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University Physics with Modern Physics

Hugh D Young; Roger A Freedman; Albert Lewis Ford

Chapter 9

Rotation of Rigid Bodies - all with Video Answers

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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 $0.700 \mathrm{rad}$. 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 $1900 \mathrm{rpm}$ (rev/min).
(a) Compute the propeller's angular velocity in $\mathrm{rad} / \mathrm{s}$. (b) How many seconds does it take for the propeller to turn through $35^{\circ}$ ?

Laura Yu
Laura Yu
University of California, Irvine
04:42

Problem 3

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

Laura Yu
Laura Yu
University of California, Irvine
04:07

Problem 4

. CALC A fan blade rotates with angular velocity given by $\omega_{z}(t)=\gamma-\beta t^{2}$, where $\gamma=5.00 \mathrm{rad} / \mathrm{s}$ and $\beta=0.800 \mathrm{rad} / \mathrm{s}^{3} .$
(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 are they different?

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
02:39

Problem 5

* CALC A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to $\theta(t)=\gamma t+\beta t^{3}$, where $\gamma=0.400 \mathrm{rad} / \mathrm{s}$ and $\beta=$ $0.0120 \mathrm{rad} / \mathrm{s}^{3}$. (a) Calculate the angular velocity of the merry-goround 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 $\omega_{\mathrm{av}-\tau}$ is not equal to the average of the instantaneous angular velocities at, $t=0$ and $t=5.00 \mathrm{~s}$, and explain why it is not.

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
05:10

Problem 6

- CALC 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) $\mathrm{At}$ 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

- CALC 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 $2.00 \mathrm{rad} / \mathrm{s}$, and 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 $\theta=\pi / 4 \mathrm{rad} ?$ (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
03:10

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 $+8.00 \mathrm{rad} / \mathrm{s}$ 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 $t=7.00 \mathrm{~s} ?$

Mayukh Banik
Mayukh Banik
Numerade Educator
02:59

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 $0.300 \mathrm{rad} / \mathrm{s}^{2}$, what is its angular velocity at $t=2.50 \mathrm{~s}$ ? (b) Through what angle has the wheel turned between $t=0$ and $t=2.50 \mathrm{~s}$ ?

Susan Hallstrom
Susan Hallstrom
Numerade Educator
04:39

Problem 10

* An electric fan is turned off, and its angular velocity decreases uniformly from $500 \mathrm{rev} / \mathrm{min}$ to $200 \mathrm{rev} / \mathrm{min}$ in $4.00 \mathrm{~s}$. (a) Find the angular acceleration in rev/s $^{2}$ and the number of revolutions made by the motor in the $4.00-\mathrm{s}$ 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)?

Averell Hause
Averell Hause
Carnegie Mellon University
08:12

Problem 11

"* The rotating blade of a blender tums 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?

Jason Bane
Jason Bane
Numerade Educator
03:15

Problem 12

- (a) Derive Eq. (9.12) by combining Eqs. (9.7) and $(9.11)$ to eliminate $t$. (b) The angular velocity of an airplane propeller increases from $12.0 \mathrm{rad} / \mathrm{s}$ to $16.0 \mathrm{rad} / \mathrm{s}$ while turning through $7.00 \mathrm{rad}$. What is the angular acceleration in $\mathrm{rad} / \mathrm{s}^{2} ?$

Averell Hause
Averell Hause
Carnegie Mellon University
01:04

Problem 13

" A turntable rotates with a constant $2.25 \mathrm{rad} / \mathrm{s}^{2}$ angular acceleration. After $4.00 \mathrm{~s}$ it has rotated through an angle of $60.0 \mathrm{rad}$. What was the angular velocity of the wheel at the beginning of the 4.00-s interval?

Sarah Mccrumb
Sarah Mccrumb
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 $500 \mathrm{mpm}$ 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

$\cdots$ 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 $432 \mathrm{rad}$ 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
03:15

Problem 18

- In a charming 19th- Figurs century 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 accelElevator eration of the disk, in $\mathrm{rad} / \mathrm{s}^{2} ?$
(c) Through what angle (in radians and degresy) turned when it has raised the elevator $3.25 \mathrm{~m}$ between floors?

