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 8.1 Rotational Motion and Angular Displacement,
Section 8.2 Angular Velocity and Angular Acceleration

| 1 | ssm In Europe, surveyors often measure angles in grads. There are 100 grads in one-quarter of a circle. How many grads are there in one radian?

| 2 | The earth spins on its axis once a day and orbits the sun once a year (365¼ days). Determine the average angular velocity (in rad/s) of the earth as it (a) spins on its axis and (b) orbits the sun. In each case, take the positive direction for the angular displacement to be the direction of the earth’s motion.

| 3 | ssm A pitcher throws a curveball that reaches the catcher in 0.60 s. The ball curves because it is spinning at an average angular velocity of 330 rev/min (assumed constant) on its way to the catcher’s mitt. What is the angular displacement of the baseball (in radians) as it travels from the pitcher to the catcher?

| 4 | A pulsar is a rapidly rotating neutron star that continuously emits a beam of radio waves in a searchlight manner. Each time the pulsar makes one revolution, the rotating beam sweeps across the earth, and the earth receives a pulse of radio waves. For one particular pulsar, the time between two successive pulses is 0.033 s. Determine the average angular speed (in rad/s) of this pulsar.

| 5 | ssm An electric circular saw is designed to reach its final angular speed, starting from rest, in 1.50 s. Its average angular acceleration is 328 rad/s2. Obtain its final angular speed.

| 6 | A Ferris wheel rotates at an angular velocity of 0.24 rad/s. Starting from rest, it reaches its operating speed with an average angular acceleration of 0.030 rad/s2. How long does it take the wheel to come up to operating speed?

| 7 | A CD has a playing time of 74 minutes. When the music starts, the CD is rotating at an angular speed of 480 revolutions per minute (rpm). At the end of the music, the CD is rotating at 210 rpm. Find the magnitude of the average angular acceleration of the CD. Express your answer in rad/s2.

* | 8 | The sun appears to move across the sky, because the earth spins on its axis. To a person standing on the earth, the sun subtends an angle of (see Conceptual Example 2). How much time (in seconds) does it take for the sun to move a distance equal to its own diameter?

Problem 8

* | 9 | A space station consists of two donut-shaped living chambers, A and B, that have the radii shown in the drawing. As the station rotates, an astronaut in chamber A is moved 2.40 × 102 m along a circular arc. How far along a circular arc is an astronaut in chamber B moved during the same time?

* | 10 | Two people start at the same place and walk around a circular lake in opposite directions. One has an angular speed of 1.7 × 10−3 rad/s, while the other has an angular speed of 3.4 × 10−3 rad/s. How long will it be before they meet?

* | 11 | A baton twirler throws a spinning baton directly upward. As it goes up and returns to the twirler’s hand, the baton turns through four revolutions. Ignoring air resistance and assuming that the average angular speed of the baton is 1.80 rev/s, determine the height to which the center of the baton travels above the point of release.

* | 12 | A stroboscope is a light that flashes on and off at a constant rate. It can be used to illuminate a rotating object, and if the flashing rate is adjusted properly, the object can be made to appear stationary. (a) What is the shortest time between flashes of light that will make a three-bladed propeller appear stationary when it is rotating with an angular speed of 16.7 rev/s? (b) What is the next shortest time?

* | 13 | ssm The drawing shows a device that can be used to measure the speed of a bullet. The device consists of two rotating disks, separated by a distance of d = 0.850 m, and rotating with an angular speed of 95.0 rad/s. The bullet first passes through the left disk and then through the right disk. It is found that the angular displacement between the two bullet holes is . From these data, determine the speed of the bullet.

** | 14 | The drawing shows a golf ball passing through a windmill at a miniature golf course. The windmill has 8 blades and rotates at an angular speed of 1.25 rad/s. The opening between successive blades is equal to the width of a blade. A golf ball of diameter 4.50 × 10−2 m is just passing by one of the rotating blades. What must be the minimum speed of the ball so that it will not be hit by the next blade?

