Physics for Scientists and Engineers with Modern Physics

Douglas C. Giancoli

Chapter 11

Angular Momentum; General Rotation - all with Video Answers

Educators

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Chapter Questions

Problem 1

If there were a great migration of people toward the Earth's equator, would the length of the day $(a)$ get longer because of conservation of angular momentum; (b) get shorter because of conservation of angular momentum; (c) get shorter because of conservation of energy; $(d)$ get longer because of conservation of energy; or (e) remain unaffected?

James Wait

James Wait

Numerade Educator

Problem 2

Can the diver of Fig. $11-2$ do a somersault without having any initial rotation when she leaves the board?

Lainey Roebuck

Numerade Educator

Problem 3

A person stands, hands at his side, on a platform that is rotating at a rate of $0.90 \mathrm{rev} / \mathrm{s}$. If he raises his arms to a horizontal position, Fig. $11-30,$ the speed of rotation decreases to $0.70 \mathrm{rev} / \mathrm{s}$
(a) Why?
(b) By what factor has his moment of inertia changed?

Satpal Satpal

Numerade Educator

Problem 4

A figure skater can increase her spin rotation rate from an initial rate of 1.0 rev every $1.5 \mathrm{~s}$ to a final rate of $2.5 \mathrm{rev} / \mathrm{s} .$ If her initial moment of inertia was $4.6 \mathrm{~kg} \cdot \mathrm{m}^{2}$ what is her final moment of inertia? How does she physically accomplish this change?

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 5

A diver (such as the one shown in Fig. $11-2$ ) can reduce her moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in $1.5 \mathrm{~s}$ when in the tuck position, what is her angular speed (rev/s) when in the straight position?

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 6

A uniform horizontal rod of mass $M$ and length $\ell$ rotates with angular velocity $\omega$ about a vertical axis through its center. Attached to each end of the rod is a small mass $m$. Determine the angular momentum of the system about the axis.

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 7

Determine the angular momentum of the Earth (a) about its rotation axis (assume the Earth is a uniform sphere), and $(b)$ in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $=6.0 \times 10^{24} \mathrm{~kg}$ and radius $=6.4 \times 10^{6} \mathrm{~m},$ and is $1.5 \times 10^{8} \mathrm{~km}$ from the Sun.

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 8

( $a$ ) What is the angular momentum of a figure skater spinning at $2.8 \mathrm{rev} / \mathrm{s}$ with arms in close to her body, assuming her to be a uniform cylinder with a height of $1.5 \mathrm{~m},$ a radius of $15 \mathrm{~cm},$ and a mass of $48 \mathrm{~kg} ?(b)$ How much torque is required to slow her to a stop in $5.0 \mathrm{~s}$, assuming she does not move her arms?

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 9

A person stands on a platform, initially at rest, that can rotate freely without friction. The moment of inertia of the person plus the platform is $I_{P}$. The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_{\mathrm{W}}$ and angular velocity $\omega_{\mathrm{W}}$. What will be the angular velocity $\omega_{\mathrm{P}}$ of the platform if the person moves the axis of the wheel so that it points $(a)$ vertically upward, (b) at a $60^{\circ}$ angle to the vertical, $(c)$ vertically downward? (d) What will $\omega_{\mathrm{P}}$ be if the person reaches up and stops the wheel in part $(a) ?$

Lainey Roebuck

Numerade Educator

Problem 10

A uniform disk turns at 3.7 rev/s around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk's diameter, is dropped onto the freely spinning disk, Fig. $11-31 .$ They then turn together around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 11

A person of mass $75 \mathrm{~kg}$ stands at the center of a rotating merry-go-round platform of radius $3.0 \mathrm{~m}$ and moment of inertia $920 \mathrm{~kg} \cdot \mathrm{m}^{2} .$ The platform rotates without friction with angular velocity $0.95 \mathrm{rad} / \mathrm{s}$. The person walks radially to the edge of the platform. (a) Calculate the angular velocity when the person reaches the edge. ( $b$ ) Calculate the rotational kinetic energy of the system of platform plus person before and after the person's walk.

