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**Hugh D. Young, College Physics, 9th edition, Addison Wesley 2011**

**Chapter 5**

**Multiple-Choice Peoblems**

1. A horizontal force accelerates a box across a rough horizontal floor with friction present, producing an acceleration a. If the force is now tripled, but all other conditions remain the same, the acceleration will become

A. greater than 3a.

B. equal to 3a.

C. less than 3a.

2. You slide an 800-N table across the kitchen floor by pushing with a force of 100 N. If the table moves at a constant speed, the friction force with the floor must be

A. 100 N B. more than 100 N

C. 800 N D. more than 800 N

3. An artist wearing spiked shoes pushes two crates across her frictionless horizontal studio floor. (See Figure 5.26.) If she exerts a horizontal 36 N force on the smaller crate, then the smaller crate exerts a force on the larger crate that is closest to

A. 36 N. B. 30 N. C. 200 N. D. 240 N

4. A horizontal pull P pulls two wagons over a horizontal frictionless floor, as shown in Figure 5.27. The tension in the light horizontal rope connecting the wagons is A. equal to P, by Newton’s third law.

B. equal to 2000 N.

C. greater than P.

D. less than P.

5. A horizontal pull P drags two boxes connected by a horizontal rope having tension T, as shown in Figure 5.28. The floor is horizontal and frictionless. Decide which of the following statements is true without doing any calculations:

A. P > T B. P = T C. P < T D. We need more information to decide which is greater.

6. A crate slides up an inclined ramp and then slides down the ramp after momentarily stopping near the top. This crate is acted upon by friction on the ramp and accelerates both ways. Which statement about this crate’s acceleration is correct?

A. The acceleration going up the ramp is greater than the acceleration going down.

B. The acceleration going down the ramp is greater than the acceleration going up.

C. The acceleration is the same in both directions.

7. A bungee cord with a normal length of 2 ft requires a force of 30 lb to stretch it to a total length of 5 ft. The force constant of the bungee cord is

A. 6 lb ft B. 10 lb ft C. 15 lb ft D. 30 lb ft

8. A weightless spring scale is attached to two equal weights as shown in Figure 5.29. The reading in the scale will be

A. 0. B. W. C. 2W.

9. Two objects are connected by a light wire as shown in Figure 5.30, with the wire pulling horizontally on the 400 N object. After this system is released from rest, the tension in the wire will be

A. less than 300 N.

B. 300 N.

C. 200 N.

D. 100 N.

10. A 100 N weight is supported by two weightless wires A and B as shown in Figure 5.31. What can you conclude about the tensions in these wires?

A. The tensions are equal to 50 N each.

B. The tensions are equal, but less than 50 N each.

C. The tensions are equal, but greater than 50 N each.

D. The tensions are equal to 100 N each.

11. The two blocks in Figure 5.32 are in balance, and there is no friction on the surface of the incline. If a slight vibration occurs, what will M do after that?

A. It will accelerate up the ramp.

B. It will accelerate down the ramp.

C. It will not accelerate.

12. The system shown in Figure 5.33 is released from rest, there is no friction between B and the tabletop, and all of the objects move together. What must be true about the friction force on A?

A. It is zero.

B. It acts to the right on A.

C. It acts to the left on A.

D. We cannot tell whether there is any friction force on A because we do not know the coefficients of friction between A and B.

13. In the system shown in Figure 5.34, M > m, the surface of the bench is horizontal and frictionless, and the connecting string pulls horizontally on m. As more and more weight is gradually added to m, which of the following statements best describes the behavior of the system after it is released?

A. The acceleration remains the same in all cases, since there is no friction and the pull of gravity on M is the same.

B. The acceleration becomes zero when enough weight is added so that m = M.

C. The velocity becomes zero when m = M.

D. None of the preceding statements is correct.

**Section 5.1**

1. A 15.0 N bucket is to be raised at a constant speed of 50.0 cm/s by a rope. According to the information in Table 5.1, how many kilograms of cement can be put into this bucket without breaking the rope if it is made of (a) thin white string, (b) 1/4 in. nylon clothesline, (c) in. manila climbing rope?

2. In a museum exhibit, three equal weights are hung with identical wires, as shown in Figure 5.35. Each wire can support a tension of no more than 75.0 N without breaking. Start each of the following parts with an appropriate free-body diagram. (a) What is the maximum value that W can be without breaking any wires? (b) Under these conditions, what is the tension in each wire?

3. 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 is fastened to the ceiling. (See Figure 5.36.) Start solving this problem by making a free-body diagram of each weight. (a) What is the tension in the rope? (b) What is the tension in the chain?

4. Two weights are hanging as shown in Figure 5.37. (a) Draw a free-body diagram of each weight. (b) Find the tension in cable A. (c) Find the tension in cables B and C.

5. 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 (Figure 5.38). The rope will break if the tension in it exceeds 2.50 ×104 N. Our hero’s mass is 90.0 kg. (a) If the angle is find the tension in the rope. Start with a free-body diagram of the archaeologist. (b) What is the smallest value the angle can have if the rope is not to break?

6. A 1130-kg car is held in place by a light cable on a very smooth (frictionless) ramp, as shown in Fig. 5.39. The cable makes an angle of 31.0o above the surface of the ramp, and the ramp itself rises at 25.0o 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?

7. Tension in a muscle. Muscles are attached to bones by means of tendons. The maximum force that a muscle can exert is directly proportional to its cross-sectional area A at the widest point. We can express this relationship mathematically as where Fmax = σA, where σ (sigma) is a proportionality constant. Surprisingly, σ is about the same for the muscles of all animals and has the numerical value of in SI units. (a) What are the SI units of σ in terms of newtons and meters and also in terms of the fundamental quantities (b) In one set of experiments, the average maximum force that the gastrocnemius muscle in the back of the lower leg could exert was measured to be 755 N for healthy males in their midtwenties. What does this result tell us was the average cross-sectional area, in cm2, of that muscle for the people in the study?

8. Muscles and tendons. The gastrocnemius muscle, in the back of the leg, has two portions, known as the medial and lateral heads. Assume that they attach to the Achilles tendon as shown in Figure 5.40. The cross-sectional area of each of these two muscles is typically 30 cm2 for many adults. What is the maximum tension they can produce in the Achilles tendon? (See the previous problem for some useful information.)