** | 15 | A quarterback throws a pass that is a perfect spiral. In other words, the football does not wobble, but spins smoothly about an axis passing through each end of the ball. Suppose the ball spins at 7.7 rev/s. In addition, the ball is thrown with a linear speed of 19 m/s at an angle of 55° with respect to the ground. If the ball is caught at the same height at which it left the quarterback’s hand, how many revolutions has the ball made while in the air?
Section 8.3 The Equations of Rotational Kinematics

| 16 | A basketball player is balancing a spinning basketball on the tip of his finger. The angular velocity of the ball slows down from 18.5 to 14.1 rad/s. During the slow-down, the angular displacement is 85.1 rad. Determine the time it takes for the ball to slow down.

| 17 | ssm The drill bit of a variable-speed electric drill has a constant angular acceleration of 2.50 rad/s2. The initial angular speed of the bit is 5.00 rad/s. After 4.00 s, (a) what angle has the bit turned through and (b) what is the bit’s angular speed?

| 18 | The shaft of a pump starts from rest and has an angular acceleration of 3.00 rad/s2 for 18.0 s. At the end of this interval, what is (a) the shaft’s angular speed and (b) the angle through which the shaft has turned?

| 19 | ssm www A flywheel has a constant angular deceleration of 2.0 rad/s2. (a) Find the angle through which the flywheel turns as it comes to rest from an angular speed of 220 rad/s. (b) Find the time required for the flywheel to come to rest.

| 20 | A person is riding a bicycle, and its wheels have an angular velocity of +20.0 rad/s. Then, the brakes are applied and the bike is brought to a uniform stop. During braking, the angular displacement of each wheel is +15.92 revolutions. (a) How much time does it take for the bike to come to rest? (b) What is the angular acceleration (in rad/s2) of each wheel?

* | 21 | A top is a toy that is made to spin on its pointed end by pulling on a string wrapped around the body of the top. The string has a length of 64 cm and is wound around the top at a spot where its radius is 2.0 cm. The thickness of the string is negligible. The top is initially at rest. Someone pulls the free end of the string, thereby unwinding it and giving the top an angular acceleration of +12 rad/s2. What is the final angular velocity of the top when the string is completely unwound?

* | 22 | Interactive LearningWare 8.1 at reviews the approach that is necessary for solving problems such as this one. A motorcyclist is traveling along a road and accelerates for 4.50 s to pass another cyclist. The angular acceleration of each wheel is +6.70 rad/s2, and, just after passing, the angular velocity of each is +74.5 rad/s, where the plus signs indicate counterclockwise directions. What is the angular displacement of each wheel during this time?

* | 23 | Interactive Solution 8.23 at offers a model for this problem. The drive propeller of a ship starts from rest and accelerates at 2.90 × 10−3 rad/s2 for 2.10 × 103s. For the next 1.40 × 103 s the propeller rotates at a constant angular speed. Then it decelerates at 2.30 × 10−3 rad/s2 until it slows (without reversing direction) to an angular speed of 4.00 rad/s. Find the total angular displacement of the propeller.

* | 24 | After 10.0 s, a spinning roulette wheel at a casino has slowed down to an angular velocity of +1.88 rad/s. During this time, the wheel has an angular acceleration of −5.04 rad/s2. Determine the angular displacement of the wheel.

* | 25 | ssm A spinning wheel on a fireworks display is initially rotating in a counterclockwise direction. The wheel has an angular acceleration of −4.00 rad/s2. Because of this acceleration, the angular velocity of the wheel changes from its initial value to a final value of −25.0 rad/s. While this change occurs, the angular displacement of the wheel is zero. (Note the similarity to that of a ball being thrown vertically upward, coming to a momentary halt, and then falling downward to its initial position.) Find the time required for the change in the angular velocity to occur.

* | 26 | At the local swimming hole, a favorite trick is to run horizontally off a cliff that is 8.3 m above the water. One diver runs off the edge of the cliff, tucks into a “ball”, and rotates on the way down with an average angular speed of 1.6 rev/s. Ignore air resistance and determine the number of revolutions she makes while on the way down.

** | 27 | ssm www A child, hunting for his favorite wooden horse, is running on the ground around the edge of a stationary merry-go-round. The angular speed of the child has a constant value of 0.250 rad/s. At the instant the child spots the horse, one-quarter of a turn away, the merry-go-round begins to move (in the direction the child is running) with a constant angular acceleration of 0.0100 rad/s2. What is the shortest time it takes for the child to catch up with the horse?
Section 8.4 Angular Variables and Tangential Variables

| 28 | Professional recording studios often use reel-to-reel tape decks. On such a tape deck, the tape is pulled past the playback head at a constant linear speed of 0.381 m/s. (a) Using the data in part a of the drawing, find the angular speed of the take-up reel. (b) After 2.40 × 103 s, the takeup reel is almost full, as part b of the drawing indicates. Find the average angular acceleration of the reel and specify whether the acceleration indicates an increasing or decreasing angular velocity.

