Physics

John D. Cutnell, Kenneth W. Johnson

Chapter 8

Rotational Kinematics - all with Video Answers

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Chapter Questions

Problem 1

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?

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 2

The table that follows lists four pairs of initial and final angles of a wheel on a moving car. The elapsed time for each pair of angles is 2.0 s. For each of the four pairs, determine the average angular
velocity (magnitude and direction as given by the algebraic sign of your answer).

Anita Gordon

Numerade Educator

Problem 3

The earth spins on its axis once a day and orbits the sun once a year $\left(365_{4}^{1} \text { days). Determine the average angular velocity (in rad/s) of the }\right.$ earth as it $(\text { a) spins on its axis and }(b) \text { orbits the sun. In each case, }$ take the positive direction for the angular displacement to be the direction of the earth's motion.

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 4

Our sun rotates in a circular orbit about the center of the Milky Way galaxy. The radius of the orbit is $2.2 \times 10^{30} \mathrm{m},$ and the angular speed of the sun is $1.1 \times 10^{-15}$ rad/s. How long (in years) does it take for the sun to make one revolution around the center?

Anita Gordon

Numerade Educator

Problem 5

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

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 6

The initial angular velocity and the angular acceleration of four rotating objects at the same instant in time are listed in the table that follows. For each of the objects (a), (b), (c), and (d), determine the final
angular speed after an elapsed time of 2.0 s.

Anita Gordon

Numerade Educator

Problem 7

The table that follows lists four pairs of initial and final angular velocities for a rotating fan blade. The elapsed time for each of the four pairs of angular velocities is 4.0 s. For each of the four pairs, find the
average angular acceleration (magnitude and direction as given by the algebraic sign of your answer).

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 8

Conceptual Example 2 provides some relevant background for this problem. A jet is circling an airport control tower at a distance of 18.0 $\mathrm{km}$ . An observer in the tower watches the jet cross in front of the moon. As seen from the tower, the moon subtends an angle of $9.04 \times 10^{-3}$ radians. Find the distance traveled (in meters) by the jet as the observer watches the nose of the jet cross from one side of the moon to the other.

Anita Gordon

Numerade Educator

Problem 9

A Ferris wheel rotates at an angular velocity of 0.24 $\mathrm{rad} / \mathrm{s}$ Starting from rest, it reaches its operating speed with an average angular acceleration of 0.030 $\mathrm{rad} / \mathrm{s}^{2}$ . How long does it take the wheel to come up to operating speed?

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 10

A floor polisher has a rotating disk that has a 15-cm radius. The disk rotates at a constant angular velocity of 1.4 rev/s and is covered with a soft material that does the polishing. An operator holds the polisher in one place for 45 s, in order to buff an especially scuffed area of the floor. How far (in meters) does a spot on the outer edge of the disk move during this time?

Anita Gordon

Numerade Educator

Problem 11

The sun appears to move across the sky, becausc the carth spins on its axis. To a person standing on the earth, the sun subtends an angle of $\theta_{\text { sun }}=9.28 \times 10^{-3}$ rad (see Conceptual Example 2). How much time (in seconds) does it take for the sun to move a distance equal to its own diameter?

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 12

A propeller is rotating about an axis perpendicular to its center, as the drawing shows. The axis is parallel to the ground. An arrow is fired at the propeller, travels parallel to the axis, and passes through one of the open spaces between the propeller blades. The angular open spaces between the three propeller blades are each $\pi / 3$ rad $\left(60.0^{\circ}\right) .$ The vertical drop of the arrow may be ignored. There is a maximum value $\omega$ for the angular speed of the propeller, beyond which the arrow cannot pass through an open space without being struck by one of the blades. Find this maximum value when the arrow has the lengths $L$ and speeds $v$ shown in the following table.

Anita Gordon

Numerade Educator

Problem 13

Two people start at the same place and walk around a circular lake in opposite directions. One walks with an angular speed of $1.7 \times 10^{-3} \mathrm{rad} / \mathrm{s},$ while the other has an angular speed of $3.4 \times 10^{3} \mathrm{rad} / \mathrm{s}$ . How long will it be before they meet?