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
03:15

Problem 19

- In a charming 19th- Figurs century 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 accelElevator eration of the disk, in $\mathrm{rad} / \mathrm{s}^{2} ?$
(c) Through what angle (in radians and degresy) turned when it has raised the elevator $3.25 \mathrm{~m}$ between floors?

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
02:14

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 CD when the innermost part of the track is scanned? The outermost part of the track? (b) The maximum playing time of a $\mathrm{CD}$ is $74.0 \mathrm{~min}$. What would be the length of the track on such a maximum-duration CD if it were stretched out in a straight line? (c) What is the average angular acceleration of a maximumduration CD during its $74.0$-min playing time? Take the direction of rotation of the disc to be positive.

Mayukh Banik
Mayukh Banik
Numerade Educator
03:36

Problem 21

w A wheel of diameter $40.0 \mathrm{~cm}$ starts from rest and rotates with a constant angular acceleration of $3.00 \mathrm{rad} / \mathrm{s}^{2}$. At the instant the wheel has computed its second revolution, compute the radial acceleration of a point on the rim in two ways: (a) using the relationship $a_{\mathrm{rad}}=\omega^{2} r$ and (b) from the relationship $a_{\mathrm{rad}}=v^{2} / r$

Sarah Mccrumb
Sarah Mccrumb
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
04:06

Problem 23

- A flywheel with a radius of $0.300 \mathrm{~m}$ starts from rest and accelerates with a constant angular acceleration of $0.600 \mathrm{rad} / \mathrm{s}^{2}$. Compute the magnitude of the tangential acceleration, the radial acceleration, and the resultant acceleration of a point on its rim (a) at the start; (b) after it has turned through $60.0^{\circ} ;$ (c) after it has turned through $120.0^{\circ}$

Zhuxi Luo
Zhuxi Luo
Numerade Educator
04:47

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

Averell Hause
Averell Hause
Carnegie Mellon University
03:09

Problem 25

Centrifuge. 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?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
00:56

Problem 26

$\mathbf{}$ - (a) Derive an equation for the radial acceleration that includes $v$ and $\omega$, but not $r$. (b) You are designing a merry-go-round for which a point on the rim will have a radial acceleration of $0.500 \mathrm{~m} / \mathrm{s}^{2}$ when the tangential velocity of that point has magnitude $2.00 \mathrm{~m} / \mathrm{s}$. What angular velocity is required to achieve these values?

Mayukh Banik
Mayukh Banik
Numerade Educator
02:05

Problem 27

- Electric Drill. According to the shop manual, when drilling a $12.7-\mathrm{mm}$-diameter hole in wood, plastic, or aluminum, a drill should have a speed of $1250 \mathrm{rev} / \mathrm{min} .$ For a $12.7-\mathrm{mm}-$ diameter drill bit turning at a constant $1250 \mathrm{rev} / \mathrm{min}$, find (a) the maximum linear speed of any part of the bit and (b) the maximum radial acceleration of any part of the bit.

Shoukat Ali
Shoukat Ali
Other Schools
03:53

Problem 28

At $t=3.00 \mathrm{~s}$ a point on the rim of a $0.200$-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
04:10

Problem 29

- The spin cycles of a washing machine have two angular speeds, $423 \mathrm{rev} / \mathrm{min}$ and $640 \mathrm{rev} / \mathrm{min} .$ The internal diameter of the drum is $0.470 \mathrm{~m}$. (a) What is the ratio of the maximum radial force on the laundry for the higher angular speed to that for the lower speed? (b) What is the ratio of the maximum tangential speed of the laundry for the higher angular speed to that for the lower speed? (c) Find the laundry's maximum tangential speed and the maximum radial acceleration, in terms of $g$.

