Solved: The equilibrium length of a certain spring with a

Predict/Explain You push two identical bricks across a tabletop with constant speed, v, as shown in Figure 616. In case 1, you place the bricks end to end; in case 2, you stack the bricks I. The normal force in case 2 is larger, and hence the bricks press down more rmly against the tabletop. II. The normal force is the same in the two cases, and friction is independent of surface area. III. Case 1 has more surface area in contact with the tabletop, and this leads to more friction.

Predict/ExplainTwo drivers traveling side-by-side at the same speed suddenly see a deer in the road ahead of them and begin braking. Driver 1 stops by locking up his brakes and screeching to a halt; driver 2 stops by applying her brakes just to the verge of locking, so that the wheels continue to turn until her car comes to a complete stop. (a) All other factors being equal, is the stopping distance of driver 1 greater than, less than, or equal to the stopping distance of driver 2? (b) Choose the best explanation from among the following: I. Locking up the brakes gives the greatest possible braking force. II. The same tires on the same road result in the same force of friction. III. Locked-up brakes lead to sliding (kinetic) friction, which is less than rolling (static) friction.

Abaseball player slides into third base with an initial speed of 4.0 m/s. If the coefcient of kinetic friction between the player and the ground is 0.46, how far does the player slide before coming to rest?

Achild goes down a playground slide with an acceleration of . Find the coefcient of kinetic friction between the child and the slide if the slide is inclined at an angle of 33.0 below the horizontal.

Hopping into your Porsche, you oor it and accelerate at without spinning the tires. Determine the minimum coefcient of static friction between the tires and the road needed to make this possible.

When you push a 1.80-kg book resting on a tabletop, it takes 2.25 N to start the book sliding. Once it is sliding, however, it takes only 1.50 N to keep the book moving with constant speed. What are the coefcients of static and kinetic friction between the book and the tabletop?

In Problem 6, what is the frictional force exerted on the book when you push on it with a force of 0.75 N?

The three identical boxes shown in Figure 617 remain at rest on a rough, horizontal surface, even though they are acted on by two different forces, and . All of the forces labeled F ! 1F have the same magnitude; all of the forces labeled are identical to one another. Rank the boxes in order of increasing magnitude of the force static friction between them and the surface. Indicate ties where appropriate.

Atie of uniform width is laid out on a table, with a fraction of its length hanging over the edge. Initially, the tie is at rest. (a) If the fraction hanging from the table is increased, the tie eventually slides to the ground. Explain. (b)What is the coefcient of static friction between the tie and the table if the tie begins to slide when one-fourth of its length hangs over the edge?

To move a large crate across a rough oor, you push on it with a force Fat an angle of 21below the horizontal, as shown inFigure 618. Find the force necessary to start the crate moving, given that the mass of the crate is 32 kg and the coefcient of static friction between the crate and the oor is 0.57.

In Problem 10, nd the acceleration of the crate if the applied force is 330 N and the coefcient of kinetic friction is 0.45.

A 48-kg crate is placed on an inclined ramp. When the angle the ramp makes with the horizontal is increased to 26, the crate begins to slide downward. (a) What is the coefcient of static friction between the crate and the ramp? (b) At what angle does the crate begin to slide if its mass is doubled?

A97-kg sprinter wishes to accelerate from rest to a speed of 13 m/s in a distance of 22 m. (a) What coefcient of static friction is required between the sprinters shoes and the track? (b) Explain the strategy used to nd the answer to part (a).

Coffee To Go Aperson places a cup of coffee on the roof of her car while she dashes back into the house for a forgotten item. When she returns to the car, she hops in and takes off with the coffee cup still on the roof. (a) If the coefcient of static friction between the coffee cup and the roof of the car is 0.24, what is the maximum acceleration the car can have without causing the cup to slide? Ignore the effects of air resistance. (b) What is the smallest amount of time in which the person can accelerate the car from rest to 15 m/s and still keep the coffee cup on the roof?

