Physics

Robert Coleman Richardson; Betty McCarthy Richardson; Alan Giambattista

Chapter 5

Circular Motion - all with Video Answers

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Chapter Questions

Problem 1

The seat on a carnival ride is fixed on the end of an $8.0 \mathrm{m}$ long beam, pivoted at the other end. If the beam sweeps through an angle of $120^{\circ},$ what is the distance through which the rider moves?

Narayan Hari

Narayan Hari

Numerade Educator

Problem 2

Convert these to radian measure: (a) $30.0^{\circ}$ (b) 33.3 revolutions.

Narayan Hari

Numerade Educator

Problem 3

Find the average angular speed of the second hand of an analog clock. What is its angular displacement during $5.0 \mathrm{s} ?$

Narayan Hari

Numerade Educator

Problem 4

An elevator cable winds on a drum of radius $90.0 \mathrm{cm}$ that is connected to a motor.
(a) If the elevator moves 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? (c) What is the drum's frequency of rotation?

Narayan Hari

Numerade Educator

Problem 5

A wheel of radius $30 \mathrm{cm}$ is rotating at a rate of 2.0 revolutions every 0.080 s. (a) Through what angle, in radians, does the wheel rotate in $1.0 \mathrm{s} ?$ (b) What is the linear speed of a point on the wheel's rim? (c) What is the wheel's frequency of rotation?

Narayan Hari

Numerade Educator

Problem 6

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

Narayan Hari

Numerade Educator

Problem 7

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

Jose Martinez

Numerade Educator

Problem 8

Dung beetles are renowned for building large (relative to their body size) balls of dung and rolling them on the ground. (a) If a dung beetle can roll (without slipping) a ball of dung whose radius is $2.5 \mathrm{cm}$ at a linear speed of $3.5 \mathrm{cm} / \mathrm{s},$ through what angle does the ball roll as the ball moves a distance of $15 \mathrm{cm} ?$ (b) What is the angular speed (assumed constant) of the ball's rotation?

Narayan Hari

Numerade Educator

Problem 9

In aviation, a standard rate turn proceeds at an angular speed of $180^{\circ}$ per minute. What is the radius of a standard rate turn for a plane moving at $240 \mathrm{m} / \mathrm{s} ?$

Narayan Hari

Numerade Educator

Problem 10

In the construction of railroads, curvature of the track is measured in the following way. First a $100.0 \mathrm{ft}$ long chord is measured. Then the curvature is reported as the angle subtended by two radii at the endpoints of the chord. (The angle is measured by determining the angle between two tangents $100 \mathrm{ft}$ apart; since each tangent is perpendicular to a radius, the angles are the same.) In modern railroad construction, track curvature is kept below $1.5^{\circ} .$ What is the radius of curvature of a "1.5 $^{\circ}$ curve"? [Hint: since the angle is small, the length of the chord is approximately equal to the arc length along the curve.]

Narayan Hari

Numerade Educator

Problem 11

Five flywheels are spinning as follows:(a) radius $8.0 \mathrm{cm},$ period $4.0 \mathrm{ms} ;$ (b) radius $2.0 \mathrm{cm},$ period (d) radius $2.0 \mathrm{cm}$ $4.0 \mathrm{ms} ;(\mathrm{c})$ radius $8.0 \mathrm{cm},$ period $1.0 \mathrm{ms}$
period $1.0 \mathrm{ms}$ (e) radius $1.0 \mathrm{cm},$ period $4.0 \mathrm{ms}$.
Rank the flywheels in order of angular speed, largest to smallest. Explain.

Narayan Hari

Numerade Educator

Problem 12

Five flywheels are spinning as follows:(a) radius $8.0 \mathrm{cm},$ period $4.0 \mathrm{ms} ;$ (b) radius $2.0 \mathrm{cm},$ period (d) radius $2.0 \mathrm{cm}$ $4.0 \mathrm{ms} ;(\mathrm{c})$ radius $8.0 \mathrm{cm},$ period $1.0 \mathrm{ms}$
period $1.0 \mathrm{ms}$ (e) radius $1.0 \mathrm{cm},$ period $4.0 \mathrm{ms}$.
Rank the flywheels in order of the linear speed at the rim, largest to smallest. Explain.

