Banked Curve I. A curve with a 120-m radius on a level

Problem 127CP A ball is held at rest at position A in ?Fig. P5.115? by two light strings. The horizontal string is cut, and the ball starts swinging as a pendulum. Position B is the farthest to the right that the ball can go as it swings back and forth. What is the ratio of the tension in the supporting string at B to its value at A before the string was cut?

Problem 1DQ A man sits in a seat that is hanging from a rope. The rope passes over a pulley suspended from the ceiling, and the man holds the other end of the rope in his hands. What is the tension in the rope, and what force does the seat exert on him? Draw a free-body force diagram for the man.

Problem 1E Two 25.0-N weights are suspended at opposite ends of a rope that passes over a light, frictionless pulley. The pulley is attached to a chain from the ceiling. (a) What is the tension in the rope? (b) What is the tension in the chain?

Problem 2DQ “In general, the normal force is not equal to the weight.” Give an example in which these two forces are equal in magnitude, and at least two examples in which they are not.

Problem 2E In ?Fig. E5.2? each of the suspended blocks has weight w. The pulleys are frictionless, and the ropes have negligible weight. In each case, draw a free-body diagram and calculate the tension T in the rope in terms of w.

Problem 3DQ A clothesline hangs between two poles. No matter how tightly the line is stretched, it sags a little at the center. Explain why.

Problem 4DQ You drive a car up a steep hill at constant speed. Discuss all of the forces that act on the car. What pushes it up the hill?

A 75.0-kg wrecking ball hangs from a uniform, heavy-duty chain of mass 26.0 kg. (a) Find the maximum and minimum tensions in the chain. (b) What is the tension at a point three-fourths of the way up from the bottom of the chain?

Problem 4E BIO Injuries to the Spinal Column.? In the treatment of spine injuries, it is often necessary to provide tension along the spinal column to stretch the backbone. One device for doing this is the Stryker frame (?Fig. E5.4a?). A weight W is attached to the patient (sometimes around a neck collar, Fig. E5.4b), and friction between the person’s body and the bed prevents sliding. (a) If the coefficient of static friction between a 78.5-kg patient’s body and the bed is 0.75, what is the maximum traction force along the spinal column that W can provide without causing the patient to slide? (b) Under the conditions of maximum traction, what is the tension in each cable attached to the neck collar?

For medical reasons, astronauts in outer space must determine their body mass at regular intervals. Devise a scheme for measuring body mass in an apparently weightless environment.

Problem 5E A picture frame hung against a wall is suspended by two wires attached to its upper corners. If the two wires make the same angle with the vertical, what must this angle be if the tension in each wire is equal to 0.75 of the weight of the frame? (Ignore any friction between the wall and the picture frame.)

Problem 6DQ To push a box up a ramp, which requires less force: pushing horizontally or pushing parallel to the ramp? Why?

Problem 6E A large wrecking ball is held in place by two light steel cables (Fig). If the mass ?m of the wrecking ball is 4090 kg, what are (a) the tension ?TB? in the cable that makes an angle of 40° with the vertical and (b) the tension ?TA? in the horizontal cable? Figure:

Problem 7DQ A woman in an elevator lets go of her briefcase, but it does not fall to the floor. How is the elevator moving?

Find the tension in each cord in Fig. E5.7 if the weight of the suspended object is w.

Problem 8DQ You can classify scales for weighing objects as those that use springs and those that use standard masses to balance unknown masses. Which group would be more accurate when used in an accelerating spaceship? When used on the moon?

Problem 8E A 1130-kg car is held in place by a light cable on a very smooth (frictionless) ramp (?Fig. E5.8?). The cable makes an angle of 31.0° above the sur-face of the ramp, and the ramp itself rises at 25.0° above the horizontal. (a) Draw a free-body diagram for the car. (b) Find the tension in the cable. (c) How hard does the surface of the ramp push on the car?

Problem 9DQ When you tighten a nut on a bolt, how are you increasing the frictional force? How does a lock washer work?

Problem 9E A man pushes on a piano with mass 180 kg so that it slides at constant velocity down a ramp that is inclined at 11.0° above the horizontal floor. Neglect any friction acting on the piano. Calculate the magnitude of the force applied by the man if he pushes (a) parallel to the incline and (b) parallel to the floor.

Problem 10DQ A block rests on an inclined plane with enough friction to prevent it from sliding down. To start the block moving, is it easier to push it up the plane or down the plane? Why?

In Fig. E5.10 the weight w is 60.0 N. (a) What is the tension in the diagonal string? (b) Find the magnitudes of the horizontal forces \(\overrightarrow{\boldsymbol{F}}_{1}\) and \(\overrightarrow{\boldsymbol{F}}_{2}\) that must be applied to hold the system in the position shown.

Problem 11DQ A crate of books rests on a level floor. To move it along the floor at a constant velocity, why do you exert less force if you pull it at an angle ? above the horizontal than if you push it at the same angle below the horizontal?

Problem 11E BIO Stay Awake! An astronaut is inside a 2.25 X 106 kg rocket that is blasting off vertically from the launch pad. You want this rocket to reach the speed of sound (331 m/s) as quickly as possible, but astronauts are in danger of blacking out at an acceleration greater than 4g. (a) What is the maximum initial thrust this rocket’s engines can have but just barely avoid blackout? Start with a free-body diagram of the rocket. (b) What force, in terms of the astronaut’s weight w, does the rocket exert on her? Start with a free-body diagram of the astronaut. (c) What is the shortest time it can take the rocket to reach the speed of sound?

Problem 12DQ In a world without friction, which of the following activities could you do (or not do)? Explain your reasoning. (a) Drive around an unbanked highway curve; (b) jump into the air; (c) start walking on a horizontal sidewalk; (d) climb a vertical ladder; (e) change lanes while you drive.

Problem 12E A 125-kg (including all the contents) rocket has an engine that produces a constant vertical force (the ?thrust?) of 1720 N. Inside this rocket, a 15.5-N electrical power supply rests on the floor. (a) Find the acceleration of the rocket. (b) When it has reached an altitude of 120 m, how hard does the floor push on the power supply? (?Hint:? Start with a free-body diagram for the power supply.)

Problem 13DQ Walkin g on horizontal slippery ice can be much more tiring than walking on ordinary pavement. Why?

Problem 13E CP Genesis Crash. On September 8, 2004, the ?Genesis? spacecraft crashed in the Utah desert because its parachute did not open. The 210-kg capsule hit the ground at 311 km/h and penetrated the soil to a depth of 81.0 cm. (a) What was its acceleration (in m/s2 and in g’s), assumed to be constant, during the crash? (b) What force did the ground exert on the capsule during the crash? Express the force in newtons and as a multiple of the capsule’s weight. (c) How long did this force last?

Problem 14DQ When you stand with bare feet in a wet bathtub, the grip feels fairly secure, and yet a catastrophic slip is quite possible. Explain this in terms of the two coefficients of friction.

Three sleds are being pulled horizontally on frictionless horizontal ice using horizontal ropes (Fig. E5.14). The pull is of magnitude 125 N. Find (a) the acceleration of the system and (b) the tension in ropes \(A\) and \(B\).

Problem 15DQ You are pushing a large crate from the back of a freight elevator to the front as the elevator is moving to the next floor. In which situation is the force you must apply to move the crate the least, and in which is it the greatest: when the elevator is accelerating upward, when it is accelerating downward, or when it is traveling at constant speed? Explain.