Force Times Distance IAt the local hockey rink, a puck with a mass of 0.12 kg is given an initial speed of v = 5.3 m/s. (a) If the coefcient of kinetic friction between the ice and the puck is 0.11, what distance d does the puck slide before coming to rest? (b)If the mass of the puck is doubled, does the frictional force Fexerted on the puck increase, decrease, or stay the same? Explain. (c) Does the stopping distance of the puck increase, decrease, or stay the same when its mass is doubled? Explain. (d)For the situation considered in part (a), show that Fd = 1 2mv2.(The signicance of this result will be discussed in Chapter 7, where we will see that is the kinetic energy of an object.)

Force Times TimeAt the local hockey rink, a puck with a mass of 0.12 kg is given an initial speed of = 6.7 m/s. (a) If the coefcient of kinetic friction between the ice and the puck is 0.13, how much time t does it take for the puck to come to rest? (b)If the mass of the puck is doubled, does the frictional force F exerted on the puck increase, decrease, or stay the same? Explain.(c)Does the stopping time of the puck increase, decrease, or stay the same when its mass is doubled? Explain. (d) For the situation considered in part (a), show that . (The signicance of this result will be discussed in Chapter 9, where we will see that mv is the momentum of an object.)

Force Times Distance II A block of mass slides with an initial speed on a smooth, horizontal surface. The block now encounters a rough patch with a coefcient of kinetic friction given by . The rough patch extends for a distance , after which the surface is again frictionless. (a) What is the acceleration of the block when it is in the rough patch? (b) What is the nal speed, , of the block when it exits the rough patch? (c) Show that (The signicance of this Fd =(mkmg)d = 1 2mvf2 - 1 2mvi2. vf d = 0.125 m mk = 0.260 vi = 4.33 m/s m = 1.95 kg Ft = mv0 v0 1 2mv2 29. IP Your friends 13.6-g graduation tassel hangs on a string from his rearview mirror. (a) When he accelerates from a stoplight, the tassel deects backward toward the rear of the car. Explain. (b)If the tassel hangs at an angle of 6.44relative to the vertical, what is the acceleration of the car? 30. In Problem 29, (a) nd the tension in the string holding the tassel. (b) At what angle to the vertical will the tension in the string be twice the weight of the tassel? FIGURE 620 Problem 28 result will be discussed in Chapter 7, where we will see that is the kinetic energy of an object.)

The coefcient of kinetic friction between the tires of your car and the roadway is (a) If your initial speed is v and you lock your tires during braking, how far do you skid? Give your answer in terms of v, , and m, the mass of your car. (b) If you double your speed, what happens to the stopping distance? (c) What is the stopping distance for a truck with twice the mass of your car, assuming the same initial speed and coefcient of kinetic friction?

Acertain spring has a force constant k. (a) If this spring is cut in half, does the resulting half spring have a force constant that is greater than, less than, or equal to k? (b) If two of the original full-length springs are connected end to end, does the resulting double spring have a force constant that is greater than, less than, or equal to k?

Pulling up on a rope, you lift a 4.35-kg bucket of water from a well with an acceleration of . What is the tension in the rope?

When a 9.09-kg mass is placed on top of a vertical spring, the spring compresses 4.18 cm. Find the force constant of the spring.

A110-kg box is loaded into the trunk of a car. If the height of the cars bumper decreases by 13 cm, what is the force constant of its rear suspension?

A 50.0-kg person takes a nap in a backyard hammock. Both ropes supporting the hammock are at an angle of 15.0 above the horizontal. Find the tension in the ropes.

Abackpack full of books weighing 52.0 N rests on a table in a physics laboratory classroom. A spring with a force constant of 150 N/m is attached to the backpack and pulled horizontally, as indicated in Figure 619. (a) If the spring is pulled until it stretches 2.00 cm and the pack remains at rest, what is the force of friction exerted on the backpack by the table? (b)Does your answer to part (a) change if the mass of the backpack is doubled? Explain.

If the 52.0-N backpack in Problem 24 begins to slide when the spring stretches by 2.50 cm, what is the coefcient of static friction between the backpack and the table?

The equilibrium length of a certain spring with a force constant of is 0.18 m. (a)What is the magnitude of the force that is required to hold this spring at twice its equilibrium length? (b) Is the magnitude of the force required to keep the spring compressed to half its equilibrium length greater than, less than, or equal to the force found in part (a)? Explain.