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 12

A potter's wheel is rotating around a vertical axis through its center at a frequency of $1.5 \mathrm{rev} / \mathrm{s} .$ The wheel can be considered a uniform disk of mass $5.0 \mathrm{~kg}$ and diameter $0.40 \mathrm{~m} .$ The potter then throws a $2.6-\mathrm{kg}$ chunk of clay, approximately shaped as a flat disk of radius $8.0 \mathrm{~cm},$ onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?

Satpal Satpal

Numerade Educator

Problem 13

A 4.2-m-diameter merry-go-round is rotating freely with an angular velocity of $0.80 \mathrm{rad} / \mathrm{s}$. Its total moment of inertia is $1760 \mathrm{~kg} \cdot \mathrm{m}^{2}$. Four people standing on the ground, each of mass $65 \mathrm{~kg}$, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now? What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 14

(II) A woman of mass $m$ stands at the edge of a solid cylindrical platform of mass $M$ and radius $R .$ At $t=0,$ the platform is rotating with negligible friction at angular velocity $\omega_{0}$ about a vertical axis through its center, and the woman begins walking with speed $v$ (relative to the platform) toward the center of the platform.
(a) Determine the angular velocity of the system as a function of time.
(b) What will be the angular velocity when the woman reaches the center?

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 15

A nonrotating cylindrical disk of moment of inertia $I$ is dropped onto an identical disk rotating at angular speed $\omega$. Assuming no external torques, what is the final common angular speed of the two disks?

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 16

Suppose our Sun eventually collapses into a white dwarf, losing about half its mass in the process, and winding up with a radius $1.0 \%$ of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun's new rotation rate be? (Take the Sun's current period to be about 30 days.) What would be its final kinetic energy in terms of its initial kinetic energy of today?

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 17

Hurricanes can involve winds in excess of $120 \mathrm{~km} / \mathrm{h}$ at the outer edge. Make a crude estimate of $(a)$ the energy, and $(b)$ the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density $1.3 \mathrm{~kg} / \mathrm{m}^{3}$ ) of radius $85 \mathrm{~km}$ and height $4.5 \mathrm{~km}$

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 18

An asteroid of mass $1.0 \times 10^{5} \mathrm{~kg}$, traveling at a speed of $35 \mathrm{~km} / \mathrm{s}$ relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth's rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.

Lainey Roebuck

Numerade Educator

Problem 19

The axis of the Earth precesses with a period of about 25,000 years. This is much like the precession of a top. Explain how the Earth's equatorial bulge gives rise to a torque exerted by the Sun and Moon on the Earth; see Fig. $11-29$, which is drawn for the winter solstice (December 21). About what axis would you expect the Earth's rotation axis to precess as a result of the torque due to the Sun? Does the torque exist three months later? Explain.

Lainey Roebuck

Numerade Educator

Problem 20

If vector $\overrightarrow{\mathbf{A}}$ points along the negative $x$ axis and vector $\overrightarrow{\mathbf{B}}$ along the positive $z$ axis, what is the direction of $(a) \overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}}$ and (b) $\overrightarrow{\mathbf{B}} \times \overrightarrow{\mathbf{A}}$ ? (c) What is the magnitude of $\overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}}$ and $\overrightarrow{\mathbf{B}} \times \overrightarrow{\mathbf{A}} ?$

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 21

Show that $(a) \hat{\mathbf{i}} \times \hat{\mathbf{i}}=\hat{\mathbf{j}} \times \hat{\mathbf{j}}=\hat{\mathbf{k}} \times \hat{\mathbf{k}}=0$ (b) $\hat{\mathbf{i}} \times \hat{\mathbf{j}}=\hat{\mathbf{k}}$ $\hat{\mathbf{i}} \times \hat{\mathbf{k}}=-\hat{\mathbf{j}},$ and $\hat{\mathbf{j}} \times\hat{\mathbf{k}}=\hat{\mathbf{i}}$

