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 that goes to the ceiling. (a) What is the tension in the rope? (b) What is the tension in the chain?

In Fig. E5.2 each of the suspended blocks has weight w. The pulleys are frictionless and the ropes have negligible weight. Calculate, in each case, the tension T in the rope in terms of the weight w. In each case, include the free-body diagram or diagrams you used to determine the answer.

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

Injuries to the Spinal Column. In the treatment of spine injuries, it is often necessary to provide some tension along the spinal column to stretch the backbone. One device for doing this is the Stryker frame, illustrated in Fig. E5.4a. A weight W is attached to the patient (sometimes around a neck collar, as shown in Fig. E5.4b), and friction between the persons body and the bed prevents sliding. (a) If the coefficient of static friction between a 78.5-kg patients 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?

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.)

A large wrecking ball is held in place by two light steel cables (Fig. E5.6). If the mass m of the wrecking ball is 4090 kg, what are (a) the tension in the cable that makes an angle of with the vertical and (b) the tension in the horizontal cable?

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

. A 1130-kg car is held in place by a light cable on a very smooth (frictionless) ramp, as shown in Fig. E5.8. The cable makes an angle of 31.0 above the surface 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?

A man pushes on a piano with mass 180 kg so that it slides at constant velocity down a ramp that is inclined at 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

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 \(\vec{F}_{1}\) and \(\vec{F}_{2}\) that must be applied to hold the system in the position shown. Equation Transcription: Text Transcription: Vec F_1 Vec F_2

Stay Awake! An astronaut is inside a rocket that is blasting off vertically from the launch pad. You want this rocket to reach the speed of sound as quickly as possible, but you also do not want the astronaut to black out. Medical tests have shown that astronauts are in danger of blacking out at an acceleration greater than 4g. (a) What is the maximum thrust the engines of the rocket can have to just barely avoid blackout? Start with a free-body diagram of the rocket. (b) What force, in terms of her weight w, does the rocket exert on the astronaut? 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?

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.

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) Assuming it to be constant, what was its acceleration (in m/\(s^2\) and in g’s) 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) For how long did this force last?

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.

Atwoods 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?

A 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 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?

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?

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 is required for takeoff, what minimum length of runway is needed? (b) What is the tension in the towrope between the two gliders while they are accelerating for the takeoff?

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?

Apparent Weight. A 550-N physics student stands on a bathroom scale in an 850-kg (including the student) elevator that is supported by a cable. 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)?

BIO Force During a Jump. An average person can reach a maximum height of about 60 cm when jumping straight up from a crouched position. During the jump itself, the persons 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 jumpers weight w, what force does the ground exert on him or her during the jump?

A 2540-kg test rocket is launched vertically from the launch pad. Its fuel (of negligible mass) provides a thrust force so that its vertical velocity as a function of time is given by where A and B are constants and time is measured from the instant the fuel is ignited. At the instant of ignition, the rocket has an upward acceleration of and later an upward velocity of (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 rockets weight. (d) What was the initial thrust due to the fuel?

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 \(F(t)=\left(6.00\mathrm{\ N}/\mathrm{s}^2\right)t^2\). (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?

A 5.00-kg crate is suspended from the end of a short vertical rope of negligible mass. An upward force is applied to the end of the rope, and the height of the crate above its initial position is given by . What is the magnitude of the force F when ?

The Trendelenburg Position. In emergencies with major blood loss, the doctor will order the patient 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 the 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?

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 and kinetic friction occur. (b) Find the coefficients of static and kinetic friction between the block and the table. (c) Why does the graph slant upward in the first part but then level out? (d) What would the graph look like if a 135-N brick were placed on the box, and what would the coeffi- cients of friction be in that case?

A stockroom worker pushes a box with mass 11.2 kg on a horizontal surface with a constant speed of 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?

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 the box? (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 boxs acceleration?

A 45.0-kg crate of tools rests on a horizontal floor. You exert a gradually increasing horizontal push on it and observe that the crate just begins to move when your force exceeds 313 N. After that you must reduce your push to 208 N to keep it moving at a steady (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 (c) Suppose you were performing the same experiment on this crate but were doing it on the moon instead, where the acceleration due to gravity is (i) What magnitude push would cause it to move? (ii) What would its acceleration be if you maintained the push in part (b)?

Some sliding rocks approach the base of a hill with a speed of The hill rises at 36 above the horizontal and has coefficients of kinetic 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 there, show why. If it slides down, find its acceleration on the way down.

You are lowering two boxes, one on top of the other, down the ramp shown in Fig. E5.31 by pulling on a rope parallel to the surface of the ramp. 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?

A pickup truck is carrying a toolbox, but the rear gate of the truck is missing, so the box will slide out if it is set moving. The coefficients of kinetic and static friction between the box and the 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 without causing the box to slide? Include a free-body diagram of the toolbox as part of your solution.

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 an automobile by locking the brakes when traveling at about ? (b) On wet pavement the coefficient of kinetic friction may be only 0.25. How fast should you drive on wet pavement in order 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)?

Two crates connected by a rope lie on a horizontal surface (Fig. E5.35). Crate A has mass and crate B has mass The coefficient of kinetic friction between each crate and the surface is The crates are pulled to the right at constant velocity by a horizontal force In terms of and calculate (a) the magnitude of the force and (b) the tension in the rope connecting the blocks. Include the free-body diagram or diagrams you used to determine each answer.

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 the angle 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?

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. After the blocks are released from rest, find (a) the speed of each block after moving 3.00 cm and (b) the tension in the cord. Include the free-body diagram or diagrams you used to determine the answers.