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 14

A space station consists of two donul-shaped living chambers, A and $\mathrm{B}$ , that have the radii shown in the drawing. As the station rotates, an astronaut in chamber A is moved $2.40 \times 10^{2} \mathrm{m}$ along a circu- lar arc. How far along a circular arc is an astronaut in chamber $\mathrm{B}$ moved during the same time'?

Anita Gordon

Numerade Educator

Problem 15

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 $d=0.850 \mathrm{m},$ and rotating with an angular speed of 95.0 $\mathrm{rad} / \mathrm{s}$ . 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 $\theta=0.240$ rad. From these data, determine the speed of the bullet.

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 16

An automatic dryer spins wet clothes at an angular speed of 5.2 rad/s. Starting from rest, the dryer reaches its operating speed with an average angular acceleration of 4.0 $\mathrm{rad} / \mathrm{s}^{2}$ . How long does it take the dryer to come up to speed?

Anita Gordon

Numerade Educator

Problem 17

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?

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 18

Review Conceptual Example 2 before attempting to work this problem. The moon has a diameter of $3.48 \times 10^{6} \mathrm{m}$ and is a distance of $3.85 \times 10^{8} \mathrm{m}$ from the earth. The sun has a diameter of $1.39 \times 10^{9} \mathrm{m}$ and is $1.50 \times 10^{11} \mathrm{m}$ from the earth. (a) Determine (in radians) the angles subtended by the moon and the sun, as measured by a person standing on the carth. $(b)$ Bascd on your answers to part (a), decide whether a total eclipse of the sun is really "total." Give your reasoning. (c) Determine the ratio, expressed as a percentage, of the apparent circular area of the moon to the apparent circular area of the sun.

Anita Gordon

Numerade Educator

Problem 19

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 $1.25 \mathrm{rad} / \mathrm{s}$. The opening between successive blades is equal to the width of a blade.A golf ball (diameter $4.50 \times 10^{-2} \mathrm{~m}$) has just reached the edge of one of the
rotating blades (see the drawing). Ignoring the thickness of the blades, find the $minimum$ linear speed with
which the ball moves along the ground,
such that the ball will not be hit by the next blade.

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 20

A figure skater is spinning with an angular velocity of $+15 \mathrm{rad} / \mathrm{s}$ . She
then comes to a stop over a brief period of time. During this time, her angular displacement is $+5.1$ rad. Determine (a) her average angular acceleration and (b) the time during which she comes to rest.

Anita Gordon

Numerade Educator

Problem 21

ssm A gymnast is performing a floor routine. In a tumbling run she spins through the air, increasing her angular velocity from 3.00 to 5.00 rev/s while rotating through one-half of a revolution. How much
time does this maneuver take?

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 22

The angular speed of the rotor in a centrifuge increases from 420 to 1420 rad/s in a time of 5.00 s. (a) Obtain the angle through which the rotor turns. (b) What is the magnitude of the angular acceleration?

Anita Gordon

Numerade Educator

Problem 23

A wind turbine is initially spinning at a constant angular speed.
As the wind’s strength gradually increases, the turbine experiences a
constant angular acceleration of $0.140 \mathrm{rad} / \mathrm{s}^{2}$. After making 2870 revolutions, its angular speed is $137 \mathrm{rad} / \mathrm{s}$
(a) What is the initial angularvelocity of the turbine?
(b) How much time elapses while the turbine is speeding up?

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 24

A car is traveling along a road, and its engine is turning over with an angular velocity of 220 rad/s. The driver steps on the accelerator, and in a time of 10.0 s the angular velocity increases to 280 rad/s.
(a) What would have been the angular displacement of the engine if its angular velocity had remained constant at the initial value of 220 rad/s during the entire 10.0-s interval? (b) What would have been the angular displacement if the angular velocity had been equal to its final value of 280 rad/s during the entire 10.0-s interval? (c) Determine the actual value of the angular displacement during the 10.0-s interval.

Anita Gordon

Numerade Educator

Problem 24

The wheels of a bicycle have an angular velocity of 20.0 rad/s. Then, the brakes are applied. In coming to rest, each wheel makes an angular displacement of 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?