Jose Carlos

Numerade Educator

Problem 13

Rank the flywheels of Problems 11 and 12 in order of the radial acceleration of a point on the rim, largest to smallest.

Jose Carlos

Numerade Educator

Problem 14

An apparatus is designed to study insects at an acceleration of magnitude $980 \mathrm{m} / \mathrm{s}^{2}(=100 \mathrm{g}) .$ 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?

Narayan Hari

Numerade Educator

Problem 15

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 16

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 static friction between a person and the wall of the cylinder 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.)

Narayan Hari

Numerade Educator

Problem 17

Medical testing has established that the maximum acceleration a pilot can be subjected to without losing consciousness is approximately $5.0 \mathrm{g}$ if the axis of acceleration is aligned with the spine. (See Example $5.4 .)$ A. pilot can avoid "blackout" at accelerations up to approximately 9.0 $g$ by wearing special "g-suits" that help keep blood pressure in the brain at a sufficient level. (a) Assuming this to be the case, what is the minimum safe radius of curvature for an unprotected pilot flying an $\mathrm{F}-15$ in a horizontal circular loop at $750 \mathrm{km} / \mathrm{h} ?$
(b) What does this radius become if the pilot is wearing a g-suit?

Narayan Hari

Numerade Educator

Problem 18

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 a constant angle of $70.0^{\circ}$ with respect to the vertical. What is the tangential speed of the ball?

Prabhu Ramji

Numerade Educator

Problem 19

A child's toy has a $0.100 \mathrm{kg}$ ball attached to two strings, $A$ and $B$. The strings are also attached to a stick and the ball 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 $\mathrm{FBD}$ 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 20

A child swings a rock of mass $m$ in a horizontal circle using a rope of length $L$. The rope makes a constant angle $\theta$ with the horizontal. The rock moves at constant speed $v .$ What is the tension in the rope? Express the tension in terms of $m, g, v, L,$ and $\theta$

Narayan Hari

Numerade Educator

Problem 21

A conical pendulum (see Example 5.6 ) has a bob of mass $m$ and a string of length $L$. It is swinging in a horizontal circle. The angle that the string makes with the vertical is $\phi .$ Find (a) the tension in the string and (b) the period of the pendulum in terms of $m, L, \phi,$ and $g,$ as needed.

Narayan Hari

Numerade Educator

Problem 22

A curve in a stretch of highway has radius $512 \mathrm{m}$ The road is unbanked. The coefficient of static friction between the tires and road is 0.70. (a) What is the maximum speed that a car can travel around the curve without skidding? (b) Explain what happens when a car enters the curve at a speed greater than this maximum safe speed. Illustrate with an $\mathrm{FBD}$.

Surjit Tewari

Numerade Educator

Problem 23

A roller coaster car of mass $320 \mathrm{kg}$ (including passengers) travels around a horizontal curve of radius $35 \mathrm{m}$. Its speed is $16 \mathrm{m} / \mathrm{s}$. (a) What are the magnitude and direction of the total force exerted on the car by the track?
(b) What is the banking angle of the track if the frictional force is zero, so that the track exerts only a normal force on the car?

Narayan Hari

Numerade Educator

Problem 24

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 25

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

Narayan Hari

Numerade Educator

Problem 26

A curve in a highway has radius of curvature $320 \mathrm{m}$ 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?

Narayan Hari

Numerade Educator

Problem 27

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 $\mu_{\mathrm{s}} N ?$ Explain.

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 28

An airplane is flying at constant speed $740 \mathrm{km} / \mathrm{h}$ in a horizontal circle of radius 4.1 km. 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?

Narayan Hari

Numerade Educator

Problem 29

A road with a radius of $75.0 \mathrm{m}$ is banked so that a car The road is banked at angle $5.8^{\circ}$ to the horizontal. The coefficient of static friction between the tires and road is
$0.50 .$ What is the fastest speed that a car can travel through the curve without skidding?

