(Answers to odd-numbered Conceptual Questions

Problem 3CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) When a traffic accident is investigated, it is common for the length of the skid marks to be measured. How could this information be used to estimate the initial speed of the vehicle that left the skid marks?

Problem 1CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) A clothesline always sags a little, even if nothing hangs from it. Explain.

Problem 2CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) In the Jurassic Park sequel, The Lost World, a man tries to keep a large vehicle from going over a cliff by connecting a cable from his Jeep to the vehicle. The man then puts the Jeep in gear and spins the rear wheels. Do you expect that spinning the tires will increase the force exerted by the Jeep on the vehicle? Why or why not?

Problem 118IP Referring to Example 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 m = 1200 kg, the radius of the corner is r = 45 m, and the coefficient of static friction between the tires and the road is µs = 0.82. (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 car’s weight. Is this new speed greater than, less than, or equal to the speed in part (a)?

Problem 2P Predict/Explain Two 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.

CE Predict/Explain You push two identical bricks across a tabletop with constant speed, v, as shown in Figure 6–16. In case 1, you place the bricks end to end; in case 2, you stack the bricks one on top of the other. (a) Is the force of kinetic friction in case 1 greater than, less than, or equal to the force of kinetic friction in case 2? (b) Choose the best explanation from among the following: I. The normal force in case 2 is larger, and hence the bricks press down more firmly 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.

Problem 3P A baseball player slides into third base with an initial speed of 4.0 m/s. If the coefficient of kinetic friction between the player and the ground is 0.46, how far does the player slide before coming to rest?

Problem 4CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) In a car with rear-wheel drive, the maximum acceleration is often less than the maximum deceleration. Why?

Problem 6P 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 coefficients of static and kinetic friction between the book and the tabletop?

Problem 4P A child goes down a playground slide with an acceleration of 1.26 m/s2. Find the coefficient of kinetic friction between the child and the slide if the slide is inclined at an angle of 33.0° below the horizontal.

Problem 5CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) A train typically requires a much greater distance to come to rest, for a given initial speed, than does a car. Why?

Problem 5P Hopping into your Porsche, you floor it and accelerate at 12 m/s2 without spinning the tires. Determine the minimum coefficient of static friction between the tires and the road needed to make this possible.

Problem 6CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) Give some everyday examples of situations in which friction is beneficial.

Problem 7CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) At the local farm, you buy a flat of strawberries and place them on the backseat of the car. On the way home, you begin to brake as you approach a stop sign. At first the strawberries stay put, but as you brake a bit harder, they begin to slide off the seat. Explain.

Problem 8CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) It is possible to spin a bucket of water in a vertical circle and have none of the water spill when the bucket is upside down. How would you explain this to members of your family?

Problem 9CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) Water sprays off a rapidly turning bicycle wheel. Why?

Problem 9P A tie 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 coefficient 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?

Problem 7P In Problem, what is the frictional force exerted on the book when you push on it with a force of 0.75 N? 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 coefficients of static and kinetic friction between the book and the tabletop?

CE The three identical boxes shown in Figure remain at rest on a rough, horizontal surface, even though they are acted on by two different forces, \(\mathrm {\vec{F}_1}\) and \(\mathrm {\vec{F}_2}\). All of the forces labeled \(\mathrm {\vec{F}_1}\) have the same magnitude; all of the forces labeled \(\mathrm {\vec{F}_2}\) 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. ________________ Equation Transcription: Text Transcription: F_1 F_2 F_1 F_1 F_2 F_1 F_2 F_1 F_2 F_2

Problem 10CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) Can an object be in equilibrium if it is moving? Explain.

Problem 11CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) In a dramatic circus act, a motorcyclist drives his bike around the inside of a vertical circle. How is this possible, considering that the motorcycle is upside down at the top of the circle?

