University Physics with Modern Physics

Wolfgang Bauer, Gary D. Westfall

Chapter 9

Circular Motion - all with Video Answers

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Chapter Questions

Problem 1

An object is moving in a circular path. If the centripetal force is suddenly removed, how will the object move?
a) It will move radially outward.
b) It will move radially inward.
c) It will move vertically downward.
d) It will move in the direction in which its velocity vector points at the instant the centripetal force vanishes.

Prabhu Ramji

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Problem 2

The angular acceleration for an object undergoing circular motion is plotted versus time in the figure. If the object started from rest at $t=0 \mathrm{~s}$, the net angular displacement of the object at $t=t_{\mathrm{f}}$
a) is in the clockwise direction.
b) is in the counterclockwise direction.
c) is zero.
d) cannot be determined.

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Problem 3

The latitude of Lubbock, Texas (known as the Hub City of the South Plains), is $33^{\circ} \mathrm{N}$. What is its rotational speed, assuming the radius of the Earth at the Equator to be $6380 \mathrm{~km} ?$
a) $464 \mathrm{~m} / \mathrm{s}$
b) $389 \mathrm{~m} / \mathrm{s}$
c) $253 \mathrm{~m} / \mathrm{s}$
d) $0.464 \mathrm{~m} / \mathrm{s}$
e) $0.389 \mathrm{~m} / \mathrm{s}$

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Problem 4

A rock attached to a string moves clockwise in uniform circular motion. In which direction from point $A$ is the rock thrown off when the string is cut?

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Problem 5

A rock attached to a string moves clockwise in uniform circular motion. In which direction from point $A$ is the rock thrown off when the string is cut?
a) centrifugal
b) normal
c) gravity
d) tension

Averell Hause

Carnegie Mellon University

Problem 6

In a conical pendulum, a bob moves in a horizontal circle, as shown in the figure. The period of the pendulum (the time it takes for the bob to perform a complete revolution) is
a) $T=2 \pi \sqrt{L \cos \theta / g}$.
b) $T=2 \pi \sqrt{g \cos \theta / L}$
c) $T=2 \pi \sqrt{L g \sin \theta}$
d) $T=2 \pi \sqrt{L \sin \theta / g}$.
e) $T=2 \pi \sqrt{L / g}$.

Prabhu Ramji

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Problem 7

A ball attached to the end of a string is swung around in a circular path of radius $r$. If the radius is doubled and the linear speed is kept constant, the centripetal acceleration
a) remains the same.
b) increases by a factor of 2 .
c) decreases by a factor of 2 .
d) increases by a factor of 4
e) decreases by a factor of 4 .

Prabhu Ramji

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Problem 8

The angular speed of the hour hand of a clock (in radians per second) is
a) $\frac{\pi}{7200}$
b) $\frac{\pi}{3600}$
c) $\frac{\pi}{1800}$
d) $\frac{\pi}{60}$
e) $\frac{\pi}{30}$
f) The correct value is not shown.

Prabhu Ramji

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Problem 9

You put three identical coins on a turntable at different distances from the center and then turn the motor on. As the turntable speeds up, the outermost coin slides off first, followed by the one at the middle distance, and, finally, when the turntable is going the fastest, the innermost one. Why is this?
a) For greater distances from the center the centripetal acceleration is higher, and so the force of friction becomes unable to hold the coin in place.
b) The weight of the coin causes the turntable to flex downward, so the coin nearest the edge falls off first.
c) Because of the way the turntable is made, the coefficient of static friction decreases with distance from the center.
d) For smaller distances from the center, the centripetal acceleration is higher.

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Problem 10

A point on a Blu-ray disc is a distance $R / 4$ from the axis of rotation. How far from the axis of rotation is a second point that has, at any instant, a linear velocity twice that of the first point?
a) $R / 16$
b) $R / 8$
c) $R / 2$
d) $R$

Prabhu Ramji

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Problem 11

The figure shows a rider stuck to the wall in the Barrel of Fun at a carnival. Which diagram correctly shows the forces acting on the rider?
a.
b.
c.
d.
e.

