College Physics With an Integrated Approach to Forces and Kinematics

Alan Giambattista, Betty McCarthy Richardson , Robert C. Richardson

Chapter 5

Circular Motion - all with Video Answers

Educators

Chapter Questions

Problem 1

A carnival swing is fixed on the end of an $8.0$ -m-long beam. If the swing and beam sweep through an angle of $120^{\circ}$, what is the distance through which the riders move?

Jose Martinez

Jose Martinez

Numerade Educator

Problem 2

A soccer ball of diameter $31 \mathrm{~cm}$ rolls without slipping at a linear speed of $2.8 \mathrm{~m} / \mathrm{s}$. Through how many revolutions has the soccer ball turned as it moves a linear distance of $18 \mathrm{~m} ?$

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 3

Find the average angular speed of the second hand of a clock.

Jose Martinez

Numerade Educator

Problem 4

Convert these to radian measure: (a) $30.0^{\circ}$, (b) $135^{\circ}$,
(c) $\frac{1}{4}$ revolution, (d) $33.3$ revolutions.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 5

A bicycle is moving at $9.0 \mathrm{~m} / \mathrm{s}$. What is the angular speed of its tires if their radius is $35 \mathrm{~cm}$ ? ( W tutorial:
car tire.

Jose Martinez

Numerade Educator

Problem 6

An elevator cable winds on a drum of radius $90.0 \mathrm{~cm}$ that is connected to a motor. (a) If the elevator is moving down at $0.50 \mathrm{~m} / \mathrm{s}$, what is the angular speed of the drum? (b) If the elevator moves down $6.0 \mathrm{~m}$, how many revolutions has the drum made?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 7

Grace is playing with her dolls and decides to give them a ride on a merry-go-round. She places one of them on an old record player turntable and sets the angular speed at $33.3$ rpm. (a) What is their angular speed in $\mathrm{rad} / \mathrm{s}$ ?
(b) If the doll is $13 \mathrm{~cm}$ from the center of the spinning turntable platform, how fast (in $\mathrm{m} / \mathrm{s}$ ) is the doll moving?

Jose Martinez

Numerade Educator

Problem 8

A wheel is rotating at a rate of $2.0$ revolutions every $3.0 \mathrm{~s}$. Through what angle, in radians, does the wheel rotate in $1.0 \mathrm{~s}$ ?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 9

In the construction of railroads, it is important that curves be gentle, so as not to damage passengers or. freight. Curvature is not measured by the radius of curvature, but in the following way. First a $100.0$ -ft-long chord is measured. Then the curvature is reported as the angle subtended by two radii at the endpoints of the

Jose Martinez

Numerade Educator

Problem 10

Verify that all three expressions for radial acceleration $\left(v \omega, v^{2} / r\right.$, and $\left.\omega^{2} r\right)$ have the correct dimensions for an acceleration.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 11

An apparatus is designed to study insects at an acceleration of magnitude $980 \mathrm{~m} / \mathrm{s}^{2}(=100 g)$. The apparatus consists of a $2.0-\mathrm{m}$ rod with insect containers at either end. The rod rotates about an axis perpendicular to the rod and at its center. (a) How fast does an insect move. when it experiences a radial acceleration of $980 \mathrm{~m} / \mathrm{s}^{2}$ ?
(b) What is the angular speed of the insect? ( $\mathrm{W}$ tutorial: centrifuge)

Narayan Hari

Numerade Educator

Problem 12

The rotor is an amusement park ride where people stand against the inside of a cylinder. Once the cylinder is spinning fast enough, the floor drops out. (a) What force keeps the people from falling out the bottom of the cylinder? (b) If the coefficient of friction is $0.40$ and the cylinder has a radius of $2.5 \mathrm{~m}$, what is the minimum angular speed of the cylinder so that the people don't fall out? (Normally the operator runs it considerably faster as a safety measure.)

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 13

Objects that are at rest relative to Earth's surface are in circular motion due to Earth's rotation. What is the radial acceleration of an African baobab tree located at the equator?