9. Traction apparatus. In order to prevent muscle contraction from misaligning bones during healing (which can cause a permanent limp), injured or broken legs must be supported horizontally and at the same time kept under tension (traction) directed along the leg. One version of a device to accomplish this aim, the Russell traction apparatus, is shown in Figure 5.41. This system allows the apparatus to support the full weight of the injured leg and at the same time provide the traction along the leg. If the leg to be supported weighs 47.0 N, (a) what must be the weight of W and (b) what traction force does this system produce along the leg?

10. A broken thigh bone. When the thigh is fractured, the patient’s leg must be kept under traction. One method of doing so is a variation on the Russell traction apparatus. (See Figure 5.42.) If the physical therapist specifies that the traction force directed along the leg must be 25 N, what must W be?

11. Two artifacts in a museum display are hung from vertical walls by very light wires, as shown in Figure 5.43. (a) Draw a free-body diagram of each artifact. (b) Find the tension in each of the three wires. (c) Would the answers be different if each wire were twice as long, but the angles were unchanged? Why or why not?

12. In a rescue, the 73 kg police officer is suspended by two cables, as shown in Figure 5.44. (a) Sketch a free-body diagram of him. (b) Find the tension in each cable.

13. A tetherball leans against the smooth, frictionless post to which it is attached. The string is attached to the ball such that a line along the string passes through the center of the ball. If the string is 1.40 m long and the ball has a radius of 0.110 m with mass 0.270 kg, (a) make a free-body diagram of the ball. (b) What is the tension in the rope? (c) What is the force the pole exerts on the ball?

14. Find the tension in each cord in Figure 5.46 if the weight of the suspended object is 250 N.

15. Two blocks, each with weight w, are held in place on a frictionless incline as shown in Figure 5.47. In terms of w and the angle α of the incline, calculate the tension in (a) the rope connecting the blocks and (b) the rope that connects block A to the wall. (c) Calculate the magnitude of the force that the incline exerts on each block. (d) Interpret your answers for the cases α = 0 and α = 90o.

16. A man pushes on a piano of mass 180 kg so that it slides at a constant velocity of 12 cm/s down a ramp that is inclined at 11.0° above the horizontal. No appreciable friction is acting on the piano. Calculate the magnitude and direction of this push (a) if the man pushes parallel to the incline, (b) if the man pushes the piano up the plane instead, also at 12.0 cm/s parallel to the incline, and (c) if the man pushes horizontally, but still with a speed of 12.0 cm/s.

17. Air-bag safety. According to safety standards for air bags, the maximum acceleration during a car crash should not exceed 60g and should last for no more than 36 ms. (a) In such a case, what force does the air bag exert on a 75 kg person? Start with a free-body diagram. (b) Express the force in part (a) in terms of the person’s weight.

18. Forces during chin-ups. People who do chin-ups raise their chin just over a bar (the chinning bar), supporting themselves only by 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 who is chinning 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. Make a free-body diagram of the person’s body, and then apply it to find the force his arms must exert on him during the accelerating part of the chin-up.

19. Force on a tennis ball. The record speed for a tennis ball that is served is 73.14 m/s. During a serve, the ball typically starts from rest and is in contact with the tennis racquet for 30 ms. Assuming constant acceleration, what was the average force exerted on the tennis ball during this record serve, expressed in terms of the ball’s weight W?

20. A 75,600 N spaceship comes in for a vertical landing. From an initial speed of 1.00 km/s, it comes to rest in 2.00 min with uniform acceleration. (a) Make a free-body diagram of this ship as it is coming in. (b) What braking force must its rockets provide? Ignore air resistance.

21. 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 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) Make 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?

22. A short train (an engine plus four cars) is accelerating at 1.10 m/s2. If the mass of each car is 38,000 kg, and if each car has negligible frictional forces acting on it, what are (a) the force of the engine on the first car, (b) the force of the first car on the second car, (c) the force of the second car on the third car, and (d) the force of the third car on the fourth car? In solving this problem, note the importance of selecting the correct set of cars to isolate as your object.

23. A large fish hangs from a spring balance supported from the roof of an elevator. (a) If the elevator has an upward acceleration of 2.45 m/s2 and the balance reads 60.0 N, what is the true weight of the fish? (b) Under what circumstances will the balance read 35.0 N? (c) What will the balance read if the elevator cable breaks?

24. 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?

25. The TGV, France’s high-speed train, pulls out of the Lyons station and is accelerating uniformly to its cruising speed. Inside one of the cars, a 3.00 N digital camera is hanging from the luggage compartment by a light, flexible strap that makes a 12.0° angle with the vertical. (a) Make a free-body diagram of this camera. (b) Apply Newton’s second law to the camera, and find the acceleration of the train and the tension in the strap.

26. Which way and by what angle does the accelerometer in Figure 5.48 deflect under the following conditions? (a) The cart is moving toward the right with speed increasing at 3.0 m/s2. (b) The cart is moving toward the left with speed decreasing at 4.5 m/s2. (c) The cart is moving toward the left with a constant speed of 4.0 m/s.

27. A skier approaches the base of an icy hill with a speed of 12.5 m/s. The hill slopes upward at 24° above the horizontal. Ignoring all friction forces, find the acceleration of this skier (a) when she is going up the hill, (b) when she has reached her highest point, and (c) after she has started sliding down the hill. In each case, start with a free-body diagram of the skier.

28. At a construction site, a 22.0 kg bucket of concrete is connected over a very light frictionless pulley to a 375 N box on the roof of a building. (See Figure 5.49.) There is no appreciable friction on the box, since it is on roller bearings. The box starts from rest. (a) Make free-body diagrams of the bucket and the box. (b) Find the acceleration of the bucket. (c) How fast is the bucket moving after it has fallen 1.50 m (assuming that the box has not yet reached the edge of the roof)?

29. Two boxes are connected by a light string that passes over a light, frictionless pulley. One box rests on a frictionless ramp that rises at 30.0° above the horizontal (see Figure 5.50), and the system is released from rest. (a) Make free-body diagrams of each box. (b) Which way will the 50.0 kg box move, up the plane or down the plane? Or will it even move at all? Show why or why not. (c) Find the acceleration of each box.

30. An 80 N box initially at rest is pulled by a horizontal rope on a horizontal table. The coefficients of kinetic and static friction between the box and the table are 1/4 and 1/2 respectively. What is the friction force on this box if the pull is (a) 0 N, (b) 25 N, (c) 39 N, (d) 41 N, (e) 150 N?

31. 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?

