Skip to main content

Bartleby Sitemap - Textbook Solutions

All Textbook Solutions for Physics for Scientists and Engineers

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 1 200 ft. (a) What is the pilots 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.A basin surrounding a drain has the shape of a circular 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 interval 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.Review. While learning to drive, you arc in a 1 200-kg 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 do not turn the steering wheel. Find the minimum distance you must be from the wall to avoid a collision. (c) If you do 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. (c) Does the conclusion in part (d) depend on the numerical values given in this problem, or is it true in general? Explain.A truck is moving with constant acceleration a up a hill that makes an angle with the horizontal as in Figure P6.36. A small sphere of mass m is suspended from the ceiling of the truck by a light cord. If the pendulum makes a constant angle with the perpendicular to the ceiling, what is a? Figure P6.36 Because the Earth rotates about its axis, a point on 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.A puck of mass m1 is tied to a string and allowed to revolve in a circle of radius R on a frictionless, horizontal table. The other end of the string passes through a small hole in the center of the table, and an object of mass m2 is tied to it (Fig. P6.38). 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. Figure P6.38 Galileo thought about whether acceleration should be defined as the rate of change of velocity over time or as the rate of change in velocity over distance. He chose the former, so lets 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 particles linear motion with constant vroomosity k, the equation v = vi + kx gives the velocity as a function of the position x if the particles speed is vi at x = 0. (a) Find the law describing the total force acting on this object of mass m. Describe an example of such a motion or explain why it is unrealistic for (b) the possibility of k positive and (c) the possibility of k negative.Members of a skydiving club were given the following data to use in planning their jumps. In the table, d is the distance fallen from rest by a skydiver in a free-fall stable spread position versus the time of fall t. (a) Convert the distances in feet into meters, (b) Graph d (in meters) versus t. (c) Determine the value of the terminal speed vt by finding the slope of the straight portion of the curve. Use a least-squares fit to determine this slope.A car rounds a banked curve as discussed in Example 6.4 and shown in Figure 6.5. The radius of curvature of the road is R, the banking angle is , and the coefficient of static friction is s. (a) Determine the range of speeds the car can have without slipping up or down the road. (b) Find the minimum value for s such that the minimum speed is zero.In Example 6.5, we investigated the forces a child experiences on a Ferris wheel. Assume the data in that example applies to this problem. What force (magnitude and direction) does the seat exert on a 40.0-kg child when the child is halfway between top and bottom?Review. A piece of putty is initially located at point A on the rim of a grinding wheel rotating at constant angular speed about a horizontal axis. The putty is dislodged from point A when the diameter through A is horizontal. It then rises vertically and returns to A at the instant the wheel completes one revolution. From this information, we wish to find the speed v of the putty when it leaves the wheel and the force holding it to the wheel. (a) What analysis model is appropriate for the motion of the putty as it rises and falls? (b) Use this model to find a symbolic expression for the time interval between when the putty leaves point A and when it arrives back at A, in terms of v and g. (c) What is the appropriate analysis model to describe point A on the wheel? (d) Find the period of the motion of point A in terms of the tangential speed v and the radius R of the wheel. (e) Set the time interval from part (b) equal to the period from part (d) and solve for the speed v of the putty as it leaves the wheel. (f) If the mass of the putty is m, what is the magnitude of the force that held it to the wheel before it was released?A model airplane of mass 0.750 kg flies with a speed of 35.0 m/s in a horizontal circle at the end of a 60.0-m-long control wire as shown in Figure P6.44a. The forces exerted on the airplane are shown in Figure P6.44b: the tension in the control wire, the gravitational force, and aerodynamic lift that acts at = 20.0 inward from the vertical. Compute the tension in the wire, assuming it makes a constant angle of = 20.0 with the horizontal. Figure P6.44 A 9.00-kg object starting from rest falls through a viscous medium and experiences a resistive force given by Equation 6.2. The object reaches one half its terminal speed in 5.54 s. (a) Determine the terminal speed. (b) At what time is the speed of the object three-fourths the terminal speed? (c) How far has the object traveled in the first 5.54 s of motion?For t 0, an object of mass m experiences no force and moves in the positive x direction with a constant speed vi. Beginning at t = 0, when the object passes position x = 0, it experiences a net resistive force proportional to the square of its speed: Fnet=mkv2i, where k is a constant. The speed of the object after t = 0 is given by v = vi/(1 + kvit). (a) Find the position x of the object as a function of time. (b) Find the objects velocity as a function of position.A golfer tees off from a location precisely at i = 35.0 north latitude. He hits the ball due south, with range 285 m. The balls initial velocity is at 48.0 above the horizontal. Suppose air resistance is negligible for the golf ball. (a) For how long is the ball in flight? The cup is due south of the golfers location, and the golfer would have a hole-in-one if the Earth were not rotating. The Earths rotation makes the tee move in a circle of radius RE cos i = (6.37 106 m) cos 35.0 as shown in Figure P6.47. The tee completes one revolution each day. (b) Find the eastward speed of the tee relative to the stars. The hole is also moving cast, but it is 285 m farther south and thus at a slightly lower latitude f. Because the hole moves in a slightly larger circle, its speed must he greater than that of the tee. (c) By how much does the holes speed exceed that of the tee? During the time interval the ball is in flight, it moves upward and downward as well as southward with the projectile motion you studied in Chapter 4, but it also moves eastward with the speed you found in part (b). The hole moves to the east at a faster speed, however, pulling ahead of the ball with the relative speed you found in part (c). (d) How far to the west of the hole does the ball land? Figure P6.47 A single bead can slide with negligible friction on a stiff wire that has been bent into a circular loop of radius 15.0 cm as shown in Figure P6.48. The circle is always in a vertical plane and rotates steadily about its vertical diameter with a period of 0.450 s. The position of the bead is described by the angle that the radial line, from the center of the loop to the bead, makes with the vertical. (a) At what angle up from the bottom of the circle can the bead slay motionless relative to the turning circle? (b) What If? Repeat the problem, this time taking the period of the circles rotation as 0.850 s. (c) Describe how the solution to part (b) is different from the solution to part (a). (d) For any period or loop size, is there always an angle at which the bead can stand still relative to the loop? (e) Are there ever more than two angles? Arnold Arons suggested the idea for this problem. Figure P6.48 Because of the Earths rotation, a plumb bob does not hang exactly along a line directed to the center of the Earth. How much does the plumb bob deviate from a radial line at 35.0 north latitude? Assume the Earth is spherical.You have a great job working at a major league baseball stadium for the summer! At this stadium, the speed of every pitch is measured using a radar gun aimed at the pitcher by an operator behind home plate. The operator has so much experience with this job that he has perfected a technique by which he can make each measurement at the exact instant at which the ball leaves the pitchers hand. Your supervisor asks you to construct an algorithm that will provide the speed of the ball as it crosses home plate, 18.3 m from the pitcher, based on the measured speed vi of the ball as it leaves the pitchers hand. The speed at home plate will be lower due to the resistive force of the air on the baseball. The vertical motion of the ball is small, so, to a good approximation, we can consider only the horizontal motion of the ball. You begin to develop your algorithm by applying the particle under a net force to the baseball in the horizontal direction. A pitch is measured to have a speed of 40.2 m/s as it leaves the pitchers hand. You need to tell your supervisor how fast it was traveling as it crossed home plate. (Hint: Use the chain rule to express acceleration in terms of a derivative with respect to x, and then solve a differential equation for v to find an expression for the speed of the baseball as a function of its position. The function will involve an exponential. Also make use of Table 6.1.)7.1QQ Figure 7.4 shows four situations in which a force is applied to an object. In all four cases, the force has the same magnitude, and the displacement of the object is to the right and of the same magnitude. Rank the situations in order of the work done by the force on the object, from most positive to most negative.Which of the following statements is true about the relationship between the dot product of two vectors and the product of the magnitudes of the vectors? (a) AB is larger than AB. (b) AB is smaller than AB. (c) AB could be larger or smaller than AB, depending on the angle between the vectors. (d) AB could be equal to AB.A dart is inserted into a spring-loaded dart gun by pushing the spring in by a distance x. For the next loading, the spring is compressed a distance 2x. How much work is required to load the second dart compared with that required to load the first? (a) four times as much (b) two times as much (c) the same (d) half as much (e) one-fourth as much A dart is inserted into a spring-loaded dart gun by pushing the spring in by a distance x. For the next loading, the spring is compressed a distance 2x. How much faster does the second dart leave the gun compared with the first? (a) four times as fast (b) two times as fast (c) the same (d) half as fast (e) one-fourth as fast Choose the correct answer. The gravitational potential energy of a system (a) is always positive (b) is always negative (c) can be negative or positive A ball is connected to a light spring suspended vertically as shown in Figure 7.18. When pulled downward from its equilibrium position and released, the ball oscillates up and down. (i) In the system of the ball, the spring, and the Earth, what forms of energy are there during the motion? (a) kinetic and elastic potential (b) kinetic and gravitational potential (c) kinetic, elastic potential, and gravitational potential (d) elastic potential and gravitational potential (ii) In the system of the ball and the spring, what forms of energy are there during the motion? Choose from the same possibilities (a) through (d).What does the slope of a graph of U(x) versus x represent? (a) the magnitude of the force on the object (b) the negative of the magnitude of the force on the object (c) the x component of the force on the object (d) the negative of the x component of the force on the object A shopper in a supermarket pushes a cart with a force of 35.0 N directed at an angle of 25.0 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 doesnt change, would the shoppers applied force be larger, smaller, or the same? (c) What about the work done on the cart by the shopper?The record number of boat lifts, including the boat and its ten crew members, was achieved by Sami Heinonen and Juha Rsnen of Sweden in 2000. They lifted a total mass of 653.2 kg approximately 4 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 hack to the ground.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 Arfeuilles teeth during the lift?Spiderman, whose mass is 80.0 kg, is dangling on the 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 60.0 angle with the vertical. How much work was done by the gravitational force on Spiderman in this maneuver?5P Vector A has a magnitude of 5.00 units, and vector B has a magnitude of 9.00 units. The two vectors make an angle of 50.0 with each other. Find AB.Find the scalar product of the vectors in Figure P7.7. Figure P7.7 Using the definition of the scalar product, find the angles between (a) A=3i2j and B=4i4j, (b) A=2i+4j and B=3i4j+2k, and (c) A=i2j+2k and B=3j+4k.A particle is subject to a force Fx that varies with position as shown in Figure P7.9. Find the work done by the force on the particle as it moves (a) from x = 0 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 = 0 to x = 15.0 m?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.When a 4.00-kg object is hung vertically on a certain light spring that obeys Hookes law, the spring stretches 2.50 cm. If the 4.00-kg object is removed, (a) how tar will the spring stretch if a 1.50-kg block is hung on it? (b) How much work must an external agent do to stretch the same spring 4.00 cm from its unstretched position?Express the units of the force constant of a spring in SI fundamental units.The tray dispenser in your cafeteria has broken and is not repairable. The custodian knows that you are good at designing things and asks you to help him build a new dispenser out of spare parts he has on his workbench. The tray dispenser supports a stack of trays on a shelf that is supported by four springs, one at each corner of the shelf. Each tray is rectangular, with dimensions 45.3 cm by 35.6 cm. Each tray is 0.450 cm thick and has a mass of 580 g. The custodian asks you to design a new four-spring dispenser such that when a tray is removed, the dispenser pushes up the remaining stack so that the top tray is at the same position as the just-removed tray was. He has a wide variety of springs that he can use to build the dispenser. Which springs should he use?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.A small particle of mass m is pulled to the top of a friction less half-cylinder (of radius R) by a light cord that passes over the top of the cylinder as illustrated in Figure P7.15. (a) Assuming the particle moves at a constant speed, show that F = mg cos . Note: If the particle moves at constant speed, the component of its acceleration tangent to the cylinder must be zero at all times. (b) By directly integrating W=Fdr, find the work done in moving the particle at constant speed from the bottom to the top of the hall-cylinder. Figure P7.15 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 = 0 to x = 3.00 m. (b) Front your graph, find the net work done by this force on the particle as it moves from x = 0 to x = 3.00 m.When different loads hang on a spring, the spring stretches to different lengths as shown in the following 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), do 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. (c) If the spring is extended to 105 mm, what force does it exert on the suspended object?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 000x2, 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)?(a) A force F=(4xi+3yj), where F is in newtons and 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. Find the work W=Fdr done by the force on the object. (b) What If? Find the work W=Fdr done by the force on the object if it moves from the origin to (5.00 m, 5.00 m) along a straightline path making an angle of 45.0 with the positive x axis. Is the work done by this force dependent on the path taken between the initial and final points?Review. The graph in Figure P7.20 specifies a functional relationship between the two variables u and v. (a) Find abudv. (b) Find baudv. (c) Find abvdu. Figure P7.20 A 0.600-kg particle has a speed of 2.00 m/s at point and kinetic energy of 7.50 J at point . What is (a) its kinetic energy at , (b) its speed at , and (c) the net work done on the particle by external forces as it moves from to ?A 4.00-kg particle is subject to a net force that varies with position as shown in Figure P7.9. The particle starts from rest at x = 0. What is its speed at (a) x = 5.00 m, (b) x = 10.0 m, and (c) x = 15.0 m?A 2 100-kg pile driver is used to drive a steel I-beam into 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.Review. In an electron microscope, there is an electron 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 microscopes 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.Review. You can think of the workkinetic energy theorem as a second theory of motion, parallel to Newtons 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 workkinetic 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)?You are lying in your bedroom, resting after doing your physics homework. As you stare at your ceiling, you come up with the idea for a new game. You grab a dart with a sticky nose and a mass of 19.0 g. You also grab a spring that has been lying on your desk from some previous project. You paint a target pattern on your ceiling. Your new game is to place the spring vertically on the floor, place the sticky-nose dart facing upward on the spring, and push the spring downward until the coils all press together, as on the right in Figure P7.26. You will then release the spring, firing the dart up toward the target on your ceiling, where its sticky nose will make it hang from the ceiling. The spring has an uncompressed end-to-end length of 5.00 cm, as shown on the left in Figure P7.26, and can be compressed to an end-to-end length of 1.00 cm when the coils are all pressed together. Before trying the game, you hold the upper end of the spring in one hand and hang a bundle of ten identical darts from the lower end of the spring. The spring extends by 1.00 cm due to the weight of the darts. You are so excited about the new game that, before doing a test of the game, you run out to gather your friends to show them. When your friends are in your room watching and you show them the first firing of your new game, why are you embarrassed? Figure P7.26 Review. A 5.75-kg object passes through the origin at time t = 0 such that its x component of velocity is 5.00 m/s and its y component of velocity is 3.00 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 on the object during this time interval? (c) What is the speed of the particle at t = 2.00 s?