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Essential University Physics

Richard Wolfson

Chapter 11

Rotational Vectors and Angular Momentum - all with Video Answers

Educators

Chapter Questions

00:56

Problem 1

Does Earth's angular velocity vector point north or south?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
06:41

Problem 2

Figure 11.12 shows four forces acting on a body. In what directions are the associated torques about point $O ?$ About point $P ?$

Katie Mcalpine
Katie Mcalpine
Numerade Educator
01:36

Problem 3

You stand with your right arm extended horizontally to the right. What's the direction of the gravitational torque about your shoulder?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
01:08

Problem 4

Although it contains no parentheses, the expression $\vec{A} \times \vec{B} \cdot \vec{C}$ is unambiguous. Why? Is the expression a vector or a scalar?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
02:21

Problem 5

What's the angle between two vectors if their dot product is equal to the magnitude of their cross product?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
02:40

Problem 6

Why does a tetherball move faster as it winds up its pole?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
01:06

Problem 7

Why do helicopters have two rotors?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
01:15

Problem 8

A group of polar bears is standing around the edge of a slowly rotating ice floe. If the bears all walk to the center, what happens to the rotation rate?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
01:49

Problem 9

Tornadoes in the northern hemisphere rotate counterclockwise as viewed from above. A far-fetched idea suggests that driving on the right side of the road may increase the frequency of tornadoes. Does this idea have any merit? Explain in terms of the angular momentum imparted to the air as two cars pass.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
04:14

Problem 10

Does a particle moving at constant speed in a straight line have angular momentum about a point on the line? About a point not on the line? In either case, is its angular momentum constant?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
00:32

Problem 11

When you turn on a high-speed power tool such as a router, the tool tends to twist in your hands. Why?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
02:02

Problem 12

Why is it easier to balance a basketball on your finger if it's spinning?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
00:59

Problem 13

A bug, initially at rest on a stationary, frictionless turntable, walks halfway around the turntable's circumference. Describe the motion of the turntable while the bug is walking and after the bug has stopped.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
00:44

Problem 14

If you increase the rotation rate of a precessing gyroscope, will the precession rate increase or decrease?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
04:50

Problem 15

A car is headed north at $70 \mathrm{km} / \mathrm{h}$. Give the magnitude and direction of the angular velocity of its 62 -cm-diameter wheels.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
04:42

Problem 16

If the car of Exercise 15 makes a $90^{\circ}$ left turn lasting 25 s, determine the average angular acceleration of the wheels.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
09:33

Problem 17

A wheel is spinning at 45 rpm with its axis vertical. After 15 s, it's spinning at 60 rpm with its axis horizontal. Find (a) the magnitude of its average angular acceleration and (b) the angle the average angular acceleration vector makes with the horizontal.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
11:19

Problem 18

A wheel is spinning about a horizontal axis with angular speed $140 \mathrm{rad} / \mathrm{s}$ and with its angular velocity pointing east. Find the magnitude and direction of its angular velocity after an angular acceleration of $35 \mathrm{rad} / \mathrm{s}^{2},$ pointing $68^{\circ}$ west of north, is applied for $5.0 \mathrm{s}$.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
07:29

Problem 19

A $12-\mathrm{N}$ force is applied at the point $x=3 \mathrm{m}, y=1 \mathrm{m} .$ Find the torque about the origin if the force points in (a) the $x$ -direction, (b) the $y$ -direction, and (c) the $z$ -direction.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
08:16

Problem 20

A force $\vec{F}=1.3 \hat{\imath}+2.7 \hat{\jmath} \mathrm{N}$ is applied at the point $x=3.0 \mathrm{m}, y=0 \mathrm{m} .$ Find the torque about (a) the origin and (b) the point $x=-1.3 \mathrm{m}, y=2.4 \mathrm{m}$.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
01:27

Problem 21

When you hold your arm outstretched, it's supported primarily by the deltoid muscle. Figure 11.13 shows a case in which the deltoid exerts a $67-\mathrm{N}$ force at $15^{\circ}$ to the horizontal. If the force-application point is $18 \mathrm{cm}$ horizontally from the shoulder joint, what torque does the deltoid exert about the shoulder?