Jose Martinez

Numerade Educator

Problem 14

Earth's orbit around the Sun is nearly circular. The period is $1 \mathrm{yr}=365.25 \mathrm{~d}$. (a) In an elapsed time of $1 \mathrm{~d}$, what is Earth's angular displacement? (b) What is the change in Earth's velocity, $\Delta \overrightarrow{\mathbf{v}} ?$ (c) What is Earth's average acceleration during $1 \mathrm{~d} ?$ (d) Compare your answer for (c) to the magnitude of Earth's instantaneous radial acceleration. Explain.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 15

A $0.700-\mathrm{kg}$ ball is on the end of a rope that is $1.30 \mathrm{~m}$ in length. The ball and rope are attached to a pole and the entire apparatus, including the pole, rotates about the pole's symmetry axis. The rope makes an angle of $70.0^{\circ}$ with respect to the vertical. What is the tangen- $\mathrm{A}$ :
tial speed of the ball?

Jose Martinez

Numerade Educator

Problem 16

A child's toy has a $0.100-\mathrm{kg}$ ball attache and $B$. The strings are also attached to swings around the stick along a circular path in a horizontal plane. Both strings are $15.0 \mathrm{~cm}$ long and make an angle of $30.0^{\circ}$ with respect to the horizontal. (a) Draw an FBD for the ball showing the tension forces and the gravitational force. (b) Find the magnitude of the tension in each string when the ball's angular speed is $6.00 \pi \mathrm{rad} / \mathrm{s}$.

Narayan Hari

Numerade Educator

Problem 17

A child swings a rock of mass $m$ in a horizontal circle using a rope of length $L .$ The rock moves at constant speed $v$. (a) Ignoring gravity, find the tension in the rope.
(b) Now include gravity (the weight of the rock is no longer negligible, although the weight of the rope still is negligible). What is the tension in the rope? Express the tension in terms of $m, g, v, L$, and the angle $\theta$ that the rope makes with the horizontal. ( Wutorial: skip rope)

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 18

A conical pendulum consists of a bob (mass $m$ ) attached to a string (length $L$ ) swinging in a horizontal circle (Fig. 5.11). As the string moves, it sweeps out the area of a cone. The angle that the string makes with the vertical is $\phi$. (a) What is the tension in the string? (b) What is the period of the pendulum?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 19

A curve in a stretch of highway has radius $R$. The road is unbanked. The coefficient of static friction between the tires and road is $\mu_{\mathrm{s}} .$ (a) What is the fastest speed that a car can safely travel around the curve? (b) Explain

Jose Martinez

Numerade Educator

Problem 20

A highway curve has a radius of $825 \mathrm{~m}$. At what angle should the road be banked so that a car traveling at $26.8 \mathrm{~m} / \mathrm{s}$ ( $60 \mathrm{mph}$ ) has no tendency to skid sideways on the road? [Hint: No tendency to skid means the frictional force is zero.]

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 21

A curve in a highway has radius of curvature $120 \mathrm{~m}$ and is banked at $3.0^{\circ} .$ On a day when the road is icy, what is the safest speed to go around the curve?

Jose Martinez

Numerade Educator

Problem 22

A roller coaster car of mass $320 \mathrm{~kg}$ (including passengers) travels around a horizontal curve of radius $35 \mathrm{~m}$. Its speed is $16 \mathrm{~m} / \mathrm{s}$. What is the magnitude and direction of the total force exerted on the car by the track?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 23

A velodrome is built for use in the Olympics. The radius of curvature of the surface is $20.0 \mathrm{~m}$. At what angle should the surface be banked for cyclists moving at $18 \mathrm{~m} / \mathrm{s} ?$ (Choose an angle so that no frictional force is needed to keep the cyclists in their circular path. Large banking angles are used in velodromes.)