Surjit Tewari

Numerade Educator

Problem 30

A curve in a stretch of highway has radius $610 \mathrm{m}$. The road is banked at angle $5.8^{\circ}$ to the horizontal. The coefficient of static friction between the tires and road is $0.50 .$ What is the fastest speed that a car can travel through the curve without skidding?

Surjit Tewari

Numerade Educator

Problem 31

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 32

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 33

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

Jose Martinez

Numerade Educator

Problem 34

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 35

Io, one of Jupiter's satellites, has an orbital period of 1.77 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 36

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?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 37

Two satellites are in circular orbits around Jupiter. One, with orbital radius $r$, makes one revolution every 16 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 38

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

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 39

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 40

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 41

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 42

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 43

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

Narayan Hari

Numerade Educator

Problem 44

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 45

During normal operation, a computer's hard disk spins at 7200 rev/min. 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 $\mathrm{rad} / \mathrm{s}^{2} ?$

Narayan Hari

Numerade Educator

Problem 46

A hamster of mass $0.100 \mathrm{kg}$ gets into its exercise wheel and starts to run at $t=0 .$ After $t=0.800 \mathrm{s},$ the wheel turns with a constant rotational frequency of $1.00 \mathrm{Hz} .$ What is the tangential acceleration of the inner surface of the wheel between $t=0$ and $t=0.800 \mathrm{s}$ assuming it is constant? The wheel's inner diameter is $20.0 \mathrm{cm}$

Narayan Hari

Numerade Educator

Problem 47

A clothes washer reaches an angular speed of 1400 rev/min in $2.0 \mathrm{s}$, starting from rest, during the spin cycle. (a) Assuming the angular acceleration is constant, what is its magnitude? (b) How many revolutions does the washer make during this time interval?

Narayan Hari

Numerade Educator

Problem 48

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 s time interval? (b) Through what angle during the third 1.0 s time interval?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 49

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 one fourth 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?

Narayan Hari

Numerade Educator

Problem 50

A disk rotates with constant angular acceleration. The initial angular speed of the disk is $2.0 \pi \mathrm{rad} / \mathrm{s}$. After the disk rotates through $10.0 \pi$ radians, the angular speed is $7.0 \pi \mathrm{rad} / \mathrm{s}$
(a) What is the magnitude of the angular acceleration?
(b) How much time did it take for the disk to rotate through $10.0 \pi$ 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?

Narayan Hari

Numerade Educator

Problem 51

A "blink of an eye" is a time interval of about $150 \mathrm{ms}$ for an average adult. The "closure" portion of the blink takes only about $55 \mathrm{ms}$. Let us model the closure of the upper eyelid as uniform angular acceleration through an angular displacement of $15^{\circ}$. (a) What is the value of the angular acceleration the eyelid undergoes while closing? (b) What is the tangential acceleration of the edge of the eyelid while closing if the radius of the eyeball is $1.25 \mathrm{cm} ?$

Narayan Hari

Numerade Educator

Problem 52

A study was done observing the ability of the eye to rapidly rotate in order to follow a moving object by placing contact lenses that contain accelerometers on a subject's eye. The eyeball has radius $1.25 \mathrm{cm} .$ Suppose that, while the subject watches a moving object, the eyeball rotates through $20.0^{\circ}$ in a time interval of 75 ms. (a) What is the magnitude of the average angular velocity of the eye? (b) Assume that the eye starts at rest, rotates with a constant angular acceleration during the first half of the interval, and then the rotation slows with a constant angular acceleration during the second half until it comes to rest. What is the magnitude of the angular acceleration of the eye? (c) What tangential acceleration would the contact-lens accelerometers record in this case?

Dominador Tan

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 magnitude $0.40 \mathrm{rad} / \mathrm{s}^{2}$. (a) How long does the rotor spin before coming to rest? (b) During this time, through what angular displacement does the rotor turn?