32. Friction at the hip joint. Figure 5.51 shows the bone structure at the hip joint. The bones are normally not in direct contact, but instead are covered with cartilage to reduce friction. The space between them is filled with water like synovial fluid, which further decreases friction. Due to this fluid, the coefficient of kinetic friction between the bones can range from 0.0050 to 0.020. Typically, approximately 65% of a person’s weight is above the hip—we’ll call this the upper weight. (a) Show that when a person is simply standing upright, each hip supports half of his upper weight. (b) When a person is walking, each hip now supports up to 2.5 times his upper weight, depending on how fast he is walking. For a 65 kg person, what is the maximum kinetic friction force at the hip joint if mk has its minimum value of 0.0050? (c) As a person gets older, the aging process, as well as osteoarthritis, can alter the composition of the synovial fluid. In the worst case, this fluid could disappear, leaving bone-on-bone contact with a coefficient of kinetic friction of 0.30. What would be the greatest friction force for the walking person in part (b)? The increased friction causes pain and, in addition, wears down the joint even more.

33. At a construction site, a pallet of bricks is to be suspended by attaching a rope to it and connecting the other end to a couple of heavy crates on the roof of a building, as shown in Figure 5.52. The rope pulls horizontally on the lower crate, and the coefficient of static friction between the lower crate and the roof is 0.666. (a) What is the weight of the heaviest pallet of bricks that can be supported this way? Start with appropriate free-body diagrams. b) What is the friction force on the upper crate under the conditions of part (a)?

34. Two crates connected by a rope of negligible mass lie on a horizontal surface. Crate A has mass mA and crate B has mass mB. The coefficient of kinetic friction between each crate and the surface is μk. The crates are pulled to the right at a constant velocity of 3.20 cm/s by a horizontal force F. In terms of mA, mB and μk calculate (a) the magnitude of the force F and (b) the tension in the rope connecting the blocks. Include the free-body diagram or diagrams you used to determine each answer.

35. A hockey puck leaves a player’s stick with a speed of 9.9 m/s and slides 32.0 m before coming to rest. Find the coefficient of friction between the puck and the ice.

36. Stopping distance of a car. (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 29.1 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 in order to be able to stop in the same distance as in part (a)?

37. An 85-N box of oranges is being pushed across a horizontal floor. As it moves, it is slowing at a constant rate of each second. The push force has a horizontal component of 20 N and a vertical component of 25 N downward. Calculate the coefficient of kinetic friction between the box and floor.

38. 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 μr for each? Assume that the net horizontal force is due to rolling friction only.

39. 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 coefficients of kinetic and static friction between the box and the surface are 0.200 and 0.450, respectively. a) What horizontal force must the worker apply to maintain the motion of the box? (b) If the worker stops pushing, what will be the acceleration of the box?

40. The coefficients of static and kinetic friction between a 476 N crate and the warehouse floor are 0.615 and 0.420, respectively. A worker gradually increases his horizontal push against this crate until it just begins to move and from then on maintains that same maximum push. What is the acceleration of the crate after it has begun to move? Start with a free-body diagram of the crate.

41. Measuring the coefficients of friction. One straightforward way to measure the coefficients of friction between a box and a wooden surface is illustrated in Figure 5.54. The sheet of wood can be raised by pivoting it about one edge. It is first raised to an angle θ1 (which is measured) for which the box just begins to slide downward. The sheet is then immediately lowered to an angle θ2 (which is also measured) for which the box slides with constant speed down the sheet. Apply Newton’s second law to the box in both cases to find the coefficients of kinetic and static friction between it and the wooden sheet in terms of the measured angles θ1 and θ2

42. With its wheels locked, a van slides down a hill inclined at 40.0° to the horizontal. Find the acceleration of this van a) if the hill is icy and frictionless, and b) if the coefficient of kinetic friction is 0.20.

43. The Trendelberg position. In emergencies with major blood loss, the doctor will order the patient placed in the Trendelberg position, which is to raise the foot of the bed to get maximum blood flow to the brain. If the coefficient of static friction between the typical patient and the bedsheets is 1.2, what is the maximum angle at which the bed can be tilted with respect to the floor before the patient begins to slide?

44. Injuries to the spinal column. In treating spinal 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 part (a) of Figure 5.55. A weight W is attached to the patient (sometimes around a neck collar, as shown in part (b) of the figure), and friction between the person’s body and the bed prevents sliding. (a) If the coefficient of static friction between a 78.5 kg patient’s body and the bed is 0.75, what is the maximum traction force along the spinal column that W can provide without causing the patient to slide? (b) Under the conditions of maximum traction, what is the tension in each cable attached to the neck collar?

45. A winch is used to drag a 375 N crate up a ramp at a constant speed of 75 cm/s by means of a rope that pulls parallel to the surface of the ramp. The rope slopes upward at 33° above the horizontal, and the coefficient of kinetic friction between the ramp and the crate is 0.25. (a) What is the tension in the rope? (b) If the rope were suddenly to snap, what would be the acceleration of the crate immediately after the rope broke?

46. A toboggan approaches a snowy hill moving at 11.0 m/s. The coefficients of static and kinetic friction between the snow and the toboggan are 0.40 and 0.30, respectively, and the hill slopes upward at 40.0° above the horizontal. Find the acceleration of the toboggan (a) as it is going up the hill and (b) after it has reached its highest point and is sliding down the hill.

47. A 25.0-kg box of textbooks rests on a loading ramp that makes an angle a with the horizontal. The coefficient of kinetic friction is 0.25, and the coefficient of static friction is 0.35. a) As the angle α 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?

48. A. person pushes on a stationary 125 N box with 75 N at 30° below the horizontal, as shown in Figure 5.56. The coefficient of static friction between the box and the horizontal floor is 0.80. (a) Make a free-body diagram of the box. (b) What is the normal force on the box? (c) What is the friction force on the box? (d) What is the largest the friction force could be? (e) The person now replaces his push with a 75 N pull at 30° above the horizontal. Find the normal force on the box in this case.

49. A crate of 45.0-kg 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 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 performing the same experiment on this crate but were doing it on the moon instead, 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)?

50. You are working for a shipping company. Your job is to stand at the bottom of an 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 μk = 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?

51. The drag coefficient for a spherical raindrop with a radius of 0.415 cm falling at its terminal velocity is 2.43 ×10–5 kg/m. Calculate the raindrop’s terminal velocity in m/s.

52. What is the acceleration of a raindrop that has reached half of its terminal velocity? Give your answer in terms of g.