Anita Gordon

Numerade Educator

Problem 26

A dentist causes the bit of a high-speed drill to accelerate from an angular speed of $1.05 \times 10^{4}$ rad/s to an angular speed of $3.14 \times 10^{4} \mathrm{rad} / \mathrm{s}$ . In the process, the bit turns through $1.88 \times 10^{4} \mathrm{rad}$ . the rin Assuming a constant angular acceleration, how long would it take the bit to reach its maximum speed of $7.85 \times 10^{4}$ rad/s, starting from rest?

Anita Gordon

Numerade Educator

Problem 27

A motorcyclist is traveling along a road and accelerates for 4.50 s to pass another cyclist. The angular acceleration of each wheel is $+6.70 \mathrm{rad} / \mathrm{s}^{2},$ and, just after passing, the angular velocity of each wheel is $+74.5 \mathrm{rad} / \mathrm{s}$ , where the plus signs indicate counterclockwise directions. What is the angular displacement of each wheel during this time?

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 28

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 $\mathrm{cm}$ and is wound around the top at a spot where its radius is 2.0 $\mathrm{cm} .$ 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 $+12 \mathrm{rad} / \mathrm{s}^{2}$ . What is the final angular
velocity of the top when the string is completely unwound?

Anita Gordon

Numerade Educator

Problem 29

The drive propeller of a ship starts from rest and accelcrates at $2.90 \times 10^{-3} \mathrm{rad} / \mathrm{s}^{2}$ for $2.10 \times 10^{3} \mathrm{~s}$. For the next $1.40 \times 10^{3} \mathrm{~s}$ the propeller rotates at a constant angular spced. Then it decelerates at $2.30 \times 10^{-3} \mathrm{rad} / \mathrm{s}^{2}$ until it slows (without reversing direction) to an angular speed of 4.00 rad/s. Find the total angular displacement of the propeller.

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 30

The drawing shows a graph of the angular velocity of a rotating wheel as a function of time. Although not shown in the graph, the angular velocity continues to increase at the same rate until $t=8.0$ s. What is the angular displacement of the wheel from 0 to 8.0 s?

Anita Gordon

Numerade Educator

Problem 31

At the local swimming hole, a favorite trick is to run horizontally off a cliff that is 8.3 $\mathrm{m}$ 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.

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 32

A spinning whecl on a fircworks display is initially rotating in a counterclockwise direction. The wheel has an angular acceleration of $-4.00 \mathrm{rad} / \mathrm{s}^{2}$ . Because of this acceleration, the angular velocity of the wheel changes from its initial value to a final value of $-25.0 \mathrm{rad} / \mathrm{s}$ . 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.

Anita Gordon

Numerade Educator

Problem 33

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 valuc of 0.250 $\mathrm{rad} / \mathrm{s}$ . At the instant the child spots the horse, onc-quarter of a turn away, the merry-go-round begins to move (in the direction the child is runing) with a constant angular acceleration of 0.0100 $\mathrm{rad} / \mathrm{s}^{2}$ . What is the shortest time it takes for the child to catch up with the horse?

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 34

A fan blade is rotating with a constant angular acceleration of $+12.0 \mathrm{rad} / \mathrm{s}^{2}$ . At what point on the blade, as measured from the axis of rotation, docs the magnitude of the tangcntial acccleration cqual that of the acceleration due to gravity?

Anita Gordon

Numerade Educator

Problem 35

Some bacteria are propelled by biological motors that spin hair- like flagella. A typical bacterial motor turning at a constant angular velocity has a radius of $1.5 \times 10^{-8} \mathrm{m}$ , and a tangential speed at the rim of $2.3 \times 10^{-5} \mathrm{m} / \mathrm{s}$ . ( a) What is the angular speed (the magnitude of the angular velocity) of this bacterial motor? (b) How long does it take the motor to make one revolution?

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 36

An auto race takes place 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.

Anita Gordon

Numerade Educator

Problem 37

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?

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 38

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

Anita Gordon

Numerade Educator

Problem 39

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.