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Problem 12

A string is tied to a rock, and the rock is twirled around in a circle at a constant speed. If gravity is ignored and the period of the circular motion is doubled, the tension in the string is
a) reduced to $\frac{1}{4}$ of its original value.
b) reduced to $\frac{1}{2}$ of its original value.
c) unchanged.
d) increased to 2 times its original value.
e) increased to 4 times its original value.

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Problem 13

A ceiling fan is rotating in clockwise direction (as viewed from below) but it is slowing down. What are the directions of $\omega$ and $\alpha ?$

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Problem 14

A hook above a stage is rated to support $150 .$ lb. A 3 -lb rope is attached to the hook, and a 147 -lb actor is going to attempt to swing across the stage on the rope. Will the hook hold the actor up during the swing?

Prabhu Ramji

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Problem 15

A popular carnival ride consists of seats attached to a central disk through cables, as shown in the figure. The passengers travel in uniform circular motion. The mass of one of the passengers (including the chair he is sitting on) is $65 \mathrm{~kg}$; the mass of an empty chair on the opposite side of the central disk is $5 \mathrm{~kg}$. If $\theta_{1}$ and $\theta_{2}$ are the angles that the cables attached to the two chairs make with respect to vertical, how do these two angles compare qualitatively? Is $\theta_{2}$ larger than, smaller than, or equal to $\theta_{1} ?$

Prabhu Ramji

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Problem 16

A person rides on a Ferris wheel of radius $R$, which is rotating at a constant angular velocity $\omega$. Compare the normal force of the seat pushing up on the person at point $A$ to that at point $B$ in the figure. Which force is greater, or are they the same?

Prabhu Ramji

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Problem 17

Bicycle tires range in size from about $25 \mathrm{~cm}$ in diameter to about $70 \mathrm{~cm}$ in diameter. Why is it impractical to make tires much smaller than $25 \mathrm{~cm}$ in diameter? (You will learn why bicycle tires can't be too large in Chapter $10 .)$

Averell Hause

Carnegie Mellon University

Problem 18

A CD starts from rest and speeds up to the operating angular frequency of the CD player. Compare the angular velocity and acceleration of a point on the edge of the CD to those of a point halfway between the center and the edge of the CD. Do the same for the linear velocity and acceleration.

Prabhu Ramji

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Problem 19

A car is traveling around an unbanked curve at a maximum speed. Which force(s) is(are) responsible for keeping it on the road?

Prabhu Ramji

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Problem 20

Two masses hang from two strings of equal length that are attached to the ceiling of a car. One mass is over the driver's seat; the other is over the passenger's seat. As the car makes a sharp turn, both masses swing away from the center of the turn. In their resulting positions, will they be farther apart, closer together, or the same distance apart as they were when the car wasn't turning?

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Problem 21

A point mass $m$ starts sliding from a height $h$ along the frictionless surface shown in the figure. What is the mini-
mum value of $h$ in order for the mass to complete the loop of radius $R ?$

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Problem 22

In a conical pendulum, the bob attached to the string (which can be considered massless) moves in a horizontal circle at constant speed. The string sweeps out a cone as the bob rotates. What forces are acting on the bob?

Prabhu Ramji

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Problem 23

Is it possible to swing a mass attached to a string in a perfectly horizontal circle (with the mass and the string parallel to the ground)?

Prabhu Ramji

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Problem 24

A small ice block of mass $m$ starts from rest from the top of an inverted bowl in the shape of a hemisphere, as shown in the figure. The hemisphere is fixed to the ground, and the block slides without friction along the surface of the hemisphere. Find the normal force exerted by the block on the sphere when the line between the block and the center of the sphere makes an angle $\theta$ with the horizontal. Discuss the result.

Prabhu Ramji

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Problem 25

Suppose you are riding on a roller coaster, which moves through a vertical circular loop. Show that your apparent weight at the bottom of the loop is six times your weight when you experience weightlessness at the top, independent of the size of the loop. Assume that friction is negligible.

