Problems - Birmingham Schools



Chapter 7 Problems

1, 2, 3 = straightforward, intermediate, challenging

Section 7.1 Angular Speed and Angular Acceleration

1. The tires on a new compact car have a diameter of 2.0 ft and are warranted for 60 000 miles. (a) Determine the angle (in radians) through which one of these tires will rotate during the warranty period. (b) How many revolutions of the tire are equivalent to your answer in (a) above?

2. A wheel has a radius of 4.1 m. How far (path length) does a point on the circumference travel if the wheel is rotated through angles of 30°, 30 rad, and 30 rev, respectively?

3. Find the angular speed of Earth about the Sun in radians per second and degrees per day.

4. A potter’s wheel moves from rest to an angular speed of 0.20 rev/s in 30 s. Find its angular acceleration in radians per second per second.

Section 7.2 Rotational Motion Under Constant Angular Acceleration

Section 7.3 Relations Between Angular and Linear Quantities

5. A dentist’s drill starts from rest. After 3.20 s of constant angular acceleration, it turns at a rate of 2.51 x 104 rev/min. (a) Find the drill’s angular acceleration. (b) Determine the angle (in radians) through which the drill rotates during this period.

6. An electric motor rotating a workshop grinding wheel at a rate of 100 rev/min is switched off. Assume constant negative angular acceleration of magnitude 2.00 rad/s2. (a) How long does it take for the grinding wheel to stop? (b) Through how many radians has the wheel turned during the interval found in (a)?

7. A car is traveling with a velocity of 17.0 m/s on a straight horizontal highway. The wheels of the car have a radius of 48.0 cm. If the car then speeds up with an acceleration of 2.00 m/s2 for 5.00 s, find the number of revolutions of the wheels during this period.

8. The car in the preceding problem runs out of gas and the rotation of the wheels slows with an angular acceleration of magnitude 1.35 rad/s 2 . If the car has a speed of 24.0 m/s when this happens, through how many revolutions do the wheels turn as the car comes to rest?

9. The diameters of the main rotor and tail rotor of a single-engine helicopter are 7.60 m and 1.02 m, respectively. The respective rotational speeds are 450 rev/min and 4 138 rev/min. Calculate the speeds of the tips of both rotors. Compare these speeds with the speed of sound, 343 m/s.

10. The tub of a washer goes into its spin-dry cycle, starting from rest and reaching an angular speed of 5.0 rev/s in 8.0 s. At this point the person doing the laundry opens the lid, and a safety switch turns off the washer. The tub slows to rest in 12.0 s. Through how many revolutions does the tub turn during this 20-s interval? Assume constant angular acceleration while it is starting and stopping.

11. A standard cassette tape is placed in a standard cassette player. Each side lasts for 30 minutes. The two tape wheels of the cassette fit onto two spindles in the player. Suppose that a motor drives one spindle at constant angular velocity ~1 rad/s and the other spindle is free to rotate at any angular speed. Find the order of magnitude of the tape’s thickness. Specify any other quantities you estimate and the values you take for them.

12. A coin with a diameter of 2.40 cm is dropped on edge onto a horizontal surface. The coin starts out with an initial angular speed of 18.0 rad/s and rolls in a straight line without slipping. If the rotation slows with an angular acceleration of magnitude 1.90 rad/s2, how far does the coin roll before coming to rest?

13. A pulsar is a celestial object that emits light in short bursts. A pulsar in the Crab Nebula flashes at a rate of 30 times/s. Suppose the light pulses are caused by the rotation of a spherical object that emits light from a pair of diametrically opposed “flashlights” on its equator. What is the maximum radius of the pulsar if no part of its surface can move faster than the speed of light (3.00 x 108 m/s)?

Section 7.4 Centripetal Acceleration

14. As noted in Conceptual Question 11, it has been suggested that rotating cylinders about 10 mi long and 5.0 mi in diameter be placed in space and used as colonies. What angular speed must such a cylinder have so that the centripetal acceleration at its surface equals the free-fall acceleration?