Justin Swantek
Justin Swantek
Numerade Educator
03:56

Problem 22

Express the units of angular momentum (a) using only the fundamental units kilogram, meter, and second; (b) in a form involving newtons; (c) in a form involving joules.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
03:32

Problem 23

In the Olympic hammer throw, a contestant whirls a 7.3 -kg steel ball on the end of a 1.2 -m cable. If the contestant's arms reach an additional $90 \mathrm{cm}$ from his rotation axis and if the ball's speed just prior to release is $27 \mathrm{m} / \mathrm{s},$ what's the magnitude of the ball's angular momentum?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
01:08

Problem 24

A gymnast of rotational inertia $62 \mathrm{kg} \cdot \mathrm{m}^{2}$ is tumbling head over heels with angular momentum $470 \mathrm{kg} \cdot \mathrm{m}^{2} / \mathrm{s} .$ What's her angular speed?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
03:52

Problem 25

A $640-\mathrm{g}$ hoop $90 \mathrm{cm}$ in diameter is rotating at 170 rpm about its central axis. What's its angular momentum?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
02:55

Problem 26

A 7.4 -cm-diameter baseball has mass $145 \mathrm{g}$ and is spinning at 2000 rpm. Treating the baseball as a uniform solid sphere, what's its angular momentum?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
01:29

Problem 27

A potter's wheel with rotational inertia $6.40 \mathrm{kg} \cdot \mathrm{m}^{2}$ is spinning freely at 19.0 rpm. The potter drops a 2.70 -kg lump of clay onto the wheel, where it sticks $46.0 \mathrm{cm}$ from the rotation axis. What's the wheel's subsequent angular speed?

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
04:19

Problem 28

A 3.0 -m-diameter merry-go-round with rotational inertia $120 \mathrm{kg} \cdot \mathrm{m}^{2}$ is spinning freely at 0.50 rev/s. Four 25 -kg children sit suddenly on the edge of the merry-go-round. (a) Find the new angular speed, and (b) determine the total energy lost to friction between children and merry-go-round.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
07:40

Problem 29

A uniform, spherical cloud of interstellar gas has mass $2.0 \times 10^{30} \mathrm{kg},$ has radius $1.0 \times 10^{13} \mathrm{m},$ and is rotating with period $1.4 \times 10^{6}$ years. The cloud collapses to form a star $7.0 \times 10^{8} \mathrm{m}$ in radius. Find the star's rotation period.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
02:59

Problem 30

A skater has rotational inertia $4.2 \mathrm{kg} \cdot \mathrm{m}^{2}$ with his fists held to his chest and $5.7 \mathrm{kg} \cdot \mathrm{m}^{2}$ with his arms outstretched. The skater is spinning at 3.0 rev/s while holding a 2.5 -kg weight in each outstretched hand; the weights are $76 \mathrm{cm}$ from his rotation axis. If he pulls his hands in to his chest, so they're essentially on his rotation axis, how fast will he be spinning?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
03:49

Problem 31

You slip a wrench over a bolt. Taking the origin at the bolt, the other end of the wrench is at $x=18 \mathrm{cm}, y=5.5 \mathrm{cm} .$ You apply a force $\vec{F}=88 \hat{\imath}-23 \hat{\jmath} \mathrm{N}$ to the end of the wrench. What's the torque on the bolt?

Vishal Gupta
Vishal Gupta
Numerade Educator
06:32

Problem 32

Vector $\vec{A}$ points $30^{\circ}$ counterclockwise from the $x$ -axis. Vector $\vec{B}$ has twice the magnitude of $\vec{A}$. Their product $\vec{A} \times \vec{B}$ has magnitude $A^{2}$ and points in the negative $z$ -direction. What's the direction of vector $\vec{B} ?$

Katie Mcalpine
Katie Mcalpine
Numerade Educator
07:51

Problem 33

A baseball player extends his arm straight up to catch a $145-\mathrm{g}$ baseball moving horizontally at $42 \mathrm{m} / \mathrm{s}$. It's $63 \mathrm{cm}$ from the player's shoulder joint to the point the ball strikes his hand, and his arm remains stiff while it rotates about the shoulder during the catch. The player's hand recoils $5.00 \mathrm{cm}$ horizontally while he stops the ball. What average torque does the player's arm exert on the ball?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
02:58

Problem 34

Show that $\vec{A} \cdot(\vec{A} \times \vec{B})=0$ for any vectors $\vec{A}$ and $\vec{B}$.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
05:23