A box with mass m is dragged across a level floor having a coefficient of kinetic friction \(\mu_{\mathrm{k}}\) by a rope that is pulled upward at an angle \(\theta\) above the horizontal with a force of magnitude F. (a) In terms of m, \(\mu_{\mathrm{k}}\), \(\theta\) 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 \(25^{\circ}\) above the horizontal. By dragging some weights wrapped in an old pair of pants down the hall with a spring balance, you find that \(\mu_{\mathrm{k}}=0.35\). Use the result of part (a) to answer the instructor's question.

A large crate with mass m rests on a horizontal floor. The coefficients of friction between the crate and the floor are \(\mu_{\mathrm{s}} \text { and } \mu_{\mathrm{k}}\). A woman pushes downward at an angle \(\theta\) below the horizontal on the crate with a force \(\overrightarrow{\boldsymbol{F}}\). (a) What magnitude of force \(\overrightarrow{\boldsymbol{F}}\) is required to keep the crate moving at constant velocity? (b) If \(\mu_{\mathrm{s}}\) is greater than some critical value, the woman cannot start the crate moving no matter how hard she pushes. Calculate this critical value of \(\mu_{\mathrm{s}}\).

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

(a) In Example 5.18 (Section 5.3), what value of D is required to make for the skydiver? (b) If the skydivers daughter, whose mass is 45 kg, is falling through the air and has the same D as her father, what is the daughters terminal speed?

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. E5.42). 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 normal force on the car when it is at the bottom of the track (point A)?

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?

A flat (unbanked) curve on a highway has a radius of 220.0 m. A car rounds the curve at a speed of (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?

A 1125-kg car and a 2250-kg pickup truck approach a curve on the expressway that has a radius of 225 m. (a) At what angle should the highway engineer bank this curve so that vehicles traveling at 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 find the normal force on each one due to the highway surface.

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. E5.46). Each arm supports a seat 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 with the vertical. (b) Does the angle depend on the weight of the passenger for a given rate of revolution?

In another version of the Giant Swing (see Exercise 5.46), the seat is connected to two cables as shown in Fig. E5.47, one of which is horizontal. The seat swings in a horizontal circle at a rate of 32.0 rpm If the seat weighs 255 N and an 825-N person is sitting in it, find the tension in each cable

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 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 a 60.0 rev>min?

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 (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 How many revolutions per minute are needed in this case?

The Cosmoclock 21 Ferris wheel in Yokohama City, 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 passengers apparent weight at the lowest point?

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) At the top of the loop, the pilot feels weightless. What is the speed of the airplane at this point? (b) At the bottom of the loop, the speed of the airplane is 280 km/h. What is the apparent weight of the pilot at this point? His true weight is 700 N.

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 planes speed at the lowest point of the circle is what is the minimum radius of the circle for the acceleration at this point not to exceed 4.00g? (b) What is the apparent weight of the pilot at the lowest point of the pullout?

Stay Dry! You tie a cord to a pail of water, and you 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 if no water is to spill from it?

A bowling ball weighing 71.2 N 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 (a) What is the acceleration of the bowling ball, in magnitude and direction, at this instant? (b) What is the tension in the rope at this instant?

Effect on Blood of Walking. While a person is walking, his arms swing through approximately a 45 angle in s. As a reasonable approximation, we can 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?

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. P5.56). The rope will break if the tension in it exceeds and our heros mass is 90.0 kg. (a) If the angle is find the tension in the rope. (b) What is the smallest value the angle can have if the rope is not to break?

Two ropes are connected to a steel cable that supports a hanging weight as shown in Fig. P5.57. (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 force 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. You can ignore the weight of the ropes and the steel cable.

In Fig. P5.58 a worker lifts a weight w by pulling down on a rope with a force 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 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

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

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

Forces During Chin-ups. People who do chinups raise their chin just over a bar (the chinning bar), supporting themselves with only their 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 this distance 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 persons body, and then apply it to find the force his arms must exert on him during the accelerating part of the chin-up

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.

A 3.00-kg box that is several hundred meters above the surface of the earth 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 this results in a tension in the rope of . The box is at rest at . The only forces on the box are the tension in the rope and gravity. (a) What is the velocity of the box at (i) and (ii) ? (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?

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 \(30.0^{\circ}\) above the horizontal. The coefficient of kinetic friction is \(\mu_{\mathrm{k}}=0.40\), and the coefficient of static friction is \(\mu_{\mathrm{s}}=0.50\). 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?

Two boxes connected by a light horizontal rope are on a horizontal surface, as shown in Fig. P5.35. The coefficient of kinetic friction between each box and the surface is . One box (box B) has mass 5.00 kg, and the other box (box A) has mass m. A force F with magnitude 40.0 N and direction above the horizontal is applied to the 5.00-kg box, and both boxes move to the right with . (a) What is the tension T in the rope that connects the boxes? (b) What is the mass m of the second box?

A 6.00-kg box sits on a ramp that is inclined at \(37.0^{\circ}\) above the horizontal. The coefficient of kinetic friction between the box and the ramp is \(\mu_{\mathrm{k}}=0.30\). What horizontal force is required to move the box up the incline with a constant acceleration of \(4.20 \mathrm{\ m}/ \mathrm{s}^2\)?

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 . 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?

In Fig. P5.68 \(m_{1}=20.0 \mathrm{~kg}\) and \(\alpha=53.1^{\circ}\). The coefficient of kinetic friction between the block and the incline is \(\mu_{\mathrm{k}}=\)=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?

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 is 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 \(\mu_{r}\) for each? Assume that the net horizontal force is due to rolling friction only. Text Transcription: mu_r

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.

A block with mass is placed on an inclined plane with slope angle and is connected to a second hanging block with mass by a cord passing over a small, frictionless pulley (Fig. P5.68). The coefficient of static friction is and the coefficient of kinetic friction is (a) Find the mass for which block moves up the plane at constant speed once it is set in motion. (b) Find the mass for which block moves down the plane at constant speed once it is set in motion. (c) For what range of values of will the blocks remain at rest if they are released from rest?