15. Find the centripetal accelerations of (a) a point on the Equator and (b) at North Pole, due to the rotation of Earth about its axis.

16. In civil aviation, a “standard turn” for level flight of a propeller-driven airplane is one in which the airplane makes a complete circular turn in 2.0 min. If the plane’s speed is 180 m/s, what is the radius of the circle? What is the centripetal acceleration of the plane?

17. (a) What is the tangential acceleration of a bug on the rim of a 10-in.-diameter disk if the disk moves from rest to an angular speed of 78 rev/min in 3.0 s? (b) When the disk is at its final speed, what is the tangential velocity of the bug? (c) One second after the bug starts from rest, what are its tangential acceleration, centripetal acceleration, and total acceleration?

18. A race car starts from rest on a circular track of radius 400 m. The car’s speed increases at the constant rate of 0.500 m/s2. At the point where the magnitudes of the centripetal and tangential accelerations are equal, determine (a) the speed of the race car, (b) the distance traveled, and (c) the elapsed time.

Section 7.6 Forces Causing Centripetal Acceleration

Section 7.7 Describing Forces in Accelerated Reference Frames

19. A 55.0-kg ice skater is moving at 4.00 m/s when she grabs the loose end of a rope, the opposite end of which is tied to a pole. She then moves in a circle of radius 0.800 m around the pole. (a) Determine the force exerted by the horizontal rope on her arms. (b) Compare this force with her weight.

20. A sample of blood is placed in a centrifuge of radius 15.0 cm. The mass of a red blood cell is 3.0 x 10 –16 kg, and the magnitude of the force acting on it as it settles out of the plasma is 4.0 x 10–11 N. At how many revolutions per second should the centrifuge be operated?

21. A 2 000-kg car rounds a circular turn of radius 20 m. If the road is flat and the coefficient of friction between tires and road is 0.70, how fast can the car go without skidding?

22. The cornering performance of an automobile is evaluated on a skid-pad, where the maximum speed that a car can maintain around a circular path on a dry, flat surface is measured. Then the centripetal acceleration, also called the lateral acceleration, is calculated as a multiple of the free-fall acceleration g. The main factors affecting the performance are the tire characteristics and the suspension system of the car. A Dodge Viper GTS can negotiate a skid-pad of radius 61.0 m at 86.5 km/h. Calculate its maximum lateral acceleration.

23. A 50.0-kg child stands at the rim of a merry-go-round of radius 2.00 m, rotating with an angular speed of 3.00 rad/s. (a) What is the child’s centripetal acceleration? (b) What is the minimum force between his feet and the floor of the carousel that is required to keep him in the circular path? (c) What minimum coefficient of static friction is required? Is the answer you found reasonable? In other words, is he likely to stay on the merry-go-round?

24. An engineer wishes to design a curved exit ramp for a toll road in such a way that a car will not have to rely on friction to round the curve without skidding. She does so by banking the road in such a way that the force causing the centripetal acceleration will be supplied by the component of the normal force toward the center of the circular path. (a) Show that for a given speed v and a radius r, the curve must be banked at the angle θ such that tan θ = v2 /rg. (b) Find the angle at which the curve should be banked if a typical car rounds it at a 50.0-m radius and a speed of 13.4 m/s.

25. An air puck of mass 0.25 kg is tied to a string and allowed to revolve in a circle of radius 1.0 m on a frictionless horizontal table. The other end of the string passes through a hole in the center of the table, and a mass of 1.0 kg is tied to it (Fig. P7.25). The suspended mass remains in equilibrium while the puck on the tabletop revolves. (a) What is the tension in the string? (b) What is the horizontal force acting on the puck? (c) What is the speed of the puck?

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Figure P7.25

26. Tarzan (m = 85 kg) tries to cross a river by swinging from a 10-m-long vine. His speed at the bottom of the swing (as he just clears the water) is 8.0 m/s. Tarzan doesn’t know that the vine has a breaking strength of 1 000 N. Does he make it safely across the river? Justify your answer.