Problem 35

A weightlifter's barbell consists of two 25 -kg masses on the ends of a $15-\mathrm{kg}$ rod $1.6 \mathrm{m}$ long. The weightlifter holds the rod at its center and spins it at 10 rpm about an axis perpendicular to the rod. What's the magnitude of the barbell's angular momentum?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
02:55

Problem 36

A particle of mass $m$ moves in a straight line at constant speed $v$ Show that its angular momentum about a point located a perpendicular distance $b$ from its line of motion is mvb regardless of where the particle is on the line.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
03:44

Problem 37

Two identical $1800-\mathrm{kg}$ cars are traveling in opposite directions at $83 \mathrm{km} / \mathrm{h} .$ Each car's center of mass is $3.2 \mathrm{m}$ from the center of the highway (Fig. 11.14 ). What are the magnitude and direction of the angular momentum of the system consisting of the two cars, about a point on the centerline of the highway? (FIGURE CAN'T COPY)

Supratim Pal
Supratim Pal
Numerade Educator
03:01

Problem 38

The dot product of two vectors is half the magnitude of their cross product. What's the angle between the two vectors?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
04:14

Problem 39

Biomechanical engineers have developed micromechanical devices for measuring blood flow as an alternative to dye injection following angioplasty to remove arterial plaque. One experimental device consists of a 300 - $\mu \mathrm{m}$ -diameter, 2.0 - $\mu \mathrm{m}$ -thick silicon rotor inserted into blood vessels. Moving blood spins the rotor, whose rotation rate provides a measure of blood flow. This device exhibited an 800 -rpm rotation rate in tests with water flows at several m/s. Treating the rotor as a disk, what was its angular momentum at 800 rpm? (Hint: You'll need to find the density of silicon.)

Steven Emmel
Steven Emmel
University of California - Los Angeles
07:07

Problem 40

Figure 11.15 shows the dimensions of a $880-\mathrm{g}$ wooden baseball bat whose rotational inertia about its center of mass is $0.048 \mathrm{kg} \cdot \mathrm{m}^{2} .$ If the bat is swung so its far end moves at $50 \mathrm{m} / \mathrm{s}$ find (a) its angular momentum about the pivot $P$ and (b) the constant torque applied about $P$ to achieve this angular momentum in 0.25 s. (Hint: Remember the parallel-axis theorem.)

Katie Mcalpine
Katie Mcalpine
Numerade Educator
02:57

Problem 41

As an automotive engineer, you're charged with redesigning a car's wheels with the goal of decreasing each wheel's angular momentum by $30 \%$ for a given linear speed of the car. Other design considerations require that the wheel diameter go from $38 \mathrm{cm}$ to $35 \mathrm{cm} .$ If the old wheel had rotational inertia $0.32 \mathrm{kg} \cdot \mathrm{m}^{2},$ what do you specify for the new rotational inertia?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
11:01

Problem 42

A turntable of radius $25 \mathrm{cm}$ and rotational inertia $0.0154 \mathrm{kg} \cdot \mathrm{m}^{2}$ is spinning freely at 22.0 rpm about its central axis, with a $19.5-\mathrm{g}$ mouse on its outer edge. The mouse walks from the edge to the center. Find (a) the new rotation speed and (b) the work done by the mouse.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
19:14

Problem 43

A 17 -kg dog is standing on the edge of a stationary, frictionless turntable of rotational inertia $95 \mathrm{kg} \cdot \mathrm{m}^{2}$ and radius $1.81 \mathrm{m}$ The dog walks once around the turntable. What fraction of a full circle does the dog's motion make with respect to the ground?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
14:52

Problem 44

A physics student is standing on an initially motionless, frictionless turntable with rotational inertia $0.31 \mathrm{kg} \cdot \mathrm{m}^{2} .$ She's holding a wheel with rotational inertia $0.22 \mathrm{kg} \cdot \mathrm{m}^{2}$ spinning at $130 \mathrm{rpm}$ about a vertical axis, as in Fig. $11.8 .$ When she turns the wheel upside down, student and turntable begin rotating at 70 rpm. (a) Find the student's mass, considering her to be a 30 -cm-diameter cylinder. (b) Neglecting the distance between the axes of the turntable and wheel, determine the work she did in turning the wheel upside down.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
14:54