53. An object is dropped from rest and encounters air resistance that is proportional to the square of its speed. Sketch qualitative graphs (no numbers) showing (a) the air resistance on this object as a function of its speed, (b) the net force on the object as a function of its speed, (c) the net force on the object as a function of time, (d) the speed of the object as a function of time, and (e) the acceleration of the object as a function of time.

5.4 Elastic Forces

54. You find that if you hang a 1.25 kg weight from a vertical spring, it stretches 3.75 cm. (a) What is the force constant of this spring in N/m (b) How much mass should you hang from the spring so it will stretch by 8.13 cm from its original, unstretched length?

55. An unstretched spring is 12.00 cm long. When you hang an 875 g weight from it, it stretches to a length of 14.40 cm. (a) What is the force constant (in of this spring? (b) What total mass must you hang from the spring to stretch it to a total length of 17.72 cm?

56. Heart repair. A surgeon is using material from a donated heart to repair a patient’s damaged aorta and needs to know the elastic characteristics of this aortal material. Tests performed on a 16.0 cm strip of the donated aorta reveal that it stretches 3.75 cm when a 1.50 N pull is exerted on it. (a) What is the force constant of this strip of aortal material? (b) If the maximum distance it will be able to stretch when it replaces the aorta in the damaged heart is 1.14 cm, what is the greatest force it will be able to exert there?

57. An “extreme” pogo stick utilizes a spring whose uncompressed length is 46 cm and whose force constant is 1.4 ×104 N/m. A 60-kg enthusiast is jumping on the pogo stick, compressing the spring to a length of only 5.0 cm at the bottom of her jump. Calculate (a) the net upward force on her at the moment the spring reaches its greatest compression and (b) her upward acceleration, in m/s2 and g’s at that moment.

58. A student measures the force required to stretch a spring by various amounts and makes the graph shown in Figure 5.57, which plots this force as a function of the distance the spring has stretched. (a) Does this spring obey Hooke’s law? How do you know? (b) What is the force constant of the spring, in N/m (c) What force would be needed to stretch the spring a distance of 17 cm from its unstretched length, assuming that it continues to obey Hooke’s law?

59. A student hangs various masses from a spring and records the resulting distance the spring has stretched. She then uses her data to construct the graph shown in Figure 5.58, which plots the mass hung from the spring as a function of the distance the mass caused the spring to stretch beyond its unstretched length. (a) Does this spring obey Hooke’s law? How do you know? (b) What is the force constant of the spring, in N/m (c) By how much will the spring have stretched if you hang a 2.35 kg mass from it, assuming that it continues to obey Hooke’s law?

60. Three identical 6.40 kg masses are hung by three identical springs, as shown in Figure 5.59. Each spring has a force constant of 7.80 kN/cm and was 12.0 cm long before any masses were attached to it. (a) Make a free-body diagram of each mass. (b) How long is each spring when hanging as shown?

61. A light spring having a force constant of 125 N/m is used to pull a 9.50 kg sled on a horizontal frictionless ice rink. If the sled has an acceleration of 2.00 m/s2 by how much does the spring stretch if it pulls on the sled (a) horizontally, (b) at 30.0° above the horizontal?

62. In the previous problem, what would the answers in both cases be if there were friction and the coefficient of kinetic friction between the sled and the ice were 0.200?

63. 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 if it is likely to cause injury, calculate how long it lasts.

64. Modeling elastic hip pads. In the previous problem, we assumed constant acceleration (and hence a constant force) due to the hip pad. But of course, such a pad would most likely be somewhat elastic. Assuming that the force you found in the previous problem was the maximum force over the period of impact with the ground, and that the hip pad is elastic enough to obey Hooke’s law, calculate the force constant of this pad, using appropriate data from the previous problem.

65. You’ve attached a bungee cord to a wagon and are using it to pull your little sister while you take her for a jaunt. The bungee’s unstretched length is 1.3 m, and you happen to know that your little sister weighs 220 N and the wagon weighs 75 N. Crossing a street, you accelerate from rest to your normal walking speed of 1.5 m/s in 2.0 s, and you notice that while you’re accelerating, the bungee’s length increases to about 2.0 m. What’s the force constant of the bungee cord, assuming it obeys Hooke’s law?

66. Atwood’s Machine. A 15.0-kg load of bricks hangs from one end of a rope that passes over a small, frictionless pulley. A 28.0-kg counterweight is suspended from the other end of the rope, as shown in Fig. 5.60. 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?

67. Mountaineering. Figure 5.61 shows a technique called rappelling, used by mountaineers for descending vertical rock faces. The climber sits in a rope seat, and the rope slides through a friction device attached to the seat. Suppose that the rock is perfectly smooth (i.e., there is no friction) and that the climber’s feet push horizontally onto the rock. If the climber’s weight is 600.0 N, find (a) the tension in the rope and (b) the force the climber’s feet exert on the rock face. Start with a free-body diagram of the climber.

68. Two identical, perfectly smooth 71.2 N bowling balls 21.7 cm in diameter are hung together from the same hook in the ceiling by means of two thin, light wires, as shown in Figure 5.62, find (a) the tension in each wire and (b) the force the balls exert on each other.

69. Stay awake! An astronaut is inside a 2.25 ×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 you also do not want the astronaut to black out. Medical tests have shown that astronauts are in danger of blacking out for 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?

70. The stretchy silk of a certain species of spider has a force constant of 1.0 mN/cm. The spider, whose mass is 15 mg, has attached herself to a branch as shown in Figure 5.63. Calculate (a) the tension in each of the three strands of silk and (b) the distance each strand is stretched beyond its normal length.

71. Block A in Figure 5.64 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 remains at rest. (a) Find the friction force exerted on block A. (b) Find the maximum weight w for which the system will remain at rest.

72. Friction in an elevator. You are riding in an elevator on the way to the 18th floor of your dormitory. The elevator is accelerating upward with a = 1.90 m/s2. Beside you is the box containing your new computer; the box and its contents have a total mass of 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 μk = 0.32, what magnitude of force must you apply?

73. A student attaches a series of weights to a tendon and measures the total length of the tendon for each weight. He then uses the data he has gathered to construct the graph shown in Figure 5.65, giving the weight as a function of the length of the tendon. (a) Does this tendon obey Hooke’s law? How do you know? (b) What is the force constant (in N/m) for the tendon? (c) What weight should you hang from the tendon to make it stretch by 8.0 cm from its unstretched length?

74. A 65.0-kg parachutist falling vertically at a speed of 6.30 m/s impacts the ground, which brings him to a complete stop in a distance of 0.92 m (roughly half of his height). Assuming constant acceleration after his feet first touch the ground, what is the average force exerted on the parachutist by the ground?