To move a large crate across a rough floor, you push on it with a force F at an angle of \(21^\circ\) below the horizontal, as shown in Figure 6–18. Find the force necessary to start the crate moving, given that the mass of the crate is 32 kg and the coefficient of static friction between the crate and the floor is 0.57. ________________ Equation Transcription: Text Transcription: 21^o vec{F} 21^o

Problem 13CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) A popular carnival ride has passengers stand with their backs against the inside wall of a cylinder. As the cylinder begins to spin, the passengers feel as if they are being pushed against the wall. Explain.

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

Problem 12P 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 coefficient of static friction between the crate and the ramp? (b) At what angle does the crate begin to slide if its mass is doubled?

Problem 12CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) The gravitational attraction of the Earth is only slightly less at the altitude of an orbiting spacecraft than it is on the Earth’s surface. Why is it, then, that astronauts feel weightless?

Problem 15CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) Your car is stuck on an icy side street. Some students on their way to class see your predicament and help out by sitting on the trunk of your car to increase its traction. Why does this help?

Problem 16CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) The parking brake on a car causes the rear wheels to lock up. What would be the likely consequence of applying the parking brake in a car that is in rapid motion? (Note: Do not try this at home.)

IP A 97-kg sprinter wishes to accelerate from rest to a speed of \(13 \mathrm{~m} / \mathrm{s}\) in a distance of \(22 \mathrm{~m}\). (a) What coefficient of static friction is required between the sprinter's shoes and the track? (b) Explain the strategy used to find the answer to part (a).

Problem 14CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) Referring to Question, after the cylinder reaches operating speed, the floor is lowered away, leaving the passengers “stuck” to the wall. Explain. (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) A popular carnival ride has passengers stand with their backs against the inside wall of a cylinder. As the cylinder begins to spin, the passengers feel as if they are being pushed against the wall. Explain

Problem 14P Coffee To Go A person 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 coefficient 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?

IP Force Times Distance I At the local hockey rink, a puck with a mass of is given an initial speed of \(v=5.3\ \mathrm{ m/s}\). (a) If the coefficient of kinetic friction between the ice and the puck is , what distance does the puck slide before coming to rest? (b) If the mass of the puck is doubled, does the frictional force exerted 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 \(Fd=\frac{1}{2}mv^2\). (The significance of this result will be discussed in Chapter 7, where we will see that \(\frac{1}{2}mv^2 \) is the kinetic energy of an object.) ________________ Equation Transcription: Text Transcription: v=5.3 m/s Fd=12mv2 12mv2

Problem 16P Force Times Time At the local hockey rink, a puck with a mass of 0.12 kg is given an initial speed of v0 = 6.7 m/s. (a) If the coefficient of kinetic friction between the ice and the puck is 0.13, how much time f 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 Ft = mv0. (The significance of this result will be discussed in Chapter 9, where we will see that mv is the momentum of an object.)

Problem 17CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) The foot of your average gecko is covered with billions of tiny hair tips—called spatulae—that are made of keratin, the protein found in human hair. A subtle shift of the electron distribution in both the spatulae and the wall to which a gecko clings produces an adhesive force by means of the van der Waals interaction between molecules. Suppose a gecko uses its spatulae to cling to a vertical windowpane. If you were to describe this situation in terms of a coefficient of static friction, µs, what value would you assign to µs? Is this a sensible way to model the gecko’s feat? Explain.

Problem 19CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) The gas pedal and the brake pedal are capable of causing a car to accelerate. Can the steering wheel also produce an acceleration? Explain.

Problem 18CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) Discuss the physics involved in the spin cycle of a washing machine. In particular, how is circular motion related to the removal of water from the clothes?

Problem 18P The coefficient of kinetic friction between the tires of your car and the roadway is µ. (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, µ, 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 coefficient of kinetic friction?