Narayan Hari

Numerade Educator

Problem 54

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

Narayan Hari

Numerade Educator

Problem 55

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 $\mathrm{FBD}$ 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.]

Jose Martinez

Numerade Educator

Problem 56

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 57

If a clothes washer's drum has a radius of $25 \mathrm{cm}$ and spins at 4.0 rev/s, what is the strength of the apparent gravitational field to which the clothes are subjected? Ignore Earth's gravity and express your answer as a multiple of $g .$

Narayan Hari

Numerade Educator

Problem 58

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.]

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 59

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 60

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? $(\mathrm{b})$ 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 61

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 62

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 apparent gravitational field $\overrightarrow{\mathbf{g}}_{\text {app }}$ whose magnitude is $2.0 g ?$

Narayan Hari

Numerade Educator

Problem 63

Mars has a mass of about $6.42 \times 10^{23} \mathrm{kg} .$ The length of a day on Mars is 24 h and 37 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 64

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 65

The time to sunset can be estimated by holding out your arm with your fingers perpendicular to the path the Sun will follow to the horizon. The number of fingers that fit between the Sun and the sunset point is proportional to the time remaining. (a) What is the angular speed, in radians per second, of the Sun's apparent circular motion around 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?

Narayan Hari

Numerade Educator

Problem 66

What's the quickest 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?

Narayan Hari

Numerade Educator

Problem 67

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, as shown in the figure. While riding a roller coaster that is moving at uniform speed around a horizontal circular path, you hold up the accelerometer and notice that the string is making a constant 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 turning in a radius of $80.0 \mathrm{m},$ how fast are you moving?

Narayan Hari

Numerade Educator

Problem 68

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 69

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

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 70

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 71

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 72

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 73

Two gears $A$ and $B$ are turning in mesh. Gear $A$ 's radius to the point of contact between the gears is $8.0 \mathrm{cm}$ and that of gear $B$ is $4.0 \mathrm{cm}$.
(a) What is the linear speed of the contact point when gear $A$ 's angular velocity is $6.0 \mathrm{rad} / \mathrm{s}$ counterclockwise?
(b) What is $B$ 's angular velocity?

Narayan Hari

Numerade Educator

Problem 74

If gear $A$ in Problem 73 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 s?

Narayan Hari

Numerade Educator

Problem 75

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 moving with respect to the center of the galaxy?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 76

A small object 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 object 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 object? (b) What is the tension in the cord?

Surjit Tewari

Numerade Educator

Problem 77

Two blocks, one with mass $m_{1}=0.050 \mathrm{kg}$ and one with mass $m_{2}=0.030 \mathrm{kg},$ are connected to each other 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 $1.5 \mathrm{rev} / \mathrm{s},$ what is the tension in each of the two strings?

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 78

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 79

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 80

You place a penny on an old 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 angular acceleration is $2.00 \mathrm{rad} / \mathrm{s}^{2} .$ How long after you turn on the turntable will the penny begin to slide?

Narayan Hari

Numerade Educator

Problem 81

A coin is placed on an old turntable. If the coefficient of static friction between the coin and the turntable is 0.10 , how far from the center of the turntable can the coin be placed without having it slip off when the turntable rotates at $33.3 \mathrm{rev} / \mathrm{min}$ ?

Narayan Hari

Numerade Educator

Problem 82

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 Earth.)

Jordan Vanevery

Jordan Vanevery

Numerade Educator

Problem 83

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 a constant angle of $45.0^{\circ}$ when the ride is operating. What is the angular speed of rotation?