75. Mars Exploration Rover landings. In January 2004 the Mars Exploration Rover spacecraft landed on the surface of the Red Planet, where the acceleration due to gravity is 0.379 what it is on earth. The descent of this 827 kg vehicle occurred in several stages, three of which are outlined here. In Stage I, friction with the Martian atmosphere reduced the speed from 19300 km/h to 1600 km/h in a 4.0 min interval. In Stage II, a parachute reduced the speed from 1600 km/h to 321 km/h in 94 s, and in Stage III, which lasted 2.5 s, retrorockets fired to reduce the speed from 321 km/h to zero. As part of your solution to this problem, make a free-body diagram of the rocket during each stage. Assuming constant acceleration, find the force exerted on the spacecraft (a) by the atmosphere during Stage I, (b) by the parachute during Stage II, and (c) by the retrorockets during Stage III.

76. Block A in Figure 5.66 weighs 1.20 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 F necessary to drag block B to the left at a constant speed of 2.50 cm/s (a) if A rests on B and moves with it (Figure 5.66a); (b) if A is held at rest by a string (Figure 5.66b). (c) In part (a), what is the friction force on block A?

77. Crash of the Genesis Mission. You are assigned the task of designing an accelerometer to be used inside a rocket ship in outer space. Your equipment consists of a very light spring that is 15.0 cm long when no forces act to stretch or compress it, plus a 1.10 kg weight. One end of this spring will be attached to a friction free tabletop, while the 1.10 kg weight is attached to the other end, as shown in Figure 5.67. (a) What should be the force constant of the spring so that it will stretch by 1.10 cm when the rocket accelerates forward at 2.50 m/s2? Start with a free-body diagram of the weight. (b) What is the acceleration (magnitude and direction) of the rocket if the spring is compressed by 2.30 cm?

78. A block with mass m1 is placed on an inclined plane with slope angle a and is connected to a second hanging block with mass m2 by a cord passing over a small, frictionless pulley (Fig. 5.68). The coefficient of static friction is ms and the coefficient of kinetic friction is mk. (a) Find the mass m2 for which block m1 moves up the plane at constant speed once it is set in motion. (b) Find the mass m2 for which block m1 moves down the plane at constant speed once it is set in motion. (c) For what range of values of m2 will the blocks remain at rest if they are released from rest?

79. 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 30.0 m/s without causing the box to slide. Include a free-body diagram of the toolbox as part of your solution.

80. Accident analysis. You have been called to testify as an expert witness in a trial involving an automobile accident. The speed limit on the highway where the accident occurred was 40 mph. The driver of the car slammed on his brakes, locking his wheels, and left skid marks as the car skidded to a halt. You measure the length of these skid marks to be 219 ft, 9 in., and determine that the coefficient of kinetic friction between the wheels and the pavement at the time of the accident was 0.40. How fast was this car traveling (to the nearest number of mph) just before the driver hit his brakes? Was he guilty of speeding?

81. A window washer pushes his scrub brush up a vertical window at constant speed by applying a force F as shown in Fig. 5.69. The brush weighs 12.0 N and the coefficient of kinetic friction is μk = 0.150. Calculate (a) the magnitude of the force F and (b) the normal force exerted by the window on the brush.

82. A fractured tibia. While a fractured tibia (the larger of the two major lower leg bones in mammals) is healing, it must be held horizontal and kept under some tension so that the bones will heal properly to prevent a permanent limp. One way to do this is to support the leg by using a variation of the Russell traction apparatus. The lower leg (including the foot) of a particular patient weighs 51.5 N,all of which must be supported by the traction apparatus. (a) What must be the mass of W, shown in the figure? (b) What traction force does the apparatus provide along the direction of the leg?

83. You push with a horizontal force of 50 N against a 20 N box, pressing it against a rough vertical wall to hold it in place. The coefficients of kinetic and static friction between this box and the wall are 0.20 and 0.50, respectively. (a) Make a free-body diagram of this box. (b) What is the friction force on the box? (c) How hard would you have to press for the box to slide downward with a uniform speed of 10.5 cm/s.

84. 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 and static friction of 0.45 and 0.65, respectively, with these rocks. Start each part of your solution to this problem with a free-body diagram. (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.

85. 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?

86. At night while it is dark, a driver inadvertently parks his car on a drawbridge. Some time later, the bridge must be raised to allow a boat to pass through. The coefficients of friction between the bridge and the car’s tires are μs = 0.750 and μk = 0.550. Start each part of your solution to this problem with a free-body diagram of the car. (a) At what angle will the car just start to slide? (b) If the bridge attendant sees the car suddenly start to slide and immediately turns off the bridge’s motor, what will be the car’s acceleration after it has begun to move?

87. A block is placed against the vertical front of a cart as shown in Figure 5.71. What acceleration must the cart have in order that block A does not fall? The coefficient of static friction between the block and the cart is ms.

88. The monkey and her bananas. 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 (Figure 5.72). 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, move 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?

**Chapter 6
**

**Multiple-Choice Problems
**

1. Objects in orbiting satellites are apparently weightless because

A. they are too far from earth to feel its gravity.

B. earth’s gravitational pull is balanced by the outward centrifugal force.

C. earth’s gravitational pull is balanced by the centripetal force.

D. both they and the satellite are in free fall toward the earth.

2. If the earth had twice its present mass, its orbital period around the sun (at our present distance from the sun) would be

A. √2 years. B. 1 year.

C. 1/√2 year. D. 1/2 year.

3. An astronaut is floating happily outside her spaceship, which is orbiting the earth at a distance above the earth’s surface equal to 1 earth radius. The astronaut’s weight is

A. zero

B. equal to her normal weight on earth

C. half her normal weight on earth

D. one-fourth her normal weight on earth

4. A stone of mass m is attached to a strong string and whirled in a vertical circle of radius R. At the exact top of the path, the tension in the string is three times the stone’s weight. At this point, the stone’s speed is

A. 2√(gR) B. √(2gR)

C. 3√(gR) D. √(3gR)

5. A frictional force provides the centripetal force as a car goes around an unbanked curve of radius R at speed V. Later, the car encounters a similar curve, except of radius 2R, and the driver continues around this curve at the same speed V. In order to make this second curve, the frictional force on the car must be equal to

A. 2*f* B. *f* C. (1/2)* f * D. (1/4)* f*

6. In the 1960s, during the Cold War, the Soviet Union put a rocket into a nearly circular orbit around the earth. One U.S. senator expressed concern that a nuclear bomb could be dropped on our nation if it were released when the rocket was over the United States. If such a bomb were, in fact, released when the rocket was directly over the United States, where would it land?