Prabhu Ramji

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Problem 26

The following event actually occurred on the Sunshine Skyway Bridge near St. Petersburg, Florida, in $1997 .$ Five daredevils tied a $55-\mathrm{m}$ -long cable to the center of the bridge. They hoped to swing back and forth under the bridge at the end of this cable. The five people (total weight $=W$ ) attached themselves to the end of the cable, at the same level and $55 \mathrm{~m}$ away from where it was attached to the bridge and dropped straight down from the bridge, following the dashed circular path indicated in the figure. Unfortunately, the daredevils were not well versed in the laws of physics, and the cable broke (at the point it was linked to their seats) at the bottom of their swing. Determine how strong the cable (and all the links where the seats and the bridge are attached to it) would have to be in order to support the five people at the bottom of the swing. Express your result in terms of their total weight, $W$.

Prabhu Ramji

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Problem 27

What is the angle in radians that the Earth sweeps out in its orbit during winter?

Prabhu Ramji

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Problem 28

Assuming that the Earth is spherical and recalling that latitudes range from $0^{\circ}$ at the Equator to $90^{\circ} \mathrm{N}$ at the North Pole, how far apart, measured on the Earth's surface, are Dubuque, Iowa $\left(42.50^{\circ} \mathrm{N}\right.$ latitude $)$, and Guatemala City $\left(14.62^{\circ} \mathrm{N}\right.$ latitude $) ?$ The two cities lie on approximately the same longitude. Do not neglect the curvature of the Earth in determining this distance.

Prabhu Ramji

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Problem 29

Refer to the information given in Problem $9.28 .$ If one could burrow through the Earth and dig a straight-line tunnel from Dubuque to Guatemala City, how long would the tunnel be? From the point of view of the digger, at what angle below the horizontal would the tunnel be directed?

Averell Hause

Carnegie Mellon University

Problem 30

A typical Major League fastball is thrown at approximately $88 \mathrm{mph}$ and with a spin rate of $110 \mathrm{rpm} .$ If the distance between the pitcher's point of release and the catcher's glove is exactly $60.5 \mathrm{ft},$ how many full turns does the ball make between release and catch? Neglect any effect of gravity or air resistance on the ball's flight.

Prabhu Ramji

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Problem 31

A vinyl record plays at 33.3 rpm. Assume it takes 5.00 s for it to reach this full speed, starting from rest.
a) What is its angular acceleration during the $5.00 \mathrm{~s} ?$
b) How many revolutions does the record make before reaching its final angular speed?

Prabhu Ramji

Numerade Educator

Problem 32

At a county fair, a boy takes his teddy bear on the giant Ferris wheel. Unfortunately, at the top of the ride, he accidentally drops his stuffed buddy. The wheel has a diameter of $12.0 \mathrm{~m}$, the bottom of the wheel is $2.0 \mathrm{~m}$ above the ground and its rim is moving at a speed of $1.0 \mathrm{~m} / \mathrm{s}$. How far from the base of the Ferris wheel will the teddy bear land?

Prabhu Ramji

Numerade Educator

Problem 33

Having developed a taste for experimentation, the boy in Problem 9.32 invites two friends to bring their teddy bears on the same Ferris wheel. The boys are seated in positions $45^{\circ}$ from each other. When the wheel brings the second boy to the maximum height, they all drop their stuffed animals. How far apart will the three teddy bears land?

Averell Hause

Carnegie Mellon University

Problem 34

Mars orbits the Sun at a mean distance of 228 million $\mathrm{km},$ in a period of 687 days. The Earth orbits at a mean distance of 149.6 million $\mathrm{km},$ in a period of 365.26 days.
a) Suppose Earth and Mars are positioned such that Earth lies on a straight line between Mars and the Sun. Exactly 365.26 days later, when the Earth has completed one orbit, what is the angle between the Earth-Sun line and the Mars-Sun line?
b) The initial situation in part (a) is a closest approach of Mars to Earth. What is the time, in days, between two closest approaches? Assume constant speed and circular orbits for both Mars and Earth.
c) Another way of expressing the answer to part (b) is in terms of the angle between the lines drawn through the Sun, Earth, and Mars in the two closest approach situations. What is that angle?