Force Times Distance II block of mass \(m=1.95\ \mathrm{ kg}\) slides with an initial speed \(v_i=4.33\ \mathrm {m/s}\) on a smooth, horizontal surface. The block now encounters a rough patch with a coefficient of kinetic friction given by \(\mu_k=0.260\). The rough patch extends for a distance \(d=0.125\ \mathrm m\), 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 final speed, \(v_f\), of the block when it exits the rough patch? (c) Show that \(-Fd=-(\mu_{k}mg)d=\frac{1}{2}mv_f^2-\frac{1}{2}mv_i^2\). (The significance of this result will be discussed in Chapter 7, where we will see that \(\frac{1}{2}mv^2\) is the kinetic energy of an object.) ________________ Equation Transcription: Text Transcription: m=1.95 kg v_i=4.33 m/s mu_k=0.260 d=0.125 m v_f -Fd=-(mu_{k}mg)d=frac{1}{2}mv_f^2-frac{1}{2}mv_i^2 frac{1}{2}mv^2

Problem 19P A certain 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?

In the movie 2001: A Space Odyssey, a rotating space station provides “artificial gravity” for its inhabitants. How does this work?

Problem 21CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) When rounding a corner on a bicycle or a motorcycle, the driver leans inward, toward the center of the circle. Why?

Problem 20P Pulling up on a rope, you lift a 4.35-kg bucket of water from a well with an acceleration of 1.78 m/s2. What is the tension in the rope?

Problem 21P 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.

Problem 22CQ (Answers to odd-numbered Conceptual Questions can be found in the back of the book.) In Robin Hood: Prince of Thieves, starring Kevin Costner, Robin swings between trees on a vine that is on fire. At the lowest point of his swing, the vine bums through and Robin begins to fall. The next shot, from high up in the trees, shows Robin falling straight downward. Would you rate the physics of this scene “Good,” “Bad,” or “Ugly”? Explain. PROBLEMS AND CONCEPTUAL EXERCISES Note: Answers to odd-numbered Problems and Conceptual Exercises can be found in the back of the book. IP denotes an integrated problem, with both conceptual and numerical parts; BIO identifies problems of biological or medical interest; CE indicates a conceptual exercise, Predict/Explain problems ask for two responses: (a) your prediction of a physical outcome, and (b) the best explanation among three provided. On all problems, red bullets (,,) are used to indicate the level of difficulty. SECTION 6-1 FRICTIONAL FORCES

Problem 22P A 110-kg box is loaded into the trunk of a car. If the height of the car’s bumper decreases by 13 cm, what is the force constant of its rear suspension?

Problem 23P 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.

Problem 26P The equilibrium length of a certain spring with a force constant of k = 250 N/m 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.

IP A backpack 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 6–19. (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. ________________ Equation Transcription: Text Transcription: vec{F} vec{f}_s

Problem 27P 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 polled from the pit in less time than that calculated in part (a).

IP A spring 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 6–20. (a) Find the minimum compression of the spring needed to keep the block from falling, given that the coefficient 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. ________________ Equation Transcription: Text Transcription: vec{F} vec{f}_s vec{N} m vec{g}

If the 52.0-N backpack in Problem 24 begins to slide when the spring stretches by 2.50 cm, what is the coefficient of static friction between the backpack and the table? ________________ Equation Transcription: Text Transcription: vec{F} vec{f}_s

Problem 29P Your friend’s 13.6-g graduation tassel hangs from his rearview mirror. (a) When he acceleration stoplight, the tassel deflects backward toward the car. Explain. (b) If the tassel hangs at an angle of 6.4 the vertical, what is the acceleration of the car?

Problem 30P In Problem 29, (a) find the tension in the siring holding the tassel. (b) At what angle to the vertical will the tension in the string be twice the weight of the tassel?

A picture hangs on the wall suspended by two strings, as shown in Figure 6–21. 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? ________________ Equation Transcription: Text Transcription: 65^o 32^o

Mechanical Advantage The pulley system shown in Figure 6–22 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. Equation Transcription: Text Transcription: vec{F}

In Problem 32, determine (a) the force \(\mathrm {\vec F}\), (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 \(2.3\ \mathrm {m/s^2}\). ________________ Equation Transcription: Text Transcription: vec{F} 2.3 m/s^2 vec{F}

Problem 34P 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 6–23 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 \(31^\circ\), find the tensions in (a) the string connecting the two blocks and (b) the string that is tied to the wall. ________________ Equation Transcription: Text Transcription: 31^o 31^o