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 bullets 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.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 stoneEarth 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?A 1 000-kg roller coaster car is initially at the top of a rise, at point . It then moves 135 ft, at an angle of 40.0 below the horizontal, to a lower point . (a) Choose the car at point to be the zero configuration for gravitational potential energy of the roller coasterEarth system. Find the potential energy of the system when the car is at points and , 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 4.00-kg particle moves from the origin to position , having coordinates x = 5.00 m and y = 5.00 m (Fig. P7.31). One force on the particle is the gravitational force acting in the negative y direction. Using Equation 7.3, calculate the work done by the gravitational force on the particle as it goes from O to along (a) the purple path, (b) the red path, and (c) the blue path, (d) Your results should all be identical. Why? Figure P7.31 (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=ifFdr and show that the force is conservative. (b) As a special case, suppose the force F=(3i+4j) N acts on a panicle that moves from O to in Figure P7.31. Calculate the work done by F on the particle as it moves along each one of the three paths shown in the figure and show that the work done along the three paths is identical. (c) What If? Is the work done also identical along the three paths for the force F=(4xi+3yj), where F is in newtons and x and y are in meters, from Problem 19? (d) What If? Suppose the force is given by F=(yi+xj), where F is in newtons and x and y are in meters. Is the work done identical along the three paths for this force?A force acting on a particle moving in the xy plane is given by F=(2yi+x2j), where F is in newtons 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 P7.31. Calculate the work done by F on the particle as it moves along (a) the purple path, (b) the red path, and (c) the blue path, (d) Is F conservative or nonconservative? (e) Explain your answer to part (d). Figure P7.31 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.A single conservative force acts on a 5.0-kg particle within a system due to its interaction with the rest of the system. The equation Fx = 2x + 4 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.A potential energy function for a system in which a two-dimensional force acts is of the form U = 3x3y 7x. Find the force that acts at the point (x, y).37P For the potential energy curve shown in Figure P7.38, (a) determine whether the force Fx is positive, negative, or zero at the five points indicated. (b) Indicate points of stable, unstable, and neutral equilibrium. (c) Sketch the curve for Fx versus x from x = 0 to x = 9.5 m. Figure P7.38 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.The potential energy function for a system of particles is given by U(x) = x3 + 2x2 + 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.You have a new internship, where you are helping to design a new freight yard for the train station in your city. There will be a number of dead-end sidings where single cars can be stored until they are needed. To keep the cars from running off the tracks at the end of the siding, you have designed a combination of two coiled springs as illustrated in Figure P7.41. When a car moves to the right in the figure and strikes the springs, they exert a force to the left on the car to slow it down. Both springs are described by Hookes law and have spring constants k1 = 1 600 N/m and k2 = 3 400 N/m. After the first spring compresses by a distance of d = 30.0 cm, the second spring acts with the first to increase the force to the left on the car in Figure P7.41. When the spring with spring constant k2 compresses by 50.0 cm, the coils of both springs are pressed together, so that the springs can no longer compress. A typical car on the siding has a mass of 6 000 kg. When you present your design to your supervisor, he asks you for the maximum speed that a car can have and be stopped by your device. Figure P7.41 When an object is displaced by an amount x from stable 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)=(k1x+k2x2+k3x3+). The first term here is just Hookes 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 = (k1x + k2x2), how much work is done on an object in displacing it from x = 0 to x = xmax by an applied force F?A particle moves along the xaxis from x = 12.8 m to x = 23.7 m under the influence of a force F=375x3+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 he accurate to within 2%.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 hall launcher in the machine sends metal halls up one side of the machine and then into play. The spring in the launcher (Fig. P7.44) has a force constant of 1.20 N/cm. The surface on which the ball moves is inclined = 10.0 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. Figure P7.44 45AP (a) Take U = 5 for a system with a particle at position x = 0 and calculate the potential energy of the system as a function of the particle position x. The force on the particle is given by (8e2x)i. (b) Explain whether the force is conservative or nonconservative and how you can tell.An inclined plane of angle = 20.0 has a spring of force constant k = 500 N/m fastened securely at the bottom so that the spring is parallel to the surface as shown in Figure P7.47. A block of mass m = 2.50 kg is placed on the plane at a distance d = 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? Figure P7.47 Problems 47 and 48.An inclined plane of angle has a spring of force constant k fastened securely at the bottom so that the 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 P7.47. By what distance is the spring compressed when the block momentarily comes to rest?Over the Christmas break, you are making some extra money for buying presents by working in a factory, helping to move crates around. At one particular time, you find that all the handtrucks, dollies, and carts are in use, so you must move a crate across the room a straight-line distance of 35.0 m without the assistance of these devices. You notice that the crate has a rope attached to the middle of one of its vertical faces. You decide to move the crate by pulling on the rope. The crate has a mass of 130 kg, and the coefficient of kinetic friction between the crate and the concrete floor is 0.350. (a) Determine the angle relative to the horizontal at which you should pull upward on the rope so that you can move the crate over the desired distance with the force of the smallest magnitude. (b) At this angle of pulling on the rope, how much work do you do in dragging the crate over the desired distanced?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 = 0. (a) The particle is pulled a distance x along a direction perpendicular to the initial configuration of the springs as shown in Figure P7.50. Show that the force exerted by the springs on the particle is F=2kx(1Lx2+L2)i (b) Show that the potential energy of the system is U(x)=kx2+2kL(Lx2+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 panicle is pulled 0.500 m to the right and then released, what is its speed when it reaches x = 0? Figure P7.50 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.A rock of mass m is dropped to the ground from a height h. A second rock, with mass 2m, is dropped from the same height. When the second rock strikes the ground, what is its kinetic energy? (a) twice that of the first rock (b) four times that of the first rock (c) the same as that of the first rock (d) half as much as that of the first rock (e) impossible to determine Three identical balls are thrown from the top of a building, all with the same initial speed. As shown in Figure 8.3, the first is thrown horizontally, the second at some angle above the horizontal, and the third at some angle below the horizontal. Neglecting air resistance, rank the speeds of the balls at the instant each hits the ground. Figure 8.3 (Quick Quiz 8.3) Three identical balls are thrown with the same initial speed from the top of a building.You are traveling along a freeway at 65 mi/h. Your car has kinetic energy. You suddenly skid to a stop because of congestion in traffic. Where is the kinetic energy your car once had? (a) It is all in internal energy in the road. (b) It is all in internal energy in the tires. (c) Some of it has transformed to internal energy and some of it transferred away by mechanical waves. (d) It is all transferred away from your car by various mechanisms.1P A 20.0-kg cannonball is fired from a cannon with muzzle speed of 1 000 m/s at an angle of 37.0 with the horizontal. A second ball is fired at an angle of 90.0. Use the isolated system model to find (a) the maximum height reached by each ball and (b) the total mechanical energy of the ballEarth system at the maximum height for each ball. Let y = 0 at the cannon.A block of mass m = 5.00 kg is released from point and slides on the frictionless track shown in Figure P8.3. Determine (a) the blocks speed at points and and (b) the net work done by the gravitational force on the block as it moves from point to point . Figure P8.3 At 11:00 a.m, on September 7, 2001, more than one million British schoolchildren jumped up and down for one minute to simulate an earthquake. (a) Find the energy stored in the childrens bodies that was converted into internal energy in the ground and their bodies and propagated into the ground by seismic waves during the experiment. Assume 1 050 000 children of average mass 36.0 kg jumped 12 times each, raising their centers of mass by 25.0 cm each time and briefly resting between one jump and the next. (b) Of the energy that propagated into the ground, most produced high-frequency microtremor vibrations that were rapidly damped and did not travel far. Assume 0.01% of the total energy was carried away by long-range seismic waves. The magnitude of an earthquake on the Richter scale is given by M=logE4.81.5 where E is the seismic wave energy in joules. According to this model, what was the magnitude of the demonstration quake?A light, rigid rod is 77.0 cm long. Its top end is pivoted on a frictionless, horizontal axle. The rod hangs straight down at rest with a small, massive ball attached to its bottom end. You strike the ball, suddenly giving it a horizontal velocity so that it swings around in a full circle. What minimum speed at the bottom is required to make the ball go over the top of the circle?6P A crate of mass 10.0 kg is pulled up a rough incline with an initial speed of 1.50 m/s. The pulling force is 100 N parallel to the incline, which makes an angle of 20.0 with the horizontal. The coefficient of kinetic friction is 0.400, and the crate is pulled 5.00 m. (a) How much work is done by the gravitational force on the crate? (b) Determine the increase in internal energy of the crateincline system owing to friction. (c) How much work is done by the 100-N force on the crate? (d) What is the change in kinetic energy of the crate? (e) What is the speed of the crate after being pulled 5.00 m?A 40.0-kg box initially at rest is pushed 5.00 m along a rough, horizontal floor with a constant applied horizontal force of 130 N. The coefficient of friction between box and floor is 0.300. Find (a) the work done by the applied force, (b) the increase in internal energy in the boxfloor system as a result of friction, (c) the work done by the normal force, (d) the work done by the gravitational force, (e) the change in kinetic energy of the box, and (f) the final speed of the box.A smooth circular hoop with a radius of 0.500 m is placed flat on the floor. A 0.400-kg particle slides around the inside edge of the hoop. The particle is given an initial speed of 8.00 m/s. After one revolution, its speed has dropped to 6.00 m/s because of friction with the floor. (a) Find the energy transformed from mechanical to internal in the particlehoopfloor system as a result of friction in one revolution. (b) What is the total number of revolutions the particle makes before stopping? Assume the friction force remains constant during the entire motion.As shown in Figure P8.10, a green bead of mass 25 g slides along a straight wire. The length of the wire from point to point is 0.600 m, and point is 0.200 m higher than point . A constant friction force of magnitude 0.025 0 N acts on the bead. (a) If the bead is released from rest at point , what is its speed at point ? (b) A red bead of mass 25 g slides along a curved wire, subject to a friction force with the same constant magnitude as that on the green bead. If the green and red beads are released simultaneously from rest at point , which bead reaches point with a higher speed? Explain. Figure P8.10 At time ti, the kinetic energy of a particle is 30.0 J and the potential energy of the system to which it belongs is 10.0 J. At some later time tf, the kinetic energy of the particle is 18.0 J. (a) If only conservative forces act on the particle, what are the potential energy and the total energy of the system at time tf? (b) If the potential energy of the system at time tf is 5.00 J, are any nonconservative forces acting on the particle? (c) Explain your answer to part (b).A 1.50-kg object is held 1.20 m above a relaxed massless, vertical spring with a force constant of 320 N/m. The object is dropped onto the spring. (a) How far does the object compress the spring? (b) What If? Repeat part (a), but this time assume a constant air-resistance force of 0.700 N acts on the object during its motion. (c) What If? How far does the object compress the spring if the same experiment is performed on the Moon, where g = 1.63 m/s2 and air resistance is neglected?13P An 80.0-kg skydiver jumps out of a balloon at an altitude of 1 000 m and opens his parachute at an altitude of 200 m. (a) Assuming the total retarding force on the skydiver is constant at 50.0 N with the parachute closed and constant at 3 600 N with the parachute open, find the speed of the skydiver when he lands on the ground. (b) Do you think the skydiver will be injured? Explain. (c) At what height should the parachute be opened so that the final speed of the skydiver when he hits the ground is 5.00 m/s? (d) How realistic is the assumption that the total retarding force is constant? Explain.You have spent a long day skiing and are tired. You are standing at the top of a hill, looking at the lodge at the bottom of the hill. You are so tired that you want to simply start from rest and coast down the slope, without pushing with your poles or doing anything else to change your motion. You want to let gravity do all the work! You have a choice of two trails to reach the lodge. Both trails have the same coefficient of friction k. In addition, both trails represent the same horizontal separation between the initial and final points. Trail A has a short, steep downslope and then a long, flat coast to the lodge. Trail B has a long, gentle downslope and then a short remaining flat coast to the lodge. Which trail will result in your arriving at the lodge with the highest final speed?The electric motor of a model train accelerates the train from rest to 0.620 m/s in 21.0 ms. The total mass of the train is 875 g. (a) Find the minimum power delivered to the train by electrical transmission from the metal rails during the acceleration. (b) Why is it the minimum power?An energy-efficient lightbulb, taking in 28.0 W of power, can produce the same level of brightness as a conventional lightbulb operating at power 100 W. The lifetime of the energy-efficient bulb is 10 000 h and its purchase price is 4.50, whereas the conventional bulb has a lifetime of 750 h and costs 0.42. Determine the total savings obtained by using one energy-efficient bulb over its lifetime as opposed to using conventional bulbs over the same time interval. Assume an energy cost of 0.200 per kilowatt-hour.An older-model car accelerates from 0 to speed v in a time interval of t. A newer, more powerful sports car accelerates from 0 to 2v in the same time period. Assuming the energy coming from the engine appears only as kinetic energy of the cars, compare the power of the two cars.Make an order-of-magnitude estimate of the power a car engine contributes to speeding the car up to highway speed. In your solution, state the physical