a system with a particle at position x Q/C and calculate the potential energy of the system as a function of the particle position x The force on the particle is given by (8e22x) ^i (b) Explain whether the force is conservative or nonconservative and how you can tell Challenge Problems 6 A particle of mass m 1.18 kg is attached between two identical springs on a frictionless, horizontal tabletop Both springs have spring constant k and are initially unstressed, and the particle is at x (a) The particle is pulled a distance x along a direction perpendicular to the initial configuration of the springs as shown in Figure P7.66 Show that the force exerted by the springs on the particle is S F 22kx a1 L "x L2 b ^i (b) Show that the potential energy of the system is U x kx 2kL L "x L2 (c) Make a plot of U(x) versus x and identify all equilibrium points Assume L 1.20 m and k 40.0 N/m (d) If the particle is pulled 0.500 m to the right and then released, what is its speed when it reaches x 0? Overhead view k L x xϭ0 L m x k Figure P7.66 67 Review A light spring Q/C has unstressed length 15.5 cm It is described by Hooke’s law with spring constant 4.30 N/m One end of the horizontal spring is held on a fixed vertical axle, and the other end is attached to a puck of mass m that can move without friction over a horizontal surface The puck is set into motion in a circle with a period of 1.30 s (a) Find the extension of the spring x as it depends on m Evaluate x for (b) m 0.070 kg, (c) m 0.140 kg, (d) m 0.180 kg, and (e) m 0.190 kg (f) Describe the pattern of variation of x as it depends on m Conservation of Energy c h a p t e r 8.1 Analysis Model: Nonisolated System (Energy) 8.2 Analysis Model: Isolated System (Energy) 8.3 Situations Involving Kinetic Friction 8.4 Changes in Mechanical Energy for Nonconservative Forces 8.5 Power In Chapter 7, we introduced three methods for storing energy in a system: kinetic energy, associated with movement of members of the system; potential energy, determined by the configuration of the system; and internal energy, which is related to the temperature of the system We now consider analyzing physical situations using the energy approach for two types of systems: nonisolated and isolated systems For nonisolated systems, we shall investigate ways that energy can cross the boundary of the system, resulting in a change in the system’s total energy This analysis leads to a critically important principle called conservation of energy The conservation of energy principle extends well beyond physics and can be applied to biological organisms, technological systems, and engineering situations In isolated systems, energy does not cross the boundary of the system For these systems, the total energy of the system is constant If no nonconservative forces act within the system, we can use conservation of mechanical energy to solve a variety of problems Three youngsters enjoy the transformation of potential energy to kinetic energy on a waterslide We can analyze processes such as these with the techniques developed in this chapter (Jade Lee/Asia Images/Getty Images) 211 212 Chapter 8 Conservation of Energy Situations involving the transformation of mechanical energy to internal energy due to nonconservative forces require special handling We investigate the procedures for these types of problems Finally, we recognize that energy can cross the boundary of a system at different rates We describe the rate of energy transfer with the quantity power 8.1 Analysis Model: Nonisolated System (Energy) As we have seen, an object, modeled as a particle, can be acted on by various forces, resulting in a change in its kinetic energy according to the work–kinetic energy theorem from Chapter If we choose the object as the system, this very simple situation is the first example of a nonisolated system, for which energy crosses the boundary of the system during some time interval due to an interaction with the environment This scenario is common in physics problems If a system does not interact with its environment, it is an isolated system, which we will study in Section 8.2 The work–kinetic energy theorem is our first example of an energy equation appropriate for a nonisolated system In the case of that theorem, the interaction of the system with its environment is the work done by the external force, and the quantity in the system that changes is the kinetic energy So far, we have seen only one way to transfer energy into a system: work We mention below a few other ways to transfer energy into or out of a system The details of these processes will be studied in other sections of the book We illustrate mechanisms to transfer energy in Figure 8.1 and summarize them as follows Work, as we have learned in Chapter 7, is a method of transferring energy to a system by applying a force to the system such that the point of application of the force undergoes a displacement (Fig 8.1a) Energy leaves the radio from the speaker by mechanical waves b a Energy enters the automobile gas tank by matter transfer d c Energy enters the hair dryer by electrical transmission Energy leaves the lightbulb by electromagnetic radiation © Cengage Learning/George Semple © Cengage Learning/George Semple fer mechanisms In each case, the system into which or from which energy is transferred is indicated Cocoon/Photodisc/Getty Images Figure 8.1 Energy trans- Energy transfers to the handle of the spoon by heat © Cengage Learning/George Semple © Cengage Learning/George Semple © Cengage Learning/George Semple Energy is transferred to the block by work e f 8.1 Analysis Model: Nonisolated System (Energy) 213 Mechanical waves (Chapters 16–18) are a means of transferring energy by allowing a disturbance to propagate through air or another medium It is the method by which energy (which you detect as sound) leaves the system of your clock radio through the loudspeaker and enters your ears to stimulate the hearing process (Fig 8.1b) Other examples of mechanical waves