Atwood’s Machine. A 15.0-kg load of bricks hangs from one end of a rope that passes over a small, frictionless pulley. A 28.0-kg counterweight is suspended from the other end of the rope as shown in (Fig. E5.15). The system is released from rest. (a) Draw two free-body diagrams, one for the load of bricks and one for the counterweight. (b) What is the magnitude of the upward acceleration of the load of bricks? (c) What is the tension in the rope while the load is moving? How does the tension compare to the weight of the load of bricks? To the weight of the counterweight?

Problem 16DQ The moon is accelerating toward the earth. Why isn’t it getting closer to us?

Problem 16E CP An 8.00-kg block of ice, released from rest at the top of a 1.50-m-long frictionless ramp, slides downhill, reaching a speed of 2.50 m/s at the bottom. (a) What is the angle between the ramp and the horizontal? (b) What would be the speed of the ice at the bottom if the motion were opposed by a constant friction force of 10.0 N parallel to the surface of the ramp?

Problem 17DQ An automotive magazine calls decreasing radius curves “the bane of the Sunday driver?” Explain.

Problem 17E A light rope is attached to a block with mass 4.00 kg that rests on a frictionless, horizontal surface. The horizontal rope passes over a frictionless, massless pulley, and a block with mass ?m? is suspended from the other end. When the blocks are released, the tension in the rope is 10.0 N. (a) Draw two free-body diagrams, one for the 4.00-kg block and one for the block with mass ?m?. (b) What is the acceleration of either block? (c) Find the mass ?m? of the hanging block. (d) How does the tension compare to the weight of the hanging block?

Problem 18DQ It is often said that “friction always opposes motion.” Give at least one example in which (a) static friction ?causes? motion, and (b) kinetic friction ?causes? motion.

Problem 18E CP Runway Design.? A transport plane takes off from a level landing field with two gliders in tow, one behind the other. The mass of each glider is 700 kg, and the total resistance (air drag plus friction with the runway) on each may be assumed constant and equal to 2500 N. The tension in the towrope between the transport plane and the first glider is not to exceed 12,000 N. (a) If a speed of 40 m/s is required for takeoff, what minimum length of runway is needed? (b) What is the tension in the tow-rope between the two gliders while they are accelerating for the takeoff?

Problem 19DQ If there is a net force on a particle in uniform circular motion, why doesn’t the particle’s speed change?

Problem 19E CP? A 750.0-kg boulder is raised from a quarry 125 m deep by a long uniform chain having a mass of 575 kg. This chain is of uniform strength, but at any point it can support a maximum tension no greater than 2.50 times its weight without breaking. (a) What is the maximum acceleration the boulder can have and still get out of the quarry, and (b) how long does it take to be lifted out at maximum acceleration if it started from rest?

Problem 20DQ A curve in a road has a bank angle calculated and posted for 80 km/h. However, the road is covered with ice, so you cautiously plan to drive slower than this limit. What might happen to your car? Why?

Problem 20E Apparent Weight.? A 550-N physics student stands on a bathroom scale in an elevator that is supported by a cable. The combined mass of student plus elevator is 850 kg. As the elevator starts moving, the scale reads 450 N. (a) Find the acceleration of the elevator (magnitude and direction). (b) What is the acceleration if the scale reads 670 N? (c) If the scale reads zero, should the student worry? Explain. (d) What is the tension in the cable in parts (a) and (c)?

Problem 21DQ You swing a ball on the end of a lightweight string in a horizontal circle at constant speed. Can the string ever be truly horizontal? If not, would it slope above the horizontal or below the horizontal? Why?

Problem 21E BIO Force During a Jump.? When jumping straight up from a crouched position, an average person can reach a maximum height of about 60 cm. During the jump, the person’s body from the knees up typically rises a distance of around 50 cm. To keep the calculations simple and yet get a reasonable result, assume that the ?entire body? rises this much during the jump. (a) With what initial speed does the person leave the ground to reach a height of 60 cm? (b) Draw a free-body diagram of the person during the jump. (c) In terms of this jumper’s weight w, what force does the ground exert on him or her during the jump?

Problem 22DQ The centrifugal force is not included in the free-body diagrams of Figs. 5.34b and 5.35. Explain why not.

Problem 22E CP CALC? A 2540-kg test rocket is launched vertically from the launch pad. Its fuel (of negligible mass) provides a thrust force such that its vertical velocity as a function of time is given by v(t) = At + Bt2 , where A and B are constants and time is measured from the instant the fuel is ignited. The rocket has an upward acceleration of 1.50 m/s2 at the instant of ignition and, 1.00 s later, an upward velocity of 2.00 m/s. (a) Determine A and B , including their SI units. (b) At 4.00 s after fuel ignition, what is the acceleration of the rocket, and (c) what thrust force does the burning fuel exert on it, assuming no air resistance? Express the thrust in newtons and as a multiple of the rocket’s weight. (d) What was the initial thrust due to the fuel?

Problem 23DQ A professor swings a rubber stopper in a horizontal circle on the end of a string in front of his class. He tells Caroline, in the front row, that he is going to let the string go when the stopper is directly in front of her face. Should Caroline worry?

Problem 23E CP CALC? A 2.00-kg box is moving to the right with speed 9.00 m/s on a horizontal, frictionless surface. At t = 0 a horizontal force is applied to the box. The force is directed to the left and has magnitude F (t) = (6.00 N/s2)t2. (a) What distance does the box move from its position at t = 0 before its speed is reduced to zero? (b) If the force continues to be applied, what is the speed of the box at t = 3.00 s?

Problem 24DQ To keep the forces on the riders within allowable limits, many loop-the-loop roller coaster rides are designed so that the loop is not a perfect circle but instead has a larger radius of curvature at the bottom than at the top. Explain.

Problem 24E CP CALC? A 5.00-kg crate is suspended from the end of a short vertical rope of negligible mass. An upward force F(t) is applied to the end of the rope, and the height of the crate above its initial position is given by y(t) = (2.80 m/s)t + (0.610 m/s3)t3. What is the magnitude of F when t = 4.00 s?

Problem 25DQ A tennis ball drops from rest at the top of a tall glass cylinder—first with the air pumped out of the cylinder so that there is no air resistance, and again after the air has been read-mitted to the cylinder. You examine multiflash photographs of the two drops. Can you tell which photo belongs to which drop? If so, how?

Problem 25E BIO The Trendelenburg Position.? After emergencies with major blood loss, a patient is placed in the Trendelenburg position, in which the foot of the bed is raised to get maximum blood flow to the brain. If the coefficient of static friction between a typical patient and the bedsheets is 1.20, what is the maximum angle at which the bed can be tilted with respect to the floor before the patient begins to slide?

Problem 26DQ You throw a baseball straight upward with speed v0. When the ball returns to the point from where you threw it, how does its speed compare to v0 (a) in the absence of air resistance and (b) in the presence of air resistance? Explain.

Problem 28DQ You have two identical tennis balls and fill one with water. You release both balls simultaneously from the top of a tall building. If air resistance is negligible, which ball will strike the ground first? Explain. What if air resistance is ?not? negligible?

Problem 29DQ A ball is dropped from rest and feels air resistance as it falls. Which of the graphs in ?Fig. Q5.25? best represents its acceleration as a function of time?

Problem 27E A stockroom worker pushes a box with mass 11.2 kg on a horizontal surface with a constant speed of 3.50 m/s. The coefficient of kinetic friction between the box and the surface is 0.20. (a) What horizontal force must the worker apply to maintain the motion? (b) If the force calculated in part (a) is removed, how far does the box slide before coming to rest?