Jose Martinez

Numerade Educator

Problem 24

A car drives around a curve with radius $410 \mathrm{~m}$ at a speed of $32 \mathrm{~m} / \mathrm{s}$. The road is not banked. The mass of the car is $1400 \mathrm{~kg}$. (a) What is the frictional force on the car?
(b) Does the frictional force necessarily have magnitude $\mu_{s} N ?$ Explain.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 25

A car drives around a curve with radius $410 \mathrm{~m}$ at a speed of $32 \mathrm{~m} / \mathrm{s}$. The road is banked at $5.0^{\circ}$. The mass of the car is $1400 \mathrm{~kg}$. (a) What is the frictional force on the car? (b) At what speed could you drive around this curve so that the force of friction is zero?

Jose Martinez

Numerade Educator

Problem 26

A curve in a stretch of highway has radius $R$. The road is banked at angle $\theta$ to the horizontal. The coefficient of static friction between the tires and road is $\mu$. What is the fastest speed that a car can travel through the curve?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 27

An airplane is flying at constant speed $v$ in a horizontal circle of radius $r$. The lift force on the wings due to the air is perpendicular to the wings. At what angle to the vertical must the wings be banked to fly in this circle? (We tutorial: plane in turn)

Jose Martinez

Numerade Educator

Problem 28

A road with a radius of $75.0 \mathrm{~m}$ is banked so that a car can navigate the curve at a speed of $15.0 \mathrm{~m} / \mathrm{s}$ without any friction. When a car is going $20.0 \mathrm{~m} / \mathrm{s}$ on this curve, what minimum coefficient of static friction is needed if the car is to navigate the curve without slipping?

Narayan Hari

Numerade Educator

Problem 29

What is the average linear speed of the Earth about the Sun?

Jose Martinez

Numerade Educator

Problem 30

The orbital speed of Earth about the Sun is $3.0 \times 10^{4} \mathrm{~m} / \mathrm{s}$ and its distance from the Sun is $1.5 \times 10^{11} \mathrm{~m}$. The mass of Earth is approximately $6.0 \times 10^{24} \mathrm{~kg}$ and that of the Sun is $2.0 \times 10^{30} \mathrm{~kg}$. What is the magnitude of the force exerted by the Sun on Earth? [Hint: Two different methods are possible. Try both.]

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 31

Two satellites are in circular orbits around Jupiter. One, with orbital radius $r$, makes one revolution every $16 \mathrm{~h}$. The other satellite has orbital radius $4.0 r$. How long does the second satellite take to make one revolution around Jupiter?

Jose Martinez

Numerade Educator

Problem 32

The Hubble Space Telescope orbits Earth $613 \mathrm{~km}$ above Earth's surface. What is the period of the telescope's orbit?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 33

Io, one of Jupiter's satellites, has an orbital period of $1.77 \mathrm{~d}$. Europa, another of Jupiter's satellites, has an orbital period of about $3.54$ d. Both moons have nearly circular orbits. Use Kepler's third law to find the distance of each satellite from Jupiter's center. Jupiter's mass is $1.9 \times 10^{27} \mathrm{~kg}$.

Jose Martinez

Numerade Educator

Problem 34

A spy satellite is in circular orbit around Earth. It makes one revolution in $6.00 \mathrm{~h}$. (a) How high above Earth's surface is the satellite? (b) What is the satellite's acceleration?

Nishant Kumar

Numerade Educator

Problem 35

Mars has a mass of about $6.42 \times 10^{23} \mathrm{~kg}$. The length of ? day on Mars is $24 \mathrm{~h}$ and $37 \mathrm{~min}$, a little longer than the length of a day on Earth. Your task is to put a satellite into a circular orbit around Mars so that it stays above one spot on the surface, orbiting Mars once each Mars day. At what distance from the center of the planet should you place the satellite?