BIO Traction After a skiing accident, your leg is in a cast and supported in a traction device, as shown in Figure . Find the magnitude of the force \(\mathrm {\vec F}\) exerted by the leg on the small pulley. (By Newton's third law, the small pulley exerts an equal and opposite force on the leg.) Let the mass be . ________________ Equation Transcription: Text Transcription: vec{F} vec{T}_1 30.0^o 30.0^o vec{T}_1 vec{F} vec{T}_2 m=2.50 kg

Two blocks are connected by a string, as shown in Figure 6–25. The smooth inclined surface makes an angle of \(42^\circ\) 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.) ________________ Equation Transcription: Text Transcription: 42^o 42^o

A 0.15-kg ball is placed in a shallow wedge with an opening angle of \(120^\circ\), as shown in Figure 6–27. For each contact point between the wedge and the ball, determine the force exerted on the ball. Assume the system is frictionless. ________________ Equation Transcription: Text Transcription: 120^o m vec{g}

IP You want to nail a board onto the wall of a barn. To position the board before nailing,you push it against the wall with a horizontal force \(\mathrm {\vec F}\) to keep it from sliding to the ground (Figure 6-28). (a) If the coefficient of static friction between the board and the wall is , 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)? ________________ Equation Transcription: Text Transcription: vec{F} vec{f}_s vec{F} vec{N} vec{W}

CE Predict/Explain (a) Referring to the hanging planter in Example , which of the three graphs , or in Figure 6-26 shows an accurate plot of the tensions \(T_1\) and \(T_2\) as a function of the angle \(\theta\)? (b) Choose the best explanation from among the following: I. The two tensions must be equal at some angle between \(\theta=0\) and \(\theta=90^\circ\). II. \(T_2\) is greater than \(T_1\) at all angles, and is equal to at \(\(\theta=90^\circ\)\). III. \(T_2\) is less than \(T_1\) at all angles, and is equal to 0 at \(\(\theta=90^\circ\)\). ________________ Equation Transcription: Text Transcription: T_1 T_2 theta theta=0 theta=90^o T_2 T_1 theta=90^o T_2 T_1 theta=90^o T_1 T_2 90^o T_2 T_1 90^o T_2 T_1 90^o

The Russell Traction System To immobilize a fractured femur (the thigh bone), doctors often utilize the Russell traction system illustrated in Figure 6-29. Notice that one force is applied directly to the knee, \(\vec{F}_{1}\) , while two other forces, \(\vec{F}_{2}\) and \(\vec{F}_{3}\) , are applied to the foot. The latter two forces combine to give a force \(\vec{\mathbf{F}}_{2}+\vec{\mathbf{F}}_{3}\) that is transmitted through the lower leg to the knee. The result is that the knee experiences the total force \(\vec{\mathbf{F}}_{\text {total }}=\vec{\mathbf{F}}_{1}+\vec{\mathbf{F}}_{2}+\vec{\mathbf{F}}_{3}\) . The goal of this traction system is to have \(\vec{\mathbf{F}}_{\text {total }}\) directly in line with the fractured femur, at an angle of \(20.0^{\circ}\) above the horizontal. Find (a) the angle \(\theta\) required to produce this alignment of \(\vec{\mathbf{F}}_{\text {total }}\) and (b) the magnitude of the force, \(\vec{\mathbf{F}}_{\text {total }}\) that is applied to the femur in this case. (Assume the pulleys are ideal.)

In Example 6-6 (Connected Blocks), suppose \(m_1\) and \(m_2\) 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? ________________ Equation Transcription: Text Transcription: m_1 m_2

Problem 43P CE Predict/Explain Suppose m1 and m2 in Example (Atwood’s 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.