Shoukat Ali
Shoukat Ali
Other Schools
04:59

Problem 30

- Four small spheres, each Figure E9.30 of which you can regard as a point of mass $0.200 \mathrm{~kg}$, are a.dsed in a square $0.400 \mathrm{~m}$ on a arranged in a square $0.400 \mathrm{~m}$ on a sice and connected by extremely side and connected by extremely light rods (Fig. E9.30). Find the 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 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 $O .$

Averell Hause
Averell Hause
Carnegie Mellon University
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-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
03:57

Problem 32

.o 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 one-fourth of the length from one end.

Averell Hause
Averell Hause
Carnegie Mellon University
04:21

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 $0.500 \mathrm{~kg}$ and can be treated as point masses. Find the moment of inertia of this combination about each of the following axes: (a) an axis perpendicular to the bar through its center; (b) an axis perpendicular to the bar through one of the balls; (c) an axis parallel to the bar through both balls; (d) an axis parallel to the bar and $0.500 \mathrm{~m}$ from it.

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
02:40

Problem 34

- A uniform disk of radius $R$ is cut in half so that the remaining half has mass $M$ (Fig. E9.34a). (a) What is the moment of inertia of this half about an axis perpendicular to its plane through point $A$ ? (b) Why did your answer in part (a) come out the same as if this were a complete disk of mass $M$ ? (c) What would be the moment of inertia of a quarter disk of mass $M$ and radius $R$ about an axis perpendicular to its plane passing through point $B$ (Fig. E9.34b)?

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
01:37

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 the formulas given in Table $9.2 .$ )

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
03:10

Problem 36

$2.08 \mathrm{~m}$ in length (from tip to tip) with mass $117 \mathrm{~kg}$ and is rotating at $2400 \mathrm{rpm}$ (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
03:30

Problem 37

$\cdots$ A compound disk of outside diameter $140.0 \mathrm{~cm}$ is made up of a uniform solid disk of radius $50.0 \mathrm{~cm}$ and area density $3.00 \mathrm{~g} / \mathrm{cm}^{2}$ surrounded by a concentric ring of inner radius $50.0 \mathrm{~cm}$, outer radius $70.0 \mathrm{~cm}$, and area density $2.00 \mathrm{~g} / \mathrm{cm}^{2}$. Find the moment of inertia of this object about an axis perpendicular to the plane of the object and passing through its center.

Ryan Hood
Ryan Hood
Numerade Educator
03:46

Problem 38

- 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
01:54

Problem 39

- 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 $176 \mathrm{~J}$, what is the tangential velocity of a point on the rim of the sphere?

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
03:28

Problem 40

$ \quad \because$ 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
02:55

Problem 41

- Energy from the Moon? Suppose that some time in the future we decide to tap the moon's rotational energy for use on earth. In additional to the astronomical data in Appendix $\mathrm{F}$, you may need to know that the moon spins on its axis once every $27.3$ days. Assume that the moon is uniform throughout. (a) How much total energy could we get from the moon's rotation? (b) The world presently uses about $4.0 \times 10^{20} \mathrm{~J}$ of energy per year. If in the future the world uses five times as much energy yearly, for how many years would the moon's rotation provide us energy? In light of your answer, does this seem like a cost-effective energy source in which to invest?

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
02:03

Problem 42

* 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
02:52

Problem 43

The flywheel of a gasoline engine is required to give up $500 \mathrm{~J}$ of kinetic energy while its angular velocity decreases from $650 \mathrm{rev} / \mathrm{min}$ to $520 \mathrm{rev} / \mathrm{min} .$ What moment of inertia is required?

Zhuxi Luo
Zhuxi Luo
Numerade Educator
04:52

Problem 44

- A light, flexible rope is wrapped several times around a hollow cylinder, with a weight of $40.0 \mathrm{~N}$ and a radius of $0.25 \mathrm{~m}$,
that rotates without friction about a fixed horizontal axis. The cylinder is attached to the axle by spokes of a negligible moment of inertia. The cylinder is initially at rest. The free end of the rope is pulled with a constant force $P$ for a distance of $5.00 \mathrm{~m}$, at which point the end of the rope is moving at $6.00 \mathrm{~m} / \mathrm{s}$. If the rope does not slip on the cylinder, what is the value of $P ?$