Jose Martinez

Numerade Educator

Problem 36

A satellite travels around Earth in uniform circular motion at an altitude of $35800 \mathrm{~km}$ above Earth's surface. The satellite is in geosynchronous orbit (that is, the time for it to complete one orbit is exactly $1 \mathrm{~d}$ ). $\operatorname{In}$ the figure with Multiple-Choice Questions $2-5$, the satellite moves counterclockwise $(A B C D A) .$ State directions in terms of the $x$ - and $y$ -axes. (a) What is the satellite's instantaneous velocity at point $C ?$ (b) What is the satellite's average velocity for one quarter of an orbit, starting at $A$ and ending at $B ?$ (c) What is the satellite's average acceleration for one quarter of an orbit, starting at $A$ and ending at $B ?$ (d) What is the satellite's instantaneous acceleration at point $D$ ?

Dominador Tan

Numerade Educator

Problem 37

A spacecraft is in orbit around Jupiter. The radius of the orbit is $3.0$ times the radius of Jupiter (which is $R_{\mathrm{J}}=71500 \mathrm{~km}$ ). The gravitational field at the surface of Jupiter is $23 \mathrm{~N} / \mathrm{kg}$. What is the period of the spacecraft's orbit? [Hint: You don't need to look up any more data about Jupiter to solve the problem.]

Narayan Hari

Numerade Educator

Problem 38

A roller coaster has a vertical loop with radius $29.5 \mathrm{~m}$. With what minimum speed should the roller coaster car be moving at the top of the loop so that the passengers do not lose contact with the seats?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 39

A pendulum is $0.80 \mathrm{~m}$ long and the bob has a mass of 1.0 $\mathrm{kg}$. At the bottom of its swing, the bob's speed is $1.6 \mathrm{~m} / \mathrm{s}$. (a) What is the tension in the string at the bottom of the swing? (b) Explain why the tension is greater than the weight of the bob.

Jose Martinez

Numerade Educator

Problem 40

A $35.0-\mathrm{kg}$ child swings on a rope with a length of $6.50 \mathrm{~m}$ that is hanging from a tree. At the bottom of the swing, the child is moving at a speed of $4.20 \mathrm{~m} / \mathrm{s}$. What is the tension in the rope?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 41

A car approaches the top of a hill that is shaped like a vertical circle with a radius of $55.0 \mathrm{~m}$. What is the fastest speed that the car can go over the hill without losing contact with the ground?

Jose Martinez

Numerade Educator

Problem 42

A child pushes a merry-go-round from rest to a final angular speed of $0.50 \mathrm{rev} / \mathrm{s}$ with constant angular acceleration. In doing so, the child pushes the merry-goround $2.0$ revolutions. What is the angular acceleration of the merry-go-round?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 43

A cyclist starts from rest and pedals so that the wheels make $8.0$ revolutions in the first $5.0 \mathrm{~s}$. What is the angular acceleration of the wheels (assumed constant)?

Jose Martinez

Numerade Educator

Problem 44

During normal operation, a computer's hard disk spins at $7200 \mathrm{rpm}$. If it takes the hard disk $4.0 \mathrm{~s}$ to reach this angular velocity starting from rest, what is the average angular acceleration of the hard disk in $\mathrm{rad} / \mathrm{s}^{2}$ ?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 45

Derive Eq. 5-20 from Eqs. 5-18 and 5-19. [Hint: See the derivation of Eq. (4-4) in Section 4.1.]

Jose Martinez

Numerade Educator

Problem 46

Derive Eq. $5-21$ from Eqs. $5-18$ and $5-19$.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 47

A pendulum is $0.800 \mathrm{~m}$ long and the bob has a mass of $1.00 \mathrm{~kg} .$ When the string makes an angle of $\theta=15.0^{\circ}$ with the vertical, the bob is moving at $1.40 \mathrm{~m} / \mathrm{s}$. Find the tangential and radial acceleration components and the tension in the string. [Hint: Draw an FBD for the bob. Choose the $x$ -axis to be tangential to the motion of the bob and the $y$ -axis to be radial. Apply Newton's second law.] Problems 47 and 48

Jose Martinez

Numerade Educator

Problem 48

Find the tangential acceleration of a freely swinging pendulum when it makes an angle $\theta$ with the vertical.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 49

A turntable reaches an angular speed of $33.3$ rpm in $2.0 \mathrm{~s}$, starting from rest. (a) Assuming the angular acceleration is constant, what is its magnitude? (b) How many revolutions does the turntable make during this time interval?