Two blocks are connected by a string, as shown in Figure 6–31. The smooth inclined surface makes an angle of \(35^\circ\) with the horizontal, and the block on the incline has a mass of 5.7 kg. The mass of the hanging block is \(m=3.2\ \mathrm{kg}\). Find (a) the direction and (b) the magnitude of the hanging block’s acceleration. ________________ Equation Transcription: Text Transcription: 35^o m=3.2 kg 35^o

Find the acceleration of the masses shown in Figure , given that \(m_1-1.0\ \mathrm{kg}\), \(m_2-2.0\ \mathrm{kg}\), and \(m_3-3.0\ \mathrm{kg}\). Assume the table is frictionless and the masses move freely. ________________ Equation Transcription: Text Transcription: m_1-1.0 kg m_2-2.0 kg m_3-3.0 kg m_1-1.0 kg m_2-2.0 kg m_3-3.0 kg

Referring to Problem 45, find (a) the direction and (b) the magnitude of the hanging block’s acceleration if its mass is \(m=4.2\ \mathrm{kg}\). Two blocks are connected by a string, as shown in Figure. The smooth inclined surface makes an angle of \(35^\circ\) with the horizontal, and the block on the incline has a mass of 5.7 kg. The mass of the hanging block is \(m=3.2\ \mathrm{kg}\), Find (a) the direction and (b) the magnitude of the hanging block’s acceleration. ________________ Equation Transcription: Text Transcription: m=4.2 kg 35^o m=3.2 kg 35^o

Referring to Figure , find the tension in the string connecting (a) \(m_1\) and \(m_2\) and (b) \(m_2\) and \(m_3\). Assume the table is frictionless and the masses move freely. ________________ Equation Transcription: Text Transcription: m_1 m_2 m_2 m_3 m_1=1.0 kg m_2=2.0 kg m_3=3.0 kg

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 6–32. 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.

Problem 49P A 7.7-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 m2 = 0.83 kg 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?

Problem 50P Buckets and a Pulley Two 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 filled 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?

Problem 51P 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?

Problem 53P A car is driven with constant speed around a circular track. Answer the of the following questions with “Yes” or “No.” (a) Is the car’s velocity constant? (b) Is its speed constant? (c) Is the magnitude of its acceleration constant? (d) Is the direction of its acceleration constant?

CE A puck attached to a string undergoes circular motion on an air table. If the string breaks at the point indicated in Figure 6–34, is the subsequent motion of the puck best described by path A, B, C, or D?

CE A car drives with constant speed on an elliptical track, as shown in Figure 6–33. Rank the points A, B, and C in order of increasing likelihood that the car might skid. Indicate ties where appropriate.

Problem 55P 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 coefficient of static friction between the car and the road is 0.88. Assuming your car doesn’t skid, what is the force exerted on it by static friction?

Problem 57P A Human Centrifuge To 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?

Problem 56P 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.

Problem 59P 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?

Problem 58P A car 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?

Problem 60P In Problem, (a) how does the tension in the vine change if Jill’s speed is doubled? Explain. (b) How does the tension change if her mass is doubled instead? Explain. 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?

IP (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 6–35. If the radius of curvature of the bump is 35 m, find the apparent weight of a 67-kg person in your car as you pass over the top of the bump.

Problem 65GP 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?

Problem 64P 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?

Problem 66GP An object moves on a flat surface with an acceleration of constant magnitude. If the acceleration is always perpendicular to the object’s direction of motion, (a) is the shape of the object’s path circular, linear, or parabolic? (b) During its motion, does the object’s 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?

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

Problem 67GP BIO Maneuvering a Jet Humans lose consciousness if exposed to prolonged accelerations of more than about 7g. This is of concern to jet fighter 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 flight 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.

Problem 70GP 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.

Problem 68GP Gravitropism As plants grow, they tend to align their stems and roots along the direction of the gravitational field. 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 inward—toward the axis of rotation—or outward—away from the axis of rotation?

A skateboard accident leaves your leg in a cast and supported by a traction device, as in Figure 6–24. 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. ________________ Equation Transcription: Text Transcription: vec{T}_1 vec{F} 30.0^o 30.0^o vec{T}_2

Problem 72GP 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 coefficient of kinetic friction between the child and the slide is 0.315.

Problem 71GP Find the coefficient 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.