quantities you take as data and the values you measure or estimate for them. The mass of a vehicle is often given in the owners manual.There is a 5K event coming up in your town. While talking to your grandmother, who uses an electric scooter for mobility, she says that she would like to accompany you on her scooter while you walk the 5.00-km distance. The manual that came with her scooter claims that the fully charged battery is capable of providing 120 Wh of energy before being depleted. In preparation for the race, you go for a test drive: beginning with a fully charged battery, your grandmother rides beside you as you walk 5.00 km on flat ground. At the end of the walk, the battery usage indicator shows that 40.0% of the original energy in the battery remains. You also know that the combined weight of the scooter and your grandmother is 890 N. A few days later, filled with confidence that the battery has sufficient energy, you and your grandmother drive to the 5K event. Unbeknownst to you, the 5K route is not on flat ground, but is all uphill, ending at a point higher than the starting line. A race official tells you that the total amount of vertical displacement on the route is 150 m. Should your grandmother accompany you on the walk, or will she be stranded when her battery runs out of energy? Assume that the only difference between your test drive and the actual event is the vertical displacement.For saving energy, bicycling and walking are far more efficient means of transportation than is travel by automobile. For example, when riding at 10.0 mi/h, a cyclist uses food energy at a rate of about 400 kcal/h above what he would use if merely sitting still. (In exercise physiology, power is often measured in kcal/h rather than in watts. Here 1 kcal = 1 nutritionists Calorie = 4 186 J.) Walking at 3.00 mi/h requires about 220 kcal/h. It is interesting to compare these values with the energy consumption required for travel by car. Gasoline yields about 1.30 108 J/gal. Find the fuel economy in equivalent miles per gallon for a person (a) walking and (b) bicycling.Energy is conventionally measured in Calories as well as in joules. One Calorie in nutrition is one kilocalorie, defined as 1 kcal = 4 186 J. Metabolizing 1 g of fat can release 9.00 kcal. A student decides to try to lose weight by exercising. He plans to run up and down the stairs in a football stadium as fast as he can and as many times as necessary. To evaluate the program, suppose he runs up a flight of 80 steps, each 0.150 m high, in 65.0 s. For simplicity, ignore the energy he uses in coming down (which is small). Assume a typical efficiency for human muscles is 20.0%. This statement means that when your body converts 100J from metabolizing fat, 20 J goes into doing mechanical work (here, climbing stairs). The remainder goes into extra internal energy. Assume the students mass is 75.0 kg. (a) How many times must the student run the flight of stairs to lose 1.00 kg of fat? (b) What is his average power output, in watts and in horsepower, as he runs up the stairs? (c) Is this activity in itself a practical way to lose weight?A block of mass m = 200 g is released from rest at point along the horizontal diameter on the inside of hemispherical bowl of radius R = 30.0 cm, and the surface of the bowl is rough (Fig. P8.23). The blocks speed at point is 1.50 m/s. Figure P8.23 (a) What is its kinetic energy at point ? (b) How much mechanical energy is transformed into internal energy as the block moves from point to point ? (c) Is it possible to determine the coefficient of friction from these results in any simple manner? (d) Explain your answer to part (c).Make an order-of-magnitude estimate of your power out-put as you climb stairs. In your solution, state the physical quantities you take as data and the values you measure or estimate for them. Do you consider your peak power or your sustainable power?25AP Review. As shown in Figure P8.26, a light string that does not stretch changes from horizontal to vertical as it passes over the edge of a table. The string connects m1, a 3.50-kg block originally at rest on the horizontal table at a height h = 1.20 m above the floor, to m2, a hanging 1.90-kg block originally a distance d = 0.900 m above the floor. Neither the surface of the table nor its edge exerts a force of kinetic friction. The blocks start to move from rest. The sliding block m1 is projected horizontally after reaching the edge of the table. The hanging block m2 stops without bouncing when it strikes the floor. Consider the two blocks plus the Earth as the system. (a) Find the speed at which m1 leaves the edge of the table. (b) Find the impact speed of m1 on the floor. (c) What is the shortest length of the string so that it does not go taut while m1 is in flight? (d) Is the energy of the system when it is released from rest equal to the energy of the system just before m1 strikes the ground? (e) Why or why not? Figure P8.26 Consider the blockspringsurface system in part (B) of Example 8.6. (a) Using an energy approach, find the position x of the block at which its speed is a maximum. (b) In the What If? section of this example, we explored the effects of an increased friction force of 10.0 N. At what position of the block docs its maximum speed occur in this situation?Why is the following situation impossible? A softball pitcher has a strange technique: she begins with her hand at rest at the highest point she can reach and then quickly rotates her arm backward so that the ball moves through a half-circle path. She releases the ball when her hand reaches the bottom of the path. The pitcher maintains a component of force on the 0.180-kg ball of constant magnitude 12.0 N in the direction of motion around the complete path. As the ball arrives at the bottom of the path, it leaves her hand with a speed of 25.0 m/s.Jonathan is riding a bicycle and encounters a hill of height 7.30 m. At the base of the hill, he is traveling at 6.00 m/s. When he reaches the top of the hill, he is traveling at 1.00 m/s. Jonathan and his bicycle together have a mass of 85.0 kg. Ignore friction in the bicycle mechanism and between the bicycle tires and the road. (a) What is the total external work done on the system of Jonathan and the bicycle between the time he starts up the hill and the time he reaches the top? (b) What is the change in potential energy stored in Jonathans body during this process? (c) How much work does Jonathan do on the bicycle pedals within the JonathanbicycleEarth system during this process?Jonathan is riding a bicycle and encounters a hill of height h. At the base of the hill, he is traveling at a speed vi. When he reaches the top of the hill, he is traveling at a speed vf. Jonathan and his bicycle together have a mass m. Ignore friction in the bicycle mechanism and between the bicycle tires and the road. (a) What is the total external work done on the system of Jonathan and the bicycle between the time he starts up the hill and the time he reaches the top? (b) What is the change in potential energy stored in Jonathans body during this process? (c) How much work does Jonathan do on the bicycle pedals within the JonathanbicycleEarth system during this process?As the driver steps on the gas pedal, a car of mass 1 160 kg accelerates from rest. During the first few seconds of motion, the cars acceleration increases with time according to the expression a=1.16t0.210t2+0.240t3 where t is in seconds and a is in m/s2. (a) What is the change in kinetic energy of the car during the interval from t = 0 to t = 2.50 s? (b) What is the minimum average power output of the engine over this time interval? (c) Why is the value in part (b) described as the minimum value?As it plows a parking lot, a snowplow pushes an ever-growing pile of snow in front of it. Suppose a car moving through the air is similarly modeled as a cylinder of area A pushing a growing disk of air in front of it. The originally stationary air is set into motion at the constant speed v of the cylinder as shown in Figure P8.32. In a time interval t, a new disk of air of mass m must be moved a distance v t and hence must be given a kinetic energy 12(m)v2. Using this model, show that the cars power loss owing to air resistance is 12Av3 and that the resistive force acting on the car is 12Av2, where is the density of air. Compare this result with the empirical expression 12DAv2 for the resistive force. Figure P8.32 Heedless of danger, a child leaps onto a pile of old mattresses to use them as a trampoline. His motion between two particular points is described by the energy conservation equation 12(46.0kg)(2.40m/s)2+(46.0kg)(9.80m/s2)(2.80m+x)=12(1.94104N/m)x2 (a) Solve the equation for x. (b) Compose the statement of a problem, including data, for which this equation gives the solution. (c) Add the two values of x obtained in part (a) and divide by 2. (d) What is the significance of the resulting value in part (c)?Review. Why is the following situation impossible? A new high-speed roller coaster is claimed to be so safe that the passengers do not need to wear seat belts or any other restraining device. The coaster is designed with a vertical circular section over which the coaster travels on the inside of the circle so that the passengers are upside down for a short time interval. The radius of the circular section is 12.0 m, and the coaster enters the bottom of the circular section at a speed of 22.0 m/s. Assume the coaster moves without friction on the track and model the coaster as a particle.A horizontal spring attached to a wall has a force constant of k = 850 N/m. A block of mass m = 1.00 kg is attached to the spring and rests on a frictionless, horizontal surface as in Figure P8.35. (a) The block is pulled to a position xi = 6.00 cm from equilibrium and released. Find the elastic potential energy stored in the spring when the block is 6.00 cm from equilibrium and when the block passes through equilibrium. (b) Find the speed of the block as it passes through the equilibrium point. (c) What is the speed of the block when it is at a position xi/2 = 3.00 cm? (d) Why isnt the answer to part (c) half the answer to part (b)? Figure P8.35 More than 2 300 years ago, the Greek teacher Aristotle wrote the first book called Physics. Put into more precise terminology, this passage is from the end of its Section Eta: Let P be the power of an agent causing motion; w, the load moved; d, the distance covered; and t, the time interval required. Then (1) a power equal to P will in an interval of time equal to t move w/2 a distance 2d; or (2) it will move w/2 the given distance d in the time interval t /2. Also, if (3) the given power P moves the given load w a distance d/2 in time interval t/2, then (4) P/2 will move w/2 the given distance d in the given time interval t. (a) Show that Aristotles proportions are included in the equation Pt = bwd, where b is a proportionality constant. (b) Show that our theory of motion includes this part of Aristotles theory as one special case. In particular, describe a situation in which it is true, derive the equation representing Aristotles proportions, and determine the proportionality constant.Review. As a prank, someone has balanced a pumpkin at the highest point of a grain silo. The silo is topped with a hemispherical cap that is frictionless when wet. The line from the center of curvature of the cap to the pumpkin makes an angle i = 0 with the vertical. While you happen to be standing nearby in the middle of a rainy night, a breath of wind makes the pumpkin start sliding downward from rest. It loses contact with the cap when the line from the center of the hemisphere to the pumpkin makes a certain angle with the vertical. What is this angle?Review. Why is the following situation impossible? An athlete tests her hand strength by having an assistant hang weights from her belt as she hangs onto a horizontal bar with her hands. When the weights hanging on her belt have increased to 80% of her body weight, her hands can no longer support her and she drops to the floor. Frustrated at not meeting her hand-strength goal, she decides to swing on a trapeze. The trapeze consists of a bar suspended by two parallel ropes, each of length , allowing performers to swing in a vertical circular are (Fig. P8.38). The athlete holds the bar and steps off an elevated platform, starting from rest with the ropes at an angle i = 60.0 with respect to the vertical. As she swings several times back and forth in a circular are, she forgets her frustration related to the hand-strength test. Assume the size of the performers body is small compared to the length and air resistance is negligible. Figure P8.38 An airplane of mass 1.50 104 kg is in level flight, initially moving at 60.0 m/s. The resistive force exerted by air on the airplane has a magnitude of 4.0 104 N. By Newtons third law, if the engines exert a force on the exhaust gases to expel them out of the back of the engine, the exhaust gases exert a force on the engines in the direction of the airplanes travel. This force is called thrust, and the value of the thrust in this situation is 7.50 104 N. (a) Is the work done by the exhaust gases on the airplane during some time interval equal to the change in the airplanes kinetic energy? Explain. (b) Find the speed of the airplane after it has traveled 5.0 102 m.A pendulum, comprising a light string of length L and a small sphere, swings in the vertical plane. The string hits a peg located a distance d below the point of suspension (Fig. P8.40). (a) Show that if the sphere is released from a height below that of the peg, it will return to this height after the string strikes the peg. (b) Show that if the pendulum is released from rest at the horizontal position ( = 90) and is to swing in a complete circle centered on the peg, the minimum value of d must be 3L/5. Figure P8.40 A ball whirls around in a vertical circle at the end of a string. The other end of the string is fixed at the center of the circle. Assuming the total energy of the ballEarth system remains constant, show that the tension in the string at the bottom is greater than the tension at the top by six times the balls weight.You are working in the distribution center of a large online shopping site. Efforts are being made to increase the number of packages per unit time that are being loaded onto a conveyor belt to be carried to waiting trucks. But the motor driving the conveyor belt is having difficulty keeping up with the increased demands. Your supervisor has asked you to determine the requirements for a new motor that can provide enough power to keep the conveyor belt moving smoothly under the increased loading rate. You are given the following information: The design goal is to have 50.0-kg packages loaded onto the belt at several locations at an average rate of 5.00 packages per second. The bell moves at a horizontal speed of 1.35 m/s. Humans at the various locations along the belt place the package on the belt so that it is initially at rest relative to the floor of the building just before being dropped from negligible height onto the belt. Your task is to determine the minimum power the driving motor must have to accelerate these packages and keep the bell moving at constant speed.43AP Starting from rest, a 64.0-kg person bungee jumps from a tethered hot-air balloon 65.0 m above the ground. The bungee cord has negligible mass and unstretched length 25.8 m. One end is tied to the basket of the balloon and the other end to a harness around the persons body. The cord is modeled as a spring that obeys Hookes law with a spring constant of 81.0 N/m, and the persons body is modeled as a particle. The hot-air balloon does not move. (a) Express the gravitational potential energy of the personEarth system as a function of the persons variable height y above the ground. (b) Express the elastic potential energy of the cord as a function of y. (c) Express the total potential energy of the personcordEarth system as a function of y. (d) Plot a graph of the gravitational, elastic, and total potential energies as functions of y. (e) Assume air resistance is negligible. Determine the minimum height of the person above the ground during his plunge. (f) Docs the potential energy graph show any equilibrium position or positions? If so, at what elevations? Are they stable or unstable? (g) Determine the jumpers maximum speed.Review. A uniform board of length L is sliding along a smooth, frictionless, horizontal plane as shown in Figure P8.45a. The board then slides across the boundary with a rough horizontal surface. The coefficient of kinetic friction between the board and the second surface is k. (a) Find the acceleration of the board at the moment its front end has traveled a distance x beyond the boundary. (b) The board stops at the moment its back end reaches the boundary as shown in Figure P8.45b. Find the initial speed v of the board. Figure P8.45 A uniform chain of length 8.00 m initially lies stretched out on a horizontal table. (a) Assuming the coefficient of static friction between chain and table is 0.600, show that the chain will begin to slide off the table if at least 3.00 m of it hangs over the edge of the table. (b) Determine the speed of the chain as its last link leaves the table, given that the coefficient of kinetic friction between the chain and the table is 0.400.What If? Consider the roller coaster described in Problem 34. Because of some friction between the coaster and the track, the coaster enters the circular section at a speed of 15.0 m/s rather than the 22.0 m/s in Problem 34. Is this situation more or less dangerous for the passengers than