Jose Martinez

Numerade Educator

Problem 50

A wheel's angular acceleration is constant. Initially its angular velocity is zero. During the first $1.0$ -s time interval, it rotates through an angle of $90.0^{\circ}$. (a) Through what angle does it rotate during the next $1.0-\mathrm{s}$ time interval? (b) Through what angle during the third $1.0$ -s time interval?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 51

A car that is initially at rest moves along a circular path with a constant tangential acceleration component of $2.00 \mathrm{~m} / \mathrm{s}^{2}$. The circular path has a radius of $50.0 \mathrm{~m}$. The initial position of the car is at the far west location on the circle and the initial velocity is to the north. (a) After the car has traveled $\frac{1}{4}$ of the circumference, what is the speed of the car? (b) At this point, what is the radial acceleration component of the car? (c) At this same point, what is the total acceleration of the car?

Jose Martinez

Numerade Educator

Problem 52

A disk rotates with constant angular acceleration. The initial angular speed of the disk is $2 \pi \mathrm{rad} / \mathrm{s}$. After the disk rotates through $10 \pi$ radians, the angular speed is $7 \pi \mathrm{rad} / \mathrm{s}$. (a) What is the magnitude of the angular acceleration? (b) How much time did it take for the disk to rotate through $10 \pi$ radians? (c) What is the tangential acceleration of a point located at a distance of $5.0 \mathrm{~cm}$ from the center of the disk?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 53

In a Beams ultracentrifuge, the rotor is suspended magnetically in a vacuum. Since there is no mechanical connection to the rotor, the only friction is the air resistance due to the few air molecules in the vacuum. If the rotor is spinning with an angular speed of $5.0 \times 10^{5} \mathrm{rad} / \mathrm{s}$ and the driving force is turned off, its spinning slows down at an angular rate of $0.40 \mathrm{rad} / \mathrm{s}^{2} .$ (a) How long does the rotor spin before coming to rest? (b) During this time, through how many revolutions does the rotor spin?

Jose Martinez

Numerade Educator

Problem 54

The rotor of the Beams ultracentrifuge (see Problem 53 ) is $20.0 \mathrm{~cm}$ long. For a point at the end of the rotor, find the (a) initial speed, (b) tangential acceleration component, and (c) maximum radial acceleration component.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 55

If a washing machine's drum has a radius of $25 \mathrm{~cm}$ and spins at $4.0 \mathrm{rev} / \mathrm{s}$, what is the strength of the artificial gravity to which the clothes are subjected? Express your answer as a multiple of $g$.

Jose Martinez

Numerade Educator

Problem 56

A space station is shaped like a ring and rotates to simulate gravity. If the radius of the space station is $120 \mathrm{~m}$, at what frequency must it rotate so that it simulates Earth's gravity? [Hint: The apparent weight of the astronauts must be the same as their weight on Earth.] ( We tutorial: space station)

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 57

A biologist is studying growth in space. He wants to simulate Earth's
gravitational field, so he positions the plants on a rotating platform in the spaceship. The distance of each plant from the central axis of rotation is $r=0.20 \mathrm{~m}$. What angular speed is required?

Jose Martinez

Numerade Educator

Problem 58

A biologist is studying plant growth and wants to simulate a gravitational field twice as strong as Earth's. She places the plants on a horizontal rotating table in her laboratory on Earth at a distance of $12.5 \mathrm{~cm}$ from the axis of rotation. What angular speed will give the plants an effective gravitational field $\vec{g}_{e f f}$, whose magnitude is $2.0 \mathrm{~g}$ ? [Hint: Remember to account for Earth's gravitational field as well as the artificial gravity when finding the apparent weight.]