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 22

The directions of vectors $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$ are given below for several cases. For each case, state the direction of $\overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}}$.
(a) $\overrightarrow{\mathbf{A}}$ points east, $\overrightarrow{\mathbf{B}}$ points south. $(b) \overrightarrow{\mathbf{A}}$ points east, $\overrightarrow{\mathbf{B}}$ points straight down. ( $c$ ) $\overrightarrow{\mathbf{A}}$ points straight up, $\overrightarrow{\mathbf{B}}$ points north.
(d) $\overrightarrow{\mathbf{A}}$ points straight up, $\overrightarrow{\mathbf{B}}$ points straight down.

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 23

What is the angle $\theta$ between two vectors $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$ if $|\overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}}|=\overrightarrow{\mathbf{A}} \cdot \overrightarrow{\mathbf{B}} ?$

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 24

A particle is located at $\overrightarrow{\mathbf{r}}=(4.0 \hat{\mathbf{i}}+3.5 \hat{\mathbf{j}}+6.0 \hat{\mathbf{k}}) \mathrm{m} .$
A force $\overrightarrow{\mathbf{F}}=(9.0 \hat{\mathbf{j}}-4.0 \hat{\mathbf{k}}) \mathbf{N}$ acts on it. What is the torque,
calculated about the origin?

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 25

Consider a particle of a rigid object rotating about a fixed axis. Show that the tangential and radial vector components of the linear acceleration are:
$$ \overrightarrow{\mathbf{a}}_{\tan }=\overrightarrow{\boldsymbol{\alpha}} \times \overrightarrow{\mathbf{r}} \quad \text { and } \quad \overrightarrow{\mathbf{a}}_{\mathrm{R}}=\overrightarrow{\boldsymbol{\omega}} \times \overrightarrow{\mathbf{v}}
$$

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 26

(II) (a) Show that the cross product of two vectors,
$$
\begin{array}{c}
\overrightarrow{\mathbf{A}}=A_{x} \hat{\mathbf{i}}+A_{y} \hat{\mathbf{j}}+A_{z} \hat{\mathbf{k}}, \text { and } \overrightarrow{\mathbf{B}}=B_{x} \hat{\mathbf{i}}+B_{y} \hat{\mathbf{j}}+B_{z} \hat{\mathbf{k}} \text { is } \\
\overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}}=\left(A_{y} B_{z}-A_{z} B_{y}\right) \hat{\mathbf{i}}+\left(A_{z} B_{x}-A_{x} B_{z}\right) \hat{\mathbf{j}} \\
+\left(A_{x} B_{y}-A_{y} B_{x}\right) \hat{\mathbf{k}}
\end{array}
$$
(b) Then show that the cross product can be written
$$
\overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}}=\left|\begin{array}{ccc}
\hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\
A_{x} & A_{y} & A_{z} \\
B_{x} & B_{y} & B_{z}
\end{array}\right|
$$
where we use the rules for evaluating a determinant. (Note, however, that this is not really a determinant, but a memory aid.)

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 27

(II) An engineer estimates that under the most adverse expected weather conditions, the total force on the highway sign in Fig. $\quad 11-32$
be $\overrightarrow{\mathbf{F}}=(\pm 2.4 \hat{\mathbf{i}}-4.1 \hat{\mathbf{j}}) \mathbf{k N},$
acting at the CM. What torque does this force exert about the base $\mathrm{O} ?$

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 29

Use the result of Problem 26 to determine (a) the vector product $\overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}}$ and $(b)$ the angle between $\overrightarrow{\mathbf{A}}$ and $\overrightarrow{\mathbf{B}}$ if $\overrightarrow{\mathbf{A}}=5.4 \hat{\mathbf{i}}-3.5 \hat{\mathbf{j}}$ and $\overrightarrow{\mathbf{B}}=-8.5 \hat{\mathbf{i}}+5.6 \hat{\mathbf{j}}+2.0 \hat{\mathbf{k}}$