that in Problem 34? Assume the circular section is still frictionless.Two objects have equal kinetic energies. How do the magnitudes of their momenta compare? (a) p1 p2 (b) p1 = p2 (c) p1 p2 (d) not enough information to tell Your physical education teacher throws a baseball to you at a certain speed and you catch it. The teacher is next going to throw you a medicine ball whose mass is ten times the mass of the baseball. You are given the following choices: You can have the medicine ball thrown with (a) the same speed as the baseball, (b) the same momentum, or (c) the same kinetic energy. Rank these choices from easiest to hardest to catch.Two objects are at rest on a frictionless surface. Object 1 has a greater mass than object 2. (i) When a constant force is applied to object 1, it accelerates through a distance d in a straight line. The force is removed from object 1 and is applied to object 2. At the moment when object 2 has accelerated through the same distance d, which statements are true? (a) p1 p2 (b) p1 = p2 (c) p1 p2 (d) K1 K2 (e) K1 = K2 (f) K1 K2 (ii) When a force is applied to object 1, it accelerates for a time interval t. The force is removed from object 1 and is applied to object 2. From the same list of choices, which statements are true after object 2 has accelerated for the same lime interval t?Rank an automobile dashboard, seat belt, and air bag, each used alone in separate collisions from the same speed, in terms of (a) the impulse and (b) the average force each delivers to a front-seat passenger, from greatest to least.In a perfectly inelastic one-dimensional collision between two moving objects, what condition alone is necessary so that the final kinetic energy of the system is zero after the collision? (a) The objects must have initial momenta with the same magnitude but opposite directions, (b) The objects must have the same mass. (c) The objects must have the same initial velocity. (d) The objects must have the same initial speed, with velocity vectors in opposite directions.A table-tennis ball is thrown at a stationary bowling ball. The table-tennis ball makes a one-dimensional elastic collision and bounces back along the same line. Compared with the bowling ball after the collision, does the table-tennis ball have (a) a larger magnitude of momentum and more kinetic energy, (b) a smaller magnitude of momentum and more kinetic energy, (c) a larger magnitude of momentum and less kinetic energy, (d) a smaller magnitude of momentum and less kinetic energy, or (e) the same magnitude of momentum and the same kinetic energy?A baseball bat of uniform density is cut at the location of its center of mass as shown in Figure 9.18. Which piece has the smaller mass? (a) the piece on the right (b) the piece on the left (c) both pieces have the same mass (d) impossible to determine Figure 9.18 (Quick Quiz 9.7) A baseball bat cut at the location of its center of mass.A cruise ship is moving at constant speed through the water. The vacationers on the ship are eager to arrive at their next destination. They decide to try to speed up the cruise ship by gathering at the bow (the front) and running together toward the stem (the back) of the ship. (i) While they are running toward the stern, is the speed of the ship (a) higher than it was before, (b) unchanged, (c) lower than it was before, or (d) impossible to determine? (ii) The vacationers stop running when they reach the stem of the ship. After they have all stopped running, is the speed of the ship (a) higher titan it was before they started running, (b) unchanged from what it was before they started running, (c) lower than it was before they started running, or (d) impossible to determine?A particle of mass m moves with momentum of magnitude p. (a) Show that the kinetic energy of the particle is K = p2/2m. (b) Express the magnitude of the particles momentum in terms of its kinetic energy and mass.A 3.00-kg particle has a velocity of (3.00i4.00j)m/s. (a) Find its x and y components of momentum. (b) Find the magnitude and direction of its momentum.A baseball approaches home plate at a speed of 45.0 m/s, moving horizontally just before being hit by a bat. The batter hits a pop-up such that after hitting the bat, the baseball is moving at 55.0 m/s straight up. The ball has a mass of 145 g and is in contact with the bat for 2.00 ms. What is the average vector force the ball exerts on the bat during their interaction?A 65.0-kg boy and his 40.0-kg sister, both wearing roller blades, face each other at rest. The girl pushes the boy hard, sending him backward with velocity 2.90 m/s toward the west. Ignore friction. (a) Describe the subsequent motion of the girl. (b) How much potential energy in the girls body is converted into mechanical energy of the boygirl system? (c) Is the momentum of the boygirl system conserved in the pushing-apart process? If so, explain how that is possible considering (d) there are large forces acting and (e) there is no motion beforehand and plenty of motion afterward.Two blocks of masses m and 3m are placed on a frictionless, horizontal surface. A light spring is attached to the more massive block, and the blocks are pushed together with the spring between them (Fig. P9.5). A cord initially holding the blocks together is burned; after that happens, the block of mass 3m moves to the right with a speed of 2.00 m/s. (a) What is the velocity of the block of mass m? (b) Find the systems original elastic potential energy, taking m = 0.350 kg. (c) Is the original energy in the spring or in the cord? (d) Explain your answer to part (c). (e) Is the momentum of the system conserved in the bursting-apart process? Explain how that is possible considering (f) there are large forces acting and (g) there is no motion before-hand and plenty of motion afterward? Figure P9.5 When you jump straight up as high as you can, what is the order of magnitude of the maximum recoil speed that you give to the Earth? Model the Earth as a perfectly solid object. In your solution, state the physical quantities you take as data and the values you measure or estimate for them.A glider of mass m is free to slide along a horizontal air track. It is pushed against a launcher at one end of the track. Model the launcher as a light spring of force constant k compressed by a distance x. The glider is released from rest. (a) Show that the glider attains a speed of v = x(k/m)1/2. (b) Show that the magnitude of the impulse imparted to the glider is given by the expression I = x(km)1/2. (c) Is more work done on a cart with a large or a small mass?You and your brother argue often about how to safely secure a toddler in a moving car. You insist that special toddler seats are critical in improving the chances of a toddler surviving a crash. Your brother claims that, as long as his wife is buckled in next to him with a seat belt while he drives, she can hold onto their toddler on her lap in a crash. You decide to perform a calculation to try to convince your brother. Consider a hypothetical collision in which the 12-kg toddler and his parents are riding in a car traveling at 60 mi/h relative to the ground. The car strikes a wall, tree, or another car, and is brought to rest in 0.10 s. You wish to demonstrate to your brother the magnitude of the force necessary for his wife to hold onto their child during the collision.The front 1.20 m of a 1 400-kg car Ls designed as a crumple zone that collapses to absorb the shock of a collision. If a car traveling 25.0 m/s stops uniformly in 1.20 m, (a) how long does the collision last, (b) what is the magnitude of the average force on the car, and (c) what is the magnitude of the acceleration of the car? Express the acceleration as a multiple of the acceleration due to gravity.The magnitude of the net force exerted in the x direction on a 2.50-kg particle varies in time as shown in Figure P9.10 (page 244). Find (a) the impulse of the force over the 5.00-s time interval, (b) the final velocity the particle attains if it is originally at rest, (c) its final velocity if its original velocity is 2.00im/s, and (d) the average force exerted on the particle for the time interval between 0 and 5.00 s. Figure P9.10 Water falls without splashing at a rate of 0.250 L/s from a height of 2.60 m into a bucket of mass 0.750 kg on a scale. If the bucket is originally empty, what does the scale read in newtons 3.00 s after water starts to accumulate in it?A 1 200-kg car traveling initially at vCi = 25.0 m/s in an easterly direction crashes into the back of a 9 000-kg truck moving in the same direction at vTi = 20.0 m/s (Fig. P9.12). The velocity of the car immediately after the collision is vCf = 18.0 m/s to the east. (a) What is the velocity of the truck immediately after the collision? (b) What is the change in mechanical energy of the cartruck system in the collision? (c) Account for this change in mechanical energy.A railroad car of mass 2.50 104 kg is moving with a speed of 4.00 m/s. It collides and couples with three other coupled railroad cars, each of the same mass as the single car and moving in the same direction with an initial speed of 2.00 m/s. (a) What is the speed of the four cars after the collision? (b) What is the decrease in mechanical energy in the collision?Four railroad cars, each of mass 2.50 104 kg, are coupled together and coasting along horizontal tracks at speed vi toward the south. A very strong but foolish movie actor, riding on the second car, uncouples the front car and gives it a big push, increasing its speed to 4.00 m/s southward. The remaining three cars continue moving south, now at 2.0 m/s. (a) Find the initial speed of the four cars. (b) By how much did the potential energy within the body of the actor change? (c) State the relationship between the process described here and the process in Problem 13.A car of mass m moving at a speed v1 collides and couples with the back of a truck of mass 2m moving initially in the same direction as the car at a lower speed v2. (a) What is the speed vf of the two vehicles immediately after the collision? (b) What is the change in kinetic energy of the cartruck system in the collision?A 7.00-g bullet, when fired from a gun into a 1.00-kg block of wood held in a vise, penetrates the block to a depth of 8.00 cm. This block of wood is next placed on a frictionless horizontal surface, and a second 7.00-g bullet is fired from the gun into the block. To what depth will the bullet penetrate the block in this case?A tennis ball of mass 57.0 g is held just above a basketball of mass 500 g as shown in Figure P9.17. With their centers vertically aligned, both balls are released from rest at the same time, to fall through a distance of 1.20 m. (a) Find the magnitude of the downward velocity with which the basketball reaches the ground. (b) Assume that an elastic collision with the ground instantaneously reverses the velocity of the basketball while the tennis ball is still moving down. Next, the two balls meet in an elastic collision. To what height does the tennis ball rebound? Figure P9.17 (a) Three carts of masses m1 = 4.00 kg, m2 = 10.0 kg, and m3= 3.00 kg move on a frictionless, horizontal track with speeds of v1 = 5.00 m/s to the right, v2 = 3.00 m/s to the right, and v3 = 4.00 m/s to the left as shown in Figure P9.18. Velcro couplers make the carts stick together after colliding. Find the final velocity of the train of three carts. (b) What If? Does your answer in part (a) require that all the carts collide and stick together at the same moment? What if they collide in a different order?You have been hired as an expert witness by an attorney for a trial involving a traffic accident. The attorneys client, the plaintiff in this case, was traveling castbound toward an intersection at 13.0 m/s as measured just before the accident by a roadside speed meter, and as seen by a trustworthy witness. As the plaintiff entered the intersection, his car was struck by a northbound driver, the defendant in this case, driving a car with identical mass to the plaintiffs. The vehicles stuck together after the collision and left parallel skid marks at an angle of = 55.0 north of east, as measured by accident investigators. The defendant is claiming that he was traveling within the 35-mi/h speed limit. What advice do you give to the attorney?Two shuffleboard disks of equal mass, one orange and the other yellow, are involved in an elastic, glancing collision. The yellow disk is initially at rest and is struck by the orange disk moving with a speed of 5.00 m/s. After the collision, the orange disk moves along a direction that makes an angle of 37.0 with its initial direction of motion. The velocities of the two disks are perpendicular after the collision. Determine the final speed of each disk.Two shuffleboard disks of equal mass, one orange and the other yellow, are involved in an elastic, glancing collision. The yellow disk is initially at rest and is struck by the orange disk moving with a speed vi. After the collision, the orange disk moves along a direction that makes an angle with its initial direction of motion. The velocities of the two disks arc perpendicular after the collision. Determine the final speed of each disk.A 90.0-kg fullback running east with a speed of 5.00 m/s is tackled by a 95.0-kg opponent running north with a speed of 3.00 m/s. (a) Explain why the successful tackle constitutes a perfectly inelastic collision. (b) Calculate the velocity of the players immediately after the tackle. (c) Determine the decrease in mechanical energy as a result of the collision. Account for this decrease.A proton, moving with a velocity of vii, collides elastically with another proton that is initially at rest. Assuming that the two protons have equal speeds after the collision, find (a) the speed of each proton after the collision in terms of vi and (b) the direction of the velocity vectors after the collision.A uniform piece of sheet metal is shaped as shown in Figure P9.24. Compute the x and y coordinates of the center of mass of the piece. Figure P9.24 Explorers in the jungle find an ancient monument in the shape of a large isosceles triangle as shown in Figure P9.25. The monument is made from tens of thousands of small stone blocks of density 3 800 kg/m3. The monument is 15.7 m high and 64.8 m wide at its base and is everywhere 3.60 m thick from front to back. Before the monument was built many years ago, all the stone blocks lay on the ground. How much work did laborers do on the blocks to put them in position while building the entire monument? Note: The gravitational potential energy of an objectEarth system is given by Ug = MgyCM, where M is the total mass of the object and yCM is the elevation of its center of mass above the chosen reference level.A rod of length 30.0 cm has linear density (mass per length) given by =50.0+20.0x where x is the distance from one end, measured in meters, and is in grams/meter. (a) What is the mass of the rod? (b) How far from the x = 0 end is its center of mass?Consider a system of two particles in the xy plane: m1 = 2.00 kg is at the location r1=(1.00i+2.00j)m and has a velocity of (3.00i+0.500j)m/s; m2 = 3.00 kg is at r2=(4.00i+3.00j)m and has velocity (3.00i+2.00j)m/s. (a) Plot these particles on a grid or graph paper. Draw their position vectors and show their velocities. (b) Find the position of the center of mass of the system and mark it on the grid. (c) Determine the velocity of the center of mass and also show it on the diagram. (d) What is the total linear momentum of the system?The vector position of a 3.50-g particle moving in the xy plane varies in time according to r1=(3i+3j)t+2jt2, where t is in seconds and r is in centimeters. At the same time, the vector position of a 5.50 g particle varies as r2=3i2it26jt. At t = 2.50 s, determine (a) the vector position of the center of mass of the system, (b) the linear momentum of the system, (c) the velocity of the center of mass, (d) the acceleration of the center of mass, and (e) the net force exerted on the two-particle system.You have been hired as an expert witness in an investigation of a quadcopter drone incident. The incident occurred during a very rare meteor shower during which several unusually massive chunks of meteoric material were passing through the atmosphere and striking the ground. The unmanned drone was hovering at rest over the center of a house on fire, having just dropped fire retardant, when it seemed to spontaneously explode into four large pieces. The locations of the four pieces on the ground were measured as follows, relative to the center of the house over which the drone was hovering: The fire department is suggesting that the drone was defective and exploded while in use. The drone manufacturer is suggesting that the drone was struck by a meteorite, causing the explosion. Perform a calculation that will show evidence suggesting agreement with one of these positions.30P A 60.0-kg person bends his knees and then jumps straight up. After his feet leave the floor, his motion is unaffected by air resistance and his center of mass rises by a maximum of 15.0 cm. Model the floor as completely solid and motionless. (a) Does the floor impart impulse to the person? (b) Does the floor do work on the person? (c) With what momentum does the person leave the floor? (d) Does it make sense to say that this momentum came from the floor? Explain. (e) With what kinetic energy does the person leave the flow? (f) Does it make sense to say that this energy came from the floor? Explain.A garden hose is held as shown in Figure P9.32. The hose is originally full of motionless water. What additional force is necessary to hold the nozzle stationary after the water flow is turned on if the discharge rate is 0.600 kg/s with a speed of 25.0 m/s? Figure P9.32 A rocket for use in deep space is to be capable of boosting a total load (payload plus rocket frame and engine) of 3.00 metric tons to a speed of 10 000 m/s. (a) It has an engine and fuel designed to produce an exhaust speed of 2 000 m/s. How much fuel plus oxidizer is required? (b) If a different fuel and engine design could give an exhaust speed of 5 000 m/s, what amount of fuel and oxidizer would be required for the same task? (c) Noting that the exhaust speed in part (b) is 2.50 times higher than that in part (a), explain why the required fuel mass is not simply smaller by a factor of 2.50.A rocket has total mass Mi = 360 kg, including Mfuel = 330 kg of fuel and oxidizer. In interstellar space, it starts from rest at the position x = 0, turns on its engine at time t = 0, and puts out exhaust with relative speed ve = 1 500 m/s at the constant rate k = 2.50 kg/s. The fuel will last for a burn time of Tb = Mfuel/k = 330 kg/(2.5 kg/s) = 132 s. (a) Show that during the burn the velocity of the rocket as a function of time is given by v(t)=veln(1ktMi) (b) Make a graph of the velocity of the rocket as a function of time for times running from 0 to 132 s. (c) Show that the acceleration of the rocket is a(t)=kveMikt (d) Graph the acceleration as a function of time. (c) Show that the position of the rocket is x(t)=ve(Mikt)ln(1ktMi)+vet (f) Graph the position during the burn as a function of time.An amateur skater of mass M is trapped in the middle of an ice rink and is unable to return to the side where there is no ice. Every motion she makes causes her to slip on the ice and remain in the same spot. She decides to try to return to safety by throwing her gloves of mass m in the direction opposite the safe side. (a) She throws the gloves as hard as she can, and they leave her hand with a horizontal velocity vgloves. Explain whether or not she moves. (b) If she does move, calculate her velocity vgirl relative to the Earth after she throws the gloves. (c) Discuss her motion from the point of view of the forces acting on her.