are seismic waves and ocean waves Heat (Chapter 20) is a mechanism of energy transfer that is driven by a temperature difference between a system and its environment For example, imagine dividing a metal spoon into two parts: the handle, which we identify as the system, and the portion submerged in a cup of coffee, which is part of the environment (Fig 8.1c) The handle of the spoon becomes hot because fast-moving electrons and atoms in the submerged portion bump into slower ones in the nearby part of the handle These particles move faster because of the collisions and bump into the next group of slow particles Therefore, the internal energy of the spoon handle rises from energy transfer due to this collision process Matter transfer (Chapter 20) involves situations in which matter physically crosses the boundary of a system, carrying energy with it Examples include filling your automobile tank with gasoline (Fig 8.1d) and carrying energy to the rooms of your home by circulating warm air from the furnace, a process called convection Electrical transmission (Chapters 27 and 28) involves energy transfer into or out of a system by means of electric currents It is how energy transfers into your hair dryer (Fig 8.1e), home theater system, or any other electrical device Electromagnetic radiation (Chapter 34) refers to electromagnetic waves such as light (Fig 8.1f), microwaves, and radio waves crossing the boundary of a system Examples of this method of transfer include cooking a baked potato in your microwave oven and energy traveling from the Sun to the Earth by light through space.1 A central feature of the energy approach is the notion that we can neither create nor destroy energy, that energy is always conserved This feature has been tested in countless experiments, and no experiment has ever shown this statement to be incorrect Therefore, if the total amount of energy in a system changes, it can only be because energy has crossed the boundary of the system by a transfer mechanism such as one of the methods listed above Energy is one of several quantities in physics that are conserved We will see other conserved quantities in subsequent chapters There are many physical quantities that not obey a conservation principle For example, there is no conservation of force principle or conservation of velocity principle Similarly, in areas other than physical quantities, such as in everyday life, some quantities are conserved and some are not For example, the money in the system of your bank account is a conserved quantity The only way the account balance changes is if money crosses the boundary of the system by deposits or withdrawals On the other hand, the number of people in the system of a country is not conserved Although people indeed cross the boundary of the system, which changes the total population, the population can also change by people dying and by giving birth to new babies Even if no people cross the system boundary, the births and deaths will change the number of people in the system There is no equivalent in the concept of energy to dying or giving birth The general statement of the principle of conservation of energy can be described mathematically with the conservation of energy equation as follows: DE system o T (8.1) where E system is the total energy of the system, including all methods of energy storage (kinetic, potential, and internal), and T (for transfer) is the amount of energy transferred across the system boundary by some mechanism Two of our transfer mechanisms have well-established symbolic notations For work, Twork W as discussed in Chapter 7, and for heat, Theat Q as defined in Chapter 20 (Now that we 1Electromagnetic radiation and work done by field forces are the only energy transfer mechanisms that not require molecules of the environment to be available at the system boundary Therefore, systems surrounded by a vacuum (such as planets) can only exchange energy with the environment by means of these two possibilities Pitfall Prevention 8.1 Heat Is Not a Form of Energy The word heat is one of the most misused words in our popular language Heat is a method of transferring energy, not a form of storing energy Therefore, phrases such as “heat content,” “the heat of the summer,” and “the heat escaped” all represent uses of this word that are inconsistent with our physics definition See Chapter 20 WW Conservation of energy 214 Chapter 8 Conservation of Energy are familiar with work, we can simplify the appearance of equations by letting the simple symbol W represent the external work Wext on a system For internal work, we will always use W int to differentiate it from W.) The other four members of our list not have established symbols, so we will call them TMW (mechanical waves), TMT (matter transfer), TET (electrical transmission), and TER (electromagnetic radiation) The full expansion of Equation 8.1 is DK DU DE int W Q TMW TMT TET TER (8.2) which is the primary mathematical representation of the energy version of the analysis model of the nonisolated system (We will see other versions of the nonisolated system model, involving linear momentum and angular momentum, in later chapters.) In most cases, Equation 8.2 reduces to a much simpler one because some of the terms are zero for the specific situation If, for a given system, all terms on the right side of the conservation of energy equation are zero, the system is an isolated system, which we study in the next section The conservation of energy equation is no more complicated in theory than the process of balancing your checking account statement If your account is the system, the change in the account balance for a given month is the sum of all the transfers: deposits, withdrawals, fees, interest, and checks written You may find it useful to think of energy as the currency of nature! Suppose a force is applied to a nonisolated system and the point of application of the force moves through a displacement Then suppose the only effect on the system