James Wait

Numerade Educator

Problem 30

Show that the velocity $\overrightarrow{\mathbf{v}}$ of any point in an object rotating with angular velocity $\overrightarrow{\boldsymbol{\omega}}$ about a fixed axis can be written
$$
\overrightarrow{\mathbf{v}}=\overrightarrow{\boldsymbol{\omega}} \times \overrightarrow{\mathbf{r}}
$$
where $\overrightarrow{\mathbf{r}}$ is the position vector of the point relative to an origin O located on the axis of rotation. Can O be anywhere on the rotation axis? Will $\overrightarrow{\mathbf{v}}=\overrightarrow{\boldsymbol{\omega}} \times \overrightarrow{\mathbf{r}}$ if $\mathrm{O}$ is located at a point not on the axis of rotation?

Lainey Roebuck

Numerade Educator

Problem 31

Let $\overrightarrow{\mathbf{A}}, \overrightarrow{\mathbf{B}},$ and $\overrightarrow{\mathbf{C}}$ be three vectors, which for generality we assume do not all lie in the same plane. Show that $\overrightarrow{\mathbf{A}} \cdot(\overrightarrow{\mathbf{B}} \times \overrightarrow{\mathbf{c}})=\overrightarrow{\mathbf{B}} \cdot(\overrightarrow{\mathbf{C}} \times \overrightarrow{\mathbf{A}})=\overrightarrow{\mathbf{C}} \cdot(\overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}})$

Lainey Roebuck

Numerade Educator

Problem 32

What are the $x, y,$ and $z$ components of the angular momentum of a particle located at $\overrightarrow{\mathbf{r}}=x \hat{\mathbf{i}}+y \hat{\mathbf{j}}+z \hat{\mathbf{k}}$
which has momentum $\overrightarrow{\mathbf{p}}=p_{x} \hat{\mathbf{i}}+p_{y} \hat{\mathbf{j}}+p_{z} \hat{\mathbf{k}}$ ?

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 33

Show that the kinetic energy $K$ of a particle of mass $m$, moving in a circular path, is $K=L^{2} / 2 I,$ where $L$ is its angular momentum and $I$ is its moment of inertia about the center of the circle.

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 34

Calculate the angular momentum of a particle of mass $m$ moving with constant velocity $v$ for two cases (see Fig. $11-33$ ):
(a) about origin O, and $(b)$ about $\mathrm{O}^{\prime}$

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 35

Two identical particles have equal but opposite momenta, $\overrightarrow{\mathbf{p}}$ and $-\overrightarrow{\mathbf{p}},$ but they are not traveling along the same line. Show that the total angular momentum of this system does not depend on the choice of origin.

Zhaojie Xu

Numerade Educator

Problem 36

Determine the angular momentum of a 75-g particle about the origin of coordinates when the particle is at $x=4.4 \mathrm{~m}$, $y=-6.0 \mathrm{~m},$ and it has velocity $v=(3.2 \hat{\mathbf{i}}-8.0 \hat{\mathbf{k}}) \mathrm{m} / \mathrm{s}$

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 37

A particle is at the position $(x, y, z)=(1.0,2.0,3.0) \mathrm{m} .$ It is traveling with a vector velocity $(-5.0,+2.8,-3.1) \mathrm{m} / \mathrm{s}$. Its mass is $3.8 \mathrm{~kg} .$ What is its vector angular momentum about the origin?