Problem 45

You're choreographing your school's annual ice show. You call for eight $60-\mathrm{kg}$ skaters to join hands and skate side by side in a line extending $12 \mathrm{m}$. The skater at one end is to stop abruptly, so the line will rotate rigidly about that skater. For safety, you don't want the fastest skater to be moving at more than $8.0 \mathrm{m} / \mathrm{s},$ and you don't want the force on that skater's hand to exceed $300 \mathrm{N}$. What do you determine is the greatest speed the skaters can have before they execute their rotational maneuver?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
01:40

Problem 46

Find the angle between two vectors whose dot product is twice the magnitude of their cross product.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
08:18

Problem 47

A circular bird feeder $19 \mathrm{cm}$ in radius has rotational inertia $0.12 \mathrm{kg} \cdot \mathrm{m}^{2} .$ It's suspended by a thin wire and is spinning slowly at 5.6 rpm. A 140 -g bird lands on the feeder's rim, coming in tangent to the rim at $1.1 \mathrm{m} / \mathrm{s}$ in a direction opposite the feeder's rotation. What's the rotation rate after the bird lands?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
04:37

Problem 48

A force $\vec{F}$ applied at the point $x=2.0 \mathrm{m}, y=0 \mathrm{m}$ produces a torque $4.6 \hat{k} \mathrm{N} \cdot$ mabout the origin. If the $x$ -component of $\vec{F}$ is $3.1 \mathrm{N}$ what angle does it make with the $x$ -axis?

Katie Mcalpine
Katie Mcalpine
Numerade Educator
02:04

Problem 49

Show that the cross product of two vectors $\vec{A}=A_{x} \hat{i}+A_{y} \hat{j}+A_{z} \hat{k}$ and $\vec{B}=B_{x} \hat{i}+B_{y} \hat{j}+B_{z} \hat{k}$ is given by $\vec{A} \times \vec{B}=\left(A_{y} B_{z}-A_{z} B_{y}\right) \hat{i}+\left(A_{z} B_{x}-A_{x} B_{z}\right) \hat{j}+\left(A_{x} B_{y}-A_{y} B_{x}\right) \hat{k}$ (Hint: You'll need to work out cross products of all possible pairs of the unit vectors $\hat{i}, \hat{j},$ and $\hat{k}$ - including with themselves.)

Carson Merrill
Carson Merrill
Numerade Educator
03:51

Problem 50

If you're familiar with determinants, show that the cross product can be written as a determinant: $$\vec{A} \times \vec{B}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\A_{x} & A_{y} & A_{z} \\B_{x} & B_{y} & B_{z}\end{array}\right|$$ (Hint: See the preceding problem.)

Katie Mcalpine
Katie Mcalpine
Numerade Educator
13:20

Problem 51

Jumbo is back! Jumbo is the 4.8 -Mg elephant from Example 9.4 This time he's standing at the outer edge of a 15 -Mg turntable of radius $8.5 \mathrm{m},$ rotating with angular velocity $0.15 \mathrm{s}^{-1}$ on frictionless bearings. Jumbo then walks to the center of the turntable. Treating Jumbo as a point mass and the turntable as a solid disk, find (a) the angular velocity of the turntable once Jumbo reaches the center and (b) the work Jumbo does in walking to the center.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
06:10

Problem 52

An anemometer for measuring wind speeds consists of four small cups, each with mass $124 \mathrm{g}$, mounted a pair of $32.6-\mathrm{cm}$ -long rods with mass $75.7 \mathrm{g}$ each, as shown in Fig. $11.16 .$ Find the angular momentum of the anemometer when it's spinning at 12.4 rev/s. You can treat the cups as point masses. (FIGURE CAN'T COPY)

Katie Mcalpine
Katie Mcalpine
Numerade Educator
07:45

Problem 53

A turntable has rotational inertia $I$ and is rotating with angular speed $\omega$ about a frictionless vertical axis. A wad of clay with mass $m$ is tossed onto the turntable and sticks a distance $d$ from the rotation axis. The clay hits horizontally with its velocity $\vec{v}$ at right angles to the turntable's radius, and in the same direction as the turntable's rotation (Fig. 11.17 ). Find expressions for $v$ that will result in (a) the turntable's angular speed dropping to half its initial value, (b) no change in the turntable's angular speed, and (c) the angular speed doubling. (FIGURE CAN'T COPY)