Illinois Jones is being pulled from a snake pit with a rope that breaks if the tension in it exceeds 755 N. (a) If Illinois Jones has a mass of 70.0 kg and the snake pit is 3.40 m deep, what is the minimum time that is required to pull our intrepid explorer from the pit? (b) Explain why the rope breaks if Jones is pulled from the pit in less time than that calculated in part (a).

Aspring with a force constant of 120 N/m is used to push a 0.27-kg block of wood against a wall, as shown in Figure 620.(a) Find the minimum compression of the spring needed to keep the block from falling, given that the coefcient of static friction between the block and the wall is 0.46. (b)Does your answer to part (a) change if the mass of the block of wood is doubled? Explain.

Your friends 13.6-g graduation tassel hangs on a string from his rearview mirror. (a) When he accelerates from a stoplight, the tassel deects backward toward the rear of the car. Explain. (b)If the tassel hangs at an angle of 6.44relative to the vertical, what is the acceleration of the car?

In Problem 29, (a) nd the tension in the string holding the tassel. (b) At what angle to the vertical will the tension in the string be twice the weight of the tassel?

Apicture hangs on the wall suspended by two strings, as shown in Figure 621. The tension in string 1 is 1.7 N. (a) Is the tension in string 2 greater than, less than, or equal to 1.7 N? Explain. (b) Verify your answer to part (a) by calculating the tension in string 2. (c) What is the weight of the picture?

Mechanical AdvantageThe pulley system shown in Figure 622 is used to lift a 52-kg crate. Note that one chain connects the upper pulley to the ceiling and a second chain connects the lower pulley to the crate. Assuming the masses of the chains, pulleys, and ropes are negligible, determine (a) the force required to lift the crate with constant speed, (b)the tension in the upper chain, and (c) the tension in the lower chain.

In Problem 32, determine (a) the force , (b) the tension in the upper chain, and (c) the tension in the lower chain, given that the crate is rising with an acceleration of

Pulling the string on a bow back with a force of 28.7 lb, an archer prepares to shoot an arrow. If the archer pulls in the center of the string, and the angle between the two halves is 138, what is the tension in the string?

In Figure 623 we see two blocks connected by a string and tied to a wall. The mass of the lower block is 1.0 kg; the mass of the upper block is 2.0 kg. Given that the angle of the incline is 31, nd the tensions in (a)the string connecting the two blocks and (b) the string that is tied to the wall.

TractionAfter a skiing accident, your leg is in a cast and supported in a traction device, as shown in Figure 624. Find the magnitude of the force exerted by the leg on the small pulley. (By Newtons third law, the small pulley exerts an equal and opposite force on the leg.) Let the mass m be 2.50 kg.

Two blocks are connected by a string, as shown in Figure 625. The smooth inclined surface makes an angle of 42 with the horizontal, and the block on the incline has a mass of 6.7 kg. Find the mass of the hanging block that will cause the system to be in equilibrium. (The pulley is assumed to be ideal.)

Predict/Explain (a) Referring to the hanging planter in Example 65, which of the three graphs (A, B, or C) in Figure 626 shows an accurate plot of the tensions and as a function of the angle ? (b) Choose the best explanation from among the following: I. The two tensions must be equal at some angle between and . II. is greater than at all angles, and is equal to mg at . III. is less than at all angles, and is equal to 0 at u = 90.

A0.15-kg ball is placed in a shallow wedge with an opening angle of 120, as shown in Figure 627. For each contact point between the wedge and the ball, determine the force exerted on the ball. Assume the system is frictionless.

You want to nail a 1.6-kg board onto the wall of a barn. To position the board before nailing, you push it against the wall with a horizontal force to keep it from sliding to the ground (Figure 628). (a) If the coefcient of static friction between the board and the wall is 0.79, what is the least force you can apply and still hold the board in place? (b) What happens to the force of static friction if you push against the wall with a force greater than that found in part (a)?