27. A 40.0-kg child takes a ride on a Ferris wheel that rotates four times each minute and has a diameter of 18.0 m. (a) What is the centripetal acceleration of the child? (b) What force (magnitude and direction) does the seat exert on the child at the lowest point of the ride? (c) What force does the seat exert on the child at the highest point of the ride? (d) What force does the seat exert on the child when she is halfway between the top and bottom?

28. A roller-coaster vehicle has a mass of 500 kg when fully loaded with passengers (Fig. P7.28). (a) If the vehicle has a speed of 20.0 m/s at A, what is the magnitude of the force that the track exerts on the vehicle at this point? (b) What is the maximum speed the vehicle can have at B in order for gravity to hold it on the track?

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Figure P7.28

Section 7.8 Newton’s Law of Universal Gravitation

Section 7.9 Gravitational Potential Energy Revisited

Section 7.10 Kepler’s Laws

29. The average distance separating Earth and the Moon is 384 000 km. Use the data in Table 7.3 to find the net gravitational force exerted by Earth and the Moon on a spaceship with mass 3.00 x 104 kg located halfway between them.

30. During a solar eclipse, the Moon, Earth, and Sun all lie on the same line, with the Moon between Earth and the Sun. (a) What force is exerted by the Sun on the Moon? (b) What force is exerted by Earth on the Moon? (c) What force is exerted by the Sun on Earth? (See Table 7.3 and Problem 29 above.)

31. A coordinate system (in meters) is constructed on the surface of a pool table, and three objects are placed on the coordinate system as follows: a 2.0-kg object at the origin, a 3.0-kg object at (0, 2.0), and a 4.0-kg object at (4.0, 0). Find the resultant gravitational force exerted by the other two objects on the object at the origin.

32. Use the data of Table 7.3 to find the point between Earth and the Sun at which an object can be placed so that the net gravitational force exerted by Earth and Sun on this object is zero.

33. Objects with masses of 200 kg and 500 kg are separated by 0.400 m. (a) Find the net gravitational force exerted by these objects on a 50.0-kg object placed midway between them. (b) At what position (other than infinitely remote ones) can the 50.0-kg object be placed so as to experience a net force of zero?

34. Two objects attract each other with a gravitational force of magnitude 1.00 x 10–8 N when separated by 20.0 cm. If the total mass of the two objects is 5.00 kg, what is the mass of each?

35. A satellite moves in a circular orbit around Earth at a speed of 5 000 m/s. Determine (a) the satellite’s altitude above Earth’s surface and (b) the period of the satellite’s orbit.

36. A 600-kg satellite is in a circular orbit about Earth at a height above Earth equal to Earth’s mean radius. Find (a) the satellite’s orbital speed, (b) the period of its revolution, and (c) the gravitational force acting on it.

37. Io, a satellite of Jupiter, has an orbital period of 1.77 days and an orbital radius of 4.22 x 105 km. From these data, determine the mass of Jupiter.

38. A satellite has a mass of 100 kg and is located at 2.00 x 106 m above Earth’s surface. (a) What is the potential energy associated with the satellite at this location? (b) What is the magnitude of the gravitational force on the satellite?

39. A satellite of mass 200 kg is launched from a site on the Equator into an orbit at 200 km above Earth’s surface. (a) If the orbit is circular, what is the orbital period of this satellite? (b) What is the satellite’s speed in orbit? (c) What is the minimum energy necessary to place this satellite in orbit, assuming no air friction?

40. Neutron stars are extremely dense objects that are formed from the remnants of supernova explosions. Many rotate very rapidly. Suppose that the mass of a certain spherical neutron star is twice the mass of the Sun and its radius is 10.0 km. Determine the greatest possible angular speed it can have so that the matter at the surface of the star on its equator is just held in orbit by the gravitational force.