(a) Figure P9.36 shows three points in the operation of the ballistic pendulum discussed in Example 9.6 (and shown in Fig. 9.10b). The projectile approaches the pendulum in Figure P9.36a. Figure P9.36b shows the situation just after the projectile is captured in the pendulum. In Figure P9.36c, the pendulum arm has swung upward and come to rest momentarily at a height A above its initial position. Prove that the ratio of the kinetic energy of the projectilependulum system immediately after the collision to the kinetic energy immediately before is m1|/(m1 + m2). (b) What is the ratio of the momentum of the system immediately after the collision to the momentum immediately before? (c) A student believes that such a large decrease in mechanical energy must be accompanied by at least a small decrease in momentum. How would you convince this student of the truth? Figure P9.36 Problem. 36 and 43. (a) A metal ball moves toward the pendulum. (b) The ball is captured by the pendulum. (c) The ballpendulum combination swings up through a height h before coming to rest.Review. A 60.0-kg person running at an initial speed of 4.00 m/s jumps onto a 120-kg cart initially at rest (Fig. P9.37). The person slides on the carts top surface and finally comes to rest relative to the cart. The coefficient of kinetic friction between the person and the cart is 0.400. Friction between the cart and ground can be ignored. (a) Find the final velocity of the person and cart relative to the ground. (b) Find the friction force acting on the person while he is sliding across the top surface of the cart. (c) How long does the friction force act on the person? (d) Find the change in momentum of the person and the change in momentum of the cart. (c) Determine the displacement of the person relative to the ground while he is sliding on the cart. (f) Determine the displacement of the cart relative to the ground while the person is sliding. (g) Find the change in kinetic energy of the person. (h) Find the change in kinetic energy of the cart. (i) Explain why the answers to (g) and (h) differ. (What kind of collision is this one, and what accounts for the loss of mechanical energy) Figure P9.37 A cannon is rigidly attached to a carriage, which can move along horizontal rails but is connected to a post by a large spring, initially unstretched and with force constant k = 2.00 104 N/m, as shown in Figure P9.38. The cannon fires a 200-kg projectile at a velocity of 125 m/s directed 45.0 above the horizontal. (a) Assuming that the mass of the cannon and its carriage is 5 000 kg, find the recoil speed of the cannon. (b) Determine the maximum extension of the spring. (c) Find the maximum force the spring exerts on the carriage. (d) Consider the system consisting of the cannon, carriage, and projectile. Is the momentum of this system conserved during the firing? Why or why not? Figure P9.38 A 1.25-kg wooden block rests on a table over a large hole as in Figure P9.39. A 5.00-g bullet with an initial velocity vi is fired upward into the bottom of the block and remains in the block after the collision. The block and bullet rise to a maximum height of 22.0 cm. (a) Describe how you would find the initial velocity of the bullet using ideas you have learned in this chapter. (b) Calculate the initial velocity of the bullet from the information provided. Figure P9.39 Problems 39 and 40.A wooden block of mass M rests on a table over a large hole as in Figure P9.39. A bullet of mass m with an initial velocity of vi is fired upward into the bottom of the block and remains in the block after the collision. The block and bullet rise to a maximum height of h. (a) Describe how you would find the initial velocity of the bullet using ideas you have learned in this chapter. (b) Find an expression for the initial velocity of the bullet. Figure P9.39 Problems 39 and 40.Two gliders are set in motion on a horizontal air track. A light spring of force constant k is attached to the back end of the second glider. As shown in Figure P9.41, the first glider, of mass m1, moves to the right with speed v1, and the second glider, of mass m2, moves more slowly to the right with speed v2. When m1 collides with the spring attached to m2, the spring compresses by a distance xmax, and the gliders then move apart again. In terms of v1, v2, m1, m2, and k, find (a) the speed v at maximum compression, (b) the maximum compression xmax, and (c) the velocity of each glider after m1 has lost contact with the spring. Figure P9.41 Pursued by ferocious wolves, you are in a sleigh with no horses, gliding without friction across an ice-covered lake. You take an action described by the equations (270kg)(7.50m/s)i=(15.0kg)(v1fi)+(255kg)(v2fi)v1f+v2f=8.00m/s (a) Complete the statement of the problem, giving the data and identifying the unknowns. (b) Find the values of v1f and v2f (c) Find the amount of energy that has been transformed from potential energy stored in your body to kinetic energy of the system.Review. A student performs a ballistic pendulum experiment using an apparatus similar to that discussed in Example 9.6 and shown in Figure P9.36. She obtains the following average data: h = 8.68 cm, projectile mass m1 = 68.8 g, and pendulum mass m2 = 263 g. (a) Determine the initial speed v1A of the projectile. (b) The second part of her experiment is to obtain v1A by firing the same projectile horizontally (with the pendulum removed from the path) and measuring its final horizontal position x and distance of fall y (Fig. P9.43). What numerical value does she obtain for v1A based on her measured values of x = 257 cm and y = 85.3 cm? (c) What factors might account for the difference in this value compared with that obtained in part (a)? Figure P9.43 Why is the following situation impossible? An astronaut, together with the equipment he carries, has a mass of 150 kg. He is taking a space walk outside his spacecraft, which is drifting through space with a constant velocity. The astronaut accidentally pushes against the spacecraft and begins moving away at 20.0 m/s, relative to the spacecraft, without a tether. To return, he takes equipment off his space suit and throws it in the direction away from the spacecraft. Because of his bulky space suit, he can throw equipment at a maximum speed of 5.00 m/s relative to himself. After throwing enough equipment, he starts moving back to the spacecraft and is able to grab onto it and climb inside.Review. A bullet of mass m = 8.00 g is fired into a block of mass M = 250 g that is initially at rest at the edge of a frictionless table of height h = 1.00 m (Fig. P9.45). The bullet remains in the block, and after the impact the block lands d = 2.00 m from the bottom of the table. Determine the initial speed of the bullet. Figure P9.45 Problems 45 and 46.Review. A bullet of mass m is fired into a block of mass M initially at rest at the edge of a frictionless table of height h (Fig. P9.45). The bullet remains in the block, and after impact the block lands a distance d from the bottom of the table. Determine the initial speed of the bullet.A 0.500-kg sphere moving with a velocity expressed as (2.00i3.00j+1.00k)m/s strikes a second, lighter sphere of mass 1.50 kg moving with an initial velocity of (1.00i+2.00j3.00k)m/s. (a) The velocity of the 0.500-kg sphere after the collision is (1.00i+3.00j8.00k)m/s. Find the final velocity of the 1.50-kg sphere and identify the kind of collision (elastic, inelastic, or perfectly inelastic). (b) Now assume the velocity of the 0.500-kg sphere after the collision is (0.250i+0.750j2.00k)m/s. Find the final velocity of the 1.50-kg sphere and identify the kind of collision. (c) What If? Take the velocity of the 0.500-kg sphere after the collision as (1.00i+3.00jak)m/s. Find the value of a and the velocity of the 1.50-kg sphere after an elastic collision.48AP Review. A light spring of force constant 3.85 N/m is compressed by 8.00 cm and held between a 0.250-kg block on the left and a 0.500-kg block on the right. Both blocks are at rest on a horizontal surface. The blocks are released simultaneously so that the spring tends to push them apart. Find the maximum velocity each block attains if the coefficient of kinetic friction between each block and the surface is (a) 0, (b) 0.100, and (c) 0.462. Assume the coefficient of static friction is greater than the coefficient of kinetic friction in every case.50AP Review. There are (one can say) three coequal theories of motion for a single particle: Newtons second law, stating that the total force on the particle causes its acceleration; the workkinetic energy theorem, stating that the total work on the particle causes its change in kinetic energy; and the impulsemomentum theorem, stating that the total impulse on the panicle causes its change in momentum. In this problem, you compare predictions of the three theories in one particular case. A 3.00-kg object has velocity 7.00jm/s. Then, a constant net force 12.0iN acts on the object for 5.00 s. (a) Calculate the objects final velocity, using the impulsemomentum theorem. (b) Calculate its acceleration from a=(vfvi)/t. (c) Calculate its acceleration from a=F/m. (d) Find the objects vector displacement from r=vit+12at2 (e) Find the work done on the object from W=Fr. (f) Find the final kinetic energy from 12mvf2=12mvfvf. (g) Find the final kinetic energy from 12mvi2+W. (h) State the result of comparing the answers to parts (b) and (c), and the answers to parts (f) and (g).Sand from a stationary hopper falls onto a moving conveyor belt at the rate of 5.00 kg/s as shown in Figure P9.52. The conveyor belt is supported by frictionless rollers and moves at a constant speed of v = 0.750 m/s under the action of a constant horizontal external force Fext supplied by the motor that drives the belt. Find (a) the sands rate of change of momentum in the horizontal direction, (b) the force of friction exerted by the belt on the sand, (c) the external force Fext, (d) the work done by Fext in 1 s, and (c) the kinetic energy acquired by the falling sand each second due to the change in its horizontal motion. (f) Why are the answers to parts (d) and (e) different?Two particles with masses m and 3m are moving toward each other along the x axis with the same initial speeds vi. Particle m is traveling to the left, and particle 3m is traveling to the right. They undergo an elastic glancing collision such that particle m is moving in the negative y direction after the collision at a right angle from its initial direction. (a) Find the final speeds of the two particles in terms of vi. (b) What is the angle at which the particle 3m is scattered?On a horizontal air track, a glider of mass m carries a -shaped post. The post supports a small dense sphere, also of mass m, hanging just above the top of the glider on a cord of length L. The glider and sphere are initially at rest with the cord vertical. A constant horizontal force of magnitude F is applied to the glider, moving it through displacement x1; then the force is removed. During the time interval when the force is applied, the sphere moves through a displacement with horizontal component x2. (a) Find the horizontal component of the velocity of the center of mass of the glidersphere system when the force is removed. (b) After the force is removed, the glider continues to move on the track and the sphere swings back and forth, both without friction. Find an expression for the largest angle the cord makes with the vertical.A rigid object rotates in a counterclockwise sense around a fixed axis. Each of the following pairs of quantities represents an initial angular position and a final angular position of the rigid object. (i) Which of the sets can only occur if the rigid object rotates through more than 180? (a) 3 rad, 6 rad (b) 1 rad, 1 rad (c) 1 rad, 5 rad (ii) Suppose the change in angular position for each of these pairs of values occurs in 1 s. Which choice represents the lowest average angular speed?Consider again the pairs of angular positions for the rigid object in Quick. Quiz 10.1. If the object starts from rest at the initial angular position, moves counterclockwise with constant angular acceleration, and arrives at the final angular position with the same angular speed in all three cases, for which choice is the angular acceleration the highest?Ethan and Rebecca are riding on a merry-go-round. Ethan rides on a horse at the outer rim of the circular platform, twice as far from the center of the circular platform as Rebecca, who rides on an inner horse. (i) When the merry-go-round is rotating at a constant angular speed, what is Ethans angular speed? (a) twice Rebeccas (b) the same as Rebeccas (c) half of Rebeccas (d) impossible to determine (ii) When the merry-go-round is rotating at a constant angular speed, describe Ethans tangential speed from the same list of choices.If you are trying to loosen a stubborn screw from a piece of wood with a screwdriver and fail, should you find a screwdriver for which the handle is (a) longer or (b) fatter?You turn off your electric drill and find that the time interval for the rotating bit to come to rest due to frictional torque in the drill is t. You replace the bit with a larger one that results in a doubling of the moment of inertia of the drills entire rotating mechanism. When this larger bit is rotated at the same angular speed as the first and the drill is turned off, the frictional torque remains the same as that for the previous situation. What is the time interval for this second bit to come to rest? (a) 4t (b) 2t (c) t (d) 0.5t (e) 0.25t (f) impossible to determine A section of hollow pipe and a solid cylinder have the same radius, mass, and length. They both rotate about their long central axes with the same angular speed. Which object has the higher rotational kinetic energy? (a) The hollow pipe does. (b) The solid cylinder does. (c) They have the same rotational kinetic energy. (d) It is impossible to determine.A ball rolls without slipping down incline A, starting from rest. At the same time, a box starts from rest and slides down incline B, which is identical to incline A except that it is frictionless. Which arrives at the bottom first? (a) The ball arrives first. (b) The box arrives first. (c) Both arrive at the same time. (d) It is impossible to determine.