Katie Mcalpine
Katie Mcalpine
Numerade Educator
17:33

Problem 54

A uniform, solid, spherical asteroid with mass $1.2 \times 10^{13} \mathrm{kg}$ and radius $1.0 \mathrm{km}$ is rotating with period $4.3 \mathrm{h}$. A meteoroid moving in the asteroid's equatorial plane crashes into the equator at $8.4 \mathrm{km} / \mathrm{s} .$ It hits at a $58^{\circ}$ angle to the vertical and embeds itself at the surface. After the impact the asteroid's rotation period is $3.9 \mathrm{h} .$ Find the meteoroid's mass.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
04:07

Problem 55

About $99.9 \%$ of the solar system's total mass lies in the Sun. Using data from Appendix E, estimate what fraction of the solar system's angular momentum about its center is associated with the Sun. Where is most of the rest of the angular momentum?

Manish Jain
Manish Jain
Numerade Educator
13:16

Problem 56

You're a civil engineer for an advanced civilization on a solid spherical planet of uniform density. Running out of room for the expanding population, the government asks you to redesign your planet to give it more surface area. You recommend reshaping the planet, without adding any material or angular momentum, into a hollow shell whose thickness is one-fifth its outer radius. How much will your design increase the surface area, and how will it change the length of the day?

Eduard Sanchez
Eduard Sanchez
Numerade Educator
11:14

Problem 57

In Fig. $11.18,$ the lower disk, of mass $440 \mathrm{g}$ and radius $3.5 \mathrm{cm},$ is rotating at 180 rpm on a frictionless shaft of negligible radius. The upper disk, of mass $270 \mathrm{g}$ and radius $2.3 \mathrm{cm},$ is initially not rotating. It drops freely down onto the lower disk, and frictional forces bring the two disks to a common rotational speed. Find (a) that common speed and (b) the fraction of the initial kinetic energy lost to friction.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
07:16

Problem 58

A massless spring with constant $k$ is mounted on a turntable of rotational inertia $I$, as shown in Fig. $11.19 .$ The turntable is on a frictionless vertical axle, though initially it's not rotating. The spring is compressed a distance $\Delta x$ from its equilibrium, with a mass $m$ placed against it. When the spring is released, the mass moves at right angles to a line through the turntable's center, at a distance $b$ from the center, and slides without friction across the table and off the edge. Find expressions for (a) the linear speed of the mass and (b) the rotational speed of the turntable. (Hint: What's conserved?) (FIGURE CAN'T COPY)

Katie Mcalpine
Katie Mcalpine
Numerade Educator
08:45

Problem 59

A solid ball of mass $M$ and radius $R$ is spinning with angular velocity $\omega_{0}$ about a horizontal axis. It drops vertically onto a surface where the coefficient of kinetic friction with the ball is $\mu_{\mathrm{k}}$ (Fig. 11.20 ). Find expressions for (a) the final angular velocity once it's achieved pure rolling motion and (b) the time it takes to achieve this motion. (FIGURE CAN'T COPY)

Katie Mcalpine
Katie Mcalpine
Numerade Educator
01:54

Problem 60

A time-dependent torque given by $\tau=a+b \sin c t$ is applied to an object that's initially stationary but is free to rotate. Here $a, b$ and $c$ are constants. Find an expression for the object's angular momentum as a function of time, assuming the torque is first applied at $t=0$.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
05:45

Problem 61

Consider a rapidly spinning gyroscope whose axis is precessing uniformly in a horizontal circle of radius $r,$ as shown in Fig. 11.10 Apply $\vec{\tau}=d \vec{L} / d t$ to show that the angular speed of precession about the vertical axis through the center of the circle is $m g r / L$.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
03:45

Problem 62

When a star like our Sun exhausts its fuel, thermonuclear reactions in its core cease, and it collapses to become a white dwarf. Often the star will blow off its outer layers and lose some mass before it collapses. Suppose a star with the Sun's mass and radius is rotating with period 25 days and then it collapses to a white dwarf with $60 \%$ of the Sun's mass and a rotation period of $131 \mathrm{s}$ What's the radius of the white dwarf? Compare your answer with the radii of Sun and Earth.