Averell Hause
Averell Hause
Carnegie Mellon University
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
05:20

Problem 46

$ \cdots$ Suppose the solid cylinder in the apparatus described in Example $9.8$ (Section 9.4) is replaced by a thin-walled, hollow cylinder with the same mass $M$ and radius $R .$ The cylinder is attached to the axle by spokes of a negligible moment of inertia.
(a) Find the speed of the hanging mass $m$ just as it strikes the floor.
(b) Use energy concepts to explain why the answer to part (a) is different from the speed found in Example $9.8 .$

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
02:52

Problem 47

- A frictionless pulley has the shape of a uniform solid disk of mass $2.50 \mathrm{~kg}$ and radius $20.0 \mathrm{~cm} . \mathrm{A} 1.50-\mathrm{kg}$ 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?

Zhuxi Luo
Zhuxi Luo
Numerade Educator
02:27

Problem 48

A bucket of mass $m$ is
tied to a massless cable that is wrapped around the outer rim of a frictionless uniform pulley of radius $R$, 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?

Averell Hause
Averell Hause
Carnegie Mellon University
06:12

Problem 49

? CP A thin, light wire Figure $\mathrm{E9.49}$ is wrapped around the rim of a wheel, as shown in 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
02:51

Problem 50

* A uniform $2.00-\mathrm{m}$ ladder of mass $9.00 \mathrm{~kg}$ is leaning against a vertical wall while making an angle of $53.0^{\circ}$ with the floor. A worker pushes the ladder up against the wall until it is vertical. What is the increase in the gravitational potential energy of the ladder?

Averell Hause
Averell Hause
Carnegie Mellon University
01:53

Problem 51

How $\boldsymbol{I}$ Scales. If we multiply all the design dimensions of an object by a scaling factor $f$, its volume and mass will be multiplied by $f^{3}$. (a) By what factor will its moment of inertia be multiplied? (b) If a $\frac{1}{48}$-scale model has a rotational kinetic energy of $2.5 \mathrm{~J}$, what will be the kinetic energy for the full-scaleobject of the same material rotating at the same angular velocity?

Zhuxi Luo
Zhuxi Luo
Numerade Educator
00:54

Problem 52

. A uniform $3.00$-kg rope $24.0 \mathrm{~m}$ long lies on the ground at the top of a vertical cliff. A mountain climber at the top lets down half of it to help his partner climb up the cliff. What was the change in potential energy of the rope during this maneuver?

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
01:44

Problem 53

* 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
00:45

Problem 54

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
01:55

Problem 55

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
04:05

Problem 56

- (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 in the figure. (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).

Bret Rosen
Bret Rosen
Numerade Educator
03:46

Problem 57

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
02:26

Problem 58

- CALC 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.

Averell Hause
Averell Hause
Carnegie Mellon University
02:16

Problem 59

$\cdots$ CALC 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
05:56

Problem 60

* CALC 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 this comparison.
(c) Repé?t p?rt (b) for ?n axis át thé right end of thé rod. How dò the results for parts (b) and (c) compare? Explain this result.

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
03:29

Problem 61

- CP CALC A flywheel has angular acceleration $\alpha_{z}(t)=$ $8.60 \mathrm{rad} / \mathrm{s}^{2}-\left(2.30 \mathrm{rad} / \mathrm{s}^{3}\right) t$, where counterclockwise rotation is positive. (a) If the flywheel is at rest at $t=0$, what is its angular velocity at $5.00 \mathrm{~s}$ ? (b) Through what angle (in radians) does the flywheel turn in the time interval from $t=0$ to $t=5.00 \mathrm{~s}$ ?

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
04:43

Problem 62

* CALC 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 $\theta(t)=$ $(1.10 \mathrm{rad} / \mathrm{s}) t+\left(8.60 \mathrm{rad} / \mathrm{s}^{2}\right) t^{2}$. 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?