Block A in Fig. P5.72 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.

Block A in Fig. P5.73 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. P5.73a). (b) If A is held at rest (Fig. P5.73b).

A window washer pushes his scrub brush up a vertical window at constant speed by applying a force as shown in Fig. P5.74. The brush weighs 15.0 N and the coefficient of kinetic friction is Calculate (a) the magnitude of the force and (b) the normal force exerted by the window on the brush.

The Flying Leap of a Flea. High-speed motion pictures of a jumping flea yielded the data to plot the fleas acceleration as a function of time as shown in Fig. P5.75. (See The Flying Leap of the Flea, by M. Rothschild et al. in the November 1973 Scientific American.) This flea was about 2 mm long and jumped at a nearly vertical takeoff angle. Use the measurements shown on the graph to answer the questions. (a) Find the initial net external force on the flea. How does it compare to the fleas 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 fleas maximum speed.

A 25,000-kg rocket blasts off vertically from the earths 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 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 earths surface when it breaks the sound barrier?

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 \(v(t)=\left(3.0\mathrm{\ m}/\mathrm{s}^2\right)t+\left(0.20\mathrm{\ m}/\mathrm{s}^3\right)t^2\). When t = 4.0 s, what is the reading of the bathroom scale?

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 passengers weight. The elevator accelerates upward with constant acceleration for a distance of 3.0 m and then starts to slow down. What is the maximum speed of the elevator?

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 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 (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?

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 What is the acceleration of the bus?

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 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 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 \(20^{\circ}\) 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 at 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 before 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?

In the system shown in Fig. P5.34, block A has mass block B has mass and the rope connecting them has a nonzero mass 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 and If there is friction between block A and the tabletop, with and 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 Will the blocks move in this case?

If the coefficient of static friction between a table and a uniform massive rope is \(\mu_{s}\), what fraction of the rope can hang over the edge of the table without the rope sliding?

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 northward and (b) when it accelerates at southward.

Traffic Court. You are called as an expert witness in the trial of 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 cars wheels, the car traveled 192 ft before stopping, and the coefficient of kinetic friction between the road and his tires was 0.750. The charge is that he was speeding in a zone. He pleads innocent. What is your conclusion, guilty or innocent? How fast was he going when he hit his brakes?

Two identical 15.0-kg balls, each 25.0 cm in diameter, are suspended by two 35.0-cm wires as shown in Fig. P5.87. 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?

Losing Cargo. A 12.0-kg box rests on the flat floor of a truck. The coefficients of friction between the box and floor are \(\mu_{\mathrm{s}}=0.19\) and \(\mu_{\mathrm{k}}=0.15\). The truck stops at a stop sign and then starts to move with an acceleration of \(2.20\mathrm{\ m}/\mathrm{s}^2\). 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?

Block A in Fig. P5.89 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

You are part of a design team for future exploration of the planet Mars, where \(g=3.7\mathrm{\ m}/\mathrm{s}^2\). 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 freely 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 compo- nents of the force the PAPS must exert, and for what interval of time the PAPS must exert it. You can ignore air resistance. Text Transcription: g = 3.7 m/s^2

Block A in Fig. P5.91 has a mass of 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 (b) What is the tension in each cord when block B has this acceleration?

Two blocks connected by a cord passing over a small, frictionless pulley rest on frictionless planes (Fig. P5.92). (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?

In terms of \(m_1\), \(m_2\) and g, find the acceleration of each block in Fig. P5.93. There is no friction anywhere in the system.

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. P5.94). There is no friction between block A and the tabletop, but the coefficient of static friction between block A and block 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?

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

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 Beside you is the box containing your new computer; the box and its contents have a total mass of 28.0 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 what magnitude of force must you apply?

A block is placed against the vertical front of a cart as shown in Fig. P5.97. What acceleration must the cart have so that block A does not fall? The coefficient of static friction between the block and the cart is \(\mu_{\mathrm{s}}\). How would an observer on the cart describe the behavior of the block?

Two blocks with masses 4.00 kg and 8.00 kg are connected by a string and slide down a inclined plane (Fig. P5.98). 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. (a) Calculate the acceleration of each block. (b) Calculate the tension in the string. (c) What happens if the positions of the blocks are reversed, so the 4.00-kg block is above the 8.00-kg block?

Block A, with weight 3w, slides down an inclined plane S of slope angle 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. P5.99). (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

Accelerometer. The system shown in Fig. P5.100 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 and what is (c) If you can vary and what is the largest angle you could achieve? Explain how you need to adjust and to do this.

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?

Banked Curve II. 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 \(\beta=25^{\circ}\), 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?

Blocks A, B, and C are placed as in Fig. P5.103 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 two 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?

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 \(30.0^{\circ}\) with the vertical. In this position the lunch box is 50.0 m from the center of curvature of the curve. What is the speed \(v\) of the bus?

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. P5.105). 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?

You throw a rock downward into water with a speed of 3mg/k, where k is the coefficient in Eq. (5.7). Assume that the relationship between fluid resistance and speed is as given in Eq. (5.7), and calculate the speed of the rock as a function of time.

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 \(k=2.20 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}\) (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 \(v_t\) (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 \(0.9v_t\)

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 \(\left(F_{\mathrm{R}}=-k v^{1 / 2}\right)\). (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?

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 the car slows down at a rate of and when its speed is decreased to it slows down at (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 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

The 4.00-kg block in Fig. P5.110 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 the diagram 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 in part (c).

Equation (5.10) applies to the case where the initial velocity is zero. (a) Derive the corresponding equation for when the falling object has an initial downward velocity with magnitude (b) For the case where sketch a graph of as a function of t and label on your graph. (c) Repeat part (b) for the case where (d) Discuss what your result says about

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 The rock is projected upward at an initial speed of 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)?