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 59

Objects that are at rest relative to the Earth's surface are in circular motion due to Earth's rotation. (a) What is the radial acceleration of an object at the equator? (b) Is the object's apparent weight greater or less than its weight? Explain. (c) By what percentage does the apparent weight differ from the weight at the equator?
(d) Is there any place on Earth where a bathroom scale reading is equal to your true weight? Explain.

Jose Martinez

Numerade Educator

Problem 60

A person of mass $M$ stands on a bathroom scale inside a Ferris wheel compartment. The Ferris wheel has radius $R$ and angular velocity $\omega$. What is the apparent weight of the person (a) at the top and (b) at the bottom?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 61

A person rides a Ferris wheel that turns with constant angular velocity. Her weight is $520.0 \mathrm{~N}$. At the top of the ride her apparent weight is $1.5 \mathrm{~N}$ different from her true weight. (a) Is her apparent weight at the top $521.5 \mathrm{~N}$ or $518.5 \mathrm{~N}$ ? Why? (b) What is her apparent weight at the bottom of the ride? (c) If the angular speed of the Ferris wheel is $0.025 \mathrm{rad} / \mathrm{s}$, what is its radius?

Jose Martinez

Numerade Educator

Problem 62

Objects that are at rest relative to Earth's surface are in circular motion due to Earth's rotation. What is the radial acceleration of a painting hanging in the Prado Museum in Madrid, Spain, at a latitude of $40.2^{\circ}$ North? (Note that the object's radial acceleration is not directed toward the center of the Earth.)

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 63

A rotating flywheel slows down at a constant rate due to friction in its bearings. After $1 \mathrm{~min}$, its angular velocity has diminished to $0.80$ of its initial value $\omega$. At the end of the third minute, what is the angular velocity in terms of the initial value?

Jose Martinez

Numerade Educator

Problem 64

The Earth rotates on its own axis once per day $(24.0 \mathrm{~h})$. What is the tangential speed of the summit of $\mathrm{Mt}$. Kilimanjaro (elevation $5895 \mathrm{~m}$ above sea level $)$, which is located approximately on the equator, due to the rotation of the Earth? The equatorial radius of Earth is $6378 \mathrm{~km}$.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 65

A trimmer for cutting weeds and grass near trees and borders has a nylon cord of $0.23-\mathrm{m}$ length that whirls about an axle at $660 \mathrm{rad} / \mathrm{s}$. What is the linear speed of the tip of the nylon cord?

Jose Martinez

Numerade Educator

Problem 66

A high-speed dental drill is rotating at $3.14 \times 10^{4} \mathrm{rad} / \mathrm{s}$. Through how many degrees does the drill rotate in $1.00 \mathrm{~s}$ ?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 67

A jogger runs counterclockwise around a path of radius $90.0 \mathrm{~m}$ at constant speed. He makes $1.00$ revolution in $188.4$ s. At $t=0$, he is heading due east. (a) What is the jogger's instantaneous velocity at $t=376.8 \mathrm{~s} ?$ (b) What is his instantaneous velocity at $t=94.2 \mathrm{~s}$ ?

Jose Martinez

Numerade Educator

Problem 68

Two gears $A$ and $B$ are in contact. The radius of gear $A$ is twice that of gear $B$. (a) When A's angular velocity is $6.00 \mathrm{~Hz}$ counterclockwise, what is $B$ 's angular velocity?
(b) If $A$ 's radius to the tip of the teeth is $10.0 \mathrm{~cm}$, what is the linear speed of a point on the tip of a gear tooth? What is the linear speed of a point on the tip of $B$ 's gear tooth?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 69

If gear $A$ in Problem 68 has an initial frequency of $0.955 \mathrm{~Hz}$ and an angular acceleration of $3.0 \mathrm{rad} / \mathrm{s}^{2}$, how many rotations does each gear go through in $2.0 \mathrm{~s}$ ?