Problem 73GP 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 Code Leonardo da Vinci (1452–1519) is credited with being the first to perform quantitative experiments on friction, though his results weren’t 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 coefficient of static friction was 0.25 in his experiments. At what angle did Leonardo’s blocks begin to slide?

Problem 75GP 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 m2 = 1.92 kg on the same surface. The coefficient of kinetic friction is µk = 0.24 for both blocks. (a) What is the acceleration of the blocks? (b) What is the tension in the string?

Problem 76GP 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.

Problem 77GP A 14-g coin slides upward on a surface that is inclined at an angle of 18° above the horizontal. The coefficient of kinetic friction between the coin and the surface is 0.23; the coefficient 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.

Problem 78GP In Problem, 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. A 14-g coin slides upward on a surface that is inclined at an angle of 18° above the horizontal. The coefficient of kinetic friction between the coin and the surface is 0.23; the coefficient 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.

Problem 79GP A physics textbook weighing 22 N rests on a table. The coefficient of static friction between the book and the table is µs = 0.60; the coefficient of kinetic friction is µk = 0.40. 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 the 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. Applied force Friction force Motion 0 5.0 N 11 N 15 N 11 N 8.0 N 5.0 N

A ball of mass m is placed in a wedge, as shown in Figure 6–36, 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. ________________ Equation Transcription: Text Transcription: 20^o m vec{g}

IP The blocks shown in Figure 6–37 are at rest. (a) Find the frictional force exerted on block A given that the mass of block A is 8.82 kg, the mass of block B is 2.33 kg, and the coefficient of static friction between block A and the surface on which it rests is 0.320. (b) If the mass of block A is doubled, does the frictional force exerted on it increase, decrease, or stay the same? Explain. ________________ Equation Transcription: Text Transcription: 45^o

In part (a) of Problem 81, what is the maximum mass block B can have and the system still be in equilibrium? ________________ Equation Transcription: Text Transcription: 45^o

Problem 83GP A picture hangs on the wall suspended by two strings, as shown in Figure. 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?

IP 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? As- sume that its rotational speed has remained constant.

A Conical Pendulum A 0.075-kg toy airplane is tied to the ceiling with a string. When the airplane’s 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 6–38. Find (a) the angle the string makes with the vertical and (b) the tension in the string. ________________ Equation Transcription: Text Transcription: theta

A tugboat tows a barge at constant speed with a 3500-kg cable, as shown in Figure 6–39. If the angle the cable makes with the horizontal where it attaches to the barge and the tugboat is \(22^\circ\), find the force the cable exerts on the barge in the forward direction. ________________ Equation Transcription: Text Transcription: 22^o 22^o 22^o

Two blocks, stacked one on top of the other, can move without friction on the horizontal surface shown in Figure 6–40. The surface between the two blocks is rough, however, with a coefficient 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.

Problem 88GP Find the coefficient 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.

Problem 89GP 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 1.10 m/s2.

Problem 90GP At the airport, you pull a 18-kg suitcase across the floor 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 coefficient of kinetic friction between the suitcase and the floor is 0.38.

Problem 92GP A 0.16-g spider hangs from the middle of the first 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.

A light 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 6–41. 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 floor. (a) Which coefficient of friction (static or kinetic) can be determined from this information? Explain. (b) What is the numerical value of this coefficient of friction?

Find the acceleration the cart in Figure 6–42 must have in order for the cereal box at the front of the cart not to fall. Assume that the coefficient of static friction between the cart and the box is 0.38.

Problem 94GP 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.

Problem 95GP A pair 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?

Problem 96GP Find the tension in the 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.

Problem 98GP A block with a mass of 3.1 kg is placed at rest on a surface inclined at an angle of 45° above the horizontal. The coefficient 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 F is greater than a minimum value, Fmin, and less than a maximum value, Fmax. Explain the reasons for this behavior. (b) Calculate Fmin. (c) Calculate Fmax.