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 40

The carth has a radius of $6.38 \times 10^{6} \mathrm{m}$ and turns on its axis once every $23,9 \mathrm{h}$ . (a) What is the tangential specd (in $\mathrm{m} / \mathrm{s} )$ of a person living in Ecuador, a country that lies on the equator? (b) At what latitude (i.e., the angle $\theta$ in the drawing) is the tangential speed one-third that of a person living in Ecuador?

Anita Gordon

Numerade Educator

Problem 41

A baseball pitcher throws a base- ball horizontally at a linear speed of 42.5 $\mathrm{m} / \mathrm{s}$ (about 95 $\mathrm{mi} / \mathrm{h} ) .$ Before being caught, the bascball travels a horizontal distance of 16.5 $\mathrm{m}$ and rotates through an angle of 49.0 $\mathrm{rad.}$ The baseball has a radius of 3.67 $\mathrm{cm}$ 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 buseball?

Supratim Pal

Numerade Educator

Problem 42

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. The rope holding the bucket unwinds with- out slipping on the barrel of the crank. Find the linear speed with which the
bucket moves down the well.

Anita Gordon

Numerade Educator

Problem 43

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?

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 44

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 rap- idly 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

Anita Gordon

Numerade Educator

Problem 45

A racing car travels with a constant tangential speed of 75.0 m/ 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 rel-
ative to the radial direction.

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 46

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 the angular speed of car B.

Anita Gordon

Numerade Educator

Problem 47

The earth orbits the sun once a year $\left(3.16 \times 10^{7} \mathrm{~s}\right)$ in a nearly circular orbit of radius $1.50 \times 10^{11} \mathrm{~m} .$ With respect to the sun, determine
(a) the angular speed of the earth,
(b) the tangential speed of the earth, and $(\mathbf{c})$ the magnitude and direction of the earth's centripetal acceleration.

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 48

Rcvicw Multiple-Concept Example 7 in this chapter as an aid in solving this problem In a fast-pitch softball game the pitcher is impressive to watch, as she delivers a pitch by rapidly whirling her arm
around so that the ball in her hand moves on a circle. In one instance, the radius of the circle is 0.670 $\mathrm{m}$ . At one point on this circle, the ball has an angular acceleration of 64.0 $\mathrm{rad} / \mathrm{s}^{2}$ and an angular speed of 16.0 $\mathrm{rad} / \mathrm{s}$ . (a) Find the magnitude of the total acceleration (centripetal plus tangential) of the ball. $(\mathrm{b})$ Determine the angle of the total acceleration relative to the radial direction.

Anita Gordon

Numerade Educator

Problem 49

A rectangular plate is rotating with a constant angular speed about an axis that passes perpendicularly through one corner, as the drawing shows. The centripetal acceleration measured at corner $A$ is $n$ times as great as that measured at corner $B$. What is the ratio $L_{1} / L_{2}$ of the lengths of the sides of the rectangle when $n=2.00 ?$

Vishal Gupta

Numerade Educator

Problem 50

Multiple-concept Example 7 explores the approach taken in problems such as this one. The blades of a ceiling fan have a radius of 0.380 $\mathrm{m}$ and are rotating about a fixed axis with an angular velocity of $+1.50 \mathrm{rad} / \mathrm{s}$ . When the switch on the fan is turned to a higher speed,
the blades acquire an angular acceleration of $+2.00 \mathrm{rad} / \mathrm{s}^{2}$ . After 0.500 $\mathrm{s}$ the blades acquire an angular acceleration of $+2.00 \mathrm{rad} / \mathrm{s}^{2}$ . After 0.500 $\mathrm{s}$ has clapsed since the switch was resct, what is $\quad$ (a) the total acceleration (in $\mathrm{m} / \mathrm{s}^{2} )$ of a point on the tip of a blade and $(\mathrm{b})$ the angle $\phi$ between the total acceleration $\overrightarrow{\mathbf{a}}$ and the centripetal acceleration $\overrightarrow{\mathbf{a}}_{\mathrm{c}} ?$ (See Figure 8.12$b . )$

Anita Gordon

Numerade Educator

Problem 51

The sun has a mass of $1.99 \times 10^{30} \mathrm{kg}$ and is moving in a circular orbit about the center of our galaxy, the Milky Way. The radius of the orbit is $2.3 \times 10^{4}$ light-years (1 light-year $=9.5 \times 10^{15} \mathrm{m} )$ and the angular speed of the sun is $1.1 \times 10^{-15} \mathrm{rad} / \mathrm{s}$ . (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?