Jose Martinez

Numerade Educator

Problem 70

The time to sunset can be estimated by holding out your arm, holding your fingers horizontally in front of your eyes, and counting the number of fingers that fit between the horizon and the setting Sun. (a) What is the angular speed, in radians per second, of the Sun's apparent circular motion around the Earth? (b) Estimate the angle subtended by one finger held at arm's length. (c) How long in minutes does it take the Sun to "move" through this same angle?

Penny Riley

Numerade Educator

Problem 71

In the professional videotape recording system known as quadriplex, four tape heads are mounted on thecircumference of a drum of radius $2.5 \mathrm{~cm}$ that spins at $1500 \mathrm{rad} / \mathrm{s}$. (a) At what speed are the tape heads moving?
(b) Why are moving tape heads used instead of stationary ones, as in audiotape recorders? [Hint: How fast would the tape have to move if the heads were stationary?]

Jose Martinez

Numerade Educator

Problem 72

The Milky Way galaxy rotates about its center with a period of about 200 million yr. The Sun is $2 \times 10^{20} \mathrm{~m}$ from the center of the galaxy. How fast is the Sun moying with respect to the center of the galaxy?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 73

A small body of mass $0.50 \mathrm{~kg}$ is attached by a $0.50-\mathrm{m}$ long cord to a pin set into the surface of a frictionless table top. The body moves in a circle on the horizontal surface with a speed of $2.0 \pi \mathrm{m} / \mathrm{s}$. (a) What is the magnitude of the radial acceleration of the body? (b) What is the tension in the cord?

Jose Martinez

Numerade Educator

Problem 74

Two blocks, one with mass $m_{1}=0.050 \mathrm{~kg}$ and one with mass $m_{2}=0.030 \mathrm{~kg}$, are connected to one another by a string. The inner block is connected to a central pole by another string as shown in the figure with $r_{1}=0.40 \mathrm{~m}$ and $r_{2}=0.75 \mathrm{~m}$. When the blocks are spun around on a horizontal frictionless surface at an angular speed of $1.5$ rev $/ \mathrm{s}$, what is the tension in each of the two strings?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 75

What's the fastest way to make a U-turn at constant speed? Suppose that you need to make a $180^{\circ}$ turn on a circular path. The minimum radius (due to the car's steering system) is $5.0 \mathrm{~m}$, while the maximum (due to the width of the road) is $20.0 \mathrm{~m}$. Your acceleration must never exceed $3.0 \mathrm{~m} / \mathrm{s}^{2}$ or else you will skid. Should you use the smallest possible radius, so the distance is small, or the largest, so you can go faster without skidding, or something in between? What is the minimum possible time for this U-turn?

Jose Martinez

Numerade Educator

Problem 76

The Milky Way galaxy rotates about its center with a period of about 200 million yr. The Sun is $2 \times 10^{20} \mathrm{~m}$ from the center of the galaxy. (a) What is the Sun's radial acceleration? (b) What is the net gravitational force on the Sun due to the other stars in the Milky Way?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 77

Bacteria swim using a corkscrew-like helical flagellum that rotates. For a bacterium with a flagellum that has a pitch of $1.0 \mu \mathrm{m}$ that rotates at $110 \mathrm{rev} / \mathrm{s}$, how fast could it swim if there were no "slippage" in the medium in which it is swimming? The pitch of a helix is the distance between "threads."

Jose Martinez

Numerade Educator

Problem 78

You place a penny on a turntable at a distance of $10.0 \mathrm{~cm}$ from the center. The coefficient of static friction between the penny and the turntable is $0.350$. The turntable's

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 79

A coin is placed on a turntable that is rotating at $33.3$ rpm. If the coefficient of static friction between the coin and the turntable is $0.1$, how far from the center of the turntable can the coin be placed without haying it slip off?