| 29 | ssm A string trimmer is a tool for cutting grass and weeds; it utilizes a length of nylon “string” that rotates about an axis perpendicular to one end of the string. The string rotates at an angular speed of 47 rev/s, and its tip has a tangential speed of 54 m/s. What is the length of the rotating string?

| 30 | An auto race is held on a circular track. A car completes one lap in a time of 18.9 s, with an average tangential speed of 42.6 m/s. Find (a) the average angular speed and (b) the radius of the track.

| 31 | Refer to Interactive Solution 8.31 at to review a model for solving this problem. The take-up reel of a cassette tape has an average radius of 1.4 cm. Find the length of tape (in meters) that passes around the reel in 13 s when the reel rotates at an average angular speed of 3.4 rad/s.

| 32 | In 9.5 s a fisherman winds 2.6 m of fishing line onto a reel whose radius is 3.0 cm (assumed to be constant as an approximation). The line is being reeled in at a constant speed. Determine the angular speed of the reel.

| 33 | The earth has a radius of 6.38 × 106 m and turns on its axis once every 23.9 h. (a) What is the tangential speed (in m/s) of a person living in Ecuador, a country that lies on the equator? (b) At what latitude (i.e., the angle in the drawing) is the tangential speed onethird that of a person living in Ecuador?

Problem 33

* | 34 | A baseball pitcher throws a baseball horizontally at a linear speed of 42.5 m/s (about 95 mi/h). Before being caught, the baseball travels a horizontal distance of 16.5 m and rotates through an angle of 49.0 rad. The baseball has a radius of 3.67 cm and is rotating about an axis as it travels, much like the earth does. What is the tangential speed of a point on the “equator” of the baseball?

* | 35 | ssm A person lowers a bucket into a well by turning the hand crank, as the drawing illustrates. The crank handle moves with a constant tangential speed of 1.20 m/s on its circular path. Find the linear speed with which the bucket moves down the well.

Problem 35

* | 36 | A thin rod (length = 1.50 m) is oriented vertically, with its bottom end attached to the floor by means of a frictionless hinge. The mass of the rod may be ignored, compared to the mass of an object fixed to the top of the rod. The rod, starting from rest, tips over and rotates downward. (a) What is the angular speed of the rod just before it strikes the floor? (Hint: Consider using the principle of conservation of mechanical energy.) (b) What is the magnitude of the angular acceleration of the rod just before it strikes the floor?

** | 37 | One type of slingshot can be made from a length of rope and a leather pocket for holding the stone. The stone can be thrown by whirling it rapidly in a horizontal circle and releasing it at the right moment. Such a slingshot is used to throw a stone from the edge of a cliff, the point of release being 20.0 m above the base of the cliff. The stone lands on the ground below the cliff at a point X. The horizontal distance of point X from the base of the cliff (directly beneath the point of release) is thirty times the radius of the circle on which the stone is whirled. Determine the angular speed of the stone at the moment of release.
Section 8.5 Centripetal Acceleration and Tangential Acceleration

| 38 | Two Formula One racing cars are negotiating a circular turn, and they have the same centripetal acceleration. However, the path of car A has a radius of 48 m, while that of car B is 36 m. Determine the ratio of the angular speed of car A to that of car B.

| 39 | ssm A race car travels with a constant tangential speed of 75.0 m/s around a circular track of radius 625 m. Find (a) the magnitude of the car’s total acceleration and (b) the direction of its total acceleration relative to the radial direction.

| 40 | The earth orbits the sun once a year (3.16 × 107 s) in a nearly circular orbit of radius 1.50 × 1011 m. With respect to the sun, determine (a) the angular speed of the earth, (b) the tangential speed of the earth, and (c) the magnitude and direction of the earth’s centripetal acceleration.

| 41 | ssm Refer to Multiple-Concept Example 7 for insight into this problem. During a tennis serve, a racket is given an angular acceleration of magnitude 160 rad/s2. At the top of the serve, the racket has an angular speed of 14 rad/s. If the distance between the top of the racket and the shoulder is 1.5 m, find the magnitude of the total acceleration of the top of the racket.

* | 42 | The front and rear sprockets on a bicycle have radii of 9.00 and 5.10 cm, respectively. The angular speed of the front sprocket is 9.40 rad/s. Determine (a) the linear speed (in cm/s) of the chain as it moves between the sprockets and (b) the centripetal acceleration (in cm/s2) of the chain as it passes around the rear sprocket.