The Russell Traction System To immobilize a fractured femur (the thigh bone), doctors often utilize the Russell traction system illustrated in Figure 629. Notice that one force is applied directly to the knee, , while two other forces, and , are applied to the foot. The latter two forces combine to give a force that is transmitted through the lower leg to the knee. The result is that the knee experiences the total force . The goal of this traction system is to have directly in line with the fractured femur, at an angle of 20.0 above the horizontal. Find (a) the angle required to produce this alignment of and (b) the magnitude of the force, that is applied to the femur in this case. (Assume the pulleys are ideal.)

In Example 66 (Connected Blocks), suppose and are both increased by a factor of 2. (a) Does the acceleration of the blocks increase, decrease, or stay the same? (b)Does the tension in the string increase, decrease, or stay the same?

Predict/Explain Suppose and in Example 67 (Atwoods Machine) are both increased by 1 kg. Does the acceleration of the blocks increase, decrease, or stay the same? (b) Choose the best explanation from among the following: I. The net force acting on the blocks is the same, but the total mass that must be accelerated is greater. II. The difference in the masses is the same, and this is what determines the net force on the system. III. The force exerted on each block is greater, leading to an increased acceleration.

Find the acceleration of the masses shown in Figure 630, given that , and . Assume the table is frictionless and the masses move freely.

Two blocks are connected by a string, as shown in Figure 631. The smooth inclined surface makes an angle of 35 with the horizontal, and the block on the incline has a mass of 5.7 kg. The mass of the hanging block is . Find (a) the direction and (b) the magnitude of the hanging blocks acceleration.

Referring to Problem 45, nd (a) the direction and (b) the magnitude of the hanging blocks acceleration if its mass is .

Referring to Figure 630, nd the tension in the string connecting (a) and and (b) and . Assume the table is frictionless and the masses move freely

A 3.50-kg block on a smooth tabletop is attached by a string to a hanging block of mass 2.80 kg, as shown in Figure 632. The blocks are released from rest and allowed to move freely. (a)Is the tension in the string greater than, less than, or equal to the weight of the hanging mass? Find (b)the acceleration of the blocks and (c) the tension in the string.

A7.7-N force pulls horizontally on a 1.6-kg block that slides on a smooth horizontal surface. This block is connected by a horizontal string to a second block of mass on the same surface. (a) What is the acceleration of the blocks? (b) What is the tension in the string? (c) If the mass of block 1 is increased, does the tension in the string increase, decrease, or stay the same?

Buckets and a PulleyTwo buckets of sand hang from opposite ends of a rope that passes over an ideal pulley. One bucket is full and weighs 120 N; the other bucket is only partly lled and weighs 63 N. (a) Initially, you hold onto the lighter bucket to keep it from moving. What is the tension in the rope? (b)You release the lighter bucket and the heavier one descends. What is the tension in the rope now? (c) Eventually the heavier bucket lands and the two buckets come to rest. What is the tension in the rope now?

Suppose you stand on a bathroom scale and get a reading of 700 N. In principle, would the scale read more, less, or the same if the Earth did not rotate?

Acar drives with constant speed on an elliptical track, as shown in Figure 633. Rank the points A, B, and C in order of increasing likelihood that the car might skid. Indicate ties where appropriate.

A car is driven with constant speed around a circular track. Answer each of the following questions with Yes or No. (a) Is the cars velocity constant? (b) Is its speed constant?(c)Is the magnitude of its acceleration constant? (d)Is the direction of its acceleration constant?

Apuck attached to a string undergoes circular motion on an air table. If the string breaks at the point indicated in Figure 634, is the subsequent motion of the puck best described by path A, B, C, or D?

When you take your 1300-kg car out for a spin, you go around a corner of radius 59 m with a speed of 16 m/s. The coefcient of static friction between the car and the road is 0.88. Assuming your car doesnt skid, what is the force exerted on it by static friction?

Find the linear speed of the bottom of a test tube in a centrifuge if the centripetal acceleration there is 52,000 times the acceleration of gravity. The distance from the axis of rotation to the bottom of the test tube is 7.5 cm.