Problem 28E A box of bananas weighing 40.0 N rests on a horizontal surface. The coefficient of static friction between the box and the surface is 0.40, and the coefficient of kinetic friction is 0.20. (a) If no horizontal force is applied to the box and the box is at rest, how large is the friction force exerted on it? (b) What is the magnitude of the friction force if a monkey applies a horizontal force of 6.0 N to the box and the box is initially at rest? (c) What minimum horizontal force must the monkey apply to start the box in motion? (d) What minimum horizontal force must the monkey apply to keep the box moving at constant velocity once it has been started? (e) If the monkey applies a horizontal force of 18.0 N, what is the magnitude of the friction force and what is the box’s acceleration?

Problem 27DQ You throw a baseball straight upward. If you do ?not? ignore air resistance, how does the time required for the ball to reach its maximum height compare to the time required for it to fall from its maximum height back down to the height from which you threw it? Explain.

In a laboratory experiment on friction, a 135-N block resting on a rough horizontal table is pulled by a horizontal wire. The pull gradually increases until the block begins to move and continues to increase thereafter. Figure E5.26 shows a graph of the friction force on this block as a function of the pull. (a) Identify the regions of the graph where static friction and kinetic friction occur. (b) Find the coefficients of static friction and kinetic friction between the block and the table. (c) Why does the graph slant upward at first but then level out? (d) What would the graph look like if a 135-N brick were placed on the block, and what would the coefficients of friction be?

Problem 29E A 45.0-kg crate of tools rests on a horizontal floor. You exert a gradually increasing horizontal push on it, and the crate just begins to move when your force exceeds 313 N. Then you must reduce your push to 208 N to keep it moving at a steady 25.0 cm/s. (a) What are the coefficients of static and kinetic friction between the crate and the floor? (b) What push must you exert to give it an acceleration of 1.10 m/s2? (c) Suppose you were per-forming the same experiment on the moon, where the acceleration due to gravity is 1.62 m/s2. (i) What magnitude push would cause it to move? (ii) What would its acceleration be if you maintained the push in part (b)?

Problem 30DQ A ball is dropped from rest and feels air resistance as it falls. Which of the graphs in ?Fig. Q5.26? best represents its vertical velocity component as a function of time?

Problem 30E Some sliding rocks approach the base of a hill with a speed of 12 m/s. The hill rises at 36° above the horizontal and has coefficients of kinetic friction and static friction of 0.45 and 0.65, respectively, with these rocks. (a) Find the acceleration of the rocks as they slide up the hill. (b) Once a rock reaches its highest point, will it stay there or slide down the hill? If it stays, show why. If it slides, find its acceleration on the way down.

Problem 32DQ When a balled baseball moves with air drag, does it travel a greater horizontal distance while climbing to its maximum height or while descending from its maximum height back to the ground? Or is the horizontal distance traveled the same for both? Explain in terms of the forces acting on the ball.

Problem 31DQ When does a baseball in light have an acceleration with a positive upward component? Explain in terms of the forces on the ball and also in terms of the velocity components compared to the terminal speed. Do ?not? ignore air resistance.

Problem 31E You are lowering two boxes, one on top of the other, down a ramp by pulling on a rope parallel to the surface of the ramp (?Fig. E5.33?). Both boxes move together at a constant speed of 15.0 cm/s. The coefficient of kinetic friction between the ramp and the lower box is 0.444, and the coefficient of static friction between the two boxes is 0.800. (a) What force do you need to exert to accomplish this? (b) What are the magnitude and direction of the friction force on the upper box?

Problem 32E A pickup truck is carrying a toolbox, but the rear gate of the truck is missing. The toolbox will slide out if it is set moving. The coefficients of kinetic friction and static friction between the box and the level bed of the truck are 0.355 and 0.650, respectively. Starting from rest, what is the shortest time this truck could accelerate uniformly to 30.0 m/s without causing the box to slide? Draw a free-body diagram of the toolbox.

Problem 33DQ “A ball is thrown from the edge of a high cliff. Regardless of the angle at which it is thrown, due to air resistance, the ball will eventually end up moving vertically downward.” Justify this statement.

Problem 33E CP Stopping Distance.? (a) If the coefficient of kinetic friction between tires and dry pavement is 0.80, what is the shortest distance in which you can stop a car by locking the brakes when the car is traveling at 28.7 m/s (about 65 mi/h)? (b) On wet pavement the coefficient of kinetic friction may be only 0.25. How fast should you drive on wet pavement to be able to stop in the same distance as in part (a)? (?Note:? Locking the brakes is ?not? the safest way to stop.)

Consider the system shown in Fig. E5.34. Block A weighs 45.0 N, and block B weighs 25.0 N. Once block B is set into downward motion, it descends at a constant speed. (a) Calculate the coefficient of kinetic friction between block A and the tabletop. (b) A cat, also of weight 45.0 N, falls asleep on top of block A . If block B is now set into downward motion, what is its acceleration (magnitude and direction)?

Problem 35E Two crates connected by a rope lie on a horizontal surface (?Fig. E5.37?). Crate A has mass mA, and crate B has mass mB. The coefficient of kinetic friction between each crate and the surface is µk . The crates are pulled to the right at constant velocity by a horizontal force Draw one or more free-body diagrams to calculate the following in terms of mA, mB, and µk: (a) the magnitude of and (b) the tension in the rope connecting the blocks.

Problem 37E CP? As shown in Fig. E5.34, block A (mass 2.25 kg) rests on a tabletop. It is connected by a horizontal cord passing over a light, frictionless pulley to a hanging block B (mass 1.30 kg). The coefficient of kinetic friction between block A and the tabletop is 0.450. The blocks are released then from rest. Draw one or more free-body diagrams to find (a) the speed of each block after they move 3.00 cm and (b) the tension in the cord.

Problem 36E CP? A 25.0-kg box of textbooks rests on a loading ramp that makes an angle ? with the horizontal. The coefficient of kinetic friction is 0.25, and the coefficient of static friction is 0.35. (a) As ? is increased, find the minimum angle at which the box starts to slip. (b) At this angle, find the acceleration once the box has begun to move. (c) At this angle, how fast will the box be moving after it has slid 5.0 m along the loading ramp?

Problem 38E A box with mass m is dragged across a level floor with coefficient of kinetic friction µk by a rope that is pulled upward at an angle ? above the horizontal with a force of magnitude F. (a) In terms of m, µk, ?, and g , obtain an expression for the magnitude of the force required to move the box with constant speed. (b) Knowing that you are studying physics, a CPR instructor asks you how much force it would take to slide a 90-kg patient across a floor at constant speed by pulling on him at an angle of 25o above the horizontal. By dragging weights wrapped in an old pair of pants down the hall with a spring balance, you find that µk = 0.35. Use the result of part (a) to answer the instructor’s question.

Problem 39E A large crate with mass m rests on a horizontal floor. The coefficients of friction between the crate and the floor are µs and µk. A woman pushes downward with a force on the crate at an angle ? below the horizontal. (a) What magnitude of force is required to keep the crate moving at constant velocity? (b) If µs is greater than some critical value, the woman cannot start the crate moving no matter how hard she pushes. Calculate this critical value of µs.

Problem 40E You throw a baseball straight upward. The drag force is proportional to v2. In terms of g, what is the y-component of the ball’s acceleration when the ball’s speed is half its terminal speed and (a) it is moving up? (b) It is moving back down?

Problem 41E (a) In Example 5.18 (Section 5.3), what value of D is required to make vt = 42 m/s for the skydiver? (b) If the skydiver’s daughter, whose mass is 45 kg, is falling through the air and has the same D (0.25 kg/m) as her father, what is the daughter’s terminal speed?

Problem 42E A small car with mass 0.800 kg travels at constant speed on the inside of a track that is a vertical circle with radius 5.00 m (Fig. E5.45). If the normal force exerted by the track on the car when it is at the top of the track (point B ) is 6.00 N, what is the nor-mal force on the car when it is at the bottom of the track (point A )?