A. Directly below, on the United States.

B. It would follow a curved path and most likely land somewhere in the ocean.

C. It would not hit the United States, because it would remain in orbit with the rocket.

D. It would move in a straight line out into space.

7. Two masses m and 2m are each forced to go around a curve of radius R at the same constant speed. If, as they move around this curve, the smaller mass is acted upon by a net force F, then the larger one is acted upon by a net force of

A. (1/2)F B. F C. 2F D. 4F

8. A weight W is swung in a vertical circle at constant speed at the end of a rigid bar that pivots about the opposite end (like the spoke of a wheel). If the tension in this bar is equal to 2W when the weight is at its lowest point, then the tension when the weight is at the highest point will be

A. 0 B. W C. 2W D. 3W

9. If a planet had twice the earth’s radius, but only half its mass, the acceleration due to gravity at its surface would be

A. g. B. (1/2)g C. (1/4)g D. (1/8)g

10. When a mass goes in a horizontal circle with speed V at the end of a string of length L on a frictionless table, the tension in the string is T. If the speed of this mass were doubled, but all else remained the same, the tension would be

A. 2T B. 4T C. T√2 D. T/√2

11. In the previous problem, if both the speed and the length were doubled, the tension in the string would be

A. T B. 2T C. 4T D. T√2

12. Two 1.0 kg point masses a distance D apart each exert a gravitational attraction F on the other one. If 1.0 kg is now added to each mass, the gravitational attraction exerted by each would be

A. 2F. B. 3F. C. 4F.

13. Two massless bags contain identical bricks, each brick having a mass M. Initially, each bag contains four bricks, and the bags mutually exert a gravitational attraction F1 on each other. You
now take two bricks from one bag and add them to the other bag, causing the bags to attract each other with a force F2. What is the closest expression for F2 in terms of F1?

A. F2 = (3/4)F1

B. F2 = (1/2)F1

C. F2 = (1/4)F1

D. F2 = F1 because neither the total mass nor the distance between the bags has changed.

14. When two point masses are a distance D apart, each exerts a gravitational attraction F on the other mass. To reduce this force to (1/3)F you would have to separate the masses to a distance of

A. D√3 B. 3D.

C. 9D. D. (1/3)D

15. If human beings ever travel to a planet whose mass and radius are both twice that of the earth, their weight on that planet will be

A. equal to their weight on earth

B. twice their weight on earth

C. half their weight on earth

D. one-fourth their weight on earth

**Problems**

1. A racing car drives at constant speed around the horizontal track shown in Figure 6.25. At points A, B, and C, draw a vector showing the magnitude and direction of the net force on this car. Make sure that the lengths of your arrows represent the relative magnitudes of the three forces.

2. A stone with a mass of 0.80 kg is attached to one end of a string 0.90 m long. The string will break if its tension exceeds 60.0 N. The stone is whirled in a horizontal circle on a frictionless tabletop; the other end of the string remains fixed. (a) Make a free-body diagram of the stone. (b) Find the maximum speed the stone can attain without breaking the string.

3. Force on a skater’s wrist. A 52 kg ice skater spins about a vertical axis through her body with her arms horizontally outstretched, making 2.0 turns each second. The distance from one hand to the other is 1.50 m. Biometric measurements indicate that each hand typically makes up about 1.25% of body weight. (a) Draw a free-body diagram of one of her hands. (b) What horizontal force must her wrist exert on her hand? (c) Express the force in part (b) as a multiple of the weight of her hand.

4. A flat (unbanked) curve on a highway has a radius of 220 m. A car rounds the curve at a speed of 25.0 m/s. (a) Make a free-body diagram of the car as it rounds this curve. (b) What is the minimum coefficient of friction that will prevent sliding?

5. The “Giant Swing” at a county fair consists of a vertical central shaft with a number of horizontal arms attached at its upper end. Each arm supports a seat suspended from a 5.00-m-long rod, the upper end of which is fastened to the arm at a point 3.00 m from the central shaft. (a) Make a free-body diagram of the seat, including the person in it. (b) Find the time of one revolution of the swing if the rod supporting the seat makes an angle of 30.0° with the vertical. (c) Does the angle depend on the weight of the passenger for a given rate of revolution?

6. 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.

7. Using only astronomical data from Appendix E, calculate (a) the speed of the planet Venus in its essentially circular orbit around the sun and (b) the gravitational force that the sun must be exerting on Venus. Start with a free-body diagram of Venus.

8. A highway curve with radius 900.0 ft is to be banked so that a car traveling 55.0 mph will not skid sideways even in the absence of friction. (a) Make a free-body diagram of this car. (b) At what angle should the curve be banked?

9. The Indy 500. The Indianapolis Speedway (home of the Indy 500) consists of a 2.5 mile track having four turns, each 0.25 mile long and banked at 9°12'. What is the no-friction-needed speed (in m/s and mph) for these turns?

10. 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 like a pendulum. As the rope swings through its lowest point, the speed of the bowling ball is measured at 4.20 m/s. At that instant, find (a) the magnitude and direction of the acceleration of the bowling ball and (b) the tension in the rope. Be sure to start with a free-body diagram.

11. 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 passenger’s apparent weight at the lowest point?

12. 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 should the minimum radius of the circle be in order for the centripetal 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?

13. Effect on blood of walking. While a person is walking, his arms swing through approximately a 45° angle in 1/2 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 gram drop of blood in the fingertips at the bottom of the swing? (b) Make 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 (b). Which way does this force point? (d) What force would the blood vessel exert if the arm were not swinging?

14. 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? Start with a free-body diagram of the water at its highest point.

15. Stunt pilots and fighter pilots who fly at high speeds in a downward-curving arc may experience a “red out,” in which blood is forced upward into the flier’s head, potentially swelling or breaking capillaries in the eyes and leading to a reddening of vision and even loss of consciousness. This effect can occur at centripetal accelerations of about 2.5 g’s. For a stunt plane flying at a speed of 320 km/h, what is the minimum radius of downward curve a pilot can achieve without experiencing a red out at the top of the arc?

16. If two tiny identical spheres attract each other with a force of 3.0 nN when they are 25 cm apart, what is the mass of each sphere?

17. What gravitational force do the two protons in the helium nucleus exert on each other? Their separation is approximately 1.0 fm.