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
02:59

Problem 63

$\cdots$ CP 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 $3.00 \mathrm{rev} / \mathrm{s}^{2}$. After the blade has turned through 155 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

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
04:52

Problem 64

- CALC A roller in a printing press turns through an angle $\theta(t)$ given by $\theta(t)=\gamma t^{2}-\beta t^{3}$, where $\gamma=3.20 \mathrm{rad} / \mathrm{s}^{2}$ and $\beta=0.500 \mathrm{rad} / \mathrm{s}^{3}$. (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?

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
01:03

Problem 65

. CP CALC 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. P9.65). 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.

Dominador Tan
Dominador Tan
Numerade Educator
01:42

Problem 66

$\cdots$ When a toy car is rapidly scooted across the floor, it stores energy in a flywheel. The car has mass $0.180 \mathrm{~kg}$, and its flywheel has moment of inertia $4.00 \times 10^{-5} \mathrm{~kg} \cdot \mathrm{m}^{2}$. The car is $15.0 \mathrm{~cm}$ long. An advertisement claims that the car can travel at a scale speed of up to $700 \mathrm{~km} / \mathrm{h}$ ( $440 \mathrm{mi} / \mathrm{h})$. The scale speed is the speed of the toy car multiplied by the ratio of the length of an actual car to the length of the toy. Assume a length of $3.0 \mathrm{~m}$ for a real car. (a) For a scale speed of $700 \mathrm{~km} / \mathrm{h}$, what is the actual translational speed of the car? (b) If all the kinetic energy that is initially in the flywheel is converted to the translational kinetic energy of the toy, how much energy is originally stored in the flywheel? (c) What initial angular velocity of the flywheel was needed to store the amount of energy calculated in part (b)?

Penny Riley
Penny Riley
Numerade Educator
03:46

Problem 67

A classic 1957 Chevrolet Corvette of mass $1240 \mathrm{~kg}$ starts from rest and speeds up with a constant tangential acceleration of $2.00 \mathrm{~m} / \mathrm{s}^{2}$ on a circular test track of radius $60.0 \mathrm{~m}$. Treat the car as a particle. (a) What is its angular acceleration? (b) What is its angular speed $6.00 \mathrm{~s}$ after it starts? (c) What is its radial acceleration at this time? (d) Sketch a view from above showing the circular track, the car, the velocity vector, and the acceleration component vectors $6.00 \mathrm{~s}$ after the car starts. (e) What are the magnitudes of the total acceleration and net force for the car at this time? (f) What

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
04:18

Problem 68

Engineers are designing a Figu\Gammae-4.68 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. P9.68). 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?

Averell Hause
Averell Hause
Carnegie Mellon University
02:05

Problem 69

- A vacuum cleaner belt is looped over a shaft of radius $0.45 \mathrm{~cm}$ and a wheel of radius $1.80 \mathrm{~cm}$. The arrangement of the belt, shaft, and wheel is similar to that of the chain and sprockets in Fig. Q9.4. The motor turns the shaft at $60.0 \mathrm{rev} / \mathrm{s}$ and the moving belt turns the wheel, which in turn is connected by another shaft to the roller that beats the dirt out of the rug being vacuumed. Assume that the belt doesn't slip on either the shaft or the wheel. (a) What is the speed of a point on the belt? (b) What is the angular velocity of the wheel in rad $/ \mathrm{s}$ ?

Zhuxi Luo
Zhuxi Luo
Numerade Educator
04:47

Problem 70

o The motor of a table saw is rotating at $3450 \mathrm{rev} / \mathrm{min} . \mathrm{A}$ 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 71

?. 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 $0.600 \mathrm{rev} / \mathrm{s}$, 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
08:05

Problem 72

o. A computer disk drive is turned on starting from rest and has constant angular acceleration. If it took $0.750 \mathrm{~s}$ 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 $\mathrm{rad} / \mathrm{s}^{2} ?$