Problem 43E A machine part consists of a thin 40.0-cm-long bar with small 1.15-kg masses fastened by screws to its ends. The screws can support a maximum force of 75.0 N without pulling out. This bar rotates about an axis perpendicular to it at its center. (a) As the bar is turning at a constant rate on a horizontal, frictionless surface, what is the maximum speed the masses can have without pulling out the screws? (b) Suppose the machine is redesigned so that the bar turns at a constant rate in a vertical circle. Will one of the screws be more likely to pull out when the mass is at the top of the circle or at the bottom? Use a free-body diagram to see why. (c) Using the result of part (b), what is the greatest speed the masses can have without pulling a screw?

Problem 44E A ftat (unbanked) curve on a highway has a radius of 220.0 m. A car rounds the curve at a speed of 25.0 m/s. (a) What is the minimum coefficient of friction that will prevent sliding? (b) Suppose the highway is icy and the coefficient of friction between the tires and pavement is only one-third what you found in part (a). What should be the maximum speed of the car so it can round the curve safely?

Problem 45E A 1125-kg car and a 2250-kg pickup truck approach a curve on a highway that has a radius of 225 m. (a) At what angle should the highway engineer bank this curve so that vehicles traveling at 65.0 mi/h can safely round it regardless of the condition of their tires? Should the heavy truck go slower than the lighter car? (b) As the car and truck round the curve at 65.0 mi/h, find the normal force on each one due to the highway surface.

Problem 46E The “Giant Swing” at a county fair consists of a vertical central shaft with a number of horizontal arms attached at its upper end. Each arm supports a seat suspended from a cable 5.00 m long, and the upper end of the cable is fastened to the arm at a point 3.00 m from the central shaft (?Fig. E5.50?). (a) Find the time of one revolution of the swing if the cable supporting a seat makes an angle of 30.0o with the vertical. (b) Does the angle depend on the weight of the passenger for a given rate of revolution?

Problem 48E A small button placed on a horizontal rotating platform with diameter 0.320 m will revolve with the platform when it is brought up to a speed of 40.0 rev/min, provided the button is no more than 0.150 m from the axis. (a) What is the coefficient of static friction between the button and the platform? (b) How far from the axis can the button be placed, without slipping, if the platform rotates at 60.0 rev/min?

Problem 50E The Cosmo Clock 21 Ferris wheel in Yokohama, Japan, has a diameter of 100 m. Its name comes from its 60 arms, each of which can function as a second hand (so that it makes one revolution every 60.0 s). (a) Find the speed of the passengers when the Ferris wheel is rotating at this rate. (b) A passenger weighs 882 N at the weight-guessing booth on the ground. What is his apparent weight at the highest and at the lowest point on the Ferris wheel? (c) What would be the time for one revolution if the passenger’s apparent weight at the highest point were zero? (d) What then would be the passenger’s apparent weight at the lowest point?

Problem 47E In another version of the “Giant Swing” (see Exercise). the scat is connected to two cables as shown in Fig 1, one of which is horizontal. The seat swings in a horizontal circle at a rate of 32.0 rpm (rev/min). If the seat weighs 255 N and an 825-N person is sitting in it, find the tension in each cable. Figure 1: Exercise: The “Giant Swing” at a county fair consists of a vertical central shaft with a number of horizontal arms attached at its upper end (Fig 2). Each arm supports a seal suspended from a cable 5.00 m long, the upper end of the cable being fastened to the arm at a point 3.00 m from the central shaft. (a) Find the time of one revolution of the swing if the cable supporting a seat makes an angle of 30.0° with the vertical. (b) Does the angle depend on the weight of the passenger for a given rate of revolution? Figure 2:

Problem 49E Rotating Space Stations.? One problem for humans living in outer space is that they are apparently weightless. One way around this problem is to design a space station that spins about its center at a constant rate. This creates “artificial gravity” at the outside rim of the station. (a) If the diameter of the space station is 800 m, how many revolutions per minute are needed for the “artificial gravity” acceleration to be 9.80 m/s2? (b) If the space station is a waiting area for travelers going to Mars, it might be desirable to simulate the acceleration due to gravity on the Martian surface (3.70 m/s2). How many revolutions per minute are needed in this case?

An airplane flies in a loop (a circular path in a vertical plane) of radius 150 m. The pilot’s head always points toward the center of the loop. The speed of the airplane is not constant; the airplane goes slowest at the top of the loop and fastest at the bottom. (a) What is the speed of the airplane at the top of the loop, where the pilot feels weightless? (b) What is the apparent weight of the pilot at the bottom of the loop, where the speed of the air-plane is 280 km/h? His true weight is 700 N.

.Problem 52E A 50.0-kg stunt pilot who has been diving her airplane vertically pulls out of the dive by changing her course to a circle in a vertical plane. (a) If the plane’s speed at the lowest point of the circle is 95.0 m/s, what is the minimum radius of the circle so that the acceleration at this point will not exceed 4.00 g? (b) What is the apparent weight of the pilot at the lowest point of the pullout?

Problem 53E Stay Dry!? You tie a cord to a pail of water and swing the pail in a vertical circle of radius 0.600 m. What minimum speed must you give the pail at the highest point of the circle to avoid spilling water?

Problem 54E A bowling ball weighing 71.2 N (16.0 lb) is attached to the ceiling by a 3.80-m rope. The ball is pulled to one side and released; it then swings back and forth as a pendulum. As the rope swings through the vertical, the speed of the bowling ball is 4.20 m/s. At this instant, what are (a) the acceleration of the bowling ball, in magnitude and direction, and (b) the tension in the rope?

Problem 55E BIO Effect on Blood of Walking.? While a person is walking, his arms swing through approximately a 45° angle in As a reasonable approximation, assume that the arm moves with constant speed during each swing. A typical arm is 70.0 cm long, measured from the shoulder joint. (a) What is the acceleration of a 1.0-g drop of blood in the fingertips at the bottom of the swing? (b) Draw a free-body diagram of the drop of blood in part (a). (c) Find the force that the blood vessel must exert on the drop of blood in part (a). Which way does this force point? (d) What force would the blood vessel exert if the arm were not swinging?

Problem 56P An adventurous archaeologist crosses between two rock cliffs by slowly going hand over hand along a rope stretched between the cliffs. He stops to rest at the middle of the rope (?Fig. P5.60?). The rope will break if the tension in it exceeds 2.50 X 104 N, and our hero’s mass is 90.0 kg. (a) If the angle ? is 10.0o, what is the tension in the rope? (b) What is the smallest value ? can have if the rope is not to break?

Problem 57P Two ropes are connected to a steel cable that supports a hanging weight (?Fig. P5.61?). (a) Draw a free-body diagram showing all of the forces acting at the knot that connects the two ropes to the steel cable. Based on your diagram, which of the two ropes will have the greater tension? (b) If the maximum tension either rope can sustain without breaking is 5000 N, determine the maximum value of the hanging weight that these ropes can safely support. Ignore the weight of the ropes and of the steel cable.

In Fig. P5.58 a worker lifts a weight w by pulling down on a rope with a force \(\overrightarrow{\boldsymbol{F}}\). The upper pulley is attached to the ceiling by a chain, and the lower pulley is attached to the weight by another chain. In terms of w, find the tension in each chain and the magnitude of the force \(\overrightarrow{\boldsymbol{F}}\) if the weight is lifted at constant speed. Include the free-body diagram or diagrams you used to determine your answers. Assume that the rope, pulleys, and chains all have negligible weights.

Problem 59P A solid uniform 45.0-kg ball of diameter 32.0 cm is supported against a vertical, frictionless wall by a thin 30.0-cm wire of negligible mass (?Fig. P5.65?). (a) Draw a free-body diagram for the ball, and use the diagram to find the tension in the wire. (b) How hard does the ball push against the wall?