Additional Problems

41. (a) Find the angular speed of Earth’s rotation on its axis. As Earth turns toward the east, we see the sky turning toward the west at this same rate.

(b) The rainy Pleiads wester

And seek beyond the sea

The head that I shall dream of

That shall not dream of me.

—A. E. Housman (© Robert E. Symons)

Cambridge, England, is at longitude 0° and Saskatoon, Saskatchewan, is at longitude 107° west. How much time elapses after the Pleiades set in Cambridge until these stars fall below the western horizon in Saskatoon?

42. The Mars probe Pathfinder is designed to drop the instrument package from a height of 20 m above the surface, after the speed of the probe has been brought to zero by a combination parachute-rocket system at that height. To cushion the landing, giant airbags surround the package. The mass of Mars is 0.107 4 times that of Earth and the radius of Mars is 0.528 2 that of Earth. Find (a) the acceleration due to gravity at the surface of Mars and (b) how long it takes for the instrument package to fall the last 20 meters.

43. An athlete swings a 5.00-kg ball horizontally on the end of a rope. The ball moves in a circle of radius 0.800 m at an angular speed of 0.500 rev/s. What are (a) the tangential speed of the ball and (b) its centripetal acceleration? (c) If the maximum tension the rope can withstand before breaking is 100 N, what is the maximum tangential speed the ball can have?

44. A digital audio compact disc carries data, with each bit occupying 0.6 μm, along a continuous spiral track from the inner circumference of the disc to the outside edge. A CD player turns the disc to carry the track counterclockwise above a lens at a constant speed of 1.30 m/s. Find the required angular speed (a) at the beginning of the recording, where the spiral has a radius of 2.30 cm and (b) at the end of the recording, where the spiral has a radius of 5.80 cm. (c) A full-length recording lasts for 74 min 33 s. Find the average angular acceleration of the disc. (d) Assuming that the acceleration is constant, find the total angular displacement of the disc as it plays. (e) Find the total length of the track.

45. The Solar Maximum Mission Satellite was placed in a circular orbit about 150 mi above Earth. Determine (a) the orbital speed of the satellite and (b) the time required for one complete revolution.

46. A car rounds a banked curve where the radius of curvature of the road is R, the banking angle is θ, and the coefficient of static friction is μ. (a) Determine the range of speeds the car can have without slipping up or down the road. (b) What is the range of speeds possible if R = 100 m, θ = 10°, and μ = 0.10 (slippery conditions)?

47. A car moves at speed v across a bridge made in the shape of a circular arc of radius r. (a) Find an expression for the normal force acting on the car when it is at the top of the arc. (b) At what minimum speed will the normal force become zero (causing occupants of the car to seem weightless) if r = 30.0 m?

48. A 0.400-kg pendulum bob passes through the lowest part of its path at a speed of 3.00 m/s. (a) What is the tension in the pendulum cable at this point if the pendulum is 80.0 cm long? (b) When the pendulum reaches its highest point, what angle does the cable make with the vertical? (c) What is the tension in the pendulum cable when the pendulum reaches its highest point?

49. Because of Earth’s rotation about its axis, a point on the Equator experiences a centripetal acceleration of 0.034 0 m/s2 while a point at the poles experiences no centripetal acceleration. (a) Show that at the Equator the gravitational force on an object (the true weight) must exceed the object’s apparent weight. (b) What are the apparent weights at the Equator and at the poles of a 75.0-kg person? (Assume Earth is a uniform sphere, and take g = 9.800 m/s2.)

50. A stunt man whose mass is 70 kg swings from the end of a 4.0-m-long rope along the arc of a vertical circle. Assuming he starts from rest when the rope is horizontal, find the tensions in the rope that are required to make him follow his circular path, (a) at the beginning of his motion, (b) at a height of 1.5 m above the bottom of the circular arc, and (c) at the bottom of his arc.