Narayan Hari

Numerade Educator

Problem 84

Centrifuges are commonly used in biological laboratories for the isolation and maintenance of cell preparations. For cell separation, the centrifugation conditions are typically $1.0 \times 10^{3} \mathrm{rev} / \mathrm{min}$ using an $8.0 \mathrm{cm}$ 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 $1.0 \times 10^{3}$ rev/min (and with an $8.0 \mathrm{cm}$ 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, super-high-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 rev/min with an $8.0 \mathrm{cm}$ rotor? Express your answer as a multiple of $g .$

Narayan Hari

Numerade Educator

Problem 85

A proposed "space elevator" consists of a cable going all the way from the ground to a space station in geostationary orbit (always above the same point on Earth's surface). Elevator "cars" would climb the cable to transport cargo to outer space. Consider a cable connected between the equator and a space station at height $H$ above the surface. Ignore the mass of the cable*. Find the height $H .$ (b) Suppose there is an elevator car of mass $100 \mathrm{kg}$ sitting halfway up at height $H / 2 .$ What tension $T$ would be required in the cable to hold the car in place? Which part of the cable would be under tension (above the car or below it)?

Narayan Hari

Numerade Educator

Problem 86

A star near the visible edge of a galaxy travels in a uniform circular orbit. It is 40000 ly (light-years) from the galactic center and has a speed of $275 \mathrm{km} / \mathrm{s}$.
(a) Estimate the total mass of the galaxy based on the motion of the star. [Hint: For this estimate, assume the total mass to be concentrated at the galactic center and relate it to the gravitational force on the star.] (b) The total visible mass (i.e., matter we can detect via electromagnetic radiation) of the galaxy is $10^{11}$ solar masses. What fraction of the total mass of the galaxy is visible $^{\dagger},$ according to this estimate?

Narayan Hari

Numerade Educator

Problem 87

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 in./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 88

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 89

A rotating flywheel slows down with constant angular acceleration due to friction in its bearings. At $t=0$, its angular velocity is $420 \mathrm{rad} / \mathrm{s}$. At $t=60 \mathrm{s}$, its angular velocity is $340 \mathrm{rad} / \mathrm{s}$. (a) What is the angular velocity at $t=180 \mathrm{s} ?$ (b) Through how many revolutions has it turned at $t=180 \mathrm{s} ?$

Narayan Hari

Numerade Educator

Problem 90

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. In the figure with Multiple-Choice Questions $1-4,$ 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 91

Objects that are at rest relative to 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 92

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 in radians? (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.

DM

Debra Mangion

Numerade Educator

Problem 93

Find the orbital radius of a geostationary satellite without using 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.

Narayan Hari

Numerade Educator

Problem 94

Two blocks are connected by a light string passing over an ideal pulley. The block with mass $m_{1}=20.0 \mathrm{kg}$ slides on a frictionless horizontal surface, while the block with mass $m_{2}=1.0 \mathrm{kg}$ hangs vertically. The radius of the pulley is $6.0 \mathrm{cm}$. (a) Assuming that the pulley rotates such that the string doesn't slip, find the angular acceleration of the pulley. (b) If the block on the table is released from rest, calculate how many revolutions the pulley has made 2.0 s later, assuming the other block hasn't reached the floor.

Narayan Hari

Numerade Educator

Problem 95

A ball weighing $20.0 \mathrm{N}$ is tied to a string fixed to the ceiling. The string makes a $30.0^{\circ}$ angle with the ceiling. Initially, the ball is held in place by a force $\overrightarrow{\mathbf{F}}$ that is perpendicular to the string. (a) What is the magnitude of the force $\overrightarrow{\mathbf{F}} ?$ (b) What is the tension in the string? (c) Just after the ball is released and allowed to start swinging back and forth, what are the tension
in the string, the radial acceleration of the ball, and the tangential acceleration of the ball?

Ozenc Gungor

Numerade Educator

Problem 96

A wheel of radius $r$ rolls to the right without slipping on a horizontal road. Its axle moves at a constant speed $v_{\text {axle }}$ (a) Find the velocities of points $A, B,$ and $C$ with respect to the axle. Express your answers in terms of $v_{\text {axle }}$ and $r,$ as needed. [Hint: In the reference frame of the axle, the wheel is rotating in place at a constant angular speed $\omega .]$ (b) Find the velocities of points $A, B,$ and $C$ with respect to the road.
(c) Comment on the velocity of point $C$ with respect to the road.

DM

Debra Mangion

Numerade Educator