Dominador Tan

Numerade Educator

Problem 35

Consider a large simple pendulum that is located at
a latitude of $55.0^{\circ} \mathrm{N}$ and is swinging in a north-south direction with points $A$ and $B$ being the northernmost and the southernmost points of the swing, respectively. A stationary (with respect to the fixed stars) observer is looking directly down on the pendulum at the moment shown in the figure. The Earth is rotating once every $23 \mathrm{~h}$ and $56 \mathrm{~min}$.
a) What are the directions (in terms of $\mathrm{N}, \mathrm{E}, \mathrm{W},$ and $\mathrm{S})$ and the magnitudes of the velocities of the surface of the Earth at points $A$ and $B$ as seen by the observer? Note: You will need to calculate answers to at least seven significant figures to see a difference.
b) What is the angular speed with which the $20.0-\mathrm{m}$ diameter circle under the pendulum appears to rotate?
c) What is the period of this rotation?
d) What would happen to a pendulum swinging at the Equator?

Dominador Tan

Numerade Educator

Problem 36

What is the centripetal acceleration of the Moon? The period of the Moon's orbit about the Earth is 27.3 days, measured with respect to the fixed stars. The radius of the Moon's orbit is $R_{M}=3.85 \cdot 10^{8} \mathrm{~m}$.

Prabhu Ramji

Numerade Educator

Problem 37

You are holding the axle of a bicycle wheel with radius $35.0 \mathrm{~cm}$ and mass $1.00 \mathrm{~kg}$. You get the wheel spinning at a rate of 75.0 rpm and then stop it by pressing the tire against the pavement. You notice that it takes $1.20 \mathrm{~s}$ for the wheel to come to a complete stop. What is the angular acceleration of the wheel?

Prabhu Ramji

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Problem 38

Life scientists use ultracentrifuges to separate biological components or to remove molecules from suspension. Samples in a symmetric array of containers are spun rapidly about a central axis. The centrifugal acceleration they experience in their moving reference frame acts as "artificial gravity" to effect a rapid separation. If the sample containers are $10.0 \mathrm{~cm}$ from the rotation axis, what rotation frequency is required to produce an acceleration of $1.00 \cdot 10^{5} g ?$

Prabhu Ramji

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Problem 39

A centrifuge in a medical laboratory rotates at an angular speed of 3600 rpm (revolutions per minute). When switched off, it rotates 60.0 times before coming to rest. Find the constant angular acceleration of the centrifuge.

Prabhu Ramji

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Problem 40

A discus thrower (with arm length of $1.2 \mathrm{~m}$ ) starts from rest and begins to rotate counterclockwise with an angular acceleration of $2.5 \mathrm{rad} / \mathrm{s}^{2}$
a) How long does it take the discus thrower's speed to get to $4.7 \mathrm{rad} / \mathrm{s} ?$
b) How many revolutions does the thrower make to reach the speed of $4.7 \mathrm{rad} / \mathrm{s} ?$
c) What is the linear speed of the discus at $4.7 \mathrm{rad} / \mathrm{s} ?$
d) What is the linear acceleration of the discus thrower at this point?
e) What is the magnitude of the centripetal acceleration of the discus thrown?
f) What is the magnitude of the discus's total acceleration?

Prabhu Ramji

Numerade Educator

Problem 41

In a department store toy display, a small disk (disk 1) of radius $0.100 \mathrm{~m}$ is driven by a motor and turns a larger disk (disk 2) of radius $0.500 \mathrm{~m}$. Disk 2 , in turn, drives disk 3 , whose radius is $1.00 \mathrm{~m}$. The three disks are in contact and there is no slipping. Disk 3 is observed to sweep through one complete revolution every $30.0 \mathrm{~s}$
a) What is the angular speed of disk $3 ?$
b) What is the ratio of the tangential velocities of the rims of the three disks?
c) What is the angular speed of disks 1 and $2 ?$
d) If the motor malfunctions, resulting in an angular acceleration of $0.100 \mathrm{rad} / \mathrm{s}^{2}$ for disk 1 , what are disks 2 and 3's angular accelerations?