Problem 97GP As your plane circles an airport, it moves in a horizontal circle of radius 2300 in with a speed of 390 km/h. If the lift of the airplane’s wings is perpendicular to the wings, at what angle should the plane be banked so that it doesn’t tend to slip sideways?

A mountain climber of mass m hangs onto a rope to keep from sliding down a smooth, ice-covered slope (Figure 6–43). Find a formula for the tension in the rope when the slope is inclined at an angle \(\theta\) above the horizontal. Check your results in the limits \(\theta=0\) and \(\theta=90^\circ\). ________________ Equation Transcription: Text Transcription: theta theta=0 theta=90^o theta theta

Problem 100GP 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 coefficient of static friction between the child and the merry-go-round that will prevent the child from slipping?

A 2.0-kg box rests on a plank that is inclined at an angle of \(65^\circ\) 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 6–44. If the coefficient 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? ________________ Equation Transcription: Text Transcription: 65^o 65^o

Problem 102GP A wood block of mass m rests on a larger wood block of mass M that rests on a wooden table. The coefficients of static and kinetic friction between all surfaces are µs and µk, 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 6-30 for general values of the masses. Your answer should be in terms of \(m_1,m_2,m_3\), and . ________________ Equation Transcription: Text Transcription: m_1,m_2,m_3

Problem 104GP The coefficient of static friction between a rope and the table on which it rests is µs. Find the fraction of the rope that can hang over the edge of the table before it begins to slip.

The Force Needed to Move a Crate To move a crate of mass \(m\) across a rough floor, you push down on it at an angle \(\theta_{f}\) as shown in Figure 6-18 for the special case of \(\theta=21^{\circ}\). (a) Find the force necessary to start the crate moving as a function of \(\theta\) given that the coefficient of static friction between the crate and the floor is \(\mu_{s}\) b) Show that it is impossible to move the crate, no matter how great the force, if the coefficient of static friction is greater than or equal to \(1 / \tan \theta\). Equation Transcription: Text Transcription: m theta_f theta=21degree theta mu _s 1/tan theta

A popular ride at amusement parks is illustrated in Figure 6–46. 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 \(\theta\) the supporting wires make with the vertical. (c) If you observe a ride like that in Figure 6–46, or as shown in the photo on page 170, you will notice that all the swings are at the same angle \(\theta\) to the vertical, regardless of the weight of the rider. Explain ________________ Equation Transcription: Text Transcription: theta theta theta

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 6–45. 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?

Problem 108GP A Conveyor Belt A box is placed on a conveyor belt that moves with a constant speed of 1.25 m/s. The coefficient 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 docs the box move in this time?

Problem 109GP You push a box along the floor against a constant force of friction. When you push with a horizontal force of 75 N, the acceleration of the box is 0.50 m/s2; when you increase the force to 81 N, the acceleration is 0.75 m/s2. Find (a) the mass of the box and (b) the coefficient of kinetic friction between the box and the floor.

As part of a circus act, a person drives a motorcycle with constant speed v around the inside of a vertical track of radius r, as indicated in Figure 6–47. If the combined mass of the motorcycle and rider is m, find the normal force exerted on the motorcycle by the track at the points (a) A, (b) B, and (c) C.

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 con- stant 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.

On the straight-line segment I in Figure 6–48 (b) we see that increasing the applied mass from 26 g to 44 g results in a

Referring to Example 6–3 Suppose the coefficients 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 \(23.2^\circ\)? (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). ________________ Equation Transcription: Text Transcription: 23.2^o

Problem 116IP Referring to The crate begins to slide when the tilt angle is 17.5°. When the crate reaches the bottom of the flatbed, after sliding a distance of 2.75 m, its speed is 3.11 m/s. Find (a) the coefficient of static friction and (b) the coefficient of kinetic friction between the crate and the flatbed.

Problem 117IP Referring to Example Suppose that the mass on the frictionless tabletop has the value m1 = 2.45 kg. (a) Find the value of m2 that gives an acceleration of 2.85 m/s2. (b) What is the corresponding tension; T, in the string? (c) Calculate the ratio T/m2g and show that it is less than 1, as expected.