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 38

An Atwood machine (Fig. $11-16$ ) consists of two masses, $m_{\mathrm{A}}=7.0 \mathrm{~kg}$ and $m_{\mathrm{B}}=8.2 \mathrm{~kg},$ connected by a cord that passes over a pulley free to rotate about a fixed axis. The pulley is a solid cylinder of radius $R_{0}=0.40 \mathrm{~m}$ and mass $0.80 \mathrm{~kg} .$ ( $a$ ) Determine the acceleration $a$ of each mass. ( $b$ ) What percentage of error in $a$ would be made if the moment of inertia of the pulley were ignored? Ignore friction in the pulley bearings.

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 40

Two lightweight rods $24 \mathrm{~cm}$ in length are mounted perpendicular to an axle and at $180^{\circ}$ to each other (Fig. $11-34)$. At the end of each rod is a 480 -g mass. The rods are spaced $42 \mathrm{~cm}$ apart along the axle. The axle rotates at $4.5 \mathrm{rad} / \mathrm{s}$. $(a)$ What is the component of the total angular momentum along the axle? (b) What angle does the vector angular momentum make with the axle? [Hint:
Remember that the vector angular momentum must be calculated about the same 480 point for both masses, which could be the CM.

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 41

Figure $11-35$ shows two masses connected by a cord passing over a pulley of radius $R_{0}$ and moment of inertia $I$. Mass $M_{\mathrm{A}}$ slides on a frictionless surface, and $M_{\mathrm{B}}$ hangs freely. Determine a formula for $(a)$ the angular momentum of the system about the pulley axis, as a function of the speed $v$ of mass $M_{\mathrm{A}}$ or $M_{\mathrm{B}}$ and (b) the acceleration of the masses.

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 42

A thin rod of length $\ell$ and mass $M$ rotates about a vertical axis through its center with angular velocity $\omega .$ The rod makes an angle $\phi$ with the rotation axis. Determine the magnitude and direction of $\overrightarrow{\mathbf{L}}$.

Prabhat Tyagi

Numerade Educator

Problem 43

Show that the total angular momentum $\overrightarrow{\mathbf{L}}=\Sigma \overrightarrow{\mathbf{r}}_{i} \times \overrightarrow{\mathbf{p}}_{i}$
of a system of particles about the origin of an inertial reference frame can be written as the sum of the angular momentum about the $\mathrm{CM}, \overrightarrow{\mathbf{L}}^{*}$ (spin angular momentum), plus the angular momentum of the $\mathrm{CM}$ about the origin (orbital angular momentum): $\overrightarrow{\mathbf{L}}=\overrightarrow{\mathbf{L}}^{*}+\overrightarrow{\mathbf{r}}_{\mathrm{CM}} \times M \overrightarrow{\mathbf{v}}_{\mathrm{CM}} .$ [Hint: See
the derivation of Eq. $11-9 \mathrm{~b} .$

Lainey Roebuck

Numerade Educator

Problem 44

What is the magnitude of the force $\overrightarrow{\mathbf{F}}$ exerted by each bearing in Fig. $11-18$ (Example $11-10) ?$ The bearings are a distance $d$ from point O. Ignore the effects of gravity.

Prabhat Tyagi

Numerade Educator

Problem 45

Suppose in Fig. $11-18$ that $m_{\mathrm{B}}=0$; that is, only one mass, $m_{\mathrm{A}}$, is actually present. If the bearings are each a distance $d$ from $\mathrm{O},$ determine the forces $F_{\mathrm{A}}$ and $F_{\mathrm{B}}$ at the upper and lower bearings respectively. [Hint: Choose an origin-different than $\mathrm{O}$ in Fig. $11-18-$ such that $\overrightarrow{\mathbf{L}}$ is parallel to $\vec{\omega}$. Ignore effects of gravity.]

Lainey Roebuck

Numerade Educator

Problem 47

(II) A thin rod of mass $M$ and length $\ell$ is suspended vertically from a frictionless pivot at its upper end. A mass $m$ of putty traveling horizontally with a speed $v$ strikes the rod at its $\mathrm{CM}$ and sticks there. How high does the bottom of the rod swing?