Problem 61P CP BIO Forces During Chin-ups.? When you do a chin-up, you raise your chin just over a bar (the chinning bar), sup-porting yourself with only your arms. Typically, the body below the arms is raised by about 30 cm in a time of 1.0 s, starting from rest. Assume that the entire body of a 680-N person doing chin-ups is raised by 30 cm, and that half the 1.0 s is spent accelerating upward and the other half accelerating downward, uniformly in both cases. Draw a free-body diagram of the person’s body, and use it to find the force his arms must exert on him during the accelerating part of the chin-up.

Problem 60P A horizontal wire holds a solid uniform ball of mass m in place on a tilted ramp that rises 35.0° above the horizontal. The surface of this ramp is perfectly smooth, and the wire is directed away from the center of the ball (?Fig. P5.64?). (a) Draw a free-body diagram of the ball. (b) How hard does the surface of the ramp push on the ball? (c) What is the tension in the wire?

Problem 62P Prevention of Hip Injuries?. People (especially the elderly) who are prone to falling can wear hip pads to cushion the impact on their hip from a fall. Experiments have shown that if the speed at impact can be reduced to 1.3 m/s or less, the hip will usually not fracture. Let us investigate the worst-case scenario in which a 55-kg person completely loses her footing (such as on icy pavement) and falls a distance of 1.0 m, the distance from her hip to the ground We shall assume that the person’s entire body has the same acceleration, which, in reality, would not quite be true. (a) With what speed does her hip reach the ground? (b) A typical hip pad can reduce the person’s speed to 1.3 m/s over a distance of 2.0 cm. Find the acceleration (assumed to be constant) of this person’s hip while she is slowing down and the force the pad exerts on it. (c) The force in part (b) is very large. To see whether it is likely to cause injury, calculate how long it lasts.

CALC A 3.00-kg box that is several hundred meters above the earth’s surface is suspended from the end of a short vertical rope of negligible mass. A time-dependent upward force is applied to the upper end of the rope and results in a tension in the rope of T(t) = (36.0 N/s)t. The box is at rest at t = 0. The only forces on the box are the tension in the rope and gravity. (a) What is the velocity of the box at (i) t = 1.00 s and (ii) t = 3.00 s? (b) What is the maximum distance that the box descends below its initial position? (c) At what value of t does the box return to its initial position?

Problem 64P A 5.00-kg box sits at rest at the bottom of a ramp that is 8.00 m long and that is inclined at 30.0° above the horizontal. The coefficient of kinetic friction is ???k = 0.40 and the coefficient or static friction is ???s = 0.50. What constant force ?F?, applied parallel to the surface of the ramp, is required to push the box to the top of the ramp in a time of 4.00 s?

Problem 65P Two boxes connected by a light horizontal rope are on a horizontal surface (Fig. E5.37). The coefficient of kinetic friction between each box and the surface is µk = 0.30. Box B has mass 5.00 kg, and box A has mass m. A force F with magnitude 40.0 N and direction 53.1o above the horizontal is applied to the 5.00-kg box, and both boxes move to the right with a = 1.50 m/s2. (a) What is the tension T in the rope that connects the boxes? (b) What is m?

Problem 66P A 6.00-kg box sits on a ramp that is inclined at 37.0° above the horizontal. The coefficient of kinetic friction between the box and the ramp is ???k = 0.30. What horizontal? force is required to move the box up the incline with a constant acceleration of 4.20 m/s2?

Problem 67P In Fig. P5.34 block ?A? has mass ?m? and block ?B? has mass 6.00 kg. The coefficient of kinetic friction between block ?A? and the tabletop is ???k = 0.40. The mass of the rope connecting the blocks can be neglected. The pulley is light and frictionless. When the system is released from rest, the hanging block descends 5.00 m in 3.00 s. What is the mass ?m? of block ?A??

Problem 68P CP? In ?Fig. P5.74?, m1 = 20.0 kg and ? = 53.1o. The coefficient of kinetic friction between the block of mass m1 and the incline is µk = 0.40. What must be the mass m2 of the hanging block if it is to descend 12.0 m in the first 3.00 s after the system is released from rest?

Problem 69P Rolling Friction?. Two bicycle tires are set rolling with the same initial speed of 3.50 m/s on a long, straight road, and the distance each travels before its speed is reduced by half measured. One tire is inflated to a pressure of 40 psi and goes 18.1 m; the other is at 105 psi and goes 92.9 m. What is the coefficient of rolling friction ???r for each? Assume that the net horizontal force is due to rolling friction only.

Problem 70P A? ?Rope with? ?Mass.? A block with mass ?M? is attached to the lower end of a vertical, uniform rope with mass ?m? and length ?L?. A constant upward force is applied to the top of the rope, causing the rope and block to accelerate upward. Find the tension in the rope at a distance ?x? from the top end of the rope, where ?x can have any value from 0 to L ? .

Problem 71P A block with mass m1 is placed on an inclined plane with slope angle ? and is connected to a hanging block with mass m2 by a cord passing over a small, frictionless pulley (Fig. P5.74). The coefficient of static friction is µs, and the coefficient of kinetic friction is µk. (a) Find the value of m2 for which the block of mass m1 moves up the plane at constant speed once it is set in motion. (b) Find the value of m2 for which the block of mass m1 moves down the plane at constant speed once it is set in motion. (c) For what range of values of m2 will the blocks remain at rest if they are released from rest?

Problem 72P Block A in ?Fig. P5.76? weighs 60.0 N. The coefficient of static friction between the block and the surface on which it rests is 0.25. The weight w is 12.0 N, and the system is in equilibrium. (a) Find the friction force exerted on block A. (b) Find the maximum weight w for which the system will remain in equilibrium.

Problem 75P BIO The Flying Leap of a Flea.? High-speed motion pictures (3500 frames/second) of a jumping 210-µg flea yielded the data to plot the flea’s acceleration as a function of time, as shown in ?Fig. P5.78?. (See “The Flying Leap of the Flea,” by M. Rothschild et al., ?Scientific American,? November 1973.) This flea was about 2 mm long and jumped at a nearly vertical takeoff angle. Using the graph, (a) find the ?initial? net external force on the flea. How does it compare to the flea’s weight? (b) Find the ?maximum? net external force on this jumping flea. When does this maximum force occur? (c) Use the graph to find the flea’s maximum speed.

Problem 76P 25,000-kg rocket blasts oil vertically from the earth’s surface with a constant acceleration. During the motion considered in the problem, assume that ?g? remains constant (see Chapter 13). Inside the rocket, a 15.0-N instrument hangs from a wire that can support a maximum tension of 45.0 N. (a) Find the minimum time for this rocket to reach the sound barrier (330 m/s) without breaking the inside wire and the maximum vertical thrust of the rocket engines under these conditions. (b) How far is the rocket above the earth’s surface when it breaks the sound barrier?

Problem 73P Block ?A? in Fig weighs 2.40 N and block ?B? weighs 3.60 N. The coefficient of kinetic friction between all surfaces is 0.300. Find the magnitude of the horizontal force necessary to drag block ?B? to the left at constant speed (a) if ?A? rests on B? and moves with it (Fig. 1). (b) If ?A? is held at rest (Fig. 2). Figure 1: Figure 2:

Problem 74P A window washer pushes his scrub brush up a vertical window at constant speed by applying a force as shown in Fig. The brush weighs 15.0 N and the coefficient of kinetic friction is ???k = 0.150. Calculate (a) the magnitude of the force and (b) the normal force exerted by the window on the brush. Figure:

Problem 77P CP CALC ?You are standing on a bathroom scale in an elevator in a tall building. Your mass is 64 kg. The elevator starts from rest and travels upward with a speed that varies with time according to v(t) = (3.0 m/s2)t + (0.20 m/s3)t2. When t = 4.0 s, what is the reading on the bathroom scale?