51. In a popular amusement park ride, a rotating cylinder of radius 3.00 m is set in rotation at an angular speed of 5.00 rad/s, as in Figure P7.51. The floor then drops away, leaving the riders suspended against the wall in a vertical position. What minimum coefficient of friction between a rider’s clothing and the wall is needed to keep the rider from slipping? (Hint: Recall that the magnitude of the maximum force of static friction is equal to μn, where n is the normal force—in this case, the force causing the centripetal acceleration.

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Figure P7.51

52. A 0.50-kg ball that is tied to the end of a 1.5-m light cord is revolved in a horizontal plane with the cord making a 30° angle with the vertical (see Fig. 7.14). (a) Determine the ball’s speed. (b) If instead the ball is revolved so that its speed is 4.0 m/s, what angle does the cord make with the vertical? (c) If the cord can withstand a maximum tension of 9.8 N, what is the highest speed at which the ball can move?

53. A skier starts at rest at the top of a large hemispherical hill (Fig. P7.53). Neglecting friction, show that the skier will leave the hill and become airborne at a distance of h = R/3 below the top of the hill. (Hint: At this point, the normal force goes to zero.)

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Figure P7.53

54. After consuming all of its nuclear fuel, a massive star can collapse to form a black hole, which is an immensely dense object for which the escape speed is greater than the speed of light. Newton’s law of universal gravitation still describes the force that a black hole exerts on objects outside it. A spacecraft in the shape of a long cylinder has a length of 100 m and its mass with occupants is 1 000 kg. It has strayed too close to a 1.0-m-radius black hole having a mass 100 times that of the Sun (Fig. P7.54). (a) If the nose of the spacecraft points toward the center of the black hole, and if distance between the nose of the spacecraft and the black hole’s center is 10 km, determine the total force on the spacecraft. (b) What is the difference in the force per kilogram of mass felt by the occupants in the nose of the ship and those in the rear of the ship farthest from the black hole?

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Figure P7.54

55. In Robert Heinlein’s The Moon Is a Harsh Mistress, the colonial inhabitants of the Moon threaten to launch rocks down onto Earth if they are not given independence (or at least representation). Assuming that a gun could launch a rock of mass m at twice the lunar escape speed, calculate the speed of the rock as it enters Earth’s atmosphere.

56. Show that the escape speed from the surface of a planet of uniform density is directly proportional to the radius of the planet.

57. A massless spring of constant k = 78.4 N/m is fixed on the left side of a level track. A block of mass m = 0.50 kg is pressed against the spring and compresses it a distance d, as in Figure P7.57. The block (initially at rest) is then released and travels toward a circular loop-the-loop of radius R = 1.5 m. The entire track and the loop-the-loop are frictionless except for the section of track between points A and B. Given that the coefficient of kinetic friction between the block and the track along AB is μk = 0.30, and that the length of AB is 2.5 m, determine the minimum compression d of the spring that enables the block to just make it through the loop-the-loop at point C. (Hint: The force exerted by the track on the block will be zero if the block barely makes it through the loop-the-loop.)

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Figure P7.57

58. A small block of mass m = 0.50 kg is fired with an initial speed of v0 = 4.0 m/s along a horizontal section of frictionless track, as shown in the top portion of Figure P7.58. The block then moves along the frictionless semicircular, vertical tracks of radius R = 1.5 m. (a) Determine the force exerted by the track on the block at points A and B. (b) The bottom of the track consists of a section (L = 0.40 m) with friction. Determine the coefficient of kinetic friction between the block and that portion of the bottom track if the block just makes it to point C on the first trip. (Hint: If the block just makes it to point C, the force of contact exerted by the track on the block at that point should be zero.)

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Figure P7.58

59. A frictionless roller coaster is given an initial velocity of v0 at height h, as in Figure P7.59. The radius of curvature of the track at point A is R. (a) Find the maximum value of v0 so that the roller coaster stays on the track at A solely because of gravity. (b) Using the value of v0 calculated in (a), determine the value of h' that is necessary if the roller coaster just makes it to point B.

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