Dading Chen
Dading Chen
Numerade Educator
04:07

Problem 73

- A wheel changes its angular velocity with a constant angular acceleration while rotating about a fixed axis through its center.
(a) Show that the change in the magnitude of the radial acceleration during any time interval of a point on the wheel is twice the product of the angular acceleration, the angular displacement, and the perpendicular distance of the point from the axis. (b) The radial acceleration of a point on the wheel that is $0.250 \mathrm{~m}$ from the axis changes from $25.0 \mathrm{~m} / \mathrm{s}^{2}$ to $85.0 \mathrm{~m} / \mathrm{s}^{2}$ as the wheel rotates through $20.0 \mathrm{rad}$. Calculate the tangential acceleration of this point. (c) Show that the change in the wheel's kinetic energy during any time interval is the product of the moment of inertia about the axis, the angular

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
02:14

Problem 74

- A sphere consists of a solid wooden ball of uniform density $800 \mathrm{~kg} / \mathrm{m}^{3}$ and radius $0.30 \mathrm{~m}$ and is covered with a thin coating of lead foil with area density $20 \mathrm{~kg} / \mathrm{m}^{2}$. Calculate the moment of inertia of this sphere about an axis passing through its center.

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
02:18

Problem 75

* It has been argued that power plants should make use of off-peak hours (such as late at night) to generate mechanical energy and store it until it is needed during peak load times, such as the middle of the day. One suggestion has been to store the energy in large flywheels spinning on nearly frictionless ball bearings. Consider a flywheel made of iron (density $7800 \mathrm{~kg} / \mathrm{m}^{3}$ ) in the shape of a $10.0$-cm-thick uniform disk. (a) What would the diameter of such a disk need to be if it is to store $10.0$ megajoules of kinetic energy when spinning at $90.0$ rpm about an axis perpendicular to the disk at its center? (b) What would be the centripetal acceleration of a point on its rim when spinning at this rate?

Zhuxi Luo
Zhuxi Luo
Numerade Educator
01:40

Problem 76

While redesigning a rocket engine, you want to reduce its weight by replacing a solid spherical part with a hollow spherical shell of the same size. The parts rotate about an axis through their center. You need to make sure that the new part always has the same rotational kinetic energy as the original part had at any given rate of rotation. If the original part had mass $M$, what must be the mass of the new part?

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
03:09

Problem 77

- The earth, which is not a uniform sphere, has a moment of inertia of $0.3308 M R^{2}$ about an axis through its north and south poles. It takes the carth $86,164 \mathrm{~s}$ to spin once about this axis. Use Appendix $\mathrm{F}$ to calculate (a) the earth's kinetic energy due to its rotation about this axis and (b) the earth's kinetic energy due to its orbital motion around the sun. (c) Explain how the value of the earth's moment of inertia tells us that the mass of the earth is concentrated toward the planet's center.

Mayukh Banik
Mayukh Banik
Numerade Educator
03:16

Problem 78

? 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
05:33

Problem 79

CALC A metal sign for a car dealership is a thin, uniform right triangle with base length $b$ and height $h .$ The sign has mass $M$. (a) What is the moment of inertia of the sign for rotation about the side of length $h ?$ (b) If $M=5.40 \mathrm{~kg}, b=1.60 \mathrm{~m}$, and $h=1.20 \mathrm{~m}$, what is the kinetic energy of the sign when it is rotating about an axis along the $1.20-\mathrm{m}$ side at $2.00 \mathrm{rev} / \mathrm{s}$ ?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:25

Problem 80

Measuring I. As an intern with an engineering firm, you are asked to measure the moment of inertia of a large wheel, for rotation about an axis through its center. Since you were a good physics student, you know what to do. You measure the diameter of the wheel to be $0.740 \mathrm{~m}$ and find that it weighs $280 \mathrm{~N}$. You mount the wheel, using frictionless bearings, on a horizontal axis through the wheel's center. You wrap a light rope around the wheel and hang an $8.00-\mathrm{kg}$ mass from the free end of the rope, as shown in Fig. 9.17. You release the mass from rest; the mass descends and the wheel turns as the rope unwinds. You find that the mass has speed $5.00 \mathrm{~m} / \mathrm{s}$ after it has descended $2.00 \mathrm{~m}$. (a) What is the moment of inertia of the wheel for an axis perpendicular to the wheel at its center? (b) Your boss tells you that a larger $I$ is needed. He asks you to design a wheel of the same mass and radius that has $I=19.0 \mathrm{~kg} \cdot \mathrm{m}^{2}$. How do you reply?