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 coating 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 from the center. (a) With what horizontal force must Jackie hold on to keep from falling off? (b) If Jackie falls off, what will be her horizontal velocity when she becomes airborne?

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 maximum speed that the cart may 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.

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 the floor on which people were standing dropped about 0.5 m. The people remained pinned against the wall. (a) Draw a force diagram for a person on this ride, after the floor has dropped. (b) What minimum coefficient of static friction is required if the person on the ride is not to slide downward to the new position of the floor? (c) Does your answer in part (b) depend on the mass of the passenger? (Note: When the ride is over, the cylinder is slowly brought to rest. As it slows down, people slide down the walls to the floor.)

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 m. 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) onequarter revolution past her highest point

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 what is the maximum radius you could make your turn and still have your friend slide your way?

A physics major is working to pay his 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 a radius of 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 if the tires of the motorcycle are not to lose 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?

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 about a vertical diameter (Fig. P5.119). (a) Find the angle at which the bead is in vertical equilibrium. (Of course, 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?

A small remote controlled car with mass 1.60 kg moves at a constant speed of v = 12.0 m/s in a vertical circle inside a hollow metal cylinder that has a radius of 5.00 m (Fig. P5.120). What is the magnitude of the normal force exerted on the car by the walls of the cylinder at (a) point A (at the bottom of the vertical circle) and (b) point B (at the top of the vertical circle)?

CALC 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 (a) In terms of and w, calculate F. (b) For and calculate F for ranging from to in increments of 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 and evaluate this optimal and compare your result to the graph you constructed in part (b).

Moving Wedge. A wedge with mass M rests on a frictionless, horizontal tabletop. A block with mass m is placed on the wedge (Fig. P5.122a). 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?

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. P5.122b). What must the magnitude of be if the block is to remain at a constant height above the tabletop?

CALC 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 balls speed (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 Newtons 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.) L dx a2 - x 2 = 1 a arctanh x a tanh1x2 = ex - e-x ex + e-x = e2x - 1 e2x +

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

The masses of blocks A and B in Fig. P5.126 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 B when F is (a) 124 N; (b) 294 N; (c) 424 N

A ball is held at rest at position A in Fig. P5.127 by two light strings. The horizontal string is cut and the ball starts swinging as a pendulum. Point B is the farthest to the right the ball goes as it swings back and forth. What is the ratio of the tension in the supporting string at position B to its value at A before the horizontal 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.

In Fig. E5.2 each of the suspended blocks has weight w. The pulleys are frictionless and the ropes have negligible weight. Calculate, in each case, the tension ???? in the rope in terms of the weight w. In each case, include the free-body diagram or diagrams you used to determine the answer. Equation Transcription: Text Transcription:

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 3E 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 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?

Injuries to the Spinal Column. In the treatment of spine injuries, it is often necessary to provide some tension along the spinal column to stretch the backbone. One device for doing this is the Stryker frame, illustrated in Fig. E5.4a. A weight \(W\) attached to the patient (sometimes around a neck collar, as shown in Fig. E5.4b), and friction between the person’s body and the bed prevents sliding. (a) If the coefficient of static friction between a \(78.5-\mathrm{kg}\) 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? Equation Transcription: Text Transcription: W 78.5-kg 0.75 W

Problem 5DQ For medical reasons, astronauts in outer space must deter-mine 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.)

To push a box up a ramp, is the force required smaller if you push horizontally or if you push parallel to the ramp? Why?

A large wrecking ball is held in place by two light steel cables (Fig. E5.6). If the mass m of the wrecking ball is 4090 kg, what are (a) the tension \(T_{\mathrm{B}}\) in the cable that makes an angle of \(40^{\circ}\) with the vertical and (b) the tension \(T_{\mathrm{B}}\) in the horizontal cable? Equation Transcription: 40° Text Transcription: T_B 40° T_B

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. Equation Transcription: 30° 45° 45° 60° Text Transcription: 30^° 45^° 45^° 60^°

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?

Find the tension in each cord in Fig. E5.7 if the weight of the suspended object is w. Equation Transcription: 30° 45° 45° 60° Text Transcription: 30^° 45^° 45^° 60^°

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 \(\vec{F}_{1}\) and \(\vec{F}_{2}\) that must be applied to hold the system in the position shown. Equation Transcription: Text Transcription: Vec F_1 Vec F_2

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?

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 on the freeway.

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. Equation Transcription: Text Transcription: 30.0 kg 20.0 kg 10.0 kg

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 \(15.0-\mathrm{kg}\) load of bricks hangs from one end of a rope that passes over a small, frictionless pulley. A \(\text { 28.0- kg }\) 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? Equation Transcription: Text Transcription: 15.0-kg 28.0-kg

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?

You often hear people say that “friction always opposes motion.” Give at least one example where (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?

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

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 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 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?

The centrifugal force is not included in the free-body diagrams of Figs. 5.34b and 5.35. Explain why not. Equation Transcription: Text Transcription: n cos beta n sin beta L cos beta L sin beta

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 \(v(t)A t+B t^{2}\), where A and B are constants and time is measured from the instant the fuel is ignited. The rocket has an upward acceleration of \(1.50 \mathrm{~m} / \mathrm{s}^{2}\) 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?

To keep the forces on the riders within allowable limits, loop-the-loop roller coaster rides are often designed so that the loop, rather than being a perfect circle, 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 then a second time after the air has been readmitted to the cylinder. You examine multiflash photographs of the two drops. From these photos how can you tell which one is which or can you?

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?