Prabhu Ramji

Numerade Educator

Problem 42

A particle is moving clockwise in a circle of radius $1.00 \mathrm{~m}$. At a certain instant, the magnitude of its acceleration is $a=|\vec{a}|=25.0 \mathrm{~m} / \mathrm{s}^{2},$ and the acceleration vector has an angle of $\theta=50.0^{\circ}$ with the position vector, as shown in the figure. At this instant, find the speed, $v=|\vec{v}|,$ of this particle.

Prabhu Ramji

Numerade Educator

Problem 43

In a tape recorder, the magnetic tape moves at a constant linear speed of $5.6 \mathrm{~cm} / \mathrm{s}$. To maintain this constant linear speed, the angular speed of the driving spool (the take-up spool) has to change accordingly.
a) What is the angular speed of the take-up spool when it is empty, with radius $r_{1}=0.80 \mathrm{~cm} ?$
b) What is the angular speed when the spool is full, with radius $r_{2}=2.20 \mathrm{~cm} ?$
c) If the total length of the tape is $100.80 \mathrm{~m}$, what is the average angular acceleration of the take-up spool while the tape is being played?

Prabhu Ramji

Numerade Educator

Problem 44

A ring is fitted loosely (with no friction) around a long, smooth rod of length $L=0.50 \mathrm{~m} .$ The rod is fixed at one end, and the other end is spun in a horizontal circle at a constant angular velocity of $\omega=4.0 \mathrm{rad} / \mathrm{s} .$ The ring has zero radial velocity at its initial position, a distance of $r_{0}=0.30 \mathrm{~m}$ from the fixed end. Determine the radial velocity of the ring as it reaches the moving end of the rod.

Supratim Pal

Numerade Educator

Problem 45

A flywheel with a diameter of $1.00 \mathrm{~m}$ is initially at rest. Its angular acceleration versus time is graphed in the figure.
a) What is the angular separation between the initial position of a fixed point on the rim of the flywheel and the point's position 8.00 s after the wheel starts rotating?
b) The point starts its motion at $\theta=0 .$ Calculate and sketch the linear position, velocity vector, and acceleration vector $8 \mathrm{~s}$ after the wheel starts rotating.

Averell Hause

Carnegie Mellon University

Problem 46

Calculate the centripetal force exerted on a vehicle of mass $m=1500 .$ kg that is moving at a speed of $15.0 \mathrm{~m} / \mathrm{s}$ around a curve of radius $R=400 . \mathrm{m} .$ Which force plays the role of the centripetal force in this case?

Prabhu Ramji

Numerade Educator

Problem 47

What is the apparent weight of a rider on the roller coaster of Solved Problem 9.1 at the bottom of the loop?

Prabhu Ramji

Numerade Educator

Problem 48

Two skaters, $A$ and $B,$ of equal mass are moving in clockwise uniform circular motion on the ice. Their motions have equal periods, but the radius of skater A's circle is half that of skater B's circle
a) What is the ratio of the speeds of the skaters?
b) What is the ratio of the magnitudes of the forces acting on each skater?

Prabhu Ramji

Numerade Educator

Problem 49

A small block of mass $m$ is in contact with the inner wall of a large hollow cylinder. Assume the coefficient of static friction between
the object and the wall of the cylinder is $\mu$. Initially, the cylinder is at rest, and the block is held in place by a peg supporting its weight. The cylinder starts rotating about its center axis, as shown in the figure, with an angular acceleration of $\alpha$. Determine the minimum time interval after the cylinder begins to rotate before the peg can be removed without the block sliding against the wall.

Prabhu Ramji

Numerade Educator

Problem 50

A race car is making a U-turn at constant speed. The coefficient of friction between the tires and the track is $\mu_{\mathrm{s}}=1.20 .$ If the radius of the curve is $10.0 \mathrm{~m},$ what is the maximum speed at which the car can turn without sliding? Assume that the car is performing uniform circular motion.