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
13:52

Problem 81

.. CP 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
09:54

Problem 82

- 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. You can ignore the thickness of the rope. [Hint: Use Eq. (9.18).]

Aaron Miller
Aaron Miller
Numerade Educator
05:00

Problem 83

" The pulley in Fig. P9.83 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
01:55

Problem 84

" The pulley in Fig. P9.84 has radius $0.160 \mathrm{~m}$ and moment of inertia $0.560 \mathrm{~kg} \cdot \mathrm{m}^{2}$. 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.

Narayan Hari
Narayan Hari
Numerade Educator
04:36

Problem 85

o You hang a thin hoop with radius $R$ over a nail at the rim of the hoop. You displace it to the side (within the plane of the hoop) through an angle $\beta$ from its equilibrium position and let it go. What is its angular speed when it returns to its equilibrium position? [Hint: Use Eq. (9.18).]

Sheh Lit Chang
Sheh Lit Chang
University of Washington
01:44

Problem 86

A passenger bus in Zurich, Switzerland, derived its motive power from the energy stored in a large flywheel. The wheel was brought up to speed periodically, when the bus stopped at a station, by an electric motor, which could then be attached to the electric power lines. The flywheel was a solid cylinder with mass $1000 \mathrm{~kg}$ and diameter $1.80 \mathrm{~m}$; its top angular speed was 3000 rev/min. (a) At this angular speed, what is the kinetic energy of the flywheel? (b) If the average power required to operate the bus is $1.86 \times 10^{4} \mathrm{~W}$, how long could it operate between stops?

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
04:17

Problem 87

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. P9.87). (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 the calculation of 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 why this is so.

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
04:12

Problem 88

? A thin, light wire is wrapped around the rim of a wheel, as shown in Fig. E9.49. The wheel rotates about a sta- $1.50 \mathrm{~kg}$ tionary 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 $-6.00 \mathrm{~J}$ 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 89

In the system shown in Fig. 9.17, a $12.0$-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
05:41

Problem 90

- In Fig. P9.90, the cylinder Figure P9.90 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
Cylinder
Box 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 $2.50 \mathrm{~m}$.

Averell Hause
Averell Hause
Carnegie Mellon University
03:23

Problem 91

$\cdots$ A thin, flat, uniform disk has mass $M$ and radius $R$. A circular hole of radius $R / 4$, centered at a point $R / 2$ from the disk's center, is then punched in the disk. (a) Find the moment of inertia of the disk with the hole about an axis through the original center of the disk, perpendicular to the plane of the disk. (Hint: Find the moment of inertia of the piece Figure P9.92 punched from the disk.) (b) Find the moment of inertia of the disk with the hole about an axis through the center of the hole, perpendicular to the plane of the disk.

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
04:26

Problem 92

? BID Human Rotational Energy. A dancer is spinning at $72 \mathrm{rpm}$ about an axis through her center with her arms outstretched, as shown in Fig. P9.92. 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 the figures in Table $9.2$ to model reasonable approximations for the pertinent parts of your body.