Jose Martinez

Numerade Educator

Problem 80

Grace, playing with her dolls, pretends the turntable of an old phonograph is a merry-go-round. The dolls are $12.7 \mathrm{~cm}$ from the central axis. She changes the setting from $33.3 \mathrm{rpm}$ to $45.0 \mathrm{rpm}$. (a) For this new setting, what is the linear speed of a point on the turntable at the location of the dolls? (b) If the coefficient of static friction between the dolls and the turntable is $0.13$, do the dolls stay on the turntable?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 81

Your car's wheels are $65 \mathrm{~cm}$ in diameter and the wheels are spinning at an angular velocity of $101 \mathrm{rad} / \mathrm{s}$. How fast is your car moving in kilometers per hour (assume no slippage)?

Jose Martinez

Numerade Educator

Problem 82

In an amusement park rocket ride, cars are suspended from $4.25-\mathrm{m}$ cables attached to rotating arms at a distance of $6.00 \mathrm{~m}$ from the axis of rotation. The cables swing out at an angle of $45.0^{\circ}$ when the ride is operating. What is the angular speed of rotation?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 83

Centrifuges are commonly used in biological laboratories for the isolation and maintenance of cell preparations. For cell separation, the centrifugation conditions are typically $1.0 \times 10^{3}$ rpm using an $8.0$ -cm-radius rotor.
(a) What is the radial acceleration of material in the centrifuge under these conditions? Express your answer as a multiple of $g$. (b) At $1.0 \times 10^{3} \mathrm{rpm}$ (and with a $8.0$ -cm rotor), what is the net force on a red blood cell whose mass is $9.0 \times 10^{-14} \mathrm{~kg}$ ? (c) What is the net force on a virus particle of mass $5.0 \times 10^{-21} \mathrm{~kg}$ under the same conditions? (d) To pellet out virus particles and even to separate large molecules such as proteins, superhigh-speed centrifuges called ultracentrifuges are used in which the rotor spins in a vacuum to reduce heating due to friction. What is the radial acceleration inside an ultracentrifuge at $75000 \mathrm{rpm}$ with an $8.0-\mathrm{cm}$ rotor? Express your answer as a multiple of $g$.

Jose Martinez

Numerade Educator

Problem 84

You take a homemade "accelerometer" to an amusement park. This accelerometer consists of a metal nut attached to a string and connected to a protractor, asshown in the figure. While riding a roller coaster that is moving at a uniform speed around a circular path, you hold up the accelerometer and notice that the string is making an angle of $55^{\circ}$ with respect to the vertical with the nut pointing away from the center of the circle, as shown. (a) What is the radial acceleration of the roller coaster? (b) What is your radial acceleration expressed as a multiple of $g$ ?
(c) If the roller coaster track is turn- Center ing in a radius of roller $80.0 \mathrm{~m}$, how fast are circular
you moving?

Narayan Hari

Numerade Educator

Problem 85

Massimo, a machinist, is cutting threads for a bolt on a lathe. He wants the bolt to have 18 threads per inch. If the cutting tool moves parallel to the axis of the wouldbe bolt at a linear velocity of $0.080 \mathrm{in} / \mathrm{s}$, what must the rotational speed of the lathe chuck be to ensure the correct thread density? [Hint: One thread is formed for each complete revolution of the chuck.]

Narayan Hari

Numerade Educator

Problem 86

In Chapter 19 we will see that a charged particle can undergo uniform circular motion when acted on by a magnetic force and no other forces. (a) For that to be true, what must be the angle between the magnetic force and the particle's velocity? (b) The magnitude of the magnetic force on a charged particle is proportional to the particle's speed, $F=k v$. Show that two identical charged particles moving in circles at different speeds in the same magnetic field must have the same period.
(c) Show that the radius of the particle's circular path is proportional to the speed.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 87

Find the orbital radius of a geosynchronous satellite. Do not assume the speed found in Example 5.9. Start by writing an equation that relates the period, radius, and speed of the orbiting satellite. Then apply Newton's second law to the satellite. You will have two equations with two unknowns (the speed and radius). Eliminate the speed algebraically and solve for the radius.

Jose Martinez

Numerade Educator