Prabhu Ramji

Numerade Educator

Problem 51

A car speeds over the top of a hill. If the radius of curvature of the hill at the top is $9.0 \mathrm{~m}$, how fast can the car be traveling and maintain constant contact with the ground?

Prabhu Ramji

Numerade Educator

Problem 52

A ball of mass $m=0.200 \mathrm{~kg}$ is attached to a (massless) string of length $L=1.00 \mathrm{~m}$ and is undergoing circular motion in the horizontal plane, as shown in the figure.
a) Draw a free-body diagram for the ball.
b) Which force plays the role of the centripetal force?
c) What should the speed of the mass be for $\theta$ to be $45.0^{\circ} ?$
d) What is the tension in the string?

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 53

You are flying to Chicago for a weekend away from the books. In your last physics class, you learned that the airflow over the wings of the plane creates a lift force, which acts perpendicular to the wings. When the plane is flying level, the upward lift force exactly balances the downward weight force. Since O'Hare is one of the busiest airports in the world, you are not surprised when the captain announces that the flight is in a holding pattern due to the heavy traffic. He informs the passengers that the plane will be flying in a circle of radius $7.00 \mathrm{mi}$ at a speed of $360 . \mathrm{mph}$ and an altitude of $2.00 \cdot 10^{4} \mathrm{ft}$. From the safety information card, you know that the total length of the wingspan of the plane is $275 \mathrm{ft}$. From this information, estimate the banking angle of the plane relative to the horizontal.

Averell Hause

Carnegie Mellon University

Problem 54

A 20.0 -g metal cylinder is placed on a turntable, with its center $80.0 \mathrm{~cm}$ from the turntable's center. The coefficient of static friction between the cylinder and the turntable's surface is $\mu_{s}=0.800$. A thin, massless string of length $80.0 \mathrm{~cm}$ connects the center of the turntable to the cylinder, and initially, the string has zero tension in it. Starting from rest, the turntable very slowly attains higher and higher angular velocities, but the turntable and the cylinder can be considered to have uniform circular motion at any instant. Calculate the tension in the string when the angular velocity of the turntable is 60.0 rpm (rotations per minute).

Averell Hause

Carnegie Mellon University

Problem 55

A speedway turn, with radius of curvature $R$, is banked at an angle $\theta$ above the horizontal.
a) What is the optimal speed at which to take the turn if the track's surface is iced over (that is, if there is very little friction between the tires and the track)?
b) If the track surface is ice-free and there is a coefficient of friction $\mu_{s}$ between the tires and the track, what are the maximum and minimum speeds at which this turn can be taken?
c) Evaluate the results of parts (a) and (b) for $R=400 . \mathrm{m}$, $\theta=45.0^{\circ},$ and $\mu_{\mathrm{s}}=0.700 .$

Averell Hause

Carnegie Mellon University

Problem 56

A particular Ferris wheel takes riders in a vertical circle of radius $9.0 \mathrm{~m}$ once every $12.0 \mathrm{~s} .$
a) Calculate the speed of the riders, assuming it to be constant.
b) Draw a free-body diagram for a rider at a time when she is at the bottom of the circle. Calculate the normal force exerted by the seat on the rider at that point in the ride.
c) Perform the same analysis as in part (b) for a point at the top of the ride.

Prabhu Ramji

Numerade Educator

Problem 57

A boy is on a Ferris wheel, which takes him in a vertical circle of radius $9.0 \mathrm{~m}$ once every $12.0 \mathrm{~s}$.
a) What is the angular speed of the Ferris wheel?
b) Suppose the wheel comes to a stop at a uniform rate during one quarter of a revolution. What is the angular acceleration of the wheel during this time?
c) Calculate the tangential acceleration of the boy during the time interval described in part (b).