(a) Find the angular speed of the Earths rotation about its axis. (b) How does this rotation affect the shape of the Earth?A bar on a hinge starts from rest and rotates with an angular acceleration = 10 + 6t, where is in rad/s2 and t is in seconds. Determine the angle in radians through which the bar turns in the first 4.00 s.A wheel starts from rest and rotates with constant angular acceleration to reach an angular speed of 12.0 rad/s in 3.00 s. Find (a) the magnitude of the angular acceleration of the wheel and (b) the angle in radians through which it rotates in this time interval.A machine part rotates at an angular speed of 0.060 rad/s; its speed is then increased to 2.2 rad/s at an angular acceleration of 0.70 rad/s2. (a) Find the angle through which the part rotates before reaching this final speed. (b) If both the initial and final angular speeds are doubled and the angular acceleration remains the same, by what factor is the angular displacement changed? Why?A dentists drill starts from rest. After 3.20 s of constant angular acceleration, it turns at a rate of 2.51 104 rev/min. (a) Find the drills angular acceleration. (b) Determine the angle (in radians) through which the drill rotates during this period.Why is the following situation impossible? Starting from rest, a disk rotates around a fixed axis through an angle of 50.0 rad in a time interval of 10.0 s. The angular acceleration of the disk is constant during the entire motion, and its final angular speed is 8.00 rad/s.Review. Consider a tall building located on the Earths equator. As the Earth rotates, a person on the top floor of the building moves faster than someone on the ground with respect to an inertial reference frame because the person on the ground is closer to the Earths axis. Consequently, if an object is dropped from the top floor to the ground a distance h below, it lands east of the point vertically below where it was dropped. (a) How far to the east will the object land? Express sour answer in terms of h, g, and the angular speed of the Earth. Ignore air resistance and assume the free-fall acceleration is constant over this range of heights. (b) Evaluate the eastward displacement for h = 50.0 m. (c) In your judgment, were we justified in ignoring this aspect of the Coriolis effect in our previous study of free fall? (d) Suppose the angular speed of the Earth were to decrease with constant angular acceleration due to tidal friction. Would the eastward displacement of the dropped object increase or decrease compared with that in part (b)?Make an order-of-magnitude estimate of the number of revolutions through which a typical automobile tire turns in one year. State the quantities you measure or estimate and their values.A discus thrower (Fig. P10.9) accelerates a discus from rest to a speed of 25.0 m/s by whirling it through 1.25 rev. Assume the discus moves on the are of a circle 1.00 m in radius. (a) Calculate the final angular speed of the discus. (b) Determine the magnitude of the angular acceleration of the discus, assuming it to be constant. (c) Calculate the time interval required for the discus to accelerate from rest to 25.0 m/s. Figure P10.9 10P A car accelerates uniformly from rest and reaches a speed of 22.0 m/s in 9.00 s. Assuming the diameter of a tire is 58.0 cm, (a) find the number of revolutions the tire makes during this motion, assuming that no slipping occurs. (b) What is the final angular speed of a tire in revolutions per second?Review. A small object with mass 4.00 kg moves counterclockwise with constant angular speed 1.50 rad/s in a circle of radius 3.00 m centered at the origin. It starts at the point with position vector 3.00im. It then undergoes an angular displacement of 9.00 rad. (a) What is its new position vector? Use unit-vector notation for all vector answers. (b) In what quadrant is the particle located, and what angle does its position vector make with the positive x axis? (c) What is its velocity? (d) In what direction is it moving? (e) What is its acceleration? (f) Make a sketch of its position, velocity, and acceleration vectors. (g) What total force is exerted on the object?In a manufacturing process, a large, cylindrical roller is used to flatten material fed beneath it. The diameter of the roller is 1.00 m, and, while being driven into rotation around a fixed axis, its angular position is expressed as =2.50t20.600t3 where is in radians and t is in seconds. (a) Find the maximum angular speed of the roller. (b) What is the maximum tangential speed of a point on the rim of the roller? (c) At what time t should the driving force be removed from the roller so that the roller does not reverse its direction of rotation? (d) Through how many rotations has the roller turned between t = 0 and the time found in part (c)?Find the net torque on the wheel in Figure P10.14 about the axle through O, taking a = 10.0 cm and b = 25.0 cm.A grinding wheel is in the form of a uniform solid disk of radius 7.00 cm and mass 2.00 kg. It starts from rest and accelerates uniformly under the action of the constant torque of 0.600 N m that the motor exerts on the wheel. (a) How long does the wheel take to reach its final operating speed of 1 200 rev/min? (b) Through how many revolutions does it turn while accelerating?Review. A block of mass m1 = 2.00 kg and a block of mass m2 = 6.00 kg are connected by a massless string over a pulley in the shape of a solid disk having radius R = 0.250 m and mass M = 10.0 kg. The fixed, wedge-shaped ramp makes an angle of = 30.0 as shown in Figure P10.16. The coefficient of kinetic friction is 0.360 for both blocks. (a) Draw force diagrams of both blocks and of the pulley. Determine (b) the acceleration of the two blocks and (c) the tensions in the string on both sides of the pulley. Figure P10.16 A model airplane with mass 0.750 kg is tethered to the ground by a wire so that it flies in a horizontal circle 30.0 m in radius. The airplane engine provides a net thrust of 0.800 N perpendicular to the tethering wire. (a) Find the torque the net thrust produces about the center of the circle. (b) Find the angular acceleration of the airplane. (c) Find the translational acceleration of the airplane tangent to its flight path.A disk having moment of inertia 100 kg m2 is free to rotate without friction, starting from rest, about a fixed axis through its center. A tangential force whose magnitude can range from F = 0 to F = 50.0 N can be applied at any distance ranging from R = 0 to R = 3.00 m from the axis of rotation. (a) Find a pair of values of F and R that cause the disk to complete 2.00 rev in 10.0 s. (b) Is your answer for part (a) a unique answer? How many answers exist?Your grandmother enjoys creating pottery as a hobby. She uses a potters wheel, which is a stone disk of radius R = 0.500 m and mass M = 100 kg. In operation, the wheel rotates at 50.0 rev/min. While the wheel is spinning, your grandmother works clay at the center of the wheel with her hands into a pot-shaped object with circular symmetry. When the correct shape is reached, she wants to stop the wheel in as short a time interval as possible, so that the shape of the pot is not further distorted by the rotation. She pushes continuously with a wet rag as hard as she can radially inward on the edge of the wheel and the wheel stops in 6.0 s. (a) You would like to build a brake to stop the wheel in a shorter time interval, but you must determine the coefficient of friction between the rag and the wheel in order to design a better system. You determine that the maximum pressing force your grandmother can sustain for 6.00 s is 70.0 N. (b) What If? If your grandmother instead chooses to press down on the upper surface of the wheel a distance r = 0.300 m from the axis of rotation, what is the force needed to stop the wheel in 6.00 s? Assume that the coefficient of kinetic friction between the wet rag and the wheel remains the same as before.At a local mine, a cave-in has trapped a number of miners. You and some classmates rush to the scene to see how you can help. The trapped miners have been able to reach a point in the mine at the bottom of a tall vertical shaft to the surface, allowing them access to fresh air. But they are in desperate need of fresh water and bandages for injuries. Some rescue workers ask you to help pack a light plastic cylindrical container with bottles of water and bandages. Simply dropping the container into the shaft risks damaging the container and contents and injuring the miners. Tying a rope to the container and lowering it on the end of the rope takes a long time. A quick and relatively safe method is to wrap a lightweight rope around the container. One end of the rope will be secured and the container will be released into the vertical shaft. The container will unroll off the rope like a falling yo-yo. (a) If immediate access to the lightweight bandages is needed due to injuries, so that you want the container to reach the bottom of the shaft in the shortest possible time interval, should you pack the heavy water bottles at the center of the container or near the outer edges? (b) If the medical necessity is not so urgent and, for safety considerations, you want the container to arrive at the bottom of the shaft with the lowest possible speed, should you pack the heavy water bottles at the center of the container or near the outer edges? Assume that the center of mass of the container is at its center.You have just bought a new bicycle. On your first riding trip, it seems that the hike comes to rest relatively quickly after you stop pedaling and let the bicycle coast on flat ground. You call the bicycle shop from which you purchased the vehicle and describe the problem. The technician says that they will replace the bearings in the wheels or do whatever else is necessary if you can prove that the frictional torque in the axle of the wheels is worse than 0.02 N m. At first, you are discouraged by the technical sound of what you have been told and by the absence of any tool to measure torque in your garage. But then you remember that you are taking a physics class! You take your bike into the garage, turn it upside down and start spinning the wheel while you think about how to determine the frictional torque. The driveway outside the garage had a small puddle, so you notice that droplets of water are flying off the edge of one point on the tire tangentially, including drops that are projected straight upward, as shown in Figure P10.21. Ah-ha! Here is your torque-measuring method! The upward-projected drops leave the rim of the wheel at the same level as the axle. You measure the height to which a drop rises from the level of the axle: h1 = 54.0 cm. The wet spot on the tire makes one revolution and another drop is projected upward. You measure its highest point: h2 = 51.0 cm. You measure the radius of the wheel: r = 0.381 m. Finally, you take the wheel off the bike and find its mass: m = 0.850 kg. Because most of the mass of the wheel is at the tire, you model the wheel as a hoop. What do you tell the technician when you call back? Figure P10.21 Imagine that you stand tall and turn about a vertical axis through the lop of your head and the point halfway between your ankles. Compute an order-of-magnitude estimate for the moment of inertia of your body for this rotation. In your solution, state the quantities you measure or estimate and their values.Following the procedure used in Example 10.7, prove that the moment of inertia about the y axis of the rigid rod in Figure 10.15 is 13ML2.Two balls with masses M and m are connected by a rigid rod of length L, and negligible mass as shown in Figure P10.24. For an axis perpendicular to the rod. (a) show that the system has the minimum moment of inertia when the axis passes through the center of mass. (b) Show that this moment of inertia is I = L2, where = mM/(m + M).Rigid rods of negligible mass lying along the y axis connect three panicles (Fig. P10.25). The system rotates about the x axis with an angular speed of 2.00 rad/s. Find (a) the moment of inertia about the x axis, (b) the total rotational kinetic energy evaluated from 12I2, (c) the tangential speed of each particle, and (d) the total kinetic energy evaluated from 12mivi2. (e) Compare the answers for kinetic energy in parts (a) and (b). Figure P10.25 A war-wolf or trebuchet is a device used during the Middle Ages to throw rocks at castles and now sometimes used to fling large vegetables and pianos as a sport. A simple trebuchet is shown in Figure P10.26. Model it as a stiff rod of negligible mass, 3.00 m long, joining particles of mass m1 = 0.120 kg and m2 = 60.0 kg at its ends. It can turn on a frictionless, horizontal axle perpendicular to the rod and 14.0 cm from the large-mass particle. The operator releases the trebuchet from rest in a horizontal orientation. (a) Find the maximum speed that the small-mass object attains. (b) While the small-mass object is gaining speed, does it move with constant acceleration? (c) Does it move with constant tangential acceleration? (d) Does the trebuchet move with constant angular acceleration? (e) Does it have constant momentum? (f) Does the trebuchetEarth system have constant mechanical energy? Figure P10.26 Big Ben, the nickname for the clock in Elizabeth Tower (named after the Queen in 2012) in London, has an hour hand 2.70 m long with a mass of 60.0 kg and a minute hand 4.50 m long with a mass of 100 kg (Fig. P10.27). Calculate the total rotational kinetic energy of the two hands about the axis of rotation. (You may model the hands as long, thin rods rotated about one end. Assume the hour and minute hands are rotating at a constant rate of one revolution per 12 hours and 60 minutes, respectively.) Figure P10.27 Problems 27 and 40.Consider two objects with m1 m2 connected by a light string that passes over a pulley having a moment of inertia of I about its axis of rotation as shown in Figure P10.28. The string does not slip on the pulley or stretch. The pulley turns without friction. The two objects are released from rest separated by a vertical distance 2h. (a) Use the principle of conservation of energy to find the translational speeds of the objects as they pass each other. (b) Find the angular speed of the pulley at this time. Figure P10.28 Review. An object with a mass of m = 5.10 kg is attached to the free end of a light string wrapped around a reel of radius R = 0.250 m and mass M = 3.00 kg. The reel is a solid disk, free to rotate in a vertical plane about the horizontal axis passing through its center as shown in Figure P10.29. The suspended object is released from rest 6.00 m above the floor. Determine (a) the tension in the string, (b) the acceleration of the object, and (c) the speed with which the object hits the floor. (d) Verify your answer to part (c) by using the isolated system (energy) model. Figure P10.29 Why is the following situation impossible? In a large city with an air-pollution problem, a bus has no combustion engine. It runs over its citywide route on energy drawn from a large, rapidly rotating flywheel under the floor of the bus. The flywheel is spun up to its maximum rotation rate of 3 000 rev/min by an electric motor at the bus terminal. Every time the bus speeds up, the flywheel slows down slightly. The bus is equipped with regenerative braking so that the flywheel can speed up when the bus slows down. The flywheel is a uniform solid cylinder with mass 1 200 kg and radius 0.500 m. The bus body does work against air resistance and rolling resistance at the average rate of 25.0 hp as it travels its route with an average speed of 35.0 km/h.A uniform solid disk of radius R and mass M is free to rotate on a frictionless pivot through a point on its rim (Fig. P10.31). If the disk is released from rest in the position shown by the copper-colored circle, (a) what is the speed of its center of mass when the disk reaches the position indicated by the dashed circle? (b) What is the speed of the lowest point on the disk in the dashed position? (c) What If? Repeat part (a) using a uniform hoop. Figure P10.31 This problem describes one experimental method for determining the moment of inertia of an irregularly shaped object such as the payload for a satellite. Figure P10.32 shows a counterweight of mass m suspended by a cord wound around a spool of radius r, forming part of a turntable supporting the object. The turntable can rotate without friction. When the counterweight is released from rest, it descends through a distance h, acquiring a speed v. Show that the moment of inertia I of the rotating apparatus (including the turntable) is mr2(2gh/v2 1). Figure P10.32 A tennis ball is a hollow sphere with a thin wall. It is set rolling without slipping at 4.03 m/s on a horizontal section of a track as shown in Figure P10.33. It rolls around the inside of a vertical circular loop of radius r = 45.0 cm. As the ball nears the bottom of the loop, the shape of the track deviates from a perfect circle so that the ball leaves the track at a point h = 20.0 cm below the horizontal section. (a) Find the balls speed at the top of the loop. (b) Demonstrate that the ball will not fall from the track at the top of the loop. (c) Find the balls speed as it leaves the track at the bottom. (d) What If? Suppose that static friction between ball and track were negligible so that the ball slid instead of rolling. Describe the speed of the ball at the top of the loop in this situation. (e) Explain your answer to part (d). Figure P10.33 A smooth cube of mass m and edge length r slides with speed v non a horizontal surface with negligible friction. The cube then moves up a smooth incline that makes an angle with the horizontal. A cylinder of mass m and radius r rolls without slipping with its center of mass moving with speed v and encounters an incline of the same angle of inclination but with sufficient friction that the cylinder continues to roll without slipping. (a) Which object will go the greater distance up the incline? (b) Find the difference between the maximum distances the objects travel up the incline. (c) Explain what accounts for this difference in distances traveled.A metal can containing condensed mushroom soup has mass 215 g, height 10.8 cm, and diameter 6.38 cm. It is placed at rest on its side at the top of a 3.00-m-long incline that is at 25.0 to the horizontal and is then released to roll straight down. It reaches the bottom of the incline after 1.50 s. (a) Assuming mechanical energy conservation, calculate the moment of inertia of the can. (b) Which pieces of data, if any, are unnecessary for calculating the solution? (c) Why cant the moment of inertia be calculated from I=12mr2 for the cylindrical can?You have been hired as an expert witness in the case of a factory owner suing a demolition company. The particular case involves a smokestack at a factory being demolished. In order to