A Human CentrifugeTo test the effects of high acceleration on the human body, the National Aeronautics and Space Administration (NASA) has constructed a large centrifuge at the Manned Spacecraft Center in Houston. In this device, astronauts are placed in a capsule that moves in a circular path with a radius of 15 m. If the astronauts in this centrifuge experience a centripetal acceleration 9.0 times that of gravity, what is the linear speed of the capsule?

Acar goes around a curve on a road that is banked at an angle of 33.5. Even though the road is slick, the car will stay on the road without any friction between its tires and the road when its speed is 22.7 m/s. What is the radius of the curve?

Jill of the Jungle swings on a vine 6.9 m long. What is the tension in the vine if Jill, whose mass is 63 kg, is moving at 2.4 m/s when the vine is vertical?

In Problem 59, (a) how does the tension in the vine change if Jills speed is doubled? Explain. (b)How does the tension change if her mass is doubled instead? Explain.

(a) As you ride on a Ferris wheel, your apparent weight is different at the top than at the bottom. Explain. (b) Calculate your apparent weight at the top and bottom of a Ferris wheel, given that the radius of the wheel is 7.2 m, it completes one revolution every 28 s, and your mass is 55 kg.

Driving in your car with a constant speed of 12 m/s, you encounter a bump in the road that has a circular cross section, as indicated in Figure 635. If the radius of curvature of the bump is 35 m, nd the apparent weight of a 67-kg person in your car as you pass over the top of the bump.

eferring to Problem 62, at what speed must you go over the bump if people in your car are to feel weightless?

You swing a 4.6-kg bucket of water in a vertical circle of radius 1.3 m. (a) What speed must the bucket have if it is to complete the circle without spilling any water? (b) How does your answer depend on the mass of the bucket?

If you weigh yourself on a bathroom scale at the equator, is the reading you get greater than, less than, or equal to the reading you get if you weigh yourself at the North Pole?

An object moves on a at surface with an acceleration of constant magnitude. If the acceleration is always perpendicular to the objects direction of motion, (a) is the shape of the objects path circular, linear, or parabolic? (b)During its motion, does the objects velocity change in direction but not magnitude, change in magnitude but not direction, or change in both magnitude and direction? (c)Does its speed increase, decrease, or stay the same?

Maneuvering a JetHumans lose consciousness if exposed to prolonged accelerations of more than about 7g. This is of concern to jet ghter pilots, who may experience centripetal accelerations of this magnitude when making high-speed turns. Suppose we would like to decrease the centripetal acceleration of a jet. Rank the following changes in ight path in order of how effective they are in decreasing the centripetal acceleration, starting with the least effective: A, decrease the turning radius by a factor of two; B, decrease the speed by a factor of three; or C, increase the turning radius by a factor of four.

Gravitropism As plants grow, they tend to align their stems and roots along the direction of the gravitational eld. This tendency, which is related to differential concentrations of plant hormones known as auxins, is referred to as gravitropism. As an illustration of gravitropism, experiments show that seedlings placed in pots on the rim of a rotating turntable do not grow in the vertical direction. Do you expect their stems to tilt inwardtoward the axis of rotationor outwardaway from the axis of rotation?

Askateboard accident leaves your leg in a cast and supported by a traction device, as in Figure 624. Find the mass m that must be attached to the rope if the net force exerted by the small pulley on the foot is to have a magnitude of 37 N.

Find the centripetal acceleration at the top of a test tube in a centrifuge, given that the top is 4.2 cm from the axis of rotation and that its linear speed is 77 m/s.

Find the coefcient of kinetic friction between a 3.85-kg block and the horizontal surface on which it rests if an 850-N/m spring must be stretched by 6.20 cm to pull it with constant speed. Assume that the spring pulls in the horizontal direction.

A child goes down a playground slide that is inclined at an angle of 26.5 below the horizontal. Find the acceleration of the child given that the coefcient of kinetic friction between the child and the slide is 0.315.

When a block is placed on top of a vertical spring, the spring compresses 3.15 cm. Find the mass of the block, given that the force constant of the spring is 1750 N/m.