If you throw a baseball straight upward with speed how does its speed, when it returns to the point from where you threw it, compare to (a) in the absence of air resistance and (b) in the presence of air resistance? 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 and kinetic friction occur. (b) Find the coefficients of static and kinetic friction between the block and the table. (c) Why does the graph slant upward in the first part but then level out? (d) What would the graph look like if a 135-N brick were placed on the box, and what would the coefficients of friction be in that case?

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.

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 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 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?

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

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)?

A ball is dropped from rest and feels air resistance as it falls. Which of the graphs in Fig. Q5.30 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.

When does a baseball in flight 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.

You are lowering two boxes, one on top of the other, down the ramp shown in Fig. E5.31 by pulling on a rope parallel to the surface of the ramp. Both boxes move together at a constant speed of 15.0 \(\mathrm{cm} / \mathrm{s}\). 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? Equation Transcription: Text Transcription: cm/s 32.0 kg 48.0 kg 2.50 m 4.75 m

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 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 \(45.0 \mathrm{~N}\) and block B weighs \(25.0 \mathrm{~N} \text {. }\). 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 \(45.0 \mathrm{~N}\), falls asleep on top of block A. If block B is now set into downward motion, what is its acceleration (magnitude and direction)? Equation Transcription: Text Transcription: 45.0 N 25.0 N 45.0 N

Two crates connected by a rope lie on a horizontal surface (Fig. E5.35). Crate A has mass \(m_{A}\) and crate \(B\) has mass \(m_{B}\). The coefficient of kinetic friction between each crate and the surface is \(\mu_{k}\). The crates are pulled to the right at constant velocity by a horizontal force \(\vec{F}\). In terms of \(m_{A}\), \(m_{B}\) and \(\mu_{\mathrm{k}}\), calculate (a) the magnitude of the force \(\vec{F}\) and (b) the tension in the rope connecting the blocks. Include the free-body diagram or diagrams you used to determine each answer. Equation Transcription: Text Transcription: m_A B m_B mu_{\{k} Vec F m_A m_B mu_{\{k} Vec F

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?

As shown in Fig. E5.34, block A (mass \(2.25 \mathrm{~kg}\)) rests on a tabletop. It is connected by a horizontal cord passing over a light, frictionless pulley to a hanging block B (mass \(1.30 \mathrm{~kg}\)). The coefficient of kinetic friction between block A and the tabletop is 0.450. After the blocks are released from rest, find (a) the speed of each block after moving \(3.00 \mathrm{~cm}\) and (b) the tension in the cord. Include the free-body diagram or diagrams you used to determine the answers. Equation Transcription: Text Transcription: 2.25 kg 1.30kg 3.00 cm

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.

A large crate with mass m rests on a horizontal floor. The coefficients of friction between the crate and the floor are \(\mu_{\mathrm{s}}\) and \(\mu_{\mathrm{k}}\). A woman pushes downward at an angle \(\theta\) below the horizontal on the crate with a force \(\overrightarrow{\boldsymbol{F}}\). (a) What magnitude of force \(\overrightarrow{\boldsymbol{F}}\) is required to keep the crate moving at constant velocity? (b) If \(\mu_{\mathrm{s}}\) is greater than some critical value, the woman cannot start the crate moving no matter how hard she pushes. Calculate this critical value of \(\mu_{\mathrm{s}}\).

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?

(a) In Example 5.18 (Section 5.3), what value of D is required to make \(v_{\mathrm{t}}=42 \mathrm{~m} / \mathrm{s}\) for the skydiver? (b) If the skydiver’s daughter, whose mass is \(45 \mathrm{kg}\), is falling through the air and has the same D (0.25 \(\mathrm{kg} / \mathrm{m}\)) as her father, what is the daughter’s terminal speed? Equation Transcription: Text Transcription: v_1=42m/s kg kg/m

A small car with mass \(0.800 \mathrm{~kg}\) travels at constant speed on the inside of a track that is a vertical circle with radius \(5.00 \mathrm{~m}\) (Fig. E5.42). 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 normal force on the car when it is at the bottom of the track (point A)? Equation Transcription: Text Transcription: 0.800 kg 5.00 m

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.

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. E5.46). Each arm supports a seat suspended from a cable \(5.00 \mathrm{~m}\) long, the upper end of the cable being fastened to the arm at a point \(3.00 \mathrm{~m}\)from the central shaft. (a) Find the time of one revolution of the swing if the cable supporting a seat makes an angle of \(30.0^{\circ}\) with the vertical. (b) Does the angle depend on the weight of the passenger for a given rate of revolution? Equation Transcription: ° Text Transcription: 5.00 m 3.00 m 30.0^°

In another version of the “Giant Swing” (see Exercise 5.46), the seat is connected to two cables as shown in Fig. E5.47, 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. Equation Transcription: 40.0° Text Transcription: 32.0 rpm 40.0^° 7.50m

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 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?

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?

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) At the top of the loop, the pilot feels weightless. What is the speed of the airplane at this point? (b) At the bottom of the loop, the speed of the airplane is 280 km/h. What is the apparent weight of the pilot at this point? 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?

Stay Dry! You tie a cord to a pail of water, and you 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 if no water is to spill from it?

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?

Effect on Blood of Walking. While a person is walking, his arms swing through approximately a \(45^{\circ}\) angle in \(\frac{1}{2} \mathrm{~S}\). As a reasonable approximation, we can 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? Equation Transcription: ° Text Transcription: 45^° \frac{1}{2} \{~S}

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. P5.56). The rope will break if the tension in it exceeds \(2.50 \times 10^{4} \mathrm{~N}\), and our hero's mass is \(90.0 \mathrm{~kg}\). (a) If the angle \(\theta\) is \(10.0^{\circ}\), find the tension in the rope. (b) What is the smallest value the angle \(\theta\) can have if the rope is not to break?