18. Rendezvous in space! A couple of astronauts agree to rendezvous in space after hours. Their plan is to let gravity bring them together. One of them has a mass of 65 kg and the other a mass of 72 kg, and they start from rest 20.0 m apart. (a) Make a free-body diagram of each astronaut, and use it to find his or her initial acceleration. As a rough approximation, we can model the astronauts as uniform spheres. (b) If the astronauts’ acceleration remained constant, how many days would they have to wait before reaching each other? (c) Would their acceleration, in fact, remain constant? If not, would it increase or decrease? Why?

19. What is the ratio of the gravitational pull of the sun on the moon to that of the earth on the moon? (Assume that the distance of the moon from the sun is approximately the same as the distance of the earth from the sun.). Use the data in Appendix E. Is it more accurate to say that the moon orbits the earth or that the moon orbits the sun? Why?

20. A 2150 kg satellite used in a cellular telephone network is in a circular orbit at a height of 780 km above the surface of the earth. What is the gravitational force on the satellite? What fraction is this force of the satellite’s weight at the surface of the earth?

21. An interplanetary spaceship passes through the point in space where the gravitational forces from the sun and the earth on the ship exactly cancel. (a) How far from the center of the earth is it? Use the data in Appendix E. (b) Once it reached the point found in part (a), could the spaceship turn off its engines and just hover there indefinitely? Explain.

22. Find the magnitude and direction of the net gravitational force on mass A due to masses B and C in Figure 6.27. Each mass is 2.00 kg.

23. How far from a very small 100-kg ball would a particle have to be placed so that the ball pulled on the particle just as hard as the earth does? Is it reasonable that you could actually set up this as an experiment? Why?

24. Each mass in Figure 6.28 is 3.00 kg. Find the magnitude and direction of the net gravitational force on mass A due to the other masses.

25. An 8.00-kg point mass and a 15.0-kg point mass are held in place 50.0 cm apart. A particle of mass m is released from a point between the two masses 20.0 cm from the 8.00-kg mass along the line connecting the two fixed masses. Find the magnitude and direction of the acceleration of the particle.

26. How many kilometers would you have to go above the surface of the earth for your weight to decrease to half of what it was at the surface?

27. Your spaceship lands on an unknown planet. To determine the characteristics of this planet, you drop a 1.30 kg wrench from 5.00 m above the ground and measure that it hits the ground 0.811s later. You also do enough surveying to determine that the circumference of the planet is 62,400 km. (a) What is the mass of the planet, in kilograms? (b) Express the planet’s mass in terms of the earth’s mass.

28. If an object’s weight is W on the earth, what would be its weight (in terms of W) if the earth had (a) twice its present mass, but was the same size, (b) half its present radius, but the same mass, (c) half its present radius and half its present mass, (d) twice its present radius and twice its present mass?

29. Huygens probe on Titan. In January 2005 the Huygens probe landed on Saturn’s moon Titan, the only satellite in the solar system having a thick atmosphere. Titan’s diameter is 5150 km, and its mass is 1.35 × 1023 kg. The probe weighed 3120 N on the earth. What did it weigh on the surface of Titan?

30. The mass of the moon is about 1/81 the mass of the earth, its radius is 1/4 that of the earth, and the acceleration due to gravity at the earth’s surface is 9.80 m/s2. Without looking up either body’s mass, use this information to compute the acceleration due to gravity on the moon’s surface.

31. Neutron stars, such as the one at the center of the Crab Nebula, have about the same mass as our sun, but a much smaller diameter. If you weigh 675 N on the earth, what would you weigh on the surface of a neutron star that has the same mass as our sun and a diameter of 20.0 km?

32. The asteroid 234 Ida has a mass of about 4.0 × 1016 kg and an average radius of about 16 km (it’s not spherical, but you can assume it is). (a) Calculate the acceleration of gravity on 234 Ida. (b) What would an astronaut whose earth weight is 650 N weigh on 234 Ida? (c) If you dropped a rock from a height of 1.0 m on 234 Ida, how long would it take to reach the ground? (d) If you can jump 60 cm straight up on earth, how high could you jump on 234 Ida? (Assume the asteroid’s gravity doesn’t weaken significantly over the distance of your jump.)

33. An earth satellite moves in a circular orbit with an orbital speed of 6200 m/s (a) Find the time of one revolution of the satellite. (b) Find the radial acceleration of the satellite in its orbit. (c) Make a free-body diagram of the satellite.

34. What is the period of revolution of a satellite with mass m that orbits the earth in a circular path of radius 7880 km (about 1500 km above the surface of the earth)?

35. What must be the orbital speed of a satellite in a circular orbit 780 km above the surface of the earth?

36. Planets beyond the solar system. On October 15, 2001, a planet was discovered orbiting around the star HD68988. Its orbital distance was measured to be 10.5 million kilometers from the center of the star, and its orbital period was estimated at 6.3 days. What is the mass of HD68988? Express your answer in kilograms and in terms of our sun’s mass.

37. Communications satellites. Communications satellites are used to bounce radio waves from the earth’s surface to send messages around the curvature of the earth. In order to be available all the time, they must remain above the same point on the earth’s surface and must move in a circle above the equator. (a) How long must it take for a communications satellite to make one complete orbit around the earth? (Such an orbit is said to be geosynchronous.) (b) Make a free-body diagram of the satellite in orbit. (c) Apply Newton’s second law to the satellite and find its altitude above the earth’s surface. (d) Draw the orbit of a communications satellite to scale on a sketch of the earth.

38. (a) Calculate the speed with which you would have to throw a rock to put it into orbit around the asteroid 234 Ida, near the surface (see Problem 32; assume 234 Ida is spherical, with the given radius and mass). (b) How long would it take for your rock to return and hit you in the back of the head?

39. In March 2006, two small satellites were discovered orbiting Pluto, one at a distance of 48,000 km and the other at 64,000 km. Pluto already was known to have a large satellite Charon, orbiting at 19,600 km with an orbital period of 6.39 days. Assuming that the satellites do not affect each other, find the orbital periods of the two small satellites without using the mass of Pluto.

40. Apparent weightlessness in a satellite. You have probably seen films of astronauts floating freely in orbiting satellites. People often think the astronauts are weightless because they are free of the gravity of the earth. Let us see if that explanation is correct. (a) Typically, such satellites orbit around 400 km above the surface of the earth. If an astronaut weighs 750 N on the ground, what will he weigh if he is 400 km above the surface? (b) Draw the orbit of the satellite in part (a) to scale on a sketch of the earth. (c) In light of your answers to parts (a) and (b), are the astronauts weightless because gravity is so weak? Why are they apparently weightless?