* | 43 | A rectangular plate is rotating with a constant angular acceleration about an axis that passes perpendicularly through one corner, as the drawing shows. The tangential acceleration measured at corner A has twice the magnitude of that measured at corner B. What is the ratio L1/L2 of the lengths of the sides of the rectangle?

* | 44 | The sun has a mass of 1.99 × 1030 kg and is moving in a circular orbit about the center of our galaxy, the Milky Way. The radius of the orbit is 2.3 × 104 light-years (1 light-year = 9.5 × 1015 m), and the angular speed of the sun is 1.1 × 10−15 rad/s. (a) Determine the tangential speed of the sun. (b) What is the magnitude of the net force that acts on the sun to keep it moving around the center of the Milky Way?

** | 45 | ssm An electric drill starts from rest and rotates with a constant angular acceleration. After the drill has rotated through a certain angle, the magnitude of the centripetal acceleration of a point on the drill is twice the magnitude of the tangential acceleration. What is the angle?
Section 8.6 Rolling Motion
Note: All problems in this section assume that there is no slipping of the surfaces in contact during the rolling motion.

| 46 | Suppose you are riding a stationary exercise bicycle, and the electronic meter indicates that the wheel is rotating at 9.1 rad/s. The wheel has a radius of 0.45 m. If you ride the bike for 35 min, how far would you have gone if the bike could move?

| 47 | ssm www A motorcycle accelerates uniformly from rest and reaches a linear speed of 22.0 m/s in a time of 9.00 s. The radius of each tire is 0.280 m. What is the magnitude of the angular acceleration of each tire?

| 48 | An automobile tire has a radius of 0.330 m, and its center moves forward with a linear speed of v = 15.0 m/s. (a) Determine the angular speed of the wheel. (b) Relative to the axle, what is the tangential speed of a point located 0.175 m from the axle?

| 49 | Refer to Interactive Solution 8.49 at in preparation for this problem. A car is traveling with a speed of 20.0 m/s along a straight horizontal road. The wheels have a radius of 0.300 m. If the car speeds up with a linear acceleration of 1.50 m/s2 for 8.00 s, find the angular displacement of each wheel during this period.

* | 50 | Multiple-Concept Example 8 provides some pertinent background for part b of this problem. A motorcycle, which has an initial linear speed of 6.6 m/s, decelerates to a speed of 2.1 m/s in 5.0 s. Each wheel has a radius of 0.65 m and is rotating in a counterclockwise (positive) direction. What is (a) the constant angular acceleration (in rad/s2) and (b) the angular displacement (in rad) of each wheel?

* | 51 | ssm www Concept Simulation 8.1 at reviews the concept that plays the central role in this problem. The warranty on a new tire says that an automobile can travel for a distance of 96 000 km before the tire wears out. The radius of the tire is 0.31 m. How many revolutions does the tire make before wearing out?

* | 52 | The two-gear combination shown in the drawing is being used to lift the load L with a constant upward speed of 2.50 m/s. The rope that is attached to the load is being wound onto a cylinder behind the big gear. The depth of the teeth of the gears is negligible compared to the radii. Determine the angular velocity (magnitude and direction) of (a) the larger gear and (b) the smaller gear.

Problem 52

* | 53 | A ball of radius 0.200 m rolls along a horizontal table top with a constant linear speed of 3.60 m/s. The ball rolls off the edge and falls a vertical distance of 2.10 m before hitting the floor. What is the angular displacement of the ball while the ball is in the air?

* | 54 | The penny-farthing is a bicycle that was popular between 1870 and 1890. As the drawing shows, this type of bicycle has a large front wheel and a small rear wheel. On a Sunday ride in the park the front wheel (radius = 1.20 m) makes 276 revolutions. How many revolutions does the rear wheel (radius = 0.340 m) make?

** | 55 | ssm Consult Concept Simulation 8.1 at for help in understanding the concepts that are important in this problem. Take two quarters and lay them on a table. Press down on one quarter so it cannot move. Then, starting at the 12:00 position, roll the other quarter along the edge of the stationary quarter, as the drawing suggests. How many revolutions does the rolling quarter make when it travels once around the circumference of the stationary quarter? Surprisingly, the answer is not one revolution. (Hint: Review the paragraph just before Equation 8.12 that discusses how the distance traveled by the axle of a wheel is related to the circular arc length along the outer edge of the wheel.)

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