Two ropes are connected to a steel cable that supports a hanging weight as shown in Fig. P5.57. (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 force 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. You can ignore the weight of the ropes and the steel cable. Equation Transcription: 60° 40° Text Transcription: 60^° 40^°

In Fig. P5.58 a worker lifts a weight w by pulling down on a rope with a force \(\vec{F}\). 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 \(\vec{F}\) 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. Equation Transcription: Text Transcription: \vec{F} \vec{F}

A solid uniform \(\text { 45.0-kg }\) ball of diameter \(32.0 \mathrm{~cm}\) is supported against a vertical, frictionless wall using a thin \(30.0-\mathrm{cm}\) wire of negligible mass, as shown in Fig. P5.59. (a) Draw a free-body diagram for the ball and use it to find the tension in the wire. (b) How hard does the ball push against the wall? Equation Transcription: Text Transcription: 45.0-kg 32.0 cm 30.0-cm

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

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.

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.

Problem 63P 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?

Two boxes connected by a light horizontal rope are on a horizontal surface, as shown in Fig. P5.35. The coefficient of kinetic friction between each box and the surface is \(\mu_{\mathrm{k}}=0.30\). One box (box B) has mass \(5.00 \mathrm{~kg}\), and the other box (box A) has mass ????. A force ???? with magnitude 40.0 N and direction \(53.1^{0}\) above the horizontal is applied to the \(\text { 5.00-kg }\) box, and both boxes move to the right with \(\mathrm{a}=\mathrm{m} / \mathrm{s}^{2}\) . (a) What is the tension ???? in the rope that connects the boxes? (b) What is the mass ???? of the second box? Equation Transcription: 53.1° Text Transcription: \mu_{\{k}}=0.30 5.00kg 53.1^° \{a}=\{m} / \{s}^{2}

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?

CP In Fig. P5.68 \(m_{1}=20.0 \mathrm{~kg} \quad\) and \(\quad \alpha=\) \(53.1^{\circ}\). The coefficient of kinetic friction between the block and the incline is \(\mu_{k}=\) \(0.40\). What must be the mass \(m_{2}\) of the hanging block if it is to descend \(12.0 \mathrm{~m}\) in the first \(3.00 \mathrm{~s}\) after the system is released from rest?

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 is 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 \(\mu_r\) for each? Assume that the net horizontal force is due to rolling friction only.

A Rope with Mass. A block with mass M is attached to the lower end of a vertical, uniform rope with mass ???? and length L. A constant upward force \(\vec{F}\) 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 ???? from the top end of the rope, where ???? can have any value from 0 to L. Equation Transcription: Text Transcription: \vec{F}

A block with mass \(m_1\) is placed on an inclined plane with slope angle \(\alpha\) and is connected to a second hanging block with mass \(m_2\) by a cord passing over a small, frictionless pulley (Fig. P5.68). The coefficient of static friction is \(\mu_{\mathrm{s}}\) and the coefficient of kinetic friction is \(\mu_{\mathrm{k}}\). (a) Find the mass \(m_2\) for which block \(m_1\) moves up the plane at constant speed once it is set in motion. (b) Find the mass \(m_2\) for which block \(m_1\) moves down the plane at constant speed once it is set in motion. (c) For what range of values of \(m_2\) will the blocks remain at rest if they are released from rest?

Block A in Fig. P5.72 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. Equation Transcription: 45.0° Text Transcription: 45.0^°

Block A in Fig. P5.73 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 \(\vec{F}\) necessary to drag block B to the left at constant speed (a) if A rests on B and moves with it (Fig. P5.73a). (b) If A is held at rest (Fig. P5.73b). Equation Transcription: Text Transcription: \vec{F}

A window washer pushes his scrub brush up a vertical window at constant speed by applying a force \(\vec{F}\) as shown in Fig. P5.74. The brush weighs 15.0 N and the coefficient of kinetic friction is \(\mu_{\mathrm{k}}=0.150\). Calculate (a) the magnitude of the force \(\vec{F}\) and (b) the normal force exerted by the window on the brush. Equation Transcription: Text Transcription: Vec F \mu_{\{k}}=0.150 Vec F

The Flying Leap of a Flea. High-speed motion pictures (3500 frames/second) of a jumping \(210-\mu_{\mathrm{g}}\)flea yielded the data to plot the flea’s acceleration as a function of time as shown in Fig. P5.75. (See “The Flying Leap of the Flea,” by M.Rothschild et al. in the November 1973 Scientific American.) This flea was about 2 mm long and jumped at a nearly vertical takeoff angle. Use the measurements shown on the graph to answer the questions. (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. Equation Transcription: Text Transcription: 210-\mu_{\{g}}

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 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?

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 maximum speed of the elevator?

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 \(37^{\circ}\) 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 \(\mu_{\mathrm{k}}=0.30\). (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?

In the system shown in Fig. P5.34, block A has mass ????A, block B has mass ????B, and the rope connecting them has a nonzero mass ????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 \(m_{A}=2.00 \mathrm{~kg}, m_{B}=0.400 \mathrm{~kg}, m_{\text {rone }}=0.160 \mathrm{~kg} \text {, and } L=1.00 \mathrm{~m}\). If there is friction between block A and the tabletop, with \(\mu_{\mathrm{k}}=0.200\) and \(\mu_{s}=0.250\), 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? Equation Transcription: Text Transcription: m_A m_B m_rope m_A=2.00kh, m_rope=0.160 kg, L=1.00 m \mu_{\{k}}=0.200 \mu_{s}=0.250 m_rope=0.040 kg

If the coefficient of static friction between a table and a uniform massive rope is \(\mu_{s}\), what fraction of the rope can hang over the edge of the table without the rope sliding?

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 \(2.20 \mathrm{\ m} / \mathrm{s}^{2}\) northward and (b) when it accelerates at \(3.40 \mathrm{\ m} / \mathrm{s}^{2}\) southward.