The da Vinci CodeLeonardo da Vinci (14521519) is credited with being the rst to perform quantitative experiments on friction, though his results werent known until centuries later, due in part to the secret code (mirror writing) he used in his notebooks. Leonardo would place a block of wood on an inclined plane and measure the angle at which the block begins to slide. He reports that the coefcient of static friction was 0.25 in his experiments. At what angle did Leonardos blocks begin to slide?

A force of 9.4 N pulls horizontally on a 1.1-kg block that slides on a rough, horizontal surface. This block is connected by a horizontal string to a second block of mass on the same surface. The coefcient of kinetic friction is for both blocks. (a) What is the acceleration of the blocks? (b) What is the tension in the string?

You swing a 3.25-kg bucket of water in a vertical circle of radius 0.950 m. At the top of the circle the speed of the bucket is 3.23 m/s; at the bottom of the circle its speed is 6.91 m/s. Find the tension in the rope tied to the bucket at (a) the top and (b) the bottom of the circle.

A 14-g coin slides upward on a surface that is inclined at an angle of 18above the horizontal. The coefcient of kinetic friction between the coin and the surface is 0.23; the coefcient of static friction is 0.35. Find the magnitude and direction of the force of friction (a) when the coin is sliding and (b) after it comes to rest.

In Problem 77, the angle of the incline is increased to 25. Find the magnitude and direction of the force of friction when the coin is (a) sliding upward initially and (b) sliding back downward later.

Aphysics textbook weighing 22 N rests on a table. The coefcient of static friction between the book and the table is ; the coefcient of kinetic friction is . You push horizontally on the book with a force that gradually increases from 0 to 15 N, and then slowly decreases to 5.0 N, as indicated in the following table. For each value of the applied force given in the table, give the magnitude of the force of friction and state whether the book is accelerating, decelerating, at rest, or moving with constant speed.

Aball of mass mis placed in a wedge, as shown in Figure 636, in which the two walls meet at a right angle. Assuming the walls of the wedge are frictionless, determine the magnitude of (a) contact force 1 and (b) contact force 2.

The blocks shown in Figure 637 are at rest. (a) Find the frictional force exerted on block Agiven that the mass of block A is 8.82 kg, the mass of block B is 2.33 kg, and the coefcient of static friction between block Aand the surface on which it rests is 0.320. (b)If the mass of block Ais doubled, does the frictional force exerted on it increase, decrease, or stay the same? Explain.

In part (a) of Problem 81, what is the maximum mass block B can have and the system still be in equilibrium?

Apicture hangs on the wall suspended by two strings, as shown in Figure 621. The tension in string 2 is 1.7 N. (a) Is the tension in string 1 greater than, less than, or equal to 1.7 N? Explain. (b) Verify your answer to part (a) by calculating the tension in string 1. (c) What is the mass of the picture?

Referring to Problem 61, suppose the Ferris wheel rotates fast enough to make you feel weightless at the top. (a) How many seconds does it take to complete one revolution in this case? (b) How does your answer to part (a) depend on your mass? Explain. (c) What are the direction and magnitude of your acceleration when you are at the bottom of the wheel? Assume that its rotational speed has remained constant.

A Conical Pendulum A0.075-kg toy airplane is tied to the ceiling with a string. When the airplanes motor is started, it moves with a constant speed of 1.21 m/s in a horizontal circle of radius 0.44 m, as illustrated in Figure 638. Find (a) the angle the string makes with the vertical and (b) the tension in the string.

Atugboat tows a barge at constant speed with a 3500-kg cable, as shown in Figure 639. If the angle the cable makes with the horizontal where it attaches to the barge and the tugboat is 22, nd the force the cable exerts on the barge in the forward direction.

Two blocks, stacked one on top of the other, can move without friction on the horizontal surface shown in Figure 640. The surface between the two blocks is rough, however, with a coefcient of static friction equal to 0.47. (a)If a horizontal force F is applied to the 5.0-kg bottom block, what is the maximum value F can have before the 2.0-kg top block begins to slip? (b) If the mass of the top block is increased, does the maximum value of F increase, decrease, or stay the same? Explain.