is to change its speed In this case, the only transfer mechanism is work (so that the right side of Eq 8.2 reduces to just W ) and the only kind of energy in the system that changes is the kinetic energy (so that the left side of Eq 8.2 reduces to just DK) Equation 8.2 then becomes DK W which is the work–kinetic energy theorem This theorem is a special case of the more general principle of conservation of energy We shall see several more special cases in future chapters Q uick Quiz 8.1 By what transfer mechanisms does energy enter and leave (a) your television set? (b) Your gasoline-powered lawn mower? (c) Your hand-cranked pencil sharpener? Q uick Quiz 8.2 Consider a block sliding over a horizontal surface with friction Ignore any sound the sliding might make (i) If the system is the block, this system is (a) isolated (b) nonisolated (c) impossible to determine (ii) If the system is the surface, describe the system from the same set of choices (iii) If the system is the block and the surface, describe the system from the same set of choices Analysis Model Nonisolated System (Energy) Imagine you have identified a system to be analyzed and have defined a system boundary Energy can exist in the system in three forms: kinetic, potential, and internal The total of that energy can be changed when energy crosses the system boundary by any of six transfer methods shown in the diagram here The total change in the energy in the system is equal to the total amount of energy that has crossed the system boundary The mathematical statement of that concept is expressed in the conservation of energy equation: DE system o T (8.1) System boundary The change in the total amount of energy in the system is equal to the total amount of energy that crosses the boundary of the system Work Heat Mechanical waves Kinetic energy Potential energy Internal energy Matter transfer Electrical Electromagnetic transmission radiation [...]... either conservative or nonconservative Of the two forces just mentioned, the gravitational force is conservative and the friction force is nonconservative Conservative Forces Conservative forces have these two equivalent properties: 1 The work done by a conservative force on a particle moving between any two points is independent of the path taken by the particle 2 The work done by a conservative force... together as a system Initially, the system has kinetic energy because the book is moving While the book is sliding, the internal energy of the system increases: the book and the surface are warmer than before When the book stops, the kinetic energy has been completely 7.7 Conservative and Nonconservative Forces 197 transformed to internal energy We can consider the nonconservative force within the system—that... (Eq 7.12) We see that the spring force is conservative because Ws depends only on the initial and final x coordinates of the object and is zero for any closed path We can associate a potential energy for a system with a force acting between members of the system, but we can do so only if the force is conservative In general, the work W int done by a conservative force on an object that is a member of... System 7.7 Conservative and Nonconservative Forces 7.8 Relationship Between Conservative Forces and Potential Energy 7.9 Energy Diagrams and Equilibrium of a System The definitions of quantities such as position, velocity, acceleration, and force and associated principles such as Newton’s second law have allowed us to solve a variety of problems Some problems that could theoretically be solved with... ideas, however, do not truly define energy They merely tell us that fuels are needed to do a job and that those fuels provide us with something we call energy Energy is present in the Universe in various forms Every physical process that occurs in the Universe involves energy and energy transfers or transformations Unfortunately, despite its extreme importance, energy cannot be easily defined The variables... develop here in our study of mechanics Our analysis models presented in earlier chapters were based on the motion of a particle or an object that could be modeled as a particle We begin our new approach by focusing our attention on a new simplification model, a system, and analysis models based on the model of a system These analysis models will be formally introduced in Chapter 8 In this chapter, we... depends on the path, so the friction force cannot be conservative 7.8 R elationship Between Conservative Forces and Potential Energy In the preceding section, we found that the work done on a member of a system by a conservative force between the members of the system does not depend on the path taken by the moving member The work depends only on the initial and final coordinates For such a system,... value we assign to Ui because any nonzero value merely shifts Uf (x) by a constant amount and only the change in potential energy is physically meaningful If the point of application of the force undergoes an infinitesimal displacement dx, we can express the infinitesimal change in the potential energy of the system dU as dU 5 2Fx dx Therefore, the conservative force is related to the potential energy... balanced on its point is in a position of unstable equilibrium If the pencil is displaced slightly from its absolutely vertical position and is then released, it will surely fall over In general, configurations of a system in unstable equilibrium correspond to those for which U(x) for the system is a maximum Finally, a configuration called neutral equilibrium arises when U is constant over some region Small... work from an internal work to be described shortly If the system cannot be modeled as a particle (for example, if the system is deformable), we cannot use Equation 7.8 because different forces on the system may move through different displacements In this case, we must evaluate the work done by each force separately and then add the works algebraically to find the net work done on the system: S Work