Traffic Court. You are called as an expert witness in the trial of 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. The charge is that he was speeding in a 45-mi/h zone. He pleads innocent. What is your conclusion, guilty or innocent? How fast was he going when he hit his brakes?

Two identical \(\text { 15.0-kg }\)balls, each \(25.0 \mathrm{~cm}\) in diameter, are suspended by two \(35.0-\mathrm{cm}\) wires as shown in Fig. P5.87. The entire apparatus is supported by a single \(\text { 18.0-cm }\) 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? Equation Transcription: Text Transcription: 15.0 kg 25.0 cm 35.0-cm 18.0-cm

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?

Block \(A\) in Fig. P5.89 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 \(\vec{F}\) 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. Equation Transcription: Text Transcription: A 1.90 N B 4.20 N 0.30 Vec F

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.

Block \(A\) in Fig. P5.91 has a mass of 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 \(2.00 \mathrm{~m} / \mathrm{s}^{2}\)? (b) What is the tension in each cord when block B has this acceleration?

Two blocks connected by a cord passing over a small, frictionless pulley rest on frictionless planes (Fig. P5.92). (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? Equation Transcription: 30.0° 53.1° Text Transcription: 100 kg 50 kg 30.0^° 53.1^°

In terms of \(m_{1}, m_{2} \text { and } g\), find the acceleration of each block in Fig. P5.93. There is no friction anywhere in the system. Equation Transcription: Text Transcription: m_1, m_2 and g

Block B, with mass \(5.00 \mathrm{~kg}\), rests on block A, with mass \(8.00 \mathrm{~kg}\), which in turn is on a horizontal tabletop (Fig. P5.94). There is no friction between block A and the tabletop, but the coefficient of static friction between block A and block 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? Equation Transcription: Text Transcription: 5.00 kg 8.00 kg

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.

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 \(a=1.90\ \frac{m}{s^2}\) . Beside you is the box containing your new computer; the box and its contents have a total mass of 28.0 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 \(\mu_k=0.32\) what magnitude of force must you apply?

A block is placed against the vertical front of a cart as shown in Fig. P5.97. What acceleration must the cart have so that block A does not fall? The coefficient of static friction between the block and the cart is \(\mu_{s}\). How would an observer on the cart describe the behavior of the block? Equation Transcription: Text Transcription: \mu_{s}

Two blocks with masses \(4.00 \mathrm{~kg}\) and \(8.00 \mathrm{~kg}\) are connected by a string and slide down a \(30.0^{\circ}\) inclined plane (Fig. P5.98). The coefficient of kinetic friction between the \(\text { 4.00-kg }\) block and the plane is 0.25; that between the \(\text { 8.00-kg }\) block and the plane is 0.35. (a) Calculate the acceleration of each block. (b) Calculate the tension in the string. (c) What happens if the positions of the blocks are reversed, so the \(\text { 4.00-kg }\) block is above the \(8.00-\mathrm{kg}\) block? Equation Transcription: 30.0° Text Transcription: 4.00 kg 8.00 kg 30.0° 4.00-kg 8.00-kg

Block A, with weight 3w, slides down an inclined plane S of slope angle 36.90 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. P5.99). (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. Equation Transcription: 36.9° Text Transcription: 36.9^°

Accelerometer. The system shown in Fig. P5.100 can be used to measure the acceleration of the system. An observer riding on the platform measures the angle \(\theta\) that the thread supporting the light ball makes with the vertical. There is no friction anywhere. (a) How is \(\theta\) related to the acceleration of the system? (b) If \(m_{1}=250 \mathrm{~kg} \text { and } m_{2}=1250 \mathrm{~kg},\), what is \(\theta\)? (c) If you can vary \(m_{1} \text { and } m_{2}\), what is the largest angle ? you could achieve? Explain how you need to adjust \(m_{1} \text { and } m_{2}\) to do this. Equation Transcription: and and Text Transcription: \theta \theta m_1=25 kg and m_2=1250 kg \theta m_1 and m_2 m_1 and m_2

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?

Banked Curve II. 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 \(\beta=25^{\circ}\) 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? Equation Transcription: ° Text Transcription: \beta=25^{\circ}

Blocks A, B, and C are placed as in Fig. P5.103 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 two 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? Equation Transcription: 36.9° Text Transcription: 36.9^°

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?

The Monkey and Bananas Problem. A \(20-\mathrm{kg}\) monkey has a firm hold on a light rope that passes over a frictionless pulley and is attached to a \(20-\mathrm{kg}\) bunch of bananas (Fig. P5.105). 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? Equation Transcription: Text Transcription:

You throw a rock downward into water with a speed of \(3mg/k\), where \(k\) is the coefficient in Eq. (5.7). Assume that the relationship between fluid resistance and speed is as given in Eq. (5.7), and calculate the speed of the rock as a function of time. Equation Transcription: Text Transcription: 3mg/k k

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 a0. (b) Find the acceleration when the speed is 3.00 m/s. (c) Find the speed when the acceleration equals 0.1a0. (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 vt.

Problem 108P A rock with mass m slides with initial velocity v0 on a horizontal surface. A retarding force FR that the surface exerts on the rock is proportional to the square root of the instantaneous velocity of the rock (FR = –kv1/2). (a) Find expressions for the velocity and position of the rock as a function of time. (b) In terms of m, k, and v0, at what time will the rock come to rest? (c) In terms of m, k, and v0, 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.

The \(4.00-\mathrm{kg}\) block in Fig. P5.110 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 the diagram 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 in part (c). Equation Transcription: Text Transcription: 4.00 kg 1.25 m 2.00 m 1.25 m

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)?

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 114P 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 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.)