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 52

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?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 53

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?

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 54

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?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 55

A car is traveling with a speed of $20.0 \mathrm{~m} / \mathrm{s}$ along a straight horizontal road. The wheels have a radius of $0.300 \mathrm{~m} .$ If the car speeds up with a linear acceleration of $1.50 \mathrm{~m} / \mathrm{s}^{2}$ for $8.00 \mathrm{~s},$ find the angular displacement of each wheel during this period.

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 56

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?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 57

Multiple-Concept Example 8 provides useful background for part b of this problem. A motorcycle, which has an initial lincar speed of 6.6 $\mathrm{m} / \mathrm{s}$ , decelerates to a speed of 2.1 $\mathrm{m} / \mathrm{s}$ in 5.0 $\mathrm{s}$ . Each wheel has a radius of 0.65 $\mathrm{m}$ and is rotating in a counterclockwisc (positive) direction. What are $(\mathrm{a})$ the constant angular acceleration (in rad/s') and (b) the angular displacement (in rad) of each wheel?

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 58

A dragster starts from rest and accelerates down a track. Each tire has a radius of 0.320 m and rolls without slipping. At a distance of 384 m, the angular speed of the wheels is 288 rad/s. Determine (a) the linear speed of the dragster and (b) the magnitude of the angular acceleration of its wheels.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 59

Over the course of a multi-stage $4520-\mathrm{km}$ bicycle race, the front wheel of an athlete's bicycle makes $2.18 \times 10^{6}$ revolutions. How many revolutions would the wheel have made during the race if its radius had been $1.2 \mathrm{~cm}$ larger?

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 60

A bicycle is rolling down a circular portion of a path; this portion of the path has a radius of 9.00 m. As the drawing illustrates, the angular displacement of the bicycle is 0.960 rad. What is the angle (in radians) through which each bicycle wheel (radius 0.400 m) rotates?

Penny Riley

Numerade Educator

Problem 61

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. During a ride, the front wheel (radius 1.20 m) makes 276 revolutions. How many revolutions does the rear wheel (radius 0.340 m) make?

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 62

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

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 63

The differential gear of a car axle allows the wheel on the left side of a car to rotate at a different angular speed than the wheel on the right side. A car is driving at a constant speed around a circular track on level ground, completing each lap in 19.5 s. The distance between the tires on the left and right sides of the car is 1.60 m, and the radius of each wheel is 0.350 m. What is the difference between the angular speeds of the wheels on the left and right sides of the car?

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 64

The trap-jaw ant can snap its mandibles shut in as little as $1.3 \times 10^{-4} \mathrm{s}$ In order to shut, cach mandible rotates through a $90^{\circ}$ angle. What is the average angular velocity of one of the mandibles of the trap-jaw ant when the mandibles snap shul?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 65

A 220-kg speedboat is negotiating a circular turn (radius 32 m) around a buoy. During the turn, the engine causes a net tangential force of magnitude 550 N to be applied to the boat. The initial tangential speed of the boat going into the turn is 5.0 m/s. (a) Find the tangential acceleration. (b) After the boat is 2.0 s into the turn, find the centripetal acceleration.

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 66

A flywheel has a constant angular deceleration of 2.0 $\mathrm{rad} / \mathrm{s}^{2}$ (a) Find the angle through which the flywheel tums as it comes to rest from an angular speed of 220 $\mathrm{rad} / \mathrm{s}$ . (b) Find the time for the flywheel to come to rest.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 67

An electric fan is running on HIGH. After the LOW button is pressed, the angular speed of the fan decreases to $83.8 \mathrm{rad} / \mathrm{s}$ in $1.75 \mathrm{~s}$. The deceleration is $42.0 \mathrm{rad} / \mathrm{s}^{2}$. Determine the initial angular speed of the fan.