save money, the factory owner wanted to move the smokestack to a nearby factory that was being built. The demolition company guaranteed to deliver the undamaged smokestack to the new factory by toppling the smokestack freely onto a huge cushioned platform lying on the ground. The then-horizontal smokestack would have been loaded onto a long truck rig for transport to the new factory. However, as the smokestack toppled, it broke apart at a point along its length. The factory owner is blaming the demolition company for the destruction of his smokestack. The demolition company is claiming that there was a defect in the smokestack and that is the reason for its destruction. What advice do you give the attorney who is handling the case on the side of the factory owner?A shaft is turning at 65.0 rad/s at time t = 0. Thereafter, its angular acceleration is given by =10.05.00t where is in rad/s2 and t is in seconds. (a) Find the angular speed of the shaft at t = 3.00 s. (b) Through what angle does it turn between t = 0 and t = 3.00 s?A shaft is turning at angular speed at time t = 0. Thereafter, its angular acceleration is given by =A=Bt (a) Find the angular speed of the shaft at time t. (b) Through what angle does it turn between t = 0 and t?An elevator system in a tall building consists of a 800-kg car and a 950-kg counterweight joined by a light cable of constant length that passers over a pulley of mass 280 kg. The pulley, called a sheave, is a solid cylinder of radius 0.700 m turning on a horizontal axle. The cable does not slip on the sheave. A number n of people, each of mass 80.0 kg, are riding in the elevator car, moving upward at 3.00 m/s and approaching the floor where the car should stop. As an energy-conservation measure, a computer disconnects the elevator motor at just the right moment so that t he sheavecarcounterweight system then coasts freely without friction and comes to rest at the floor desired. There it is caught by a simple latch rather than by a massive brake. (a) Determine the distance d the car coasts upward as a function of n. Evaluate the distance for (b) n = 2, (c) n = 12, and (d) n = 0. (e) For what integer values of n does the expression in part (a) apply? (f) Explain your answer to part (e). (g) If an infinite number of people could fit on the elevator, what is the value of d?The hour hand and the minute hand of Big Ben, the Elizabeth Tower clock in London, are 2.70 m and 4.50 m long and have masses of 60.0 kg and 100 kg, respectively (see Fig. P10.27). (a) Determine the total torque due to the weight of these hands about the axis of rotation when the lime reads (i) 3:00, (ii) 5:15, (iii) 6:00, (iv) 8:20, and (v) 9:45. (You may model the hands as long, thin, uniform rods.) (b) Determine all times when the total torque about the axis of rotation is zero. Determine the times to the nearest second, solving a transcendental equation numerically.Review. A string is wound around a uniform disk of radius R and mass M. The disk is released from rest with the string vertical and its top end tied to a fixed bar (Fig. P10.41). Show that (a) the tension in the string is one third of the weight of the disk, (b) the magnitude of the acceleration of the center of mass is 2g/3, and (c) the speed of the center of mass is (4gh/3)1/2 after the disk has descended through distance h. (d) Verify your answer to part (c) using the energy approach. Figure P10.41 Review. A spool of wire of mass M and radius R is unwound under a constant force F (Fig. P10.42). Assuming the spool is a uniform, solid cylinder that doesnt slip, show that (a) the acceleration of the center of mass is 4F/3M and (b) the force of friction is to the right and equal in magnitude to F/3. (c) If the cylinder starts from rest and rolls without slipping, what is the speed of its center of mass after it has rolled through a distance d?Review. A clown balances a small spherical grape at the top of his bald head, which also has the shape of a sphere. After drawing sufficient applause, the grape starts from rest and rolls down without slipping. It will leave contact with the clowns scalp when the radial line joining it to the center of curvature makes what angle with the vertical?As a gasoline engine operates, a flywheel turning with the crankshaft stores energy after each fuel explosion, providing the energy required to compress the next charge of fuel and air. For the engine of a certain lawn tractor, suppose a flywheel must be no more than 18.0 cm in diameter. Its thickness, measured along its axis of rotation, must be no larger than 8.00 cm. The flywheel must release energy 60.0 J when its angular speed drops from 800 rev/min to 600 rev/min. Design a sturdy steel (density 7.85 103 kg/m3) flywheel to meet these requirements with the smallest mass you can reasonably attain. Specify the shape and mass of the flywheel.A spool of thread consists of a cylinder of radius R1 with end caps of radius R2 as depicted in the end view shown in Figure P10.45. The mass of the spool, including the thread, is m, and its moment of inertia about an axis through its center is I. The spool is placed on a rough, horizontal surface so that it rolls without slipping when a force T acting to the right is applied to the free end of the thread. (a) Show that the magnitude of the friction force exerted by the surface on the spool is given by f=(I+mR1R2I+mR22)T (b) Determine the direction of the force of friction. Figure P10.45 To find the total angular displacement during the playing time of the compact disc in part (B) of Example 10.2, the disc was modeled as a rigid object under constant angular acceleration. In reality, the angular acceleration of a disc is not constant. In this problem, let us explore the actual time dependence of the angular acceleration. (a) Assume the track on the disc is a spiral such that adjacent loops of the track are separated by a small distance h. Slum that the radius r of a given portion of the track is given by r=ri+h2 where ri is the radius of the innermost portion of the track and is the angle through which the disc turns to arrive at the location of the track of radius r. (b) Show that the rate of change of the angle is given by ddt=vri+(h/2) where v is the constant speed with which the disc surface passes the laser. (c) From the result in part (b), use integration to find an expression for the angle as a function of time. (d) From the result in part (c), use differentiation to find the angular acceleration of the disc as a function of time.A uniform, hollow, cylindrical spool has inside radius R/2, outside radius R, and mass M (Fig. P10.47). It is mounted so that it rotates on a fixed, horizontal axle. A counterweight of mass m is connected to the end of a string wound around the spool. The counterweight falls from rest at t = 0 to a position y at time t. Show that the torque due to the friction forces between spool and axle is f=R[m(g2yt2)M5y4t2] Figure P10.47 A cord is wrapped around a pulley that is shaped like a disk of mass m and radius r. The cords free end is connected to a block of mass M. The block starts from rest and then slides down an incline that makes an angle with the horizontal as show n in Figure P10.48. The coefficient of kinetic friction between block and incline is . (a) Use energy methods to show that the blocks speed as a function of position d down the incline is v=4Mgd(sincos)m+2M (b) Find the magnitude of the acceleration of the block in terms of , m, M, g, and . Figure P10.48 Which of the following statements about the relationship between the magnitude of the cross product of two vectors and the product of the magnitudes of the vectors is true? (a) AB is larger than AB. (b) AB is smaller than AB. (c) AB could be larger or smaller than AB, depending on the angle between the vectors. (d) AB could he equal to AB.Recall the skater described at the beginning of this section. Let her mass be m. (i) What would be her angular momentum relative to the pole at the instant she is a distance d from the pole if she were skating directly toward it at speed v? (a) zero (b) mvd (c) impossible to determine (ii) What would be her angular momentum relative to the pole at the instant she is a distance d from the pole if she were skating at speed v along a straight path that is offset by a perpendicular distance a from the pole? (a) zero (b) mvd (c) mva (d) impossible to determine A solid sphere and a hollow sphere have the same mass and radius. They are rotating with the same angular speed. Which one has the higher angular momentum? (a) the solid sphere (b) the hollow sphere (c) both have the same angular momentum (d) impossible to determine A competitive diver leaves the diving board and falls toward the water with her body straight and rotating slowly. She pulls her arms and legs into a tight tuck position. What happens to her rotational kinetic energy? (a) It increases. (b) It decreases. (c) It stays the same. (d) It is impossible to determine.1P The displacement vectors 42.0 cm at 15.0 and 23.0 cm at 65.0 both start from the origin and form two sides of a parallelogram. Both angles are measured counterclockwise from the x axis. (a) Find the area of the parallelogram. (b) Find the length of its longer diagonal.If AB=AB, what is the angle between A and B?Use the definition of the vector product and the definitions of the unit vectors i, j, and k to prove Equations 11.7. You may assume the x axis points to the right, the y axis up, and the z axis horizontally toward you (not away from you). This choice is said to make the coordinate system a right-handed system.Two forces F1 and F2 act along the two sides of an equilateral triangle as shown in Figure P11.5. Point O is the intersection of the altitudes of the triangle. (a) Find the magnitude of a third force F3 to be applied at B and along BC that will make the total torque zero about the point O. (b) What If? Will the total torque change if F3 is applied not at B but at any other point along BC? Figure P11.5 6P A particle is located at a point described by the position vector r=(4.00i+6.00j)m, and a force exerted on it is given by F=(3.00i+2.00j)N. (a) What is the torque acting on the particle about the origin? (b) Can there be another point about which the torque caused by this force on this particle will be in the opposite direction and half as large in magnitude? (c) Can there be more than one such point? (d) Can such a point lie on the y axis? (e) Can more than one such point lie on the y axis? (f) Determine the position vector of one such point.A 1.50-kg particle moves in the xy plane with a velocity of v=(4.20i3.60j)m/s. Determine the angular momentum of the particle about the origin when its position vector r=(1.50i+2.20j)m.A particle of mass m moves in the xy plane with a velocity of v=vxi+vyj. Determine the angular momentum of the particle about the origin when its position vector is r=xi+yj.Heading straight toward the summit of Pikes Peak, an airplane of mass 12 000 kg flies over the plains of Kansas at nearly constant altitude 4.30 km with constant velocity 175 m/s west. (a) What is the airplanes vector angular momentum relative to a wheat farmer on the ground directly below the airplane? (b) Does this value change as the airplane continues its motion along a straight line? (c) What If? What is its angular momentum relative to the summit of Pikes Peak?Review. A projectile of mass m is launched with an initial velocity vi making an angle with the horizontal as shown in Figure P11.11. The projectile moves in the gravitational field of the Earth. Find the angular momentum of the projectile about the origin (a) when the projectile is at the origin, (b) when it is at the highest point of its trajectory, and (c) just before it hits the ground. (d) What torque causes its angular momentum to change? Figure P11.11 Review. A conical pendulum consists of a bob of mass m in motion in a circular path in a horizontal plane as shown in Figure P11.12. During the motion, the supporting wire of length maintains a constant angle with the vertical. Show that the magnitude of the angular momentum of the bob about the vertical dashed line is L=(m2glSsin4cos)1/2 Figure P11.12 A particle of mass m moves in a circle of radius R at a constant v speed as shown in Figure P11.13. Time t = 0 is defined as when the particle is at point Q. Determine the angular momentum of the particle about the axis perpendicular to the page through point P as a function of time. Figure P11.13 A 5.00-kg particle starts from the origin at time zero. Its velocity as a function of time is given by v=6t2i+2tj where v is in meters per second and t is in seconds. (a) Find its position as a function of time. (b) Describe its motion qualitatively. Find (c) its acceleration as a function of time, (d) the net force exerted on the particle as a function of time, (e) the net torque about the origin exerted on the particle as a function of time, (f) the angular momentum of the particle as a function of time, (g) the kinetic energy of the particle as a function of time, and (h) the power injected into the system of the particle as a function of time.A ball having mass m is fastened at the end of a flagpole that is connected to the side of a tall building at point P as shown in Figure P11.15. The length of the flagpole is , and it makes an angle with the x axis. The ball becomes loose and starts to fall with acceleration gj. (a) Determine the angular momentum of the ball about point P as a function of time. (b) For what physical reason does the angular momentum change? (c) What is the rate of change of the angular momentum of the ball about point P? Figure P11.15 A uniform solid sphere of radius r = 0.500 m and mass m = 15.0 kg turns counterclockwise about a vertical axis through its center. Find its vector angular momentum about this axis when its angular speed is 3.00 rad/s.A uniform solid disk of mass m = 3.00 kg and radius r = 0.200 m rotates about a fixed axis perpendicular to its face with angular frequency 6.00 rad/s. Calculate the magnitude of the angular momentum of the disk when the axis of rotation (a) passes through its center of mass and (b) passes through a point midway between the center and the rim.Show that the kinetic energy of an object rotating about a fixed axis with angular momentum L = I can be written as K = L2/2I.Big Ben (Fig. P10.27, page 281), the Parliament tower clock in London, has hour and minute hands with lengths of 2.70 m and 4.50 m and masses of 60.0 kg and 100 kg, respectively. Calculate the total angular momentum of these hands about the center point. (You may model the hands as long, thin rods rotating about one end. Assume the hour and minute hands are rotating at a constant rate of one revolution per 12 hours and minutes, respectively.)Model the Earth as a uniform sphere. (a) Calculate the angular momentum of the Earth due to its spinning motion about its axis. (b) Calculate the angular momentum of the Earth due to its orbital motion about the Sun. (c) Explain why the answer in part (b) is larger than that in part (a) even though it takes significantly longer for the Earth to go once around the Sun than to rotate once about its axis.The distance between the centers of the wheels of a motorcycle is 155 cm. The center of mass of the motorcycle, including the rider, is 88.0 cm above the ground and halfway between the wheels. Assume the mass of each wheel is small compared with the body of the motorcycle. The engine drives the rear wheel only. What horizontal acceleration of the motorcycle will make the front wheel rise off the ground?You are working in an observatory, taking data on electromagnetic radiation from neutron stars. You happen to be analyzing results from the neutron star in Example 11.6, verifying that the period of the 10.0-km-radius neutron star is indeed 2.6 s. You go through weeks of data showing the same period. Suddenly, as you analyze the most recent data, you notice that the period has decreased to 2.3 s and remained at that level since that time. You ask your supervisor about this, who becomes excited and says that the neutron star must have undergone a glitch, which is a sudden shrinking of the radius of the star, resulting in a higher angular speed. As she runs to her computer to start writing a paper on the glitch, she calls back to you to calculate the new radius of the planet, assuming it has remained spherical. She is also talking about vortices and a superfluid core, but you dont understand those words.A 60.0-kg woman stands at the western rim of a horizontal turntable having a moment of inertia of 500 kg m2 and a radius of 2.00 m. The turntable is initially at rest and is free to rotate about a frictionless, vertical axle through its center. The woman then starts walking around the rim clockwise (as viewed from above the system) at a constant speed of 1.50 m/s relative to the Earth. Consider the womanturntable system as motion begins. (a) Is the mechanical energy of the system constant? (b) Is the momentum of the system constant? (c) Is the angular momentum of the system constant? (d) In what direction and with what angular speed does the turntable rotate? (c) How much potential energy in the womans body is converted into mechanical energy of the womanturntable system as the woman sets herself and the turntable into motion?24P A uniform cylindrical turntable of radius 1.90 m and mass 30.0 kg rotates counterclockwise in a horizontal plane with an initial angular speed of 4 rad/s. The fixed turntable bearing is frictionless. A lump of clay of mass 2.25 kg and negligible size is dropped onto the turntable from a small distance above it and immediately sticks to the turntable at a point 1.80 