41. Baseball on Deimos! Deimos, a moon of Mars, is about 12 km in diameter, with a mass of 2.0 × 1015 kg. Suppose you are stranded alone on Deimos and want to play a one-person game of baseball. You would be the pitcher, and you would be the batter! With what speed would you have to throw a baseball so that it would go into orbit and return to you so you could hit it? Do you think you could actually throw it at that speed?

42. International Space Station. The International Space Station, launched in November 1998, makes 15.65 revolutions around the earth each day in a circular orbit. (a) Draw a free-body diagram of this satellite. (b) Apply Newton’s second law to the space station and find its height (in kilometers) above the earth’s surface.

43. One way to create artificial gravity in a space station is to spin it. If a cylindrical space station 275 m in diameter is to spin about its central axis, at how many revolutions per minute (rpm) must it turn so that the outermost points have an acceleration equal to g?

44. Hip wear on the moon. (a) Use data from Appendix E to calculate the acceleration due to gravity on the moon. (b) Calculate the friction force on a walking 65 kg astronaut carrying a 43 kg instrument pack on the moon if the coefficient of kinetic friction at her hip joint is 0.0050. (If necessary, see problem 32 in Chapter 5 and recall that approximately 65% of the body weight is above the hip.) (c) What would be the friction force on earth for this astronaut?

45. Volcanoes on Io. Jupiter’s moon Io has active volcanoes (in fact, it is the most volcanically active body in the solar system) that eject material as high as 500 km (or even higher) above the surface. Io has a mass of 8.94 ×1022 kg and a radius of 1815 km. Ignore any variation in gravity over the 500 km range of the debris. How high would this material go on earth if it were ejected with the same speed as on Io?

46. You are driving with a friend who is sitting to your right on the passenger side of the front seat of your car. You would like to be closer to your friend, so you 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 toward you? (b) If the coefficient of static friction between your friend and the car seat is 0.55 and you keep driving at a constant speed of 15 m/s, what is the maximum radius you could make your turn and still have your friend slide your way?

47. 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 on which the people were standing dropped about 0.5 m. The people remained pinned against the wall. (a) Draw a free-body 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?

48. Physical training. As part of a training program, an athlete runs while holding 8.0 kg weights in each hand. As he runs, the weights swing through a 30.0° arc in 1/3 s at essentially constant speed. His hands are 72 cm from his shoulder joint, and they are light enough that we can neglect their weight compared with that of the 8.0 kg weight he is carrying. (a) Make a free-body diagram of one of the 8.0 kg weights at the bottom of its swing. (b) What force does the runner’s hand exert on each weight at the bottom of the swing?

49. The asteroid Toro has a radius of about 5.0 km. Consult Appendix E as necessary. (a) Assuming that the density of Toro is the same as that of the earth (5.5 g/cm3) find its total mass and find the acceleration due to gravity at its surface. (b) Suppose an object is to be placed in a circular orbit around Toro, with a radius just slightly larger than the asteroid’s radius. What is the speed of the object? Could you launch yourself into orbit around Toro by running?

50. 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 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.

51. Exploring Europa. Europa, a satellite of Jupiter, is believed to have an ocean of liquid water (with the possibility of life) beneath its icy surface. (See Figure 6.29.) Europa is 3130 km in diameter and has a mass of 4.78 × 1022 kg. In the future, we will surely want to send astronauts to investigate Europa. In planning such a future mission, what is the fastest that such an astronaut could walk on the surface of Europa if her legs are 1.0 m long?

52. The star Cancri is 57 light-years from the earth and has a mass 0.85 times that of our sun. A planet has been detected in a circular orbit around Rho Cancri with an orbital radius equal to 0.11 times the radius of the earth’s orbit around the sun. What are (a) the orbital speed and (b) the orbital period of the planet of Rho Cancri?

53. Catch a piece of a comet. The Stardust spacecraft was designed to visit a comet and bring samples of its material back to the earth. The craft is 1.7 m across and has a mass of 385 kg. In January 2004, from a distance of 237 km, it took the photograph of the nucleus of comet Wild2, shown in Figure 6.30. The distance across this nucleus is 5 km, and we can model it as an approximate sphere. Since astronomers think that comets have a composition similar to that of Pluto, we can assume a density the same as Pluto’s, which is 2.1 g/cm3. The samples taken were returned to earth on Jan. 15, 2006. (a) What gravitational force did the comet exert on the spacecraft when it took the photo in the figure? (b) Compare this force with the force that the earth’s gravity exerted on it when the spacecraft was on the launch pad.

54. The 4.00 kg block in the figure 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 Figure 6.31 and the tension in the upper string is 80.0 N. (a) What is the tension in the lower cord? Start with a free-body diagram of the block. (b) What is the speed of the block?

55. As your bus rounds a flat curve at constant speed, a package with mass 0.500 kg, suspended from the luggage compartment of the bus by a string 45.0 cm long, is found to hang at rest relative to the bus, with the string making an angle of 30.0° with the vertical. In this position, the package is 50.0 m from the center of curvature of the curve. What is the speed of the bus?

56. Artificial gravity in 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 cylindrical space station that spins about an axis through its center at a constant rate. (See Figure 6.32.). This spin creates “artificial gravity” at the outside rim of the station. (a) If the diameter of the space station is 800.0 m, how fast must the rim be moving in order for the “artificial gravity” acceleration to be g at the outer rim? (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 fast must the rim move in this case? (c) Make a free-body diagram of an astronaut at the outer rim.

57. If the diameter of the space station is 1000 m, how fast must the outer edge of the space station move to give an astronaut the experience of a reduced “gravity” of 5 m/s2 (roughly 1⁄2 earth normal)? You may assume that the astronaut is standing on the inner wall at a distance of nearly 1000 m from the axis of the space station.

A. 5000 m/s

B. 2500 m/s

C. 71 m/s

D. 50 m/s

58. What would be the resulting period of rotation of the space station? You may assume that the space station has a circumference of approximately 3000 m.

A. 0.6 s

B. 3 s

C. 42 s

D. 60 s

E. 380 s

59. Under these same conditions, how much force would an astronaut need to exert on a 2-kg space helmet that she is holding to keep it moving on the same circular path that she follows? Note: This force is known as the “effective weight” of the space helmet.

A. 20 N

B. 10 N

C. 5 N

D. 0 N

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