Problem 78P CP Elevator Design.? You are designing an elevator for a hospital. The force exerted on a passenger by the floor of the elevator is not to exceed 1.60 times the passenger’s weight. The elevator accelerates upward with constant acceleration for a distance of 3.0 m and then starts to slow down. What is the maxi-mum speed of the elevator?

Problem 79P You are working for a shipping company. Your job is to stand at the bottom of a 8.0-m-long ramp that is inclined at 37° above the horizontal. You grab packages off a conveyor belt and propel them up the ramp. The coefficient of kinetic friction between the packages and the ramp is ???k = 0.30. (a) What speed do you need to give a package at the bottom of the ramp so that it has zero speed at the top of the ramp? (b) Your coworker is supposed to grab the packages as they arrive at the top of the ramp, but she misses one and it slides back down. What is its speed when it returns to you?

Problem 80P A hammer is hanging by a light rope from the ceiling of a bus. The ceiling of the bus is parallel to the roadway. The bus is traveling in a straight line on a horizontal street. You observe that the hammer hangs at rest with respect to the bus when the angle between the rope and the ceiling of the bus is 67°. What is the acceleration of the bus?

Problem 81P A steel washer is suspended inside an empty shipping crate from a light string attached to the top of the crate. The crate slides down a long ramp that is inclined at an angle of 37° above the horizontal. The crate has mass 180 kg. You are sitting inside the crate (with a flashlight); your mass is 55 kg. As the crate is sliding down the ramp, you find the washer is at rest with respect to the crate when the string makes an angle of 68° with the top of the crate. What is the coefficient of kinetic friction between the ramp and the crate?

Lunch Time!? You are riding your motorcycle one day down a wet street that slopes downward at an angle of 20° below the horizontal. As you start to ride down the hill, you notice a construction crew has dug a deep hole in the street at the bottom of the hill. A Siberian tiger, escaped from the City Zoo, has taken up residence in the hole. You apply the brakes and lock your wheels al the top of the hill, where you are moving with a speed of 20 m/s. The inclined street in front of you is 40 m long. (a) Will you plunge into the hole and become the tiger’s lunch, or do you skid to a stop befre you reach the hole? (The coefficients of friction between your motorcycle tires and the wet pavement are \(\mu_{\mathrm{s}}=0.90\) and \(\mu_{\mathrm{k}}=0.70\).) (b) What must your initial speed be if you are to stop just before reaching the hole?

Problem 84P If the coefficient of static friction between a table and a uniform, massive rope is µs, what fraction of the rope can hang over the edge of the table without the rope sliding?

In the system shown in Fig. P5.34, block A has mass \(m_A\), block B has mass \(m_B\), and the rope connecting them has a nonzero mass \(m_{\text {rope }}\). The rope has a total length L, and the pulley has a very small radius. You can ignore any sag in the horizontal part of the rope. (a) If there is no friction between block A and the tabletop, find the acceleration of the blocks at an instant when a length d of rope hangs vertically between the pulley and block B. As block B falls, will the magnitude of the acceleration of the system increase, decrease, or remain constant? Explain. (b) Let \(m_A=2.00 \mathrm{~kg}\), \(m_B=0.400 \mathrm{~kg}, m_{\text {rope }}=0.160 \mathrm{~kg}\), and \(L=1.00 \mathrm{~m}\). If there is friction between block A and the tabletop, with \(\mu_{\mathrm{k}}=0.200\) and \(\mu_{\mathrm{s}}=0.250\), find the minimum value of the distance d such that the blocks will start to move if they are initially at rest. (c) Repeat part (b) for the case \(m_{\text {rope }}=0.040 \mathrm{~kg}\). Will the blocks move in this case?

Problem 85P A 40.0-kg packing case is initially at rest on the floor of a 1500-kg pickup truck. The coefficient of static friction between the case and the truck floor is 0.30, and the coefficient of kinetic friction is 0.20. Before each acceleration given below, the truck is traveling due north at constant speed. Find the magnitude and direction of the friction force acting on the case (a) when the truck accelerates at 2.20 m/s2 northward and (b) when it accelerates at 3.40 m/s2 southward.

Problem 86P CP Traffic Court.? You are called as an expert witness in a trial for a traffic violation. The facts are these: A driver slammed on his brakes and came to a stop with constant acceleration. Measurements of his tires and the skid marks on the pavement indicate that he locked his car’s wheels, the car traveled 192 ft before stopping, and the coefficient of kinetic friction between the road and his tires was 0.750. He was charged with speeding in a 45-mi/h zone but pleads innocent. What is your conclusion: guilty or innocent? How fast was he going when he hit his brakes?

Problem 87P Two identical 15.0-kg balls, each 25.0 cm in diameter, are suspended by two 35.0-cm wires (?Fig. P5.85?). The entire apparatus is supported by a single 18.0-cm wire, and the surfaces of the balls are perfectly smooth. (a) Find the tension in each of the three wires. (b) How hard does each ball push on the other one?

Problem 88P CP Losing Cargo.? A 12.0-kg box rests on the level bed of a truck. The coefficients of friction between the box and bed are µs = 0.19 and µk = 0.15. The truck stops at a stop sign and then starts to move with an acceleration of 2.20 m/s2. If the box is 1.80 m from the rear of the truck when the truck starts, how much time elapses before the box falls off the truck? How far does the truck travel in this time?

Problem 90P You are part of a design team for future exploration of the planet Mars, where ?g? = 3.7 m/s2. An explorer is to step out of a survey vehicle traveling horizontally at 33 m/s when it is 1200 m above the surface and then fall ficely for 20 s. At that time, a portable advanced propulsion system (PAPS) is to exert a constant force that will decrease the explorer’s speed to zero at the instant she touches the surface. The total mass (explorer, suit, equipment, and PAPS) is 150 kg. Assume the change in mass of the PAPS to be negligible. Find the horizontal and vertical components of the force the PAPS must exert, and for what interval of time the PAPS must exert it. You can ignore air resistance.

Problem 89P Block A in ?Fig. P5.87? weighs 1.90 N, and block B weighs 4.20 N. The coefficient of kinetic friction between all surfaces is 0.30. Find the magnitude of the horizontal force necessary to drag block B to the left at constant speed if A and B are connected by a light, flexible cord passing around a fixed, frictionless pulley.

Problem 91P Block A in ?Fig. P5.89? has mass 4.00 kg, and block B has mass 12.0 kg. The coefficient of kinetic friction between block B and the horizontal surface is 0.25. (a) What is the mass of block C if block B is moving to the right and speeding up with an acceleration of 2.00 m/s2? (b) What is the tension in each cord when block B has this acceleration?

Problem 92P Two blocks connected by a cord passing over a small, frictionless pulley rest on frictionless planes (?Fig. P5.90?). (a) Which way will the system move when the blocks are released from rest? (b) What is the acceleration of the blocks? (c) What is the tension in the cord?

Problem 93P In terms of ?m?1, ?m?2, and g, find the acceleration of each block in ?Fig. P5.91?. There is no friction anywhere in the system.

Problem 94P Block B , with mass 5.00 kg, rests on block A , with mass 8.00 kg, which in turn is on a horizontal tabletop (?Fig. P5.92?). There is no friction between block A and the tabletop, but the coefficient of static friction between blocks A and B is 0.750. A light string attached to block A passes over a frictionless, massless pulley, and block C is suspended from the other end of the string. What is the largest mass that block C can have so that blocks A and B still slide together when the system is released from rest?