Find the coefcient of kinetic friction between a 4.7-kg block and the horizontal surface on which it rests if an 89-N/m spring must be stretched by 2.2 cm to pull the block with constant speed. Assume the spring pulls in a direction 13 above the horizontal.

In a daring rescue by helicopter, two men with a combined mass of 172 kg are lifted to safety. (a)If the helicopter lifts the men straight up with constant acceleration, is the tension in the rescue cable greater than, less than, or equal to the combined weight of the men? Explain. (b) Determine the tension in the cable if the men are lifted with a constant acceleration of

At the airport, you pull a 18-kg suitcase across the oor with a strap that is at an angle of 45 above the horizontal. Find (a) the normal force and (b) the tension in the strap, given that the suitcase moves with constant speed and that the coefcient of kinetic friction between the suitcase and the oor is 0.38.

Alight spring with a force constant of 13 N/m is connected to a wall and to a 1.2-kg toy bulldozer, as shown in Figure 641. When the electric motor in the bulldozer is turned on, it stretches the spring for a distance of 0.45 m before its tread begins to slip on the oor. (a)Which coefcient of friction (static or kinetic) can be determined from this information? Explain. (b) What is the numerical value of this coefcient of friction?

A0.16-g spider hangs from the middle of the rst thread of its future web. The thread makes an angle of 7.2 with the horizontal on both sides of the spider. (a)What is the tension in the thread? (b) If the angle made by the thread had been less than 7.2, would its tension have been greater than, less than, or the same as in part (a)? Explain.

Find the acceleration the cart in Figure 642 must have in order for the cereal box at the front of the cart not to fall. Assume that the coefcient of static friction between the cart and the box is 0.38.

Playing a Violin The tension in a violin string is 2.7 N. When pushed down against the neck of the violin, the string makes an angle of 4.1 with the horizontal. (a) With what force must you push down on the string to bring it into contact with the neck?(b)If the angle were less than 4.1, would the required force be greater than, less than, or the same as in part (a)? Explain.

Apair of fuzzy dice hangs from a string attached to your rearview mirror. As you turn a corner with a radius of 98 m and a constant speed of 27 mi/h, what angle will the dice make with the vertical? Why is it unnecessary to give the mass of the dice?

Find the tension in each of the two ropes supporting a hammock if one is at an angle of 18 above the horizontal and the other is at an angle of 35 above the horizontal. The person sleeping in the hammock (unconcerned about tensions and ropes) has a mass of 68 kg.

As your plane circles an airport, it moves in a horizontal circle of radius 2300 m with a speed of 390 km/h. If the lift of the airplanes wings is perpendicular to the wings, at what angle should the plane be banked so that it doesnt tend to slip sideways?

Ablock with a mass of 3.1 kg is placed at rest on a surface inclined at an angle of 45 above the horizontal. The coefcient of static friction between the block and the surface is 0.50, and a force of magnitude F pushes upward on the block, parallel to the inclined surface. (a)The block will remain at rest only if Fis greater than a minimum value, , and less than a maximum value, . Explain the reasons for this behavior. (b) Calculate . (c) Calculate .

A mountain climber of mass m hangs onto a rope to keep from sliding down a smooth, ice-covered slope (Figure 643). Find a formula for the tension in the rope when the slope is inclined at an angle above the horizontal. Check your results in the limits and

A child sits on a rotating merry-go-round, 2.3 m from its center. If the speed of the child is 2.2 m/s, what is the minimum coefcient of static friction between the child and the merry-go-round that will prevent the child from slipping?

A2.0-kg box rests on a plank that is inclined at an angle of 65 above the horizontal. The upper end of the box is attached to a spring with a force constant of 360 N/m, as shown in Figure 644. If the coefcient of static friction between the box and the plank is 0.22, what is the maximum amount the spring can be stretched and the box remain at rest?

Awood block of mass mrests on a larger wood block of mass M that rests on a wooden table. The coefcients of static and kinetic friction between all surfaces are and , respectively. What is the minimum horizontal force, F, applied to the lower block that will cause it to slide out from under the upper block?

Find the tension in each of the two strings shown in Figure 630for general values of the masses. Your answer should be in terms of , and g.

