Problems

Note to Instructors: Most of the homework problems in this chapter are available for assignment via an on-line homework management program such as Wiley PLUS or WebAssign, and those marked with the icon are presented in a guided tutorial format that provides enhanced interactivity. See Preface for additional details.
ssm Solution is in the Student Solutions Manual.
This icon represents a biomedical application.
Section 5.1 Uniform Circular Motion,
Section 5.2 Centripetal Acceleration

| 1 | ssm How long does it take a plane, traveling at a constant speed of 110 m/s, to fly once around a circle whose radius is 2850 m?

| 2 | Review Conceptual Example 2 in preparation for this problem. In Figure 5-4, an object, after being released from its circular path, travels the distance OA in the same time it would have moved from O to P on the circle. The speed of the object on and off the circle remains constant at the same value. Suppose that the radius of the circle in Figure 5-4 is 3.6 m and the angle is 25°. What is the distance OA?

| 3 | Interactive Solution 5.3 presents a method for modeling this problem. The blade of a windshield wiper moves through an angle of 90.0° in 0.40 s. The tip of the blade moves on the arc of a circle that has a radius of 0.45 m. What is the magnitude of the centripetal acceleration of the tip of the blade?

| 4 | A car travels at a constant speed around a circular track whose radius is 2.6 km. The car goes once around the track in 360 s. What is the magnitude of the centripetal acceleration of the car?

| 5 | ssm Speedboat A negotiates a curve whose radius is 120 m. Speedboat B negotiates a curve whose radius is 240 m. Each boat experiences the same centripetal acceleration. What is the ratio of the speeds of the boats?

| 6 | The National Aeronautics and Space Administration (NASA) studies the physiological effects of large accelerations on astronauts. Some of these studies use a machine known as a centrifuge. This machine consists of a long arm, to one end of which is attached a chamber in which the astronaut sits. The other end of the arm is connected to an axis about which the arm and chamber can be rotated. The astronaut moves on a circular path, much like a model airplane flying in a circle on a guideline. The chamber is located 15 m from the center of the circle. At what speed must the chamber move so that an astronaut is subjected to 7.5 times the acceleration due to gravity?

| 7 | There is a clever kitchen gadget for drying lettuce leaves after you wash them. It consists of a cylindrical container mounted so that it can be rotated about its axis by turning a hand crank. The outer wall of the cylinder is perforated with small holes. You put the wet leaves in the container and turn the crank to spin off the water. The radius of the container is 12 cm. When the cylinder is rotating at 2.0 revolutions per second, what is the magnitude of the centripetal acceleration at the outer wall?

* | 8 | Each of the space shuttle’s main engines is fed liquid hydrogen by a high-pressure pump. Turbine blades inside the pump rotate at 617 rev/s. A point on one of the blades traces out a circle with a radius of 0.020 m as the blade rotates. (a) What is the magnitude of the centripetal acceleration that the blade must sustain at this point? (b) Express this acceleration as a multiple of .

* | 9 | ssm The large blade of a helicopter is rotating in a horizontal circle. The length of the blade is 6.7 m, measured from its tip to the center of the circle. Find the ratio of the centripetal acceleration at the end of the blade to that which exists at a point located 3.0 m from the center of the circle.

* | 10 | A computer is reading data from a rotating CD-ROM. At a point that is 0.030 m from the center of the disc, the centripetal acceleration is 120 m/s2. What is the centripetal acceleration at a point that is 0.050 m from the center of the disc?
Section 5.3 Centripetal Force

| 11 | ssm Review Example 3, which deals with the bobsled in Figure 5-5. Also review Conceptual Example 4. The mass of the sled and its two riders in Figure 5-5 is 350 kg. Find the magnitude of the centripetal force that acts on the sled during the turn with a radius of (a) 33 m and (b) 24 m.

| 12 | In a skating stunt known as “crack-the-whip,” a number of skaters hold hands and form a straight line. They try to skate so that the line rotates about the skater at one end, who acts as the pivot. The skater farthest out has a mass of 80.0 kg and is 6.10 m from the pivot. He is skating at a speed of 6.80 m/s. Determine the magnitude of the centripetal force that acts on him.

| 13 | Concept Simulation 5.1 at reviews the concepts that are involved in this problem. A child is twirling a 0.0120-kg ball on a string in a horizontal circle whose radius is 0.100 m. The ball travels once around the circle in 0.500 s. (a) Determine the centripetal force acting on the ball. (b) If the speed is doubled, does the centripetal force double? If not, by what factor does the centripetal force increase?

| 14 | Multiple-Concept Example 7 and Concept Simulation 5.2 at review the concepts that play a role in this problem. Car A uses tires for which the coefficient of static friction is 1.1 on a particular unbanked curve. The maximum speed at which the car can negotiate this curve is 25 m/s. Car B uses tires for which the coefficient of static friction is 0.85 on the same curve. What is the maximum speed at which car B can negotiate the curve?