m to the east of the axis. (a) Find the final angular speed of the clay and turntable. (b) Is the mechanical energy of the turntableclay system constant in this process? Explain and use numerical results to verify your answer. (c) Is the momentum of the system constant in this process? Explain your answer.26P A wooden block of mass M resting on a frictionless, horizontal surface is attached to a rigid rod of length and of negligible mass (Fig. P11.27). The rod is pivoted at the other end. A bullet of mass m traveling parallel to the horizontal surface and perpendicular to the rod with speed v hits the block and becomes embedded in it. (a) What is the angular momentum of the bulletblock system about a vertical axis through the pivot? (b) What fraction of the original kinetic energy of the bullet is converted into internal energy in the system during the collision? Figure P11.27 Why is the following situation impossible? A space station shaped like a giant wheel (Fig. P11.28, page 306) has a radius of r = 100 m and a moment of inertia of 5.00 108 kg m2. A crew of 150 people of average mass 65.0 kg is living on the rim, and the stations rotation causes the crew to experience an apparent free-fall acceleration of g. A research technician is assigned to perform an experiment in which a ball is dropped at the rim of the station every 15 minutes and the time interval for the ball to drop a given distance is measured as a lest to make sure the apparent value of g is correctly maintained. One evening, 100 average people move to the center of the station for a union meeting. The research technician, who has already been performing his experiment for an hour before the meeting, is disappointed that he cannot attend the meeting, and his mood sours even further by his boring experiment in which every time interval for the dropped ball is identical for the entire evening. Figure P11.28 A wad of sticky clay with mass m and velocity vi is fired at a solid cylinder of mass M and radius R (Fig. P11.29). The cylinder is initially at rest and is mounted on a fixed horizontal axle that runs through its center of mass. The line of motion of the projectile is perpendicular to the axle and at a distance d R from the center. (a) Find the angular speed of the system just after the clay strikes and sticks to the surface of the cylinder. (b) Is the mechanical energy of the claycylinder system constant in this process? Explain your answer. (c) Is the momentum of the claycylinder system constant in this process? Explain your answer. Figure P11.29 A 0.005 00-kg bullet traveling horizontally with a speed of 1.00 103 m/s strikes an 18.0-kg door, embedding itself 10.0 cm from the side opposite the hinges as shown in Figure P11.30. The 1.00-m wide door is free to swing on its frictionless hinges. (a) Before it hits the door, does the bullet have angular momentum relative to the doors axis of rotation? (b) If so, evaluate this angular momentum. If not, explain why there is no angular momentum. (c) Is the mechanical energy of the bulletdoor system constant during this collision? Answer without doing a calculation. (d) At what angular speed does the door swing open immediately after the collision? (e) Calculate the total energy of the bulletdoor system and determine whether it is less than or equal to the kinetic energy of the bullet before the collision. (f) What If? Imagine now that the door is hanging vertically downward, hinged at the top, so that Figure P11.30 is a side view of the door and bullet during the collision. What is the maximum height that the bottom of the door will reach after the collision? Figure P11.30 An overhead view of a bullet striking a door.The angular momentum vector of a precessing gyroscope sweeps out a cone as shown in Figure P11.31. The angular speed of the tip of the angular momentum vector, called its precessional frequency, is given by p=/I, where is the magnitude of the torque on the gyroscope and L is the magnitude of its angular momentum. In the motion called precession of the equinoxes, the Earths axis of rotation processes about the perpendicular to its orbital plane with a period of 2.58 104 yr. Model the Earth as a uniform sphere and calculate the torque on the Earth that is causing this precession. Figure P11.31 A precessing angular momentum vector sweeps out a cone in space.A light rope passes over a light, frictionless pulley. One end is fastened to a bunch of bananas of mass M, and a monkey of mass M clings to the other end (Fig. P11.32). The monkey climbs the rope in an attempt to reach the bananas. (a) Treating the system as consisting of the monkey, bananas, rope, and pulley, find the net torque on the system about the pulley axis. (b) Using the result of part (a), determine the total angular momentum about the pulley axis and describe the motion of the system. (c) Will the monkey reach the bananas? Figure P11.32 Review. A thin, uniform, rectangular signboard hangs vertically above the door of a shop. The sign is hinged to a stationary horizontal rod along its top edge. The mass of the sign is 2.40 kg, and its vertical dimension is 50.0 cm. The sign is swinging without friction, so it is a tempting target for children armed with snowballs. The maximum angular displacement of the sign is 25.0 on both sides of the vertical. At a moment when the sign is vertical and moving to the left, a snowball of mass 400 g, traveling horizontally with a velocity of 160 cm/s to the right, strikes perpendicularly at the lower edge of the sign and sticks there. (a) Calculate the angular speed of the sign immediately before the impact. (b) Calculate its angular speed immediately after the impact. (c) The spattered sign will swing up through what maximum angle?34AP We have all complained that there arent enough hours in a day. In an attempt to fix that, suppose all the people in the world line up at the equator and all start running east at 2.50 m/s relative to the surface of the Earth. By how much does the length of a day increase? Assume the world population to be 7.00 109 people with an average mass of 55.0 kg each and the Earth to be a solid homogeneous sphere. In addition, depending on the details of your solution, you may need to use the approximation 1/(1 x) = 1 + x for small x.36AP A rigid, massless rod has three particles with equal masses attached to it as shown in Figure P11.37. The rod is free to rotate in a vertical plane about a frictionless axle perpendicular to the rod through the point P and is released from rest in the horizontal position at t = 0. Assuming m and d are known, find (a) the moment of inertia of the system of three particles about the pivot, (b) the torque acting on the system at t = 0, (c) the angular acceleration of the system at t = 0, (d) the linear acceleration of the particle labeled 3 at t = 0, (e) the maximum kinetic energy of the system, (f) the maximum angular speed reached by the rod, (g) the maximum angular momentum of the system, and (h) the maximum speed reached by the particle labeled 2. Figure P11.37 Review. Two boys are sliding toward each other on a frictionless, ice-covered parking lot. Jacob, mass 45.0 kg, is gliding to the right at 8.00 m/s, and Ethan, mass 31.0 kg, is gliding to the left at 11.0 m/s along the same line. When they meet, they grab each other and hang on. (a) What is their velocity immediately thereafter? (b) What fraction of their original kinetic energy is still mechanical energy after their collision? That was so much fun that the boys repeat the collision with the same original velocities, this time moving along parallel lines 1.20 m apart. At closest approach, they lock arms and start rotating about their common center of mass. Model the boys as particles and their arms as a cord that does not stretch. (c) Find the velocity of their center of mass. (d) Find their angular speed. (e) What fraction of their original kinetic energy is still mechanical energy after they link arms? (f) Why are the answers to parts (b) and (e) so different?Two astronauts (Fig. P11.39), each having a mass of 75.0 kg, are connected by a 10.0-m rope of negligible mass. They are isolated in space, orbiting their center of mass at speeds of 5.0 m/s. Treating the astronauts as particles, calculate (a) the magnitude of the angular momentum of the two-astronaut system and (b) the rotational energy of the system. By pulling on the rope, one astronaut shortens the distance between them to 5.00 m. (c) What is the new angular momentum of the system? (d) What are the astronauts new speeds? (e) What is the new rotational energy of the system? (f) How much potential energy in the body of the astronaut was converted to mechanical energy in the system when he shortened the rope? Figure P11.39 Problems 39 and 40.Two astronauts (Fig. P11.39), each having a mass M, are connected by a rope of length d having negligible mass. They are isolated in space, orbiting their center of mass at speeds v. Treating the astronauts as particles, calculate (a) the magnitude of the angular momentum of the two-astronaut system and (b) the rotational energy of the system. By pulling on the rope, one of the astronauts shortens the distance between them to d/2. (c) What is the new angular momentum of the system? (d) What are the astronauts new speeds? (e) What is the new rotational energy of the system? (f) How much potential energy in the body of the astronaut was converted to mechanical energy in the system when he shortened the rope? Figure P11.39 Problems 39 and 40.Native people throughout North and South America used a bola to hunt for birds and animals. A bola can consist of three stones, each with mass m, at the ends of three light cords, each with length . The other ends of the cords are tied together to form a Y. The hunter holds one stone and swings the other two above his head (Figure P11.41a, page 308). Both these stones move together in a horizontal circle of radius 2 with speed v0. At a moment when the horizontal component of their velocity is directed toward the quarry, the hunter releases the stone in his hand. As the bola flies through the air, the cords quickly take a stable arrangement with constant 120-degree angles between them (Fig. P11.41b). In the vertical direction, the bola is in free fall. Gravitational forces exerted by the Earth make the junction of the cords move with the downward acceleration g. You may ignore the vertical motion as you proceed to describe the horizontal motion of the bola. In terms of m, , and v0, calculate (a) the magnitude of the momentum of the bola at the moment of release and, after release, (b) the horizontal speed of the center of mass of the bola, and (c) the angular momentum of the bola about its center of mass. (d) Find the angular speed of the bola about its center of mass after it has settled into its Y shape. Calculate the kinetic energy of the bola (e) at the instant of release and (f) in its stable Y shape. (g) Explain how the conservation laws apply to the bola as its configuration changes. Robert Beichner suggested the idea for this problem. Figure P11.41 Two children are playing on stools at a restaurant counter. Their feet do not reach the footrests, and the tops of the stools are free to rotate without friction on pedestals fixed to the floor. One of the children catches a tossed ball, in a process described by the equation (0.730kgm2)(240jrad/s)+(0.120kg)(0.350im)(4.30km/s)=[0.730kgm2+(0.120kg)(0.350m)2] (a) Solve the equation for the unknown . (b) Complete the statement of the problem to which this equation applies. Your statement must include the given numerical information and specification of the unknown to be determined. (c) Could the equation equally well describe the other child throwing the ball? Explain your answer.You are attending a county fair with your friend from your physics class. While walking around the fairgrounds, you discover a new game of skill. A thin rod of mass M = 0.500 kg and length = 2.00 m hangs from a friction-free pivot at its upper end as shown in Figure P11.43. The front surface of the rod is covered with Velcro. You are to throw a Velcro-covered ball of mass m = 1.0 kg at the rod in an attempt to make it swing backward and rotate all the way across the top. The ball must stick to the rod at all times after striking it. If you cause the rod to rotate over the top position, you win a stuffed animal. Your friend volunteers to try his luck. He feels that the most torque would be applied to the rod by striking it at its lowest end. While he prepares to aim at the lowest point on the rod, you calculate how fast he must throw the ball to win the stuffed animal with this technique. Figure P11.43 A uniform rod of mass 300 g and length 50.0 cm rotates in a horizontal plane about a fixed, frictionless, vertical pin through its center. Two small, dense beads, each of mass m, are mounted on the rod so that they can slide without friction along its length. Initially, the beads are held by catches at positions 10.0 cm on each side of the center and the system is rotating at an angular speed of 36.0 rad/s. The catches are released simultaneously, and the beads slide outward along the rod. (a) Find an expression for the angular speed f of the system at the instant the beads slide off the ends of the rod as it depends on m. (b) What are the maximum and the minimum possible values for f and the values of m to which they correspond?Global warming is a cause for concern because even small changes in the Earths temperature can have significant consequences. For example, if the Earths polar ice caps were to melt entirely, the resulting additional water in the oceans would flood many coastal areas. Model the polar ice as having mass 2.30 1019 kg and forming two flat disks of radius 6.00 105 m. Assume the water spreads into an unbroken thin, spherical shell after it melts. Calculate the resulting change in the duration of one day both in seconds and as a percentage.The puck in Figure P11.46 has a mass of 0.120 kg. The distance of the puck from the center of rotation is originally 40.0 cm, and the puck is sliding with a speed of 80.0 cm/s. The string is pulled downward 15.0 cm through the hole in the frictionless table. Determine the work done on the puck. (Suggestion: Consider the change of kinetic energy.) Figure P11.46 You operate a restaurant that has many large, circular tables. At the center of each table is a Lazy Susan that can turn to deliver salt, pepper, jam, hot sauce, bread, and other items to diners on the other side of the table. A fancy flower arrangement is located at the center of each Lazy Susan, and the turning of the flower arrangement is beautiful to you. Because of your interest in model trains, you decide to replace each Lazy Susan with a circular track on the table around which a model train will run. You can load the various condiments in the cars of the train and press a button to operate the train, causing the train to begin moving around the circle and deliver the load to your fellow diners! The train is of mass 1.96 kg and moves at a speed of 0.18 m/s relative to the track. After a few days, you realize that you miss the beautiful turning flower arrangements. So you come up with a new scheme. You return the Lazy Susan to the table and mount the circular track on the platform of the Lazy Susan, which has a friction-free axle at its center. The radius of the circular track is 40.0 cm (measured halfway between the rails) and the platform of the Lazy Susan is a uniform disk of mass 3.00 kg and radius 48.0 cm. You finally equip all of your tables with the new apparatus and open your restaurant. As a demonstration to the diners, you mount one salt shaker and one pepper shaker, having a mass of 0.100 kg each, onto a flatcar and push the button to deliver the condiments to the other side of the table! How long does it take to deliver the condiments to the exact opposite side of the table? Ignore the moment of inertia of the flower arrangement, since its mass is all close to the rotation axis.A solid cube of wood of side 2a and mass M is resting on a horizontal surface. The cube is constrained to rotate about a fixed axis AB (Fig. P11.48). A bullet of mass m and speed v is shot at the face opposite ABCD at a height of 4a/3. The bullet becomes embedded in the cube. Find the minimum value of v required to tip the cube so that it falls on face ABCD. Assume m M. Figure P11.48 In Example 11.8, we investigated an elastic collision between a disk and a stick lying on a frictionless surface. Suppose everything is the same as in the example except that the collision is perfectly inelastic so that the disk adheres to the stick at the endpoint at which it strikes. Find (a) the speed of the center of mass of the system and (b) the angular speed of the system after the collision.50CP Consider the object subject to the two forces of equal magnitude in Figure 12.2. Choose the correct statement with regard to this situation. (a) The object is in force equilibrium but not torque equilibrium. (b) The object is in torque equilibrium but not force equilibrium. (c) The object is in both force equilibrium and torque equilibrium. (d) The object is in neither force equilibrium nor torque equilibrium. Figure 12.2 (Quick Quiz 12.1) Two forces of equal magnitude are applied at equal distances from the center of mass of rigid object.Consider the object subject to the three forces in Figure 12.3. Choose the correct statement with regard to this situation from the choices (a)(d) in Quick Quiz 12.1. Figure 12.3 (Quick Quiz 12.2) Three forces act on an object. Notice that the lines of action of all three forces pass through a common point.

Page: [1][2][3][4][5][6][7][8]