Zhuxi Luo
Zhuxi Luo
Numerade Educator
04:26

Problem 93

w BID The Kinetic Energy of Walking. If a person of mass $M$ simply moved forward with speed $V$, his kinetic energy would be $\frac{1}{2} M V^{2} .$ However, in addition to possessing a forward motion, various parts of his body (such as the arms and legs) undergo rotation. Therefore, his total kinetic energy is the sum of the energy from his forward motion plus the rotational kinetic energy of his arms and legs. The purpose of this problem is to see how much this rotational motion contributes to the person's kinetic energy. Biomedical measurements show that the arms and hands together typically make up $13 \%$ of a person's mass, while the legs and feet together account for $37 \%$. For a rough (but reasonable) calculation, we can model the arms and legs as thin uniform bars pivoting about the shoulder and hip, respectively. In a brisk walk, the arms and legs each move through an angle of about $\pm 30^{\circ}$ (a total of $60^{\circ}$ ) from the vertical in approximately 1 second. We shall assume that they are held straight, rather than being bent, which is not quite true. Let us consider a $75-\mathrm{kg}$ person walking at $5.0 \mathrm{~km} / \mathrm{h}$, having arms $70 \mathrm{~cm}$ long and legs $90 \mathrm{~cm}$ long. (a) What is the average angular velocity of his arms and legs? (b) Using the average angular velocity from part (a), calculate the amount of rotational kinetic energy in this person's arms and legs as he walks. (c) What is the total kinetic energy due to both his forward motion and his rotation? (d) What percentage of his kinetic energy is due to the rotation of his legs and arms?

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
02:40

Problem 94

* BID The Kinetic Energy of Running. Using Problen $9.93$ as a guide, apply it to a person running at $12 \mathrm{~km} / \mathrm{h}$, with hi? arms and legs each swinging through $\pm 30^{\circ}$ in $\frac{1}{2} \mathrm{~s}$. As before assume that the arms and legs are kept straight.

Ryan Hood
Ryan Hood
Numerade Educator
03:57

Problem 95

Perpendicular-Axis Theorem. Consider a rigid body that is a thin, plane sheet of arbitrary shape. Take the body to lie in the $x y$-plane and let the origin $O$ of coordinates be located at any point within or outside the body. Let $I_{x}$ and $I_{y}$ be the moments of inertia about the $x$-and $y$-axes, and let $I_{O}$ be the moment of inertia about an axis through $O$ perpendicular to the plane. (a) By considering mass elements $m_{i}$ with coordinates $\left(x_{i}, y_{i}\right)$, show that $I_{x}+I_{y}=I_{O}$. This is called the perpendicular-axis theorem. Note that point $O$ does not have to be the center of mass. (b) For a thin washer with mass $M$ and with inner and outer radii $R_{1}$ and $R_{2}$, use the perpendicular-axis theorem to find the moment of inertia about an axis that is in the plane of the washer and that passes through its center. You may use the information in Table 9.2. (c) Use the perpendicular-axis theorem to show that for a thin, square sheet with mass $M$ and side $L$, the moment of inertia about any axis in the plane of the sheet that passes through the center of the sheet is $\frac{1}{12} M L^{2}$. You may use the information in Table $9.2 .$

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
06:01

Problem 96

.. 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.

Averell Hause
Averell Hause
Carnegie Mellon University
09:35

Problem 97

- CALE A cylinder with radius $R$ and mass $M$ has density that increases linearly with distance $r$ from the cylinder axis, $\rho=\alpha r$, where $\alpha$ is a positive constant. (a) Calculate the moment of inertia of the cylinder about a longitudinal axis through its center in terms of $M$ and $R$. (b) Is your answer greater or smaller than the moment
of inertia of a cylinder of the same mass and radius but of uniform density? Explain why this result makes qualitative sense.

Konstantin Pavlovskii
Konstantin Pavlovskii
Numerade Educator
10:28

Problem 98

Nebula is a cloud of glowing gas about 10 light-years across, located about 6500 light-years from the earth (Fig. P9.98). It is the remnant of a star that underwent a supernova explosion, 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 increas ing 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
04:49

Problem 99

.. CALC 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.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
08:34

Problem 100

os CALC Calculate the moment of inertia of a uniform solid cone about an axis through its center (Fig. P9.100). The cone has mass $M$ and altitude $h .$ The radius of its circular base is $R$.

Averell Hause
Averell Hause
Carnegie Mellon University
01:17

Problem 101

*. CALC 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 $C D$ player, the track is scanned at a conAxis stant 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 CD, $r_{0}$ is the inner radius of the spiral track. If we take the rotation direction of the $\mathrm{CD}$ to be positive, $\beta$ must be positive so that $r$ increases as the disc turns and $\theta$ increases. (a) When the disc

Dominador Tan
Dominador Tan
Numerade Educator