| 15 | At an amusement park there is a ride in which cylindrically shaped chambers spin around a central axis. People sit in seats facing the axis, their backs against the outer wall. At one instant the outer wall moves at a speed of 3.2 m/s, and an 83-kg person feels a 560-N force pressing against his back. What is the radius of a chamber?

| 16 | Review Multiple-Concept Example 7 to see the basic ideas underlying this problem. A stone has a mass of and is wedged into the tread of an automobile tire, as the drawing shows. The coefficient of static friction between the stone and each side of the tread channel is 0.90. When the tire surface is rotating at a maximum speed of 13 m/s, the stone flies out of the tread. The magnitude FN of the normal force that each side of the tread channel exerts on the stone is 1.8 N. Assume that only static friction supplies the centripetal force, and determine the radius r of the tire.

* | 17 | ssm www The hammer throw is a track-and-field event in which a 7.3-kg ball (the “hammer”) is whirled around in a circle several times and released. It then moves upward on the familiar curving path of projectile motion and eventually returns to earth some distance away. The world record for this distance is 86.75 m, achieved in 1986 by Yuriy Sedykh. Ignore air resistance and the fact that the ball is released above the ground rather than at ground level. Furthermore, assume that the ball is whirled on a circle that has a radius of 1.8 m and that its velocity at the instant of release is directed 41° above the horizontal. Find the magnitude of the centripetal force acting on the ball just prior to the moment of release.

* | 18 | A block is hung by a string from the inside roof of a van. When the van goes straight ahead at a speed of 28 m/s, the block hangs vertically down. But when the van maintains this same speed around an unbanked curve (radius = 150 m), the block swings toward the outside of the curve. Then the string makes an angle with the vertical. Find .

* | 19 | A rigid massless rod is rotated about one end in a horizontal circle. There is a mass m1 attached to the center of the rod and a mass m2 attached to the outer end of the rod. The inner section of the rod sustains three times as much tension as the outer section. Find the ratio .
Section 5.4 Banked Curves

| 20 | Two banked curves have the same radius. Curve A is banked at an angle of 13°, and curve B is banked at an angle of 19°. A car can travel around curve A without relying on friction at a speed of 18 m/s. At what speed can this car travel around curve B without relying on friction?

| 21 | ssm A curve of radius 120 m is banked at an angle of 18°. At what speed can it be negotiated under icy conditions where friction is negligible?

| 22 | On a banked race track, the smallest circular path on which cars can move has a radius of 112 m, while the largest has a radius of 165 m, as the drawing illustrates. The height of the outer wall is 18 m. Find (a) the smallest and (b) the largest speed at which cars can move on this track without relying on friction.

| 23 | Before attempting this problem, review Examples 7 and 8 and 7 and 8. Two curves on a highway have the same radii. However, one is unbanked and the other is banked at an angle . A car can safely travel along the unbanked curve at a maximum speed v0 under conditions when the coefficient of static friction between the tires and the road is . The banked curve is frictionless, and the car can negotiate it at the same maximum speed v0. Find the angle of the banked curve.

* | 24 | As an aid in working this problem, first consult Interactive Solution 5.24 at . A racetrack has the shape of an inverted cone, as the drawing shows. On this surface the cars race in circles that are parallel to the ground. For a speed of 34.0 m/s, at what value of the distance d should a driver locate his car if he wishes to stay on a circular path without depending on friction?

* | 25 | ssm www A jet (), flying at 123 m/s, banks to make a horizontal circular turn. The radius of the turn is 3810 m. Calculate the necessary lifting force.

** | 26 | The drawing shows a baggage carousel at an airport. Your suitcase has not slid all the way down the slope and is going around at a constant speed on a circle (r = 11.0 m) as the carousel turns. The coefficient of static friction between the suitcase and the carousel is 0.760, and the angle in the drawing is 36.0°. How much time is required for your suitcase to go around once?
Section 5.5 Satellites in Circular Orbits, Section 5.6 Apparent Weightlessness and Artificial Gravity

| 27 | ssm www A satellite is in a circular orbit around an unknown planet. The satellite has a speed of , and the radius of the orbit is . A second satellite also has a circular orbit around this same planet. The orbit of this second satellite has a radius of . What is the orbital speed of the second satellite?

| 28 | Consult Multiple-Concept Example 14 for background pertinent to this problem. In designing rotating space stations to provide for artificial-gravity environments, one of the constraints that must be considered is motion sickness. Studies have shown that the negative effects of motion sickness begin to appear when the rotational motion is faster than two revolutions per minute. On the other hand, the magnitude of the centripetal acceleration at the astronauts’ feet should equal the magnitude of the acceleration due to gravity on earth. Thus, to eliminate the difficulties with motion sickness, designers must choose the distance between the astronauts’ feet and the axis about which the space station rotates to be greater than a certain minimum value. What is this minimum value?