Zulfiqar Ali
Zulfiqar Ali
Numerade Educator
10:26

Problem 63

Pulsars- -the rapidly rotating neutron stars described in Example 11.2 - have magnetic fields that interact with charged particles in the surrounding interstellar medium. The result is torque that causes the pulsar's spin rate and therefore its angular momentum to decrease very slowly. The table below gives values for the rotation period of a given pulsar as it's been observed at the same date every 5 years for two decades. The pulsar's rotational inertia is known to be $1.12 \times 10^{38} \mathrm{kg} \cdot \mathrm{m}^{2} .$ Make a plot of the pulsar's angular momentum over time, and use the associated best-fit line, along with the rotational analog of Newton's law, to find the torque acting on the pulsar.
$$\begin{array}{|l|c|c|c|c|c|}\hline \text { Year of observation } & 1995 & 2000 & 2005 & 2010 & 2015 \\\hline \begin{array}{l}
\text { Angular momentum } \\\left(10^{37} \mathrm{kg} \cdot \mathrm{m}^{2} / \mathrm{s}\right)\end{array} & 7.844 & 7.831 & 7.816 & 7.799 & 7.787 \\\hline\end{array}$$

Averell Hause
Averell Hause
Carnegie Mellon University
06:48

Problem 64

A system has total angular momentum $\vec{L}$ about an axis $O .$ Show that the system's angular momentum about a parallel axis $O^{\prime}$ is given by $\vec{L}^{\prime}=L-\vec{h} \times \vec{p},$ where $\vec{p}$ is the system's linear momentum and $\vec{h}$ is a vector from $O$ to $O^{\prime}$ (see Fig. $11.21,$ which also shows vectors $\vec{r}_{i}$ and $\vec{r}_{i}$ from each axis to the system's ith mass element $m_{i}$ ). (FIGURE CAN'T COPY)

Katie Mcalpine
Katie Mcalpine
Numerade Educator
01:53

Problem 65

Figure 11.22 shows a demonstration gyroscope, consisting of a solid disk mounted on a shaft. The disk spins about the shaft on essentially frictionless bearings. The shaft is mounted on a stand so it's free to pivot both horizontally and vertically. A weight at the far end of the shaft balances the disk, so in the configuration shown there's no torque on the system. An arrowhead mounted on the disk end of the shaft indicates the direction of the disk's angular velocity. (FIGURE CAN'T COPY)
If you push on the shaft between the arrowhead and the disk, pushing horizontally away from you (i.e., into the page in Fig.
11.22), the arrowhead end of the shaft will move
a. away from you (i.e., into the page).
b. toward you (i.e., out of the page).
c. downward.
d. upward.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
01:15

Problem 66

Figure 11.22 shows a demonstration gyroscope, consisting of a solid disk mounted on a shaft. The disk spins about the shaft on essentially frictionless bearings. The shaft is mounted on a stand so it's free to pivot both horizontally and vertically. A weight at the far end of the shaft balances the disk, so in the configuration shown there's no torque on the system. An arrowhead mounted on the disk end of the shaft indicates the direction of the disk's angular velocity. (FIGURE CAN'T COPY)
If you push on the shaft between the arrowhead and the disk, pushing directly upward on the bottom of the shaft, the arrowhead end of the shaft will move
a. away from you (i.e., into the page).
b. toward you (i.e., out of the page).
c. downward.
d. upward.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
01:21

Problem 67

Figure 11.22 shows a demonstration gyroscope, consisting of a solid disk mounted on a shaft. The disk spins about the shaft on essentially frictionless bearings. The shaft is mounted on a stand so it's free to pivot both horizontally and vertically. A weight at the far end of the shaft balances the disk, so in the configuration shown there's no torque on the system. An arrowhead mounted on the disk end of the shaft indicates the direction of the disk's angular velocity.
If an additional weight is hung on the left end of the shaft, the arrowhead will
a. pivot upward until the weighted end of the shaft hits the base.
b. pivot downward until the arrowhead hits the base.
c. precess counterclockwise when viewed from above.
d. precess clockwise when viewed from above.

Katie Mcalpine
Katie Mcalpine
Numerade Educator
00:57

Problem 68

Figure 11.22 shows a demonstration gyroscope, consisting of a solid disk mounted on a shaft. The disk spins about the shaft on essentially frictionless bearings. The shaft is mounted on a stand so it's free to pivot both horizontally and vertically. A weight at the far end of the shaft balances the disk, so in the configuration shown there's no torque on the system. An arrowhead mounted on the disk end of the shaft indicates the direction of the disk's angular velocity.
If the system is precessing, and only the disk's rotation rate is increased, the precession rate will
a. decrease.
b. increase.
c. stay the same.
d. become zero.

Katie Mcalpine
Katie Mcalpine
Numerade Educator