Averell Hause

Carnegie Mellon University

Problem 48

A uniform stick $1.0 \mathrm{~m}$ long with a total mass of $270 \mathrm{~g}$ is pivoted at its center. A 3.0 -g bullet is shot through the stick midway between the pivot and one end (Fig. $11-36$ ). The bullet approaches at $250 \mathrm{~m} / \mathrm{s}$ and leaves at $140 \mathrm{~m} / \mathrm{s}$. With what angular speed is the stick spinning after the collision?

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 49

Suppose a $5.8 \times 10^{10} \mathrm{~kg}$ meteorite struck the Earth at the equator with a speed $\quad v=2.2 \times 10^{4} \mathrm{~m} / \mathrm{s}$
as shown in Fig. $11-37$ and $\begin{array}{llll}\text { remained } & \text { stuck. } & \text { By } & \text { what }\end{array}$ factor would this affect the rotational frequency of the Earth (1 rev/day)?

Prabhat Tyagi

Numerade Educator

Problem 50

A 230 -kg beam $2.7 \mathrm{~m}$ in length slides broadside down the ice with a speed of $18 \mathrm{~m} / \mathrm{s}$ (Fig. 11-38). A 65-kg man at rest grabs one end as it goes past and hangs on as both he and the beam go spinning down the ice. Assume frictionless motion. ( $a$ ) How fast does the center of mass of the system move after the collision? (b) With what angular velocity does the system rotate about its CM?

Lainey Roebuck

Numerade Educator

Problem 51

A thin rod of mass $M$ and length $\ell$ rests on a frictionless table and is struck at a point $\ell / 4$ from its $\mathrm{CM}$ by a clay ball of mass $m$ moving at speed $v$ (Fig. $11-39$ ). The ball sticks to the rod. Determine the translational and rotational motion of the rod after the collision.

Dominador Tan

Numerade Educator

Problem 52

On a level billiards table a cue ball, initially at rest at point $\mathrm{O}$ on the table, is struck so that it leaves the cue stick with a center-of-mass speed $v_{0}$ and a "reverse" spin of angular speed $\omega_{0}$ (see Fig. $\left.11-40\right)$. A kinetic friction force acts on the ball as it initially skids across the table. (a) Explain why the ball's angular momentum is conserved about point $\mathrm{O}$.
(b) Using conservation of angular momentum, find the critical angular speed $\omega_{C}$ such that, if $\omega_{0}=\omega_{C},$ kinetic friction will bring the ball to a complete (as opposed to momentary) stop. $(c)$ If $\omega_{0}$ is $10 \%$ smaller than $\omega_{\mathrm{C}},$ i.e., $\omega_{0}=0.90 \omega_{\mathrm{C}},$ determine the ball's $\mathrm{CM}$ velocity $v_{\mathrm{CM}}$ when it starts to roll without slipping.
(d) If $\omega_{0}$ is $10 \%$ larger than $\omega_{\mathrm{C}},$ i.e., $\omega_{0}=1.10 \omega_{\mathrm{C}},$ determine the ball's CM velocity $v_{\mathrm{CM}}$ when it starts to roll without slipping. [Hint: The ball possesses two types of angular momentum, the first due to the linear speed $v_{\mathrm{CM}}$ of its $\mathrm{CM}$ relative to point $\mathrm{O}$ the second due to the spin at angular velocity $\omega$ about its own $\mathrm{CM}$. The ball's total $L$ about $\mathrm{O}$ is the sum of these two angular momenta.]

Lainey Roebuck

Numerade Educator

Problem 53

A 220-g top spinning at 15 rev/s makes an angle of $25^{\circ}$ to the vertical and precesses at a rate of 1.00 rev per $6.5 \mathrm{~s}$. If its $\mathrm{CM}$ is $3.5 \mathrm{~cm}$ from its tip along its symmetry axis, what is the moment of inertia of the top?