| 29 | A satellite is placed in orbit above the surface of the planet Jupiter. Jupiter has a mass of and a radius of . Find the orbital speed of the satellite.

| 30 | A satellite is in a circular orbit about the earth (). The period of the satellite is . What is the speed at which the satellite travels?

| 31 | Review Interactive Solution 5.31 at before beginning this problem. Two satellites, A and B, are in different circular orbits about the earth. The orbital speed of satellite A is three times that of satellite B. Find the ratio () of the periods of the satellites.

* | 32 | The earth orbits the sun once per year at the distance of . Venus orbits the sun at a distance of . These distances are between the centers of the planets and the sun. How long (in earth days) does it take for Venus to make one orbit around the sun?

* | 33 | ssm A satellite has a mass of 5850 kg and is in a circular orbit above the surface of a planet. The period of the orbit is two hours. The radius of the planet is . What is the true weight of the satellite when it is at rest on the planet’s surface?

* | 34 | A satellite moves on a circular earth orbit that has a radius of . A model airplane is flying on a 15-m guideline in a horizontal circle. The guideline is parallel to the ground. Find the speed of the plane such that the plane and the satellite have the same centripetal acceleration.

** | 35 | Multiple-Concept Example 14 deals with the issues upon which this problem focuses. To create artificial gravity, the space station shown in the drawing is rotating at a rate of 1.00 rpm. The radii of the cylindrically shaped chambers have the ratio . Each chamber A simulates an acceleration due to gravity of 10.0 m/s2. Find values for (a) rA, (b) rB, and (c) the acceleration due to gravity that is simulated in chamber B.
Section 5.7 Vertical Circular Motion

| 36 | A roller coaster at an amusement park has a dip that bottoms out in a vertical circle of radius r. A passenger feels the seat of the car pushing upward on her with a force equal to twice her weight as she goes through the dip. If r = 20.0 m, how fast is the roller coaster traveling at the bottom of the dip?

| 37 | ssm A 2100-kg demolition ball swings at the end of a 15-m cable on the arc of a vertical circle. At the lowest point of the swing, the ball is moving at a speed of 7.6 m/s. Determine the tension in the cable. (Hint: The tension serves the same purpose as the normal force at point 1 in Figure 5-21.)

| 38 | Pilots of high-performance fighter planes can be subjected to large centripetal accelerations during high-speed turns. Because of these accelerations, the pilots are subjected to forces that can be much greater than their body weight, leading to an accumulation of blood in the abdomen and legs. As a result, the brain becomes starved for blood, and the pilot can lose consciousness (“black out”). The pilots wear “anti-G suits” to help keep the blood from draining out of the brain. To appreciate the forces that a fighter pilot must endure, consider the magnitude FN of the normal force that the pilot’s seat exerts on him at the bottom of a dive. The magnitude of the pilot’s weight is W. The plane is traveling at 230 m/s on a vertical circle of radius 690 m. Determine the ratio FN/W. For comparison, note that black-out can occur for values of FN/W as small as 2 if the pilot is not wearing an anti-G suit.

| 39 | ssm For the normal force in Figure 5-21 to have the same magnitude at all points on the vertical track, the stunt driver must adjust the speed to be different at different points. Suppose, for example, that the track has a radius of 3.0 m and that the driver goes past point 1 at the bottom with a speed of 15 m/s. What speed must she have at point 3, so that the normal force at the top has the same magnitude as it did at the bottom?

* | 40 | A motorcycle is traveling up one side of a hill and down the other side. The crest is a circular arc with a radius of 45.0 m. Determine the maximum speed that the cycle can have while moving over the crest without losing contact with the road.

* | 41 | Reviewing Interactive Solution 5.41 at will help in solving this problem. A stone is tied to a string (length = 1.10 m) and whirled in a circle at the same constant speed in two different ways. First, the circle is horizontal and the string is nearly parallel to the ground. Next, the circle is vertical. In the vertical case the maximum tension in the string is 15.0% larger than the tension that exists when the circle is horizontal. Determine the speed of the stone.

** | 42 | In an automatic clothes dryer, a hollow cylinder moves the clothes on a vertical circle (radius r = 0.32 m), as the drawing shows. The appliance is designed so that the clothes tumble gently as they dry. This means that when a piece of clothing reaches an angle of above the horizontal, it loses contact with the wall of the cylinder and falls onto the clothes below. How many revolutions per second should the cylinder make in order that the clothes lose contact with the wall when ?

Copyright © 2007 John Wiley & Sons, Inc. All rights reserved.