Problem 95P Two objects, with masses 5.00 kg and 2.00 kg, hang 0.600 m above the floor from the ends of a cord that is 6.00 m long and passes over a frictionless pulley. Both objects start from rest. Find the maximum height reached by the 2.00-kg object.

Problem 97P A block is placed against the vertical front of a cart (?Fig. P5.? . What acceleration must the cart have so that block A does not fall? The coefficient of static friction be-tween the block and the cart is µs. How would an observer on the cart describe the behavior of the block?

Problem 96P Friction in an Elevator.? You are riding in an elevator on the way to the 18th floor of your dormitory. The elevator is accelerating upward with a = 1.90 m/s2. Beside you is the box containing your new computer; the box and its contents have a total mass of 36.0 kg. While the elevator is accelerating upward, you push horizontally on the box to slide it at constant speed toward the elevator door. If the coefficient of kinetic friction between the box and the elevator floor is µk = 0.32, what magnitude of force must you apply?

Problem 98P Two blocks, with masses 4.00 kg and 8.00 kg, are con-nected by a string and slide down a 30.0o inclined plane (?Fig. P5.96?). The coefficient of kinetic friction between the 4.00-kg block and the plane is 0.25; that between the 8.00-kg block and the plane is 0.35. Calculate (a) the acceleration of each block and (b) the tension in the string. (c) What happens if the positions of the blocks are reversed, so that the 4.00-kg block is uphill from the 8.00-kg block?

Problem 99P Block A, with weight 3w, slides down an inclined plane S of slope angle 36.9o at a constant speed while plank B, with weight w, rests on top of A. The plank is attached by a cord to the wall (?Fig. P5.97?). (a) Draw a diagram of all the forces acting on block A. (b) If the coefficient of kinetic friction is the same between A and B and between S and A, determine its value.

Problem 102P Banked Curve IL Consider a wet roadway banked as in Example 5.22 (Section 5.4), where there is a coefficient of static friction of 0.30 and a coefficient of kinetic friction of 0.25 between the tires and the roadway. The radius of the curve is R = 50 m. (a) If the banking angle is ? = 25°, what is the maximum speed the automobile can have before sliding up the banking? (b) What is the minimum speed the automobile can have before sliding down the banking?

Problem 100P Accelerometer?. The system shown in Fig can be used to measure the acceleration of the system. An observer riding on the platform measures the angle ??? that the thread supporting the light ball makes with the vertical. There is no friction anywhere. (a) How is ??? related to the acceleration of the system? (b) If ?m?1 = 250 kg and ?m?2 = 1250 kg, what is ???? (c) If you can vary ?m?1 and ?m?2, what is the largest angle ??? you could achieve? Explain how you need to adjust ?m?1 and ?m?2 to do this. Figure:

Problem 101P Banked Curve I.? A curve with a 120-m radius on a level road is banked at the correct angle for a speed of 20 m/s. If an automobile rounds this curve at 30 m/s, what is the minimum coefficient of static friction needed between tires and road to prevent skidding?

Problem 103P Blocks A , B , and C are placed as in ? ig. P5.101? and connected by ropes of negligible mass. Both A and B weigh 25.0 N each, and the coefficient of kinetic friction between each block and the surface is 0.35. Block C descends with constant velocity. (a) Draw separate free-body diagrams showing the forces acting on A and on B. (b) Find the tension in the rope connecting blocks A and B. (c) What is the weight of block C? (d) If the rope connecting A and B were cut, what would be the acceleration of C?

CALC You throw a rock downward into water with a speed of 3 mg/k, where k is the coefficient in Eq. (5.5). Assume that the relationship between fluid resistance and speed is as given in Eq. (5.5), and calculate the speed of the rock as a function of time. \(f=k_{v}\) (fluid resistance at low speed) (5.5)

Problem 104P You are riding in a school bus. As the bus rounds a flat curve at constant speed, a lunch box with mass 0.500 kg, suspended from the ceiling of the bus by a string 1.80 m long, is found to hang at rest relative to the bus when the string makes an angle of 30.0o with the vertical. In this position the lunch box is 50.0 m from the curve’s center of curvature. What is the speed v of the bus?

Problem 108P A rock with mass ?m? slides with initial velocity ?v?0 on a horizontal surface. A retarding force ?F?R that the surface exerts on the rock is proportional to the square root of the instantaneous velocity of the rock (?FR? = –?kv?1/2). (a) Find expressions for the velocity and position of the rock as a function of time. (b) In terms of ?m?, ?k?, and ?v?0, at what time will the rock come to rest? (c) In terms of ?m?, ?k?, and ?v?0, what is the distance of the rock from its starting point when it comes to rest?

Problem 109P You observe a 1350-kg sports car rolling along flat pavement in a straight line. The only horizontal forces acting on it are a constant rolling friction and air resistance (proportional to the square of its speed). You take the following data during a time interval of 25 s: When its speed is 32 m/s, the car slows down at a rate of –0.42 m/s2, and when its speed is decreased to 24 m/s, it slows down at –0.30 m/s2. (a) Find the coefficient of rolling friction and the air drag constant ?D?. (b) At what constant speed will this car move down an incline that makes a 2.2° angle with the horizontal? (c) How is the constant speed for an incline of angle ?? related to the: terminal speed of this sports car if the car drops off a high cliff? Assume that in both cases the air resistance force is proportional to the square of the speed, and the air drag constant is the same.

Problem 107P A rock with mass ?m? = 3.00 kg falls from rest in a viscous medium. The rock is acted on by a net constant downward force of 18.0 N (a combination of gravity and the buoyant force exerted by the medium) and by a fluid resistance force ?f? = kv?, where ?v? is the speed in m/s and ?k? = 2.20 N s/m (see Section 5.3). (a) Find the initial acceleration ?a?0. (b) Find the acceleration when the speed is 3.00 m/s. (c) Find the speed when the acceleration equals 0.1?a?0. (d) Find the terminal speed ?vt (e) Find the coordinate, speed, and acceleration 2.00 s after the start of the motion. (f) Find the time required to reach a speed of 0.9 v ? t?.

Problem 105P The Monkey and Bananas Problem?. A 20-kg monkey has a firm hold on a light rope that passes over a frictionless pulley and is attached to a 20-kg bunch of bananas (Fig). The monkey looks up, sees the bananas, and starts to climb the rope to get them. (a) As the monkey climbs, do the bananas move up, down, or remain at rest? (b) As the monkey climbs, does the distance between the monkey and the bananas decrease, increase, or remain constant? (c) The monkey releases her hold on the rope. What happens to the distance between the monkey and the bananas while she is falling? (d) Before reaching the ground, the monkey grabs the rope to stop her fall. What do the bananas do? Figure:

Problem 110P A 4.00-kg block is attached to a vertical rod by means of two strings. When the system rotates about the axis of the rod, the strings are extended as shown in ?Fig. P5.104? and the tension in the upper string is 80.0 N. (a) What is the tension in the lower cord? (b) How many revolutions per minute does the system make? (c) Find the number of revolutions per minute at which the lower cord just goes slack. (d) Explain what happens if the number of revolutions per minute is less than that in part (c).

Problem 112P A small rock moves in water, and the force exerted on it by the water is given by Eq. (5.7). The terminal speed of the rock is measured and found to be 2.0 m/s. The rock is projected ?upward? at an initial speed of 6.0 m/s. You can ignore the buoyancy force on the rock. (a) In the absence of fluid resistance, how high will the rock rise and how long will it take to reach this maximum height? (b) When the effects of fluid resistance are included, what are the answers to the questions in part (a)?