Prabhat Tyagi

Numerade Educator

Problem 54

A toy gyroscope consists of a 170 -g disk with a radius of $5.5 \mathrm{~cm}$ mounted at the center of a thin axle $21 \mathrm{~cm}$ long (Fig. 11-41). The gyroscope spins at $45 \mathrm{rev} / \mathrm{s}$. One end of its axle rests on a stand and the other end precesses horizontally about the stand. (a) How long does it take the gyroscope to precess once around? $(b)$ If all the dimensions of the gyroscope were doubled (radius $=11 \mathrm{~cm},$ axle $=42 \mathrm{~cm})$ how long would it take to precess once?

Prabhat Tyagi

Numerade Educator

Problem 55

Suppose the solid wheel of Fig. $11-41$ has a mass of $300 \mathrm{~g}$ and rotates at $85 \mathrm{rad} / \mathrm{s}$; it has radius $6.0 \mathrm{~cm}$ and is mounted at the center of a horizontal thin axle $25 \mathrm{~cm}$ long. At what rate does the axle precess?

Prabhat Tyagi

Numerade Educator

Problem 56

If a mass equal to half the mass of the wheel in Problem 55 is placed at the free end of the axle, what will be the precession rate now? Treat the extra mass as insignificant in size.

Prabhat Tyagi

Numerade Educator

Problem 58

If a plant is allowed to grow from seed on a rotating platform, it will grow at an angle, pointing inward. Calculate what this angle will be (put yourself in the rotating frame) in terms of $g, r,$ and $\omega$. Why does it grow inward rather than outward?

Prabhat Tyagi

Numerade Educator

Problem 59

Let $\overrightarrow{\mathrm{g}}^{\prime}$ be the effective acceleration of gravity at a point on the rotating Earth, equal to the vector sum of the "true" value $\overrightarrow{\mathrm{g}}$ plus the effect of the rotating reference frame $\left(m \omega^{2} r\right.$ term). See Fig. 11-42. Determine the magnitude and direction of $\overrightarrow{\mathbf{g}}^{\prime}$ relative to a radial line from the center of the Earth $(a)$ at the North Pole, $(b)$ at a latitude of $45.0^{\circ}$ north, and $(c)$ at the equator. Assume that $g$ (if $\omega$ were zero) is a constant $9.80 \mathrm{~m} / \mathrm{s}^{2}$

Lainey Roebuck

Numerade Educator

Problem 60

Suppose the man at $\mathrm{B}$ in Fig. $11-26$ throws the ball toward the woman at A. ( $a$ ) In what direction is the ball deflected as seen in the noninertial system? (b) Determine a formula for the amount of deflection and for the (Coriolis) acceleration in this case.

Lainey Roebuck

Numerade Educator

Problem 61

For what directions of velocity would the Coriolis effect on an object moving at the Earth's equator be zero?

Prabhat Tyagi

Numerade Educator

Problem 62

We can alter Eqs. $11-14$ and $11-15$ for use on Earth by considering only the component of $\overrightarrow{\mathbf{v}}$ perpendicular to the axis of rotation. From Fig. $11-43,$ we see that this is $v \cos \lambda$ for a vertically falling object, where $\lambda$ is the latitude of the place on the Earth. If a lead ball is dropped vertically from a 110-m-high tower in Florence, Italy (latitude $\left.=44^{\circ}\right),$ how far from the base of the tower is it deflected by the Coriolis force?

Lainey Roebuck

Numerade Educator

Problem 64

A thin string is wrapped around a cylindrical hoop of radius $R$ and mass $M .$ One end of the string is fixed, and the hoop is allowed to fall vertically, starting from rest, as the string unwinds. ( $a$ ) Determine the angular momentum of the hoop about its $\mathrm{CM}$ as a function of time. $(b)$ What is the tension in the string as function of time?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 65

A particle of mass $1.00 \mathrm{~kg}$ is moving with velocity $\overrightarrow