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 m. 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. Equation Transcription: Text Transcription: 85 kg 35 m 25 s

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?

Problem 118P A physics major is working to pay her college tuition by performing in a traveling carnival. She rides a motorcycle inside a hollow, transparent plastic sphere. After gaining sufficient speed, she travels in a vertical circle with radius 13.0 m. She has mass 70.0 kg, and her motorcycle has mass 40.0 kg. (a) What minimum speed must she 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, her 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?

A small bead can slide without friction on a circular hoop that is in a vertical plane and has a radius of \(0.100 \mathrm{~m}\). The hoop rotates at a constant rate of \(4.00 \mathrm{rev} / \mathrm{s}\) about a vertical diameter (Fig. P5.119). (a) Find the angle \(\beta\) at which the bead is in vertical equilibrium. (Of course, 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 \(1.00 \mathrm{rev} / \mathrm{s}\)? Equation Transcription: Text Transcription: 0.100 m 4.00 rev/s \beta 1.00 rev/s

A small remote controlled car with mass \(1.60 \mathrm{~kg}\) moves at a constant speed of \(\mathrm{v}=120 \mathrm{~m} / \mathrm{s}\) in a vertical circle inside a hollow metal cylinder that has a radius of \(5.00 \mathrm{~m}\) (Fig. P5.120). What is the magnitude of the normal force exerted on the car by the walls of the cylinder at (a) point A (at the bottom of the vertical circle) and (b) point B (at the top of the vertical circle)? Equation Transcription: Text Transcription: 1.60 kg v=120 m/s 5.00 m

Angle for Minimum Force. A box with weight w is pulled at constant speed along a level floor by a force \(\vec{F}\) that is at an angle \(\theta\) above the horizontal. The coefficient of kinetic friction between the floor and box is \(\mu_{k}\). (a) In terms of \(\theta, \mu_{k}\), and w, calculate ????. (b) For w = 400 N and \(\mu_{\mathrm{k}}=0.25\), calculate ???? for \(\theta\) ranging from \(0^{\circ} \text { to } 90^{\circ}\) in increments of \(10^{\circ}\). Graph ???? versus \(\theta\). (c) From the general expression in part (a), calculate the value of \(\theta\) for which the value of ????, 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 ???? is a function of ?. ) For the special case of w = 400 N and \(\mu_{\mathrm{k}}=0.25\), evaluate this optimal \(\theta\) and compare your result to the graph you constructed in part (b). Equation Transcription: 0° to 90° 10° Text Transcription: \vec{F} \theta \mu_{k} \theta, \mu_{k} \mu_{\{k}}=0.25 \theta 0^° to 90^° 10^° \(\theta\) \theta \mu_{\{k}}=0.25 \theta

Moving Wedge. A wedge with mass M rests on a frictionless, horizontal tabletop. A block with mass m is placed on the wedge (Fig. P5.122a). 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? Equation Transcription: Text Transcription: Vec F

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. P5.122b). What must the magnitude of be if the block is to remain at a constant height above the tabletop? Equation Transcription: Text Transcription: Vec F Vec F

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 \(\left(f=D v^{2}\right)\) (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: \(\int \frac{d x}{a^{a}-x^{2}}=\frac{1}{a} \operatorname{arctanh}\left(\frac{z}{a}\right)\) where \tanh (x)=\frac{e^{*}-e^{-z}}{e^{-4} e^{-z}}=\frac{e^{2^{2}}-1}{e^{2-1}} defines the hyperbolic tangent.) Equation Transcription: ? Text Transcription: (f=D v^2) \int frac{d x}{a^{a}-x^{2}= frac{1}{a} arctanh left (frac{z}{a} right) \tanh (x)=\frac{e^{*}-e^{-z}}{e^{-4} e^{-z}}=\frac{e^{2^{2}}-1}{e^{2-1}}

Double Atwood’s Machine. In Fig. P5.125 masses \(m_{1} \text { and } m_{2}\) are connected by a light string A over a light, frictionless pulley B. The axle of pulley B is connected by a second light string C over a second light, frictionless pulley D to a mass \(m_{3}\). Pulley D is suspended from the ceiling by an attachment to its axle. The system is released from rest. In terms of \(m_{1}, m_{2}, m_{3}\) , and g, what are (a) the acceleration of block \(m_{3}\); (b) the acceleration of pulley B; (c) the acceleration of block \(m_{1}\) (d) the acceleration of block \(m_{2}\) ; (e) the tension in string A; (f) the tension in string C? (g) What do your expressions give for the special case of \(m_{1}=m_{2} \text { and } m_{3}=m_{1}+m_{2}\)? Is this sensible? Equation Transcription: and and Text Transcription: m_1and m_2 m_3 m_1, m_2, m_3 m_3 m_1 m_2 m_1=m_2 and m_3=m_1+m_2

The masses of blocks A and B in Fig. P5.126 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 \(\vec{F}\) is applied to the pulley. Find the accelerations of \(\vec{a}_{\mathrm{A}}\) block A and \(\vec{a}_{\mathrm{B}}\) of block B when ???? is (a) 124 N; (b) 294 N; (c) 424 N. Equation Transcription: Text Transcription: Vec F Vec a_A Vec a_B

A ball is held at rest at position A in Fig. P5.127 by two light strings. The horizontal string is cut and the ball starts swinging as a pendulum. Point B is the farthest to the right the ball goes as it swings back and forth. What is the ratio of the tension in the supporting string at position B to its value at A before the horizontal string was cut? Equation Transcription: Text Transcription: \beta \beta

Problem 82P 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 ?s = 0.90 and ?k = 0.70.) (b) What must your initial speed be if you are to stop just before reaching the hole?

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 v0. (b) For the case where v0<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 v0 > vt (d) Discuss what your result says about vy(t) when v0 = vt.