Prabhu Ramji

Numerade Educator

Problem 58

Consider a $53-\mathrm{cm}$ -long lawn mower blade rotating about its center at 3400 rpm.
a) Calculate the linear speed of the tip of the blade.
b) If safety regulations require that the blade be stoppable within $3.0 \mathrm{~s}$, what minimum angular acceleration will accomplish this? Assume that the angular acceleration is constant.

Prabhu Ramji

Numerade Educator

Problem 59

A car accelerates uniformly from rest and reaches a speed of $22.0 \mathrm{~m} / \mathrm{s}$ in $9.00 \mathrm{~s}$. The diameter of a tire on this car is $58.0 \mathrm{~cm}$.
a) Find the number of revolutions the tire makes during the car's motion, assuming that no slipping occurs.
b) What is the final angular speed of a tire in revolutions per second?

Prabhu Ramji

Numerade Educator

Problem 60

Gear $A$, with a mass of $1.00 \mathrm{~kg}$ and a radius of $55.0 \mathrm{~cm}$ is in contact with gear $\mathrm{B}$, with a mass of $0.500 \mathrm{~kg}$ and a radius of $30.0 \mathrm{~cm} .$ The gears do not slip with respect to each other as they rotate. Gear A rotates at 120. rpm and slows to 60.0 rpm in $3.00 \mathrm{~s}$. How many rotations does gear B undergo during this time interval?

Prabhu Ramji

Numerade Educator

Problem 61

A top spins for 10.0 min, beginning with an angular speed of 10.0 rev/s. Determine its angular acceleration, assuming it is constant, and its total angular displacement.

Prabhu Ramji

Numerade Educator

Problem 62

A penny is sitting on the edge of an old phonograph disk that is spinning at 33 rpm and has a diameter of 12 inches. What is the minimum coefficient of static friction between the penny and the surface of the disk to ensure that the penny doesn't fly off?

Prabhu Ramji

Numerade Educator

Problem 63

A vinyl record that is initially turning at $33 \frac{1}{3}$ rpm slows uniformly to a stop in a time of $15 \mathrm{~s}$. How many rotations are made by the record while stopping?

Prabhu Ramji

Numerade Educator

Problem 64

Determine the linear and angular speeds and accelerations of a speck of dirt located $2.0 \mathrm{~cm}$ from the center of a CD rotating inside a CD player at 250 rpm.

Prabhu Ramji

Numerade Educator

Problem 65

What is the acceleration of the Earth in its orbit? (Assume the orbit is circular.)

Prabhu Ramji

Numerade Educator

Problem 66

A day on Mars is 24.6 Earth hours long. A year on Mars is 687 Earth days long. How do the angular velocities of Mars's rotation and orbit compare to the angular velocities of Earth's rotation and orbit?

Prabhu Ramji

Numerade Educator

Problem 67

A monster truck has tires with a diameter of $1.10 \mathrm{~m}$ and
is traveling at $35.8 \mathrm{~m} / \mathrm{s}$. After the brakes are applied, the truck slows uniformly and is brought to rest after the tires rotate through 40.2 turns.
a) What is the initial angular speed of the tires?
b) What is the angular acceleration of the tires?
c) What distance does the truck travel before coming to rest?

Prabhu Ramji

Numerade Educator

Problem 68

The motor of a fan turns a small wheel of radius $r_{\mathrm{m}}=$ $2.00 \mathrm{~cm} .$ This wheel turns a belt, which is attached to a wheel of radius $r_{f}=3.00 \mathrm{~cm}$ that is mounted to the axle of the fan blades. Measured from the center of this axle, the tip of the fan blades are at a distance $r_{\mathrm{b}}=15.0 \mathrm{~cm} .$ When the fan is in operation, the motor spins at an angular speed of $\omega=1200$. rpm. What is the tangential speed of the tips of the fan blades?

Prabhu Ramji

Numerade Educator

Problem 69

A car with a mass of $1000 .$ kg goes over a hill at a constant speed of $60.0 \mathrm{~m} / \mathrm{s}$. The top of the hill can be approximated as an arc length of a circle with a radius of curvature of $370 . \mathrm{m} .$ What force does the car exert on the hill as it passes over the top?