The coefcient of static friction between a rope and the table on which it rests is . Find the fraction of the rope that can hang over the edge of the table before it begins to slip.

A hockey puck of mass m is attached to a string that passes through a hole in the center of a table, as shown in Figure 645. The hockey puck moves in a circle of radius r. Tied to the other end of the string, and hanging vertically beneath the table, is a mass M. Assuming the tabletop is perfectly smooth, what speed must the hockey puck have if the mass M is to remain at rest?

The Force Needed to Move a Crate To move a crate of mass m across a rough oor, you push down on it at an angle as shown in Figure 618 for the special case of . (a) Find the force necessary to start the crate moving as a function of , given that the coefcient of static friction between the crate and the oor is . (b)Show that it is impossible to move the crate, no matter how great the force, if the coefcient of static friction is greater than or equal to .

A popular ride at amusement parks is illustrated in Figure 646. In this ride, people sit in a swing that is suspended from a rotating arm. Riders are at a distance of 12 m from the axis of rotation and move with a speed of 25 mi/h. (a)Find the centripetal acceleration of the riders. (b) Find the angle the supporting wires make with the vertical. (c) If you observe a ride like that in Figure 646, or as shown in the photo on page 170, you will notice that all the swings are at the same angle to the vertical, regardless of the weight of the rider. Explain.

A Conveyor BeltAbox is placed on a conveyor belt that moves with a constant speed of 1.25 m/s. The coefcient of kinetic friction between the box and the belt is 0.780. (a) How much time does it take for the box to stop sliding relative to the belt? (b) How far does the box move in this time?

You push a box along the oor against a constant force of friction. When you push with a horizontal force of 75 N, the acceleration of the box is ; when you increase the force to 81 N, the acceleration is . Find (a)the mass of the box and (b) the coefcient of kinetic friction between the box and the oor.

As part of a circus act, a person drives a motorcycle with constant speed varound the inside of a vertical track of radius r, as indicated in Figure 647. If the combined mass of the motorcycle and rider is m, nd the normal force exerted on the motorcycle by the track at the points (a) A, (b) B, and (c) C.

On the straight-line segment I in Figure 648(b)we see that increasing the applied mass from 26 g to 44 g results in a reduction of the end-to-end distance from 21 mm to 14 mm. What is the force constant in N/m on segment I? A. 2.6 N/m B. 3.8 N/m C. 9.8 N/m D. 25 N/m

Is the force constant on segment II greater than, less than, or equal to the force constant on segment I?

Which of the following is the best estimate for the force constant on segment II? A. 0.83 N/m B. 1.3 N/m C. 2.5 N/m D. 25 N/m

Rank the straight segments I, II, and III in order of increasing stiffness of the nasal strip.

Referring to Example 63 Suppose the coefcients of static and kinetic friction between the crate and the truck bed are 0.415 and 0.382, respectively. (a) Does the crate begin to slide at a tilt angle that is greater than, less than, or equal to 23.2? (b) Verify your answer to part (a) by determining the angle at which the crate begins to slide. (c) Find the length of time it takes for the crate to slide a distance of 2.75 m when the tilt angle has the value found in part (b).

Referring to Example 63 The crate begins to slide when the tilt angle is 17.5. When the crate reaches the bottom of the atbed, after sliding a distance of 2.75 m, its speed is 3.11 m/s. Find (a) the coefcient of static friction and (b) the coefcient of kinetic friction between the crate and the atbed.

Referring to Example 66 Suppose that the mass on the frictionless tabletop has the value m1  2.45 kg. (a) Find the value of that gives an acceleration of 2.85 m/s2. (b)What is the corresponding tension, T, in the string? (c) Calculate the ratio and show that it is less than 1, as expected.

Referring to Example 68 (a) At what speed will the force of static friction exerted on the car by the road be equal to half the weight of the car? The mass of the car is , the radius of the corner is m, and the coefcient of static friction between the tires and the road is . (b)Suppose that the mass of the car is now doubled, and that it moves with a speed that again makes the force of static friction equal to half the cars weight. Is this new speed greater than, less than, or equal to the speed in part (a)?