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 68

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 $\mathrm{rad} / \mathrm{s}^{2}$ . At the top of the serve, the racket has an angular speed of 14 $\mathrm{rad} / \mathrm{s}$ . If the distance between the top of the racket and the shoulder is $1.5 \mathrm{m},$ find the magnitude of the total acceleration of the top of the racke..

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 69

mmh The drawing shows a chain-saw blade. The rotating sprocket tip at the end of the guide bar has a radius of $4.0 \times 10^{-2} \mathrm{m} .$ The linear speed of a chain link at point A is 5.6 m/s. Find the angular speed of the sprocket tip in rev/s.

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 70

In a large centrifuge used for training pilots and astronauts, a small chamber is fixed at the end of a rigid arm that rotates in a horizontal circle. A trainee riding in the chamber of a centrifuge rotating with a constant angular speed of 2.5 rad/s experiences a centripetal acceleration of 3.2 times the acceleration due to gravity. In a second training exercise, the centrifuge speeds up from rest with a constant angular acceleration. When the centrifuge reaches an angular speed of 2.5 rad/s, the trainee experiences a total acceleration equal to 4.8 times the acceleration due to gravity. (a) How long is the arm of the centrifuge? (b) What is the angular acceleration of the centrifuge in the second training exercise?

KF

Kyle Feist

Numerade Educator

Problem 71

A compact disc (CD) contains music on a spiral track. Music is pul onto a CD with the assumption that, during playback, the music will be detected at a constant tangential speed at any point. Since $v_{\mathrm{T}}=r \omega, \mathrm{a}$ CD rotates at a smaller angular speed for music near the outer edge and a larger angular speed for music neur the inner part of the disc. For music at the outer edge (r 0.0568 m), the angular speed is 3.50 rev/s. Find (a) the constant tangential speed at which music is detected and (b) the angular speed (in rev/s) for music at a distance of 0.0249 m from the center of a CD.

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 72

After $10.0 \mathrm{s},$ a spinning roulette wheel at a casino has slowed down to an angular velocity of $+1.88 \mathrm{rad} / \mathrm{s}$ . During this time, the wheel has an angular accelcration of $-5.04 \mathrm{rad} / \mathrm{s}^{2}$ , Detcrmine the angular displacement of the wheel.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 73

At a county fair there is a betting game that involves a spinning whecl. As the drawing shows, the wheel is set into rotational motion with the bcginning of the angular section labeled "l" at the marker at the top of the wheel. The wheel then decelerates and eventually comes to a halt on one of the numbered sections. The wheel in the drawing is divided into twelve sections, each of which is an angle of $30.0^{\circ}$ . Determine the numbered when the deccleration of the whecl has a magnitude of 0.200 $\mathrm{rev} / \mathrm{s}^{2}$ and the initial angular velocity is $(\mathrm{a})+1.20 \mathrm{rev} / \mathrm{s}$ and $\quad(\mathrm{b})+1.47 \mathrm{rev} / \mathrm{s}$ .section on which the wheel comes to a halt

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 74

A racing car, starting from rest, travels around a circular turn of radius 23.5 m. At a certain instant, the car is still accelerating, and its angular speed is 0.571 rad/s. At this time, the total acceleration
(centripetal plus tangential) makes an angle of 35.0 with respect to the radius. (The situation is similar to that in Figure 8.12b.) What is the magnitude of the total acceleration?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 75

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 cach end of the ball. Suppose the ball spins at $7,7 \mathrm{rev} / \mathrm{s}$. In addition, the ball is thrown with a linear speed of $19 \mathrm{~m} / \mathrm{s}$ at an angle of $55^{\circ}$ 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?

Marshall Styczinski

Marshall Styczinski

Numerade Educator

Problem 76

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 rev-olutions does the rolling quarter make when it travels once around the circumference of the stationary quarter? Surprisingly, the answer is not one revolution

Guilherme Barros

Guilherme Barros

Numerade Educator