A 70-kg person rides in a 30-kg cart moving at 12 m/s at the top of a hill that is in the shape of an arc of a circle with a radius of 40 m. (a) What is the apparent weight of the person as the cart passes over the top of the hill? (b) Determine the maxi-mum speed that the cart can travel at the top of the hill without losing contact with the surface. Does your answer depend on the mass of the cart or the mass of the person? Explain.

Problem 113P Merry-Go-Round.? One December identical twins Jena and Jackie are playing on a large merry-go-round (a disk mounted parallel to the ground, on a vertical axle through its center) in their school playground in northern Minnesota. Each twin has mass 30.0 kg. The icy coaling on the merry-go-round surface makes it frictionless. The merry-go-round revolves at a constant rate as the twins ride on it. Jena, sitting 1.80 m from the center of the merry-go-round, must hold on to one of the metal posts attached to the merry-go-round with a horizontal force of 60.0 N to keep from sliding off. Jackie is sitting at the edge, 3.60 m f rom the center. (a) With what horizontal force must Jackie hold on to keep horn falling off? (b) If Jackie falls off, what will be her horizontal velocity when she becomes airborne?

Problem 111P Equation (5.10) applies to the case where the initial velocity is zero. (a) Derive the corresponding equation for ?vy?(t) when the falling object has an initial downward velocity with magnitude ?v?0. (b) For the case where ?v?0<?vt?, sketch a graph of ?vy? as a function of ?t? and label ?vt? on your graph. (c) Repeal part (b) for the case where ?v?0 > ?vt? (d) Discuss what your result says about v ? y?? ?) when ?v?0 = ?vt?.

Problem 116P A passenger with mass 85 kg rides in a Ferris wheel like that in Example 5.23 (Section 5.4). The seats travel in a circle of radius 35 in. The Ferris wheel rotates at constant speed and makes one complete revolution every 25 s. Calculate the magnitude and direction of the net force exerted on the passenger by the seat when she is (a) one-quarter revolution past her lowest point and (b) one-quarter revolution past her highest point.

Problem 115P On the ride “Spindletop” at the amusement park Six Flags Over Texas, people stood against the inner wall of a hollow vertical cylinder with radius 2.5 m. The cylinder started to rotate, and when it reached a constant rotation rate of 0.60 rev/s, the floor dropped about 0.5 m. The people remained pinned against the wall without touching the floor. (a) Draw a force diagram for a person on this ride after the floor has dropped. (b) What minimum coefficient of static friction was required for the person not to slide downward to the new position of the floor? (c) Does your answer in part (b) depend on the person’s mass? (?Note:? When such a ride is over, the cylinder is slowly brought to rest. As it slows down, people slide down the walls to the floor.)

Problem 117P Ulterior Motives.? You are driving a classic 1954 Nash Ambassador with a friend who is sitting to your right on the passenger side of the front seat. The Ambassador has flat bench seats. You would like to be closer to your friend and decide to use physics to achieve your romantic goal by making a quick turn. (a) Which way (to the left or to the right) should you turn the car to get your friend to slide closer to you? (b) If the coefficient of static friction between your friend and the car seat is 0.35, and you keep driving at a constant speed of 20 m/s, what is the maximum radius you could make your turn and still have your friend slide you way?

A physics major is working to pay her college tuition by performing in a traveling carnival. He rides a motorcycle inside a hollow, transparent plastic sphere. After gaining sufficient speed, he travels in a vertical circle with radius 13.0 m. The physics major has mass 70.0 kg, and his motorcycle has mass 40.0 kg. (a) What minimum speed must he have at the top of the circle for the motorcycle tires to remain in contact with the sphere? (b) At the bottom of the circle, his speed is twice the value calculated in part (a). What is the magnitude of the normal force exerted on the motorcycle by the sphere at this point?

Problem 119P A small bead can slide without friction on a circular hoop that is in a vertical plane and has a radius of 0.100 m. The hoop rotates at a constant rate of 4.00 rev/s about a vertical diameter (?Fig. P5.107?). (a) Find the angle ? at which the bead is in vertical equilibrium. (It has a radial acceleration toward the axis.) (b) Is it possible for the bead to “ride” at the same elevation as the center of the hoop? (c) What will happen if the hoop rotates at 1.00 rev/s?

Problem 120P A small remote-controlled car with mass 1.60 kg moves at a constant speed of v = 12.0 m/s in a track formed by a vertical circle inside a hollow metal cylinder that has a radius of 5.00 m (?Fig. E5.45?). What is the magnitude of the normal force exerted on the car by the walls of the cylinder at (a) point A (bottom of the track) and (b) point B (top of the track)?

Problem 122CP Moving Wedge.? A wedge with mass M rests on a frictionless, horizontal tabletop. A block with mass ?m? is placed on the wedge (?Fig. P5.112a?). There is no friction between the block and the wedge. The system is released from rest. (a) Calculate the acceleration of the wedge and the horizontal and vertical components of the acceleration of the block. (b) Do your answers to part (a) reduce to the correct results when M is very large? (c) As seen by a stationary observer, what is the shape of the trajectory of the block?

Problem 121CP Angle for Minimum Force?. A box with weight ?w? is pulled at constant speed along a level floor by a force that is at an angle ??? above the horizontal. The coefficient of kinetic friction between the floor and box is ???k. (a) In terms of ??, ??k, and ?w?, calculate ?F.? (b) For ?w? = 400 N and ???k = 0.25, calculate ?F? for ??? ranging from 0° to 90° in increments of 10°. Graph ?F? versus ???. (c) From the general expression in part (a), calculate the value of ??? for which the value of ?F?, required to maintain constant speed, is a minimum. (?Hint?: At a point where a function is minimum, what are the first and second derivatives of the function? Here ?F? is a function of ???.) For the special case of ?w? = 400 N and ???k = 0.25, evaluate this optimal ??? and compare your result to the graph you constructed in part (b).

Problem 123CP A wedge with mass M rests on a frictionless, horizontal tabletop. A block with mass m is placed on the wedge, and a horizontal force is applied to the wedge (Fig. P5.112b). What must the magnitude of be if the block is to remain at a constant height above the tabletop?

Problem 125CP Double Atwood’s Machine.? In ?Fig. P5.114? masses m1 and m2 are connected by a light string A over a light, frictionless pulley B. The axle of pulley B is connected by a light string C over a light, frictionless pulley D to a mass m3. Pulley D is suspended from the ceiling by an attachment to its axle. The system is released from rest. In terms of m1, m2, m3, and g , what are (a) the acceleration of block m3; (b) the acceleration of pulley B ; (c) the acceleration of block m1; (d) the acceleration of block m2; (e) the tension in string A; (f) the tension in string C? (g) What do your expressions give for the special case of m1 = m2 and m3 = m1 + m2? Is this reasonable?

Problem 124CP Falling Baseball.? You drop a baseball from the roof of a tall building. As the ball falls, the air exerts a drag force proportional to the square of the ball’s speed (?f? = Dv?2). (a) In a diagram, show the direction of motion and indicate, with the aid of vectors, all the forces acting on the ball. (b) Apply Newton’s second law and infer from the resulting equation the general properties of the motion. (c) Show that the ball acquires a terminal speed that is as given in Eq. (5.13). (d) Derive the equation for the speed at any time. (?Note?: where defines the hyperbolic tangent.)

Problem 126CP The masses of blocks ?A? and ?B? in Fig are 20.0 kg and 10.0 kg, respectively. The blocks are initially at rest on the floor and are connected by a massless string passing over a massless and frictionless pulley. An upward force is applied to the pulley. Find the accelerations of block ?A? and of block ? ? when F? is (a) 124 N; (b) 294 N; (c) 424 N. Figure: