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University Physics with Modern Physics

Roger A. Freedman, Hugh D. Young

Chapter 9

Rotation of Rigid Bodies - all with Video Answers

Educators

Chapter Questions

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Problem 1

(a) What angle in radians is subtended by an arc $1.50 \mathrm{~m}$ long on the circumference of a circle of radius $2.50 \mathrm{~m} ?$ What is this angle in degrees? (b) An arc $14.0 \mathrm{~cm}$ long on the circumference of a circle subtends an angle of $128^{\circ} .$ What is the radius of the circle? (c) The angle between two radii of a circle with radius $1.50 \mathrm{~m}$ is 0.700 rad. What length of arc is intercepted on the circumference of the circle by the two radii?

Zhuxi Luo
Zhuxi Luo
Numerade Educator
04:42

Problem 2

An airplane propeller is rotating at 1900 rpm (rev/min). (a) Compute the propeller's angular velocity in rad/s. (b) How many seconds does it take for the propeller to turn through $35^{\circ} ?$

Laura Yu
Laura Yu
University of California, Irvine
04:19

Problem 3

The angular velocity of a flywheel obeys the equation $\omega_{z}(t)=A+B t^{2},$ where $t$ is in seconds and $A$ and $B$ are constants having numerical values 2.75 (for $A$ ) and 1.50 (for $B$ ). (a) What are the units of $A$ and $B$ if $\omega_{z}$ is in $\mathrm{rad} / \mathrm{s} ?$ (b) What is the angular acceleration of the wheel at
(i) $t=0$ and (ii) $t=5.00 \mathrm{~s} ?$ (c) Through what angle does the flywheel turn during the first 2.00 s? (Hint: See Section $2.6 .)$

Andrew C
Andrew C
Numerade Educator
03:42

Problem 4

A fan blade rotates with angular velocity given by $\omega_{z}(t)=\gamma-\beta t^{2}, \quad$ where $\quad \gamma=5.00 \mathrm{rad} / \mathrm{s} \quad$ and $\quad \beta=0.800 \mathrm{rad} / \mathrm{s}^{3}$
(a) Calculate the angular acceleration as a function of time. (b) Calculate the instantaneous angular acceleration $\alpha_{z}$ at $t=3.00 \mathrm{~s}$ and the average angular acceleration $\alpha_{\mathrm{av}-z}$ for the time interval $t=0$ to $t=3.00 \mathrm{~s}$. How do these two quantities compare? If they are different, why?

Averell Hause
Averell Hause
Carnegie Mellon University
04:36

Problem 5

A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to $\theta(t)=\gamma t+\beta t^{3}, \quad$ where $\quad \gamma=0.400 \mathrm{rad} / \mathrm{s} \quad$ and $\quad \beta=0.0120 \mathrm{rad} / \mathrm{s}^{3}$
(a) Calculate the angular velocity of the merry-go-round as a function of time. (b) What is the initial value of the angular velocity? (c) Calculate the instantaneous value of the angular velocity $\omega_{z}$ at $t=5.00 \mathrm{~s}$ and the average angular velocity $\omega_{\mathrm{av}-z}$ for the time interval $t=0$ to $t=5.00 \mathrm{~s}$ Show that $\omega_{\mathrm{av}-z}$ is $n o t$ equal to the average of the instantaneous angular velocities at $t=0$ and $t=5.00 \mathrm{~s},$ and explain.

Andrew C
Andrew C
Numerade Educator
05:10

Problem 6

At $t=0$ the current to a dc electric motor is reversed, resulting in an angular displacement of the motor shaft given by $\theta(t)=(250 \mathrm{rad} / \mathrm{s}) t-\left(20.0 \mathrm{rad} / \mathrm{s}^{2}\right) t^{2}-\left(1.50 \mathrm{rad} / \mathrm{s}^{3}\right) t^{3} .$ (a) At what
time is the angular velocity of the motor shaft zero? (b) Calculate the angular acceleration at the instant that the motor shaft has zero angular velocity. (c) How many revolutions does the motor shaft turn through between the time when the current is reversed and the instant when the angular velocity is zero?
(d) How fast was the motor shaft rotating at $t=0,$ when the current was reversed?
(e) Calculate the average angular velocity for the time period from $t=0$ to the time calculated in part (a).

Averell Hause
Averell Hause
Carnegie Mellon University
06:04

Problem 7

The angle $\theta$ through which a disk drive turns is given by $\theta(t)=a+b t-c t^{3},$ where $a, b,$ and $c$ are constants, $t$ is in seconds, and $\theta$ is in radians. When $t=0, \theta=\pi / 4$ rad and the angular velocity is $2.00 \mathrm{rad} / \mathrm{s}$. When $t=1.50 \mathrm{~s},$ the angular acceleration is $1.25 \mathrm{rad} / \mathrm{s}^{2}$.
(a) Find $a, b,$ and $c,$ including their units. (b) What is the angular acceleration when $\theta=\pi / 4$ rad? (c) What are $\theta$ and the angular velocity when the angular acceleration is $3.50 \mathrm{rad} / \mathrm{s}^{2} ?$

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
05:07

Problem 8

A wheel is rotating about an axis that is in the $z$ -direction. The angular velocity $\omega_{z}$ is $-6.00 \mathrm{rad} / \mathrm{s}$ at $t=0,$ increases linearly with time, and is $+4.00 \mathrm{rad} / \mathrm{s}$ at $t=7.00 \mathrm{~s}$. We have taken counterclockwise rotation to be positive. (a) Is the angular acceleration during this time interval positive or negative? (b) During what time interval is the speed of the wheel increasing? Decreasing? (c) What is the angular displacement of the wheel at $t=7.00 \mathrm{~s}$ ?

Averell Hause
Averell Hause
Carnegie Mellon University
02:45

Problem 9

A bicycle wheel has an initial angular velocity of $1.50 \mathrm{rad} / \mathrm{s}$
(a) If its angular acceleration is constant and equal to $0.200 \mathrm{rad} / \mathrm{s}^{2},$ what is its angular velocity at $t=2.50 \mathrm{~s} ?$ (b) Through what angle has the wheel turned between $t=0$ and $t=2.50 \mathrm{~s} ?$

Zhuxi Luo
Zhuxi Luo
Numerade Educator
04:25

Problem 10

An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev $/$ min to 200 rev $/ \min$ in 4.00 s. (a) Find the angular acceleration in rev/s $^{2}$ and the number of revolutions made by the motor in the 4.00 s interval. (b) How many more seconds are required for the fan to come to rest if the angular acceleration remains constant at the value calculated in part (a)?

Vishal Gupta
Vishal Gupta
Numerade Educator
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Problem 11

The rotating blade of a blender turns with constant angular acceleration $1.50 \mathrm{rad} / \mathrm{s}^{2}$. (a) How much time does it take to reach an angular velocity of $36.0 \mathrm{rad} / \mathrm{s},$ starting from rest?
(b) Through how many revolutions does the blade turn in this time interval?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
02:34

Problem 12

A wheel rotates from rest with constant angular acceleration. If it rotates through 8.00 revolutions in the first $2.50 \mathrm{~s}$, how many more revolutions will it rotate through in the next 5.00 s?

Andrew C
Andrew C
Numerade Educator
02:02

Problem 13

$\mathrm{A}$ turntable rotates with a constant $2.25 \mathrm{rad} / \mathrm{s}^{2}$ clockwise angular acceleration. After $4.00 \mathrm{~s}$ it has rotated through a clockwise angle of 30.0 rad. What was the angular velocity of the wheel at the beginning of the 4.00 s interval?

Andrew C
Andrew C
Numerade Educator
02:09

Problem 14

A circular saw blade $0.200 \mathrm{~m}$ in diameter starts from rest. In $6.00 \mathrm{~s}$ it accelerates with constant angular acceleration to an angular velocity of $140 \mathrm{rad} / \mathrm{s} .$ Find the angular acceleration and the angle through which the blade has turned.

Averell Hause
Averell Hause
Carnegie Mellon University
02:55

Problem 15

A high-speed flywheel in a motor is spinning at 500 rpm when a power failure suddenly occurs. The flywheel has mass $40.0 \mathrm{~kg}$ and diameter $75.0 \mathrm{~cm}$. The power is off for $30.0 \mathrm{~s}$, and during this time the flywheel slows due to friction in its axle bearings. During the time the power is off, the flywheel makes 200 complete revolutions. (a) At what rate is the flywheel spinning when the power comes back on? (b) How long after the beginning of the power failure would it have taken the flywheel to stop if the power had not come back on, and how many revolutions would the wheel have made during this time?

Zhuxi Luo
Zhuxi Luo
Numerade Educator
05:34

Problem 16

$\mathrm{At} t=0$ a grinding wheel has an angular velocity of $24.0 \mathrm{rad} / \mathrm{s}$ It has a constant angular acceleration of $30.0 \mathrm{rad} / \mathrm{s}^{2}$ until a circuit breaker trips at $t=2.00 \mathrm{~s}$. From then on, it turns through 432 rad as it coasts to a stop at constant angular acceleration. (a) Through what total angle did the wheel turn between $t=0$ and the time it stopped?
(b) At what time did it stop? (c) What was its acceleration as it slowed down?

Averell Hause
Averell Hause
Carnegie Mellon University
09:12

Problem 17

A safety device brings the blade of a power mower from an initial angular speed of $\omega_{1}$ to rest in 1.00 revolution. At the same constant acceleration, how many revolutions would it take the blade to come to rest from an initial angular speed $\omega_{3}$ that was three times as great, $\omega_{3}=3 \omega_{1} ?$

RS
Ramsey Seweingyawma
Numerade Educator
03:20

Problem 18

$\begin{array}{llll}\mathrm{In} & \mathrm{a} & \text { charming } & 19 \text { th-century }\end{array}$
hotel, an old-style elevator is connected to a counterweight by a cable that passes over a rotating disk $2.50 \mathrm{~m}$ in diameter (Fig. E9.18). The elevator is raised and lowered by turning the disk, and the cable does not slip on the rim of the disk but turns with it. (a) At how many rpm must the disk turn to raise the elevator at $25.0 \mathrm{~cm} / \mathrm{s} ?$
(b) To start the elevator moving, it must be accelerated at $\frac{1}{8} g .$ What must be the angular acceleration of the disk, in rad/s $^{2} ?$
(c) Through what angle (in radians and degrees) has the disk turned when it has raised the elevator $3.25 \mathrm{~m}$ between floors?

Andrew C
Andrew C
Numerade Educator
03:19

Problem 19

Spin cycles of washing machines remove water from clothes by producing a large radial acceleration at the rim of the cylindrical tub that holds the water and clothes. Estimate the diameter of the tub in a typical home washing machine. (a) What is the rotation rate, in rev/min, of the tub during the spin cycle if the radial acceleration of points on the tub wall is $3 g ?$ (b) At this rotation rate, what is the tangential speed in $\mathrm{m} / \mathrm{s}$ of a point on the tub wall?

Andrew C
Andrew C
Numerade Educator
10:06

Problem 20

Compact Disc. A compact disc (CD) stores music in a coded pattern of tiny pits $10^{-7} \mathrm{~m}$ deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are $25.0 \mathrm{~mm}$ and $58.0 \mathrm{~mm}$, respectively. As the disc spins inside a CD player, the track is scanned at a constant linear speed of $1.25 \mathrm{~m} / \mathrm{s}$. (a) What is the angular speed of the $\mathrm{CD}$ when the innermost part of the track is scanned? The outermost part of the track? (b) The maximum playing time of a CD is 74.0 min. What would be the length of the track on such a maximum-duration $\mathrm{CD}$ if it were stretched out in a straight line? (c) What is the average angular acceleration of a maximum-duration CD during its 74.0 min playing time? Take the direction of rotation of the disc to be positive.

JH
Jingjing Huang
Numerade Educator
04:09

Problem 21

A wheel of diameter $40.0 \mathrm{~cm}$ starts from rest and rotates with a constant angular acceleration of $3.00 \mathrm{rad} / \mathrm{s}^{2}$. Compute the radial acceleration of a point on the rim for the instant the wheel completes its second revolution from the relationship
(a) $a_{\mathrm{rad}}=\omega^{2} r$ and
(b) $a_{\mathrm{rad}}=v^{2} / r$

Vishal Gupta
Vishal Gupta
Numerade Educator
03:00

Problem 22

You are to design a rotating cylindrical axle to lift $800 \mathrm{~N}$ buckets of cement from the ground to a rooftop $78.0 \mathrm{~m}$ above the ground. The buckets will be attached to a hook on the free end of a cable that wraps around the rim of the axle; as the axle turns, the buckets will rise.
(a) What should the diameter of the axle be in order to raise the buckets at a steady $2.00 \mathrm{~cm} / \mathrm{s}$ when it is turning at $7.5 \mathrm{rpm} ?$ (b) If instead the axle must give the buckets an upward acceleration of $0.400 \mathrm{~m} / \mathrm{s}^{2},$ what should the angular acceleration of the axle be?

Averell Hause
Averell Hause
Carnegie Mellon University
01:58

Problem 23

The blade of an electric saw rotates at $2600 \mathrm{rev} / \mathrm{min} .$ Estimate the diameter of a typical saw that is used to saw boards in home construction and renovation. What is the linear speed in $\mathrm{m} / \mathrm{s}$ of a point on the rim of the circular saw blade?

Andrew C
Andrew C
Numerade Educator
04:13

Problem 24

An electric turntable $0.750 \mathrm{~m}$ in diameter is rotating about a fixed axis with an initial angular velocity of $0.250 \mathrm{rev} / \mathrm{s}$ and a constant angular acceleration of $0.900 \mathrm{rev} / \mathrm{s}^{2}$. (a) Compute the angular velocity of the turntable after $0.200 \mathrm{~s}$. (b) Through how many revolutions has the turntable spun in this time interval? (c) What is the tangential speed of a point on the rim of the turntable at $t=0.200 \mathrm{~s} ?$ (d) What is the magnitude of the resultant acceleration of a point on the rim at $t=0.200 \mathrm{~s} ?$

Ajay Singhal
Ajay Singhal
Numerade Educator
01:13

Problem 25

An advertisement claims that a centrifuge takes up only $0.127 \mathrm{~m}$ of bench space but can produce a radial acceleration of $3000 g$ at 5000 rev $/$ min. Calculate the required radius of the centrifuge. Is the claim realistic?

Zhuxi Luo
Zhuxi Luo
Numerade Educator
03:53

Problem 26

At $t=3.00 \mathrm{~s}$ a point on the rim of a $0.200-\mathrm{m}$ -radius wheel has a tangential speed of $50.0 \mathrm{~m} / \mathrm{s}$ as the wheel slows down with a tangential acceleration of constant magnitude $10.0 \mathrm{~m} / \mathrm{s}^{2}$. (a) Calculate the wheel's constant angular acceleration. (b) Calculate the angular velocities at $t=3.00 \mathrm{~s}$ and $t=0 .$ (c) Through what angle did the wheel turn between $t=0$ and $t=3.00 \mathrm{~s} ?$ (d) At what time will the radial acceleration equal $g ?$

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
03:48

Problem 27

A rotating wheel with diameter $0.600 \mathrm{~m}$ is speeding up with constant angular acceleration. The speed of a point on the rim of the wheel increases from $3.00 \mathrm{~m} / \mathrm{s}$ to $6.00 \mathrm{~m} / \mathrm{s}$ while the wheel turns through 4.00 revolutions. What is the angular acceleration of the wheel?

Andrew C
Andrew C
Numerade Educator
03:04

Problem 28

The earth is approximately spherical, with a diameter of $1.27 \times 10^{7} \mathrm{~m} .$ It takes 24.0 hours for the earth to complete one revolution. What are the tangential speed and radial acceleration of a point on the surface of the earth, at the equator?

Andrew C
Andrew C
Numerade Educator
03:05

Problem 29

A flywheel with radius $0.300 \mathrm{~m}$ starts from rest and accelerates with a constant angular acceleration of $0.600 \mathrm{rad} / \mathrm{s}^{2} .$ For a point on the rim of the flywheel, what are the magnitudes of the tangential, radial, and resultant accelerations after $2.00 \mathrm{~s}$ of acceleration?

Andrew C
Andrew C
Numerade Educator
04:04

Problem 30

Four small spheres, each of which you can regard as a point of mass $0.200 \mathrm{~kg}$, are arranged in a square $0.400 \mathrm{~m}$ on a side and connected by extremely light rods (Fig. E9.30). Find the moment of inertia of the system about an axis
(a) through the center of the square, perpendicular to its plane (an axis through point $O$ in the figure); (b) bisecting two opposite sides of the square (an axis along the line $A B$ in the figure); (c) that passes through the centers of the upper left and lower right spheres and through point $O$.

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
00:56

Problem 31

Calculate the moment of inertia of each of the following uniform objects about the axes indicated. Consult Table 9.2 as needed.
(a) A thin $2.50 \mathrm{~kg}$ rod of length $75.0 \mathrm{~cm},$ about an axis perpendicular to it and passing through (i) one end and (ii) its center, and (iii) about an axis parallel to the rod and passing through it. (b) A $3.00 \mathrm{~kg}$ sphere $38.0 \mathrm{~cm}$ in diameter, about an axis through its center, if the sphere is (i) solid and (ii) a thin-walled hollow shell. (c) An $8.00 \mathrm{~kg}$ cylinder, of length $19.5 \mathrm{~cm}$ and diameter $12.0 \mathrm{~cm},$ about the central axis of the cylinder, if the cylinder is (i) thin-walled and hollow, and
(ii) solid.

Zhuxi Luo
Zhuxi Luo
Numerade Educator
03:26

Problem 31

Three small blocks, each with mass $m$, are clamped at the ends and at the center of a rod of length $L$ and negligible mass. Compute the moment of inertia of the system about an axis perpendicular to the rod and passing through (a) the center of the rod and (b) a point onefourth of the length from one end.

Andrew C
Andrew C
Numerade Educator
01:21

Problem 32

Three small blocks, each with mass $m,$ are clamped at the ends and at the center of a rod of length $L$ and negligible mass. Compute the moment of inertia of the system about an axis perpendicular to the rod and passing through (a) the center of the rod and (b) a point onefourth of the length from one end.

Lizandra Chagas
Lizandra Chagas
Numerade Educator
03:20

Problem 33

A uniform bar has two small balls glued to its ends. The bar is $2.00 \mathrm{~m}$ long and has mass $4.00 \mathrm{~kg},$ while the balls each have mass $0.300 \mathrm{~kg}$ and can be treated as point masses. Find the moment of inertia of this combination about an axis (a) perpendicular to the bar through its center; (b) perpendicular to the bar through one of the balls;
(c) parallel to the bar through both balls; and (d) parallel to the bar and $0.500 \mathrm{~m}$ from it.

Zhuxi Luo
Zhuxi Luo
Numerade Educator
04:48

Problem 34

You are a project manager for a manufacturing company. One of the machine parts on the assembly line is a thin, uniform rod that is $60.0 \mathrm{~cm}$ long and has mass $0.400 \mathrm{~kg}$. (a) What is the moment of inertia of this rod for an axis at its center, perpendicular to the rod? (b) One of your engineers has proposed to reduce the moment of inertia by bending the rod at its center into a V-shape, with a $60.0^{\circ}$ angle at its vertex. What would be the moment of inertia of this bent rod about an axis perpendicular to the plane of the $\mathrm{V}$ at its vertex?

Averell Hause
Averell Hause
Carnegie Mellon University
01:48

Problem 35

A wagon wheel is constructed as shown in Fig. E9.35. The radius of the wheel is $0.300 \mathrm{~m},$ and the rim has mass $1.40 \mathrm{~kg}$. Each of the eight spokes that lie along a diameter and are $0.300 \mathrm{~m}$ long has mass $0.280 \mathrm{~kg}$. What is the moment of inertia of the wheel about
an axis through its center and perpendicular to the plane of the wheel? (Use Table $9.2 .$

Zhuxi Luo
Zhuxi Luo
Numerade Educator
04:27

Problem 36

A uniform sphere made of modeling clay has radius $R$ and moment of inertia $I_{1}$ for rotation about a diameter. It is flattened to a disk with the same radius $R .$ In terms of $I_{1},$ what is the moment of inertia of the disk for rotation about an axis that is at the center of the disk
and perpendicular to its flat surface?

Andrew C
Andrew C
Numerade Educator
04:26

Problem 37

A rotating flywheel has moment of inertia $12.0 \mathrm{~kg} \cdot \mathrm{m}^{2}$ for an axis along the axle about which the wheel is rotating. Initially the flywheel has $30.0 \mathrm{~J}$ of kinetic energy. It is slowing down with an angular acceleration of magnitude $0.500 \mathrm{rev} / \mathrm{s}^{2} .$ How long does it take for the rotational kinetic energy to become half its initial value, so it is $15.0 \mathrm{~J} ?$

Andrew C
Andrew C
Numerade Educator
03:10

Problem 38

An airplane propeller is $2.08 \mathrm{~m}$ in length (from tip to tip) with mass $117 \mathrm{~kg}$ and is rotating at 2400 rpm (rev/min) about an axis through its center. You can model the propeller as a slender rod.
(a) What is its rotational kinetic energy?
(b) Suppose that, due to weight constraints, you had to reduce the propeller's mass to $75.0 \%$ of its original mass, but you still needed to keep the same size and kinetic energy. What would its angular speed have to be, in rpm?

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
03:10

Problem 39

A uniform sphere with mass $M$ and radius $R$ is rotating with angular speed $\omega_{1}$ about a frictionless axle along a diameter of the sphere. The sphere has rotational kinetic energy $K_{1}$. A thin-walled hollow sphere has the same mass and radius as the uniform sphere. It is also rotating about a fixed axis along its diameter. In terms of $\omega_{1},$ what angular speed must the hollow sphere have if its kinetic energy is also $K_{1},$ the same as for the uniform sphere?

Andrew C
Andrew C
Numerade Educator
03:46

Problem 40

A wheel is turning about an axis through its center with constant angular acceleration. Starting from rest, at $t=0,$ the wheel turns through 8.20 revolutions in $12.0 \mathrm{~s}$. At $t=12.0 \mathrm{~s}$ the kinetic energy of the wheel is $36.0 \mathrm{~J}$. For an axis through its center, what is the moment of inertia of the wheel?

Averell Hause
Averell Hause
Carnegie Mellon University
02:35

Problem 41

A uniform sphere with mass $28.0 \mathrm{~kg}$ and radius $0.380 \mathrm{~m}$ is rotating at constant angular velocity about a stationary axis that lies along a diameter of the sphere. If the kinetic energy of the sphere is $236 \mathrm{~J}$, what is the tangential velocity of a point on the rim of the sphere?

Zhuxi Luo
Zhuxi Luo
Numerade Educator
03:28

Problem 42

A hollow spherical shell has mass $8.20 \mathrm{~kg}$ and radius $0.220 \mathrm{~m} .$ It is initially at rest and then rotates about a stationary axis that lies along a diameter with a constant acceleration of $0.890 \mathrm{rad} / \mathrm{s}^{2}$. What is the kinetic energy of the shell after it has turned through 6.00 rev?

Averell Hause
Averell Hause
Carnegie Mellon University
02:42

Problem 43

Wheel $A$ has three times the moment of inertia about its axis
of rotation as wheel $B .$ Wheel $B$ 's angular speed is four times that of wheel $A$. (a) Which wheel has the greater rotational kinetic energy?
(b) If $K_{A}$ and $K_{B}$ are the rotational kinetic energies of the wheels, what is $K_{A} / K_{B} ?$

Andrew C
Andrew C
Numerade Educator
02:03

Problem 44

You need to design an industrial turntable that is $60.0 \mathrm{~cm}$ in diameter and has a kinetic energy of $0.250 \mathrm{~J}$ when turning at $45.0 \mathrm{rpm}$ (rev/min). (a) What must be the moment of inertia of the turntable about the rotation axis? (b) If your workshop makes this turntable in the shape of a uniform solid disk, what must be its mass?

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
01:14

Problem 45

Energy is to be stored in a $70.0 \mathrm{~kg}$ flywheel in the shape of a uniform solid disk with radius $R=1.20 \mathrm{~m}$. To prevent structural failure of the flywheel, the maximum allowed radial acceleration of a point on its rim is $3500 \mathrm{~m} / \mathrm{s}^{2}$. What is the maximum kinetic energy that can be stored in the flywheel?

Zhuxi Luo
Zhuxi Luo
Numerade Educator
03:49

Problem 46

You are designing a flywheel. It is to start from rest and then rotate with a constant angular acceleration of $0.200 \mathrm{rev} / \mathrm{s}^{2}$. The design specifications call for it to have a rotational kinetic energy of $240 \mathrm{~J}$ after it has turned through 30.0 revolutions. What should be the moment of inertia of the flywheel about its rotation axis?

Andrew C
Andrew C
Numerade Educator
06:35

Problem 47

A pulley on a frictionless axle has the shape of a uniform solid disk of mass $2.50 \mathrm{~kg}$ and radius $20.0 \mathrm{~cm}$. A $1.50 \mathrm{~kg}$ stone is
attached to a very light wire that is wrapped around the rim of the pulley (Fig. E9.47), and the system is released from rest. (a) How far must the stone fall so that the pulley has $4.50 \mathrm{~J}$ of kinetic energy? (b) What percent of the total kinetic energy does the pulley have?

Andrew C
Andrew C
Numerade Educator
02:20

Problem 48

A bucket of mass $m$ is tied to a massless cable that is wrapped around the outer rim of a uniform pulley of radius $R,$ on a frictionless axle, similar to the system shown in Fig. E9.47. In terms of the stated variables, what must be the moment of inertia of the pulley so that it always has half as much kinetic energy as the bucket?

Andrew C
Andrew C
Numerade Educator
06:12

Problem 49

A thin, light wire is wrapped around the rim of a wheel (Fig. E9.49). The wheel rotates without friction about a stationary horizontal axis that passes through the center of the wheel. The wheel is a uniform disk with radius $R=0.280 \mathrm{~m}$. An object of mass $m=4.20 \mathrm{~kg}$ is suspended from the free end of the wire. The system is released from rest and the suspended object descends with constant acceleration. If the suspended object moves downward a distance of $3.00 \mathrm{~m}$ in $2.00 \mathrm{~s},$ what is the mass of the wheel?

Shoukat Ali
Shoukat Ali
Other Schools
00:45

Problem 50

Find the moment of inertia of a hoop (a thin-walled, hollow ring) with mass $M$ and radius $R$ about an axis perpendicular to the hoop's plane at an edge.

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
01:44

Problem 51

About what axis will a uniform, balsa-wood sphere have the same moment of inertia as does a thin-walled, hollow, lead sphere of the same mass and radius, with the axis along a diameter?

Zhuxi Luo
Zhuxi Luo
Numerade Educator
06:52

Problem 52

(a) For the thin rectangular plate shown in part (d) of Table 9.2 , find the moment of inertia about an axis that lies in the plane of the plate, passes through the center of the plate, and is parallel to the axis shown. (b) Find the moment of inertia of the plate for an axis that lies in the plane of the plate, passes through the center of the plate, and is perpendicular to the axis in part (a).

Averell Hause
Averell Hause
Carnegie Mellon University
01:55

Problem 53

A thin, rectangular sheet of metal has mass $M$ and sides of length $a$ and $b$. Use the parallel-axis theorem to calculate the moment of inertia of the sheet for an axis that is perpendicular to the plane of the sheet and that passes through one corner of the sheet.

Zhuxi Luo
Zhuxi Luo
Numerade Educator
03:46

Problem 54

A thin uniform rod of mass $M$ and length $L$ is bent at its center so that the two segments are now perpendicular to each other. Find its moment of inertia about an axis perpendicular to its plane and passing through (a) the point where the two segments meet and (b) the midpoint of the line connecting its two ends.

Shoukat Ali
Shoukat Ali
Other Schools
02:16

Problem 55

Use Eq. (9.20) to calculate the moment of inertia of a uniform, solid disk with mass $M$ and radius $R$ for an axis perpendicular to the plane of the disk and passing through its center.

Zhuxi Luo
Zhuxi Luo
Numerade Educator
01:34

Problem 56

Use Eq. (9.20) to calculate the moment of inertia of a slender, uniform rod with mass $M$ and length $L$ about an axis at one end, perpendicular to the rod.

Narayan Hari
Narayan Hari
Numerade Educator
06:08

Problem 57

A slender rod with length $L$ has a mass per unit length that varies with distance from the left end, where $x=0,$ according to $d m / d x=\gamma x,$ where $\gamma$ has units of $\mathrm{kg} / \mathrm{m}^{2}$. (a) Calculate the total mass of the rod in terms of $\gamma$ and $L$. (b) Use Eq. (9.20) to calculate the moment of inertia of the rod for an axis at the left end, perpendicular to the rod. Use the expression you derived in part (a) to express $I$ in terms of $M$ and $L$. How does your result compare to that for a uniform rod? Explain.
(c) Repeat part (b) for an axis at the right end of the rod. How do the results for parts (b) and (c) compare? Explain.

Andrew C
Andrew C
Numerade Educator
04:34

Problem 58

A uniform disk with radius $R=0.400 \mathrm{~m}$ and mass $30.0 \mathrm{~kg}$ rotates in a horizontal plane on a frictionless vertical axle that passes through the center of the disk. The angle through which the disk has turned varies with time according to $\theta(t)=(1.10 \mathrm{rad} / \mathrm{s}) t+\left(6.30 \mathrm{rad} / \mathrm{s}^{2}\right) t^{2}$
What is the resultant linear acceleration of a point on the rim of the disk at the instant when the disk has turned through 0.100 rev?

Averell Hause
Averell Hause
Carnegie Mellon University
02:18

Problem 59

A circular saw blade with radius $0.120 \mathrm{~m}$ starts from rest and turns in a vertical plane with a constant angular acceleration of $2.00 \mathrm{rev} / \mathrm{s}^{2} .$ After the blade has turned through $155 \mathrm{rev},$ a small piece of the blade breaks loose from the top of the blade. After the piece breaks loose, it travels with a velocity that is initially horizontal and equal to the tangential velocity of the rim of the blade. The piece travels a vertical distance of $0.820 \mathrm{~m}$ to the floor. How far does the piece travel horizontally, from where it broke off the blade until it strikes the floor?

Zhuxi Luo
Zhuxi Luo
Numerade Educator
04:56

Problem 60

A roller in a printing press turns through an angle $\begin{array}{llll}\theta(t) & \text { given } & \text { by } & \theta(t)=\gamma t^{2}-\beta t^{3}, & \text { where } & \gamma=3.20 \mathrm{rad} / \mathrm{s}^{2} & \text { and }\end{array}$
$\beta=0.500 \mathrm{rad} / \mathrm{s}^{3} .$ (a) Calculate the angular velocity of the roller as a function of time. (b) Calculate the angular acceleration of the roller as a function of time. (c) What is the maximum positive angular velocity, and at what value of $t$ does it occur?

Averell Hause
Averell Hause
Carnegie Mellon University
04:56

Problem 60

A roller in a printing press turns through an angle $\begin{array}{llll}\theta(t) & \text { given } & \text { by } & \theta(t)=\gamma t^{2}-\beta t^{3}, & \text { where } & \gamma=3.20 \mathrm{rad} / \mathrm{s}^{2} & \text { and }\end{array}$
$\beta=0.500 \mathrm{rad} / \mathrm{s}^{3} .$ (a) Calculate the angular velocity of the roller as a function of time. (b) Calculate the angular acceleration of the roller as a function of time. (c) What is the maximum positive angular velocity, and at what value of $t$ does it occur?

Averell Hause
Averell Hause
Carnegie Mellon University
06:06

Problem 61

A disk of radius $25.0 \mathrm{~cm}$ is free to turn about an axle perpendicular to it through its center. It has very thin but strong string wrapped around its rim, and the string is attached to a ball that is pulled tangentially away from the rim of the disk (Fig. $\mathbf{P 9 . 6 1}$ ). The pull increases in magnitude and produces an acceleration of the ball that obeys the equation $a(t)=A t,$ where $t$ is in seconds and $A$ is a constant. The cylinder starts from rest, and at the end of the third second, the ball's acceleration is $1.80 \mathrm{~m} / \mathrm{s}^{2}$. (a) Find $A$. (b) Express the angular acceleration of the disk as a function of time. (c) How much time after the disk has begun to turn does it reach an angular speed of $15.0 \mathrm{rad} / \mathrm{s} ?$ (d) Through what angle has the disk turned just as it reaches $15.0 \mathrm{rad} / \mathrm{s} ?$ (Hint: See Section $2.6 .)$

Andrew C
Andrew C
Numerade Educator
04:20

Problem 62

You are designing a rotating metal flywheel that will be used to store energy. The flywheel is to be a uniform disk with radius $25.0 \mathrm{~cm} .$ Starting from rest at $t=0,$ the flywheel rotates with constant angular acceleration $3.00 \mathrm{rad} / \mathrm{s}^{2}$ about an axis perpendicular to the flywheel at its center. If the flywheel has a density (mass per unit volume) of $8600 \mathrm{~kg} / \mathrm{m}^{3},$ what thickness must it have to store $800 \mathrm{~J}$ of kinetic energy at $t=8.00 \mathrm{~s} ?$

Averell Hause
Averell Hause
Carnegie Mellon University
07:10

Problem 63

A uniform wheel in the shape of a solid disk is mounted on a frictionless axle at its center. The wheel has mass $5.00 \mathrm{~kg}$ and radius $0.800 \mathrm{~m} .$ A thin rope is wrapped around the wheel, and a block is suspended from the free end of the rope. The system is released from rest and the block moves downward. What is the mass of the block if the wheel turns through 8.00 revolutions in the first $5.00 \mathrm{~s}$ after the block is released?

Andrew C
Andrew C
Numerade Educator
07:33

Problem 64

Engineers are designing a system by which a falling mass $m$ imparts kinetic energy to a rotating uniform drum to which it is attached by thin, very light wire wrapped around the rim of the drum (Fig. $\mathbf{P 9 . 6 4}$ ). There is no appreciable friction in the axle of the drum, and everything starts from rest. This system is being tested on earth, but it is to be used on Mars, where the acceleration due to gravity is $3.71 \mathrm{~m} / \mathrm{s}^{2} .$ In the earth tests, when $m$ is set to $15.0 \mathrm{~kg}$ and allowed to fall through $5.00 \mathrm{~m},$ it gives $250.0 \mathrm{~J}$ of kinetic energy to the drum. (a) If the system is operated on Mars, through what distance would the $15.0 \mathrm{~kg}$ mass have to fall to give the same amount of kinetic energy to the drum? (b) How fast would the $15.0 \mathrm{~kg}$ mass be moving on Mars just as the drum gained $250.0 \mathrm{~J}$ of kinetic energy?

Andrew C
Andrew C
Numerade Educator
05:21

Problem 65

Consider the system of two blocks shown in Fig. P9.77. There is no friction between block $A$ and the tabletop. The mass of block $B$ is $5.00 \mathrm{~kg} .$ The pulley rotates about a frictionless axle, and the light rope doesn't slip on the pulley surface. The pulley has radius $0.200 \mathrm{~m}$ and moment of inertia $1.30 \mathrm{~kg} \cdot \mathrm{m}^{2}$. If the pulley is rotating with an angular speed of $8.00 \mathrm{rad} / \mathrm{s}$ after the block has descended $1.20 \mathrm{~m},$ what is the mass of block $A ?$

Andrew C
Andrew C
Numerade Educator
04:47

Problem 66

The motor of a table saw is rotating at 3450 rev $/$ min. A pulley attached to the motor shaft drives a second pulley of half the diameter by means of a V-belt. A circular saw blade of diameter $0.208 \mathrm{~m}$ is mounted on the same rotating shaft as the second pulley. (a) The operator is careless and the blade catches and throws back a small piece of wood. This piece of wood moves with linear speed equal to the tangential speed of the rim of the blade. What is this speed? (b) Calculate the radial acceleration of points on the outer edge of the blade to see why sawdust doesn't stick to its teeth.

Andrew C
Andrew C
Numerade Educator
05:34

Problem 67

While riding a multispeed bicycle, the rider can select the radius of the rear sprocket that is fixed to the rear axle. The front sprocket of a bicycle has radius $12.0 \mathrm{~cm} .$ If the angular speed of the front sprocket is 0.600 rev $/ \mathrm{s},$ what is the radius of the rear sprocket for which the tangential speed of a point on the rim of the rear wheel will be $5.00 \mathrm{~m} / \mathrm{s} ?$ The rear wheel has radius $0.330 \mathrm{~m}$.

Ankit Pandey
Ankit Pandey
Numerade Educator
03:28

Problem 68

A computer disk drive is turned on starting from rest and has constant angular acceleration. If it took $0.0865 \mathrm{~s}$ for the drive to make its second complete revolution,
(a) how long did it take to make the first complete revolution, and (b) what is its angular acceleration, in rad/s $^{2}$ ?

Averell Hause
Averell Hause
Carnegie Mellon University
06:42

Problem 69

Consider the system shown in Fig. E9.49. The suspended block has mass $1.50 \mathrm{~kg}$. The system is released from rest and the block descends as the wheel rotates on a frictionless axle. As the wheel is ro-
tating, the tension in the light wire is $9.00 \mathrm{~N}$. What is the kinetic energy of the wheel $2.00 \mathrm{~s}$ after the system is released?

Andrew C
Andrew C
Numerade Educator
05:02

Problem 70

A uniform disk has radius $R_{0}$ and mass $M_{0}$. Its moment of inertia for an axis perpendicular to the plane of the disk at the disk's center is $\frac{1}{2} M_{0} R_{0}^{2}$. You have been asked to halve the disk's moment of inertia by cutting out a circular piece at the center of the disk. In terms of $R_{0}$, what should be the radius of the circular piece that you remove?

Andrew C
Andrew C
Numerade Educator
05:12

Problem 71

Measuring $I$. As an intern at an engineering firm, you are asked to measure the moment of inertia of a large wheel for rotation about an axis perpendicular to the wheel at its center. You measure the diameter of the wheel to be $0.640 \mathrm{~m}$. Then you mount the wheel on frictionless bearings on a horizontal frictionless axle at the center of the wheel. You wrap a light rope around the wheel and hang an $8.20 \mathrm{~kg}$ block of wood from the free end of the rope, as in Fig. E9.49. You release the system from rest and find that the block descends $12.0 \mathrm{~m}$ in $4.00 \mathrm{~s}$. What is the moment of inertia of the wheel for this axis?

Andrew C
Andrew C
Numerade Educator
03:16

Problem 72

A uniform, solid disk with mass $m$ and radius $R$ is pivoted about a horizontal axis through its center. A small object of the same mass $m$ is glued to the rim of the disk. If the disk is released from rest with the small object at the end of a horizontal radius, find the angular speed when the small object is directly below the axis.

Averell Hause
Averell Hause
Carnegie Mellon University
13:52

Problem 73

A meter stick with a mass of $0.180 \mathrm{~kg}$ is pivoted about one end so it can rotate without friction about a horizontal axis. The meter stick is held in a horizontal position and released. As it swings through the vertical, calculate (a) the change in gravitational potential energy that has occurred; (b) the angular speed of the stick; (c) the linear speed of the end of the stick opposite the axis.
(d) Compare the answer in part (c) to the speed of a particle that has fallen $1.00 \mathrm{~m}$, starting from rest.

MT
Michael Thees
Numerade Educator
05:02

Problem 74

A physics student of mass $43.0 \mathrm{~kg}$ is standing at the edge of the flat roof of a building, $12.0 \mathrm{~m}$ above the sidewalk. An unfriendly $\operatorname{dog}$ is running across the roof toward her. Next to her is a large wheel mounted on a horizontal axle at its center. The wheel, used to lift objects from the ground to the roof, has a light crank attached to it and a light rope wrapped around it; the free end of the rope hangs over the edge of the roof. The student grabs the end of the rope and steps off the roof. If the wheel has radius $0.300 \mathrm{~m}$ and a moment of inertia of $9.60 \mathrm{~kg} \cdot \mathrm{m}^{2}$ for rotation about the axle, how long does it take her to reach the sidewalk, and how fast will she be moving just before she lands? Ignore friction in the axle.

Andrew C
Andrew C
Numerade Educator
01:58

Problem 75

A slender rod is $80.0 \mathrm{~cm}$ long and has mass $0.120 \mathrm{~kg} . \mathrm{A}$
small $0.0200 \mathrm{~kg}$ sphere is welded to one end of the rod, and a small $0.0500 \mathrm{~kg}$ sphere is welded to the other end. The rod, pivoting about a stationary, frictionless axis at its center, is held horizontal and released from rest. What is the linear speed of the $0.0500 \mathrm{~kg}$ sphere as it passes through its lowest point?

Zhuxi Luo
Zhuxi Luo
Numerade Educator
04:33

Problem 76

Exactly one turn of a flexible rope with mass $m$ is wrapped around a uniform cylinder with mass $M$ and radius $R .$ The cylinder rotates without friction about a horizontal axle along the cylinder axis. One end of the rope is attached to the cylinder. The cylinder starts with angular speed $\omega_{0} .$ After one revolution of the cylinder the rope has unwrapped and, at this instant, hangs vertically down, tangent to the cylinder. Find the angular speed of the cylinder and the linear speed of the lower end of the rope at this time. Ignore the thickness of the rope. [Hint: Use Eq. (9.18).]

Averell Hause
Averell Hause
Carnegie Mellon University
05:00

Problem 77

The pulley in Fig. $\mathrm{P} 9.77$ has radius $R$ and a moment of inertia $I$. The rope does not slip over the pulley, and the pulley spins on a frictionless axle. The coefficient of kinetic friction between block $A$ and the tabletop is $\mu_{\mathrm{k}}$. The system is released from rest, and block $B$ descends. Block $A$ has mass $m_{A}$ and block $B$ has mass $m_{B}$. Use energy methods to calculate the speed of block $B$ as a function of the distance $d$ that it has descended.

Zhuxi Luo
Zhuxi Luo
Numerade Educator
05:45

Problem 78

The pulley in Fig. $\mathbf{P} 9.78$ has radius $0.160 \mathrm{~m}$ and moment of inertia $0.380 \mathrm{~kg} \cdot \mathrm{m}^{2}$. The rope does not slip on the pulley rim. Use energy methods to calculate the speed of the $4.00 \mathrm{~kg}$ block just before it strikes the floor.

Andrew C
Andrew C
Numerade Educator
07:14

Problem 79

Two metal disks, one with radius $R_{1}=2.50 \mathrm{~cm}$ and mass $M_{1}=0.80 \mathrm{~kg}$ and
the other with radius $R_{2}=5.00 \mathrm{~cm}$ and mass $M_{2}=1.60 \mathrm{~kg},$ are welded together and mounted on a frictionless axis through their common center (Fig. $\mathbf{P 9 . 7 9}$ ). (a) What is the total moment of inertia of the two disks? (b) A light string is wrapped around the edge of the smaller disk, and a $1.50 \mathrm{~kg}$ block is suspended from the free end of the string. If the block is released from rest at a distance of $2.00 \mathrm{~m}$ above the floor, what is its speed just before it strikes the floor? (c) Repeat part (b), this time with the string wrapped around the edge of the larger disk. In which case is the final speed of the block greater? Explain.

Andrew C
Andrew C
Numerade Educator
04:12

Problem 80

A thin, light wire is wrapped around the rim of a wheel as shown in Fig. E9.49. The wheel rotates about a stationary horizontal axle that passes through the center of the wheel. The wheel has radius $0.180 \mathrm{~m}$ and moment of inertia for rotation about the axle of $I=0.480 \mathrm{~kg} \cdot \mathrm{m}^{2}$. A small block with mass $0.340 \mathrm{~kg}$ is suspended from the free end of the wire. When the system is released from rest, the block descends with constant acceleration. The bearings in the wheel at the axle are rusty, so friction there does $-9.00 \mathrm{~J}$ of work as the block descends $3.00 \mathrm{~m}$. What is the magnitude of the angular velocity of the wheel after the block has descended $3.00 \mathrm{~m} ?$

Andrew C
Andrew C
Numerade Educator
03:25

Problem 81

In the system shown in Fig. $9.17,$ a $12.0 \mathrm{~kg}$ mass is released from rest and falls, causing the uniform $10.0 \mathrm{~kg}$ cylinder of diameter $30.0 \mathrm{~cm}$ to turn about a frictionless axle through its center. How far will the mass have to descend to give the cylinder $480 \mathrm{~J}$ of kinetic energy?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:05

Problem 82

In Fig. $\mathbf{P 9 . 8 2},$ the cylinder
and pulley turn without friction about stationary horizontal axles that pass through their centers. A light rope is wrapped around the cylinder, passes over the pulley, and has a $3.00 \mathrm{~kg}$ box suspended from its free end. There is no slipping between the rope and the pulley surface. The uniform cylinder has mass $5.00 \mathrm{~kg}$ and radius $40.0 \mathrm{~cm} .$ The pulley is a uniform disk with mass $2.00 \mathrm{~kg}$ and radius $20.0 \mathrm{~cm} .$ The box is released from rest and descends as the rope unwraps from the cylinder. Find the speed of the box when it has fallen $2.50 \mathrm{~m}$.

Andrew C
Andrew C
Numerade Educator
06:19

Problem 83

A dancer is spinning at 72 rpm about an axis through her center with her arms outstretched (Fig. $\mathbf{P} 9.83)$. From biomedical measurements, the typical distribution of mass in a human body is as follows:
Head: $7.0 \%$ Arms: $13 \%$ (for both) Trunk and legs: $80.0 \%$
Suppose you are this dancer. Using this information plus length measurements on your own body, calculate (a) your moment of inertia about your spin axis and
(b) your rotational kinetic energy. Use Table 9.2 to model reasonable approximations for the pertinent parts of your body.

Andrew C
Andrew C
Numerade Educator
04:41

Problem 84

A thin, uniform rod is bent into a square of side length $a$. If the total mass is $M,$ find the moment of inertia about an axis through the center and perpendicular to the plane of the square. (Hint: Use the parallel-axis theorem.)

Artemisa Mazón
Artemisa Mazón
Numerade Educator
03:56

Problem 85

A sphere with radius $R=0.200 \mathrm{~m}$ has density $\rho$ that decreases with distance $r$ from the center of the sphere according to $\rho=3.00 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}-\left(9.00 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{4}\right) r .$ (a) Calculate the
total mass of the sphere. (b) Calculate the moment of inertia of the sphere for an axis along a diameter.

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
10:28

Problem 86

The Crab Nebula is a cloud of glowing gas about 10 light-years across, located about 6500 light-years from the earth (Fig. P9.86). It is the remnant of a star that underwent a supernova $e x$ plosion, seen on earth in 1054 A.D. Energy is released by the Crab Nebula at a rate of about $5 \times 10^{31} \mathrm{~W},$ about $10^{5}$ times the rate at which the sun
radiates energy. The Crab Nebula obtains its energy from the rotational kinetic energy of a rapidly spinning neutron star at its center. This object rotates once every $0.0331 \mathrm{~s},$ and this period is increasing by $4.22 \times 10^{-13} \mathrm{~s}$ for each second of time that elapses. (a) If the rate at which energy is lost by the neutron star is equal to the rate at which energy is released by the nebula, find the moment of inertia of the neutron star.
(b) Theories of supernovae predict that the neutron star in the Crab Nebula has a mass about 1.4 times that of the sun. Modeling the neutron star as a solid uniform sphere, calculate its radius in kilometers.
(c) What is the linear speed of a point on the equator of the neutron star? Compare to the speed of light. (d) Assume that the neutron star is uniform and calculate its density. Compare to the density of ordinary rock $\left(3000 \mathrm{~kg} / \mathrm{m}^{3}\right)$ and to the density of an atomic nucleus (about $\left.10^{17} \mathrm{~kg} / \mathrm{m}^{3}\right) .$ Justify the statement that a neutron star is essentially a large atomic nucleus.

Andrew C
Andrew C
Numerade Educator
09:47

Problem 87

A technician is testing a computer-controlled, variablespeed motor. She attaches a thin disk to the motor shaft, with the shaft at the center of the disk. The disk starts from rest, and sensors attached to the motor shaft measure the angular acceleration $\alpha_{z}$ of the shaft as a function of time. The results from one test run are shown in Fig. $\mathbf{P 9 . 8 7}$.
(a) Through how many revolutions has the disk turned in the first $5.0 \mathrm{~s} ?$ Can you use Eq. (9.11)? Explain. What is the angular velocity, in rad/s, of the disk (b) at $t=5.0 \mathrm{~s} ;$ (c) when it has turned through 2.00 rev?

Andrew C
Andrew C
Numerade Educator
09:53

Problem 88

$9.88^{\circ}$ DATA You are analyzing the motion of a large flywheel that has radius $0.800 \mathrm{~m}$. In one test run, the wheel starts from rest and turns in a horizontal plane with constant angular acceleration. An accelerometer on the rim of the flywheel measures the magnitude of the resultant acceleration $a$ of a point on the rim of the flywheel as a function of the angle $\theta-\theta_{0}$ through which the wheel has turned. You collect these results: $$
\begin{array}{l|cccccccc}
\boldsymbol{\theta}-\boldsymbol{\theta}_{\mathbf{0}}(\mathbf{r a d}) & 0.50 & 1.00 & 1.50 & 2.00 & 2.50 & 3.00 & 3.50 & 4.00 \\
\hline \boldsymbol{a}\left(\mathbf{m} / \mathbf{s}^{\mathbf{2}}\right) & 0.678 & 1.07 & 1.52 & 1.98 & 2.45 & 2.92 & 3.39 & 3.87
\end{array}
$$
Construct a graph of $a^{2}\left(\right.$ in $\left.\mathrm{m}^{2} / \mathrm{s}^{4}\right)$ versus $\left(\theta-\theta_{0}\right)^{2}$ (in rad $^{2}$ ). (a) What are the slope and $y$ -intercept of the straight line that gives the best fit to the data? (b) Use the slope from part (a) to find the angular acceleration of the flywheel. (c) What is the linear speed of a point on the rim of the flywheel when the wheel has turned through an angle of $135^{\circ} ?$
(d) When the flywheel has turned through an angle of $90.0^{\circ},$ what is the angle between the linear velocity of a point on its rim and the resultant acceleration of that point?

Susan Hallstrom
Susan Hallstrom
Numerade Educator
09:16

Problem 89

You are rebuilding a 1965 Chevrolet. To decide whether to replace the flywheel with a newer, lighter-weight one, you want to determine the moment of inertia of the original, 35.6 -cm-diameter flywheel. It is not a uniform disk, so you can't use $I=\frac{1}{2} M R^{2}$ to calculate the moment of inertia. You remove the flywheel from the car and use low-friction bearings to mount it on a horizontal, stationary rod that passes through the center of the flywheel, which can then rotate freely (about $2 \mathrm{~m}$ above the ground). After gluing one end of a long piece of flexible fishing line to the rim of the flywheel, you wrap the line a number of turns around the rim and suspend a $5.60 \mathrm{~kg}$ metal block from the free end of the line. When you release the block from rest, it descends as the flywheel rotates. With high-speed photography you measure the distance $d$ the block has moved downward as a function of the time since it was released. The equation for the graph shown in Fig. $\mathrm{P} 9.89$ that gives a good fit to the data points is $d=\left(165 \mathrm{~cm} / \mathrm{s}^{2}\right) t^{2}$. (a) Based on the graph, does the block fall with constant acceleration? Explain. (b) Use the graph to calculate the speed of the block when it has descended $1.50 \mathrm{~m}$.
(c) Apply conservation of mechanical energy to the system of flywheel and block to calculate the moment of inertia of the flywheel. (d) You are relieved that the fishing line doesn't break. Apply Newton's second law to the block to find the tension in the line as the block descended.

Andrew C
Andrew C
Numerade Educator
08:34

Problem 90

Calculate the moment of inertia of a uniform solid cone about an axis through its center (Fig. $\mathrm{P} 9.90$ ). The cone has mass $M$ and altitude $h .$ The radius of its circular base is $R$.

Averell Hause
Averell Hause
Carnegie Mellon University
13:01

Problem 91

On a compact disc (CD), music is coded in a pattern of tiny pits arranged in a track that spirals outward toward the rim of the disc. As the disc spins inside a CD player, the track is scanned at a constant linear speed of $v=1.25 \mathrm{~m} / \mathrm{s} .$ Because the radius of the track varies as it spirals outward, the angular speed of the disc must change as the $\mathrm{CD}$ is played. (See Exercise $9.20 .$ ) Let's see what angular acceleration is required to keep $v$ constant. The equation of a spiral is $r(\theta)=r_{0}+\beta \theta,$ where $r_{0}$ is the radius of the spiral at $\theta=0$ and $\beta$ is a constant. On a $\mathrm{CD}, r_{0}$ is the inner radius of the spiral track. If we take the rotation direction of the CD to be positive, $\beta$ must be positive so that $r$ increases as the disc turns and $\theta$ increases. (a) When the disc rotates through a small angle $d \theta,$ the distance scanned along the track is $d s=r d \theta .$ Using the above expression for $r(\theta),$ integrate $d s$ to find the total distance $s$ scanned along the track as a function of the total angle
$\theta$ through which the disc has rotated. (b) since the track is scanned at a constant linear speed $v,$ the distance $s$ found in part (a) is equal to vi. Use this to find $\theta$ as a function of time. There will be two solutions
for $\theta ;$ choose the positive one, and explain why this is the solution to choose. (c) Use your expression for $\theta(t)$ to find the angular velocity $\omega_{z}$ and the angular acceleration $\alpha_{z}$ as functions of time. Is $\alpha_{z}$ constant?
(d) On a CD, the inner radius of the track is $25.0 \mathrm{~mm}$, the track radius increases by $1.55 \mu \mathrm{m}$ per revolution, and the playing time is $74.0 \mathrm{~min} .$ Find $r_{0}, \beta,$ and the total number of revolutions made during the playing time. (e) Using your results from parts (c) and (d), make graphs of $\omega_{z}$ (in rad/s) versus $t$ and $\alpha_{z}$ (in rad/s $^{2}$ ) versus $t$ between $t=0$ and $t=74.0 \mathrm{~min}$

Andrew C
Andrew C
Numerade Educator
02:08

Problem 92

A field researcher uses the slow-motion feature on her phone's camera to shoot a video of an eel spinning at its maximum rate. The camera records at 120 frames per second. Through what angle does the eel rotate from one frame to the next?
(a) $1^{\circ} ;$ (b) $10^{\circ} ;$ (c) $22^{\circ}$;
(d) $42^{\circ}$.

Averell Hause
Averell Hause
Carnegie Mellon University
01:30

Problem 93

The eel is observed to spin at 14 spins per second clockwise, and 10 seconds later it is observed to spin at 8 spins per second counterclockwise. What is the magnitude of the eel's average angular acceleration during this time? (a) $6 / 10 \mathrm{rad} / \mathrm{s}^{2}$;
(b) $6 \pi / 10 \mathrm{rad} / \mathrm{s}^{2}$
(c) $12 \pi / 10 \mathrm{rad} / \mathrm{s}^{2} ;$ (d) $44 \pi / 10 \mathrm{rad} / \mathrm{s}^{2}$.

Zhuxi Luo
Zhuxi Luo
Numerade Educator
02:45

Problem 94

The eel has a certain amount of rotational kinetic energy when spinning at 14 spins per second. If it swam in a straight line instead, about how fast would the eel have to swim to have the same amount of kinetic energy as when it is spinning?
(a) $0.5 \mathrm{~m} / \mathrm{s} ;$ (b) $0.7 \mathrm{~m} / \mathrm{s} ;$ (c) $3 \mathrm{~m} / \mathrm{s} ;$ (d) $5 \mathrm{~m} / \mathrm{s}$.

Averell Hause
Averell Hause
Carnegie Mellon University
01:36

Problem 95

A new species of eel is found to have the same mass but onequarter the length and twice the diameter of the American eel. How does its moment of inertia for spinning around its long axis compare to that of the American eel? The new species has (a) half the moment of inertia as the American eel; (b) the same moment of inertia as the American eel; (c) twice the moment of inertia as the American eel;
(d) four times the moment of inertia as the American eel.

Andrew C
Andrew C
Numerade Educator