Prabhat Tyagi

Numerade Educator

Problem 70

Unlike a ship, an airplane does not use its rudder to turn. It turns by banking its wings: The lift force, perpendicular to the wings, has a horizontal component, which provides the centripetal acceleration for the turn, and a vertical component, which supports the plane's weight. (The rudder counteracts yaw and thus it keeps the plane pointed in the direction it is moving.) The famous spy plane, the SR-71 Blackbird, flying at $4800 \mathrm{~km} / \mathrm{h}$, has a turning radius of $290 . \mathrm{km} .$ Find its banking angle.

Averell Hause

Carnegie Mellon University

Problem 71

A $80.0-\mathrm{kg}$ pilot in an aircraft moving at a constant speed of $500 . \mathrm{m} / \mathrm{s}$ pulls out of a vertical dive along an arc of a circle of radius $4000 . \mathrm{m}$.

Prabhu Ramji

Numerade Educator

Problem 72

A ball having a mass of $1.00 \mathrm{~kg}$ is attached to a string $1.00 \mathrm{~m}$ long and is whirled in a vertical circle at a constant speed of $10.0 \mathrm{~m} / \mathrm{s}$
a) Determine the tension in the string when the ball is at the top of the circle.
b) Determine the tension in the string when the ball is at the bottom of the circle.
c) Consider the ball at some point other than the top or bottom. What can you say about the tension in the string at this point?

Prabhu Ramji

Numerade Educator

Problem 73

A car starts from rest and accelerates around a flat curve of radius $R=36 \mathrm{~m}$. The tangential component of the car's acceleration remains constant at $a_{\mathrm{t}}=3.3 \mathrm{~m} / \mathrm{s}^{2},$ while the centripetal acceleration increases to keep the car on the curve as long as possible. The coefficient of friction between the tires and the road is $\mu=0.95 .$ What distance does the car travel around the curve before it begins to skid? (Be sure to include both the tangential and centripetal components of the acceleration.)

Prabhu Ramji

Numerade Educator

Problem 74

A girl on a merry-go-round platform holds a pendulum in her hand. The pendulum is $6.0 \mathrm{~m}$ from the rotation axis of the platform. The rotational speed of the platform is 0.020 rev/s. It is found that the pendulum hangs at an angle $\theta$ to the vertical. Find $\theta$

Prabhu Ramji

Numerade Educator

Problem 75

A carousel at a carnival has a diameter of $6.00 \mathrm{~m}$. The
ride starts from rest and accelerates at a constant angular acceleration to an angular speed of 0.600 rev/s in $8.00 \mathrm{~s}$.
a) What is the value of the angular acceleration?
b) What are the centripetal and angular accelerations of a seat on the carousel that is $2.75 \mathrm{~m}$ from the rotation axis?
c) What is the total acceleration, magnitude and direction, $8.00 \mathrm{~s}$ after the angular acceleration starts?

Prabhu Ramji

Numerade Educator

Problem 76

A car of weight $W=$ $10.0 \mathrm{kN}$ makes a turn on a track that is banked at an angle of $\theta=20.0^{\circ} .$ Inside the car, hanging from a short string tied to the rear-view mirror, is an ornament. As the car turns, the ornament swings out at an angle of $\varphi=30.0^{\circ}$ measured from the vertical inside the car. What is the force of static friction between the car and the road?

Averell Hause

Carnegie Mellon University

Problem 77

A popular carnival ride consists of seats attached to a central disk through cables. The passengers travel in uniform circular motion. As shown in the figure, the radius of the central disk is $R_{0}=3.00 \mathrm{~m},$ and the length of the cable is $L=3.20 \mathrm{~m}$ The mass of one of the passengers (including the chair he is sitting on) is $65.0 \mathrm{~kg}$.
a) If the angle $\theta$ that the cable makes with respect to the vertical is $30.0^{\circ},$ what is the speed, $v$, of this passenger?
b) What is the magnitude of the force exerted by the cable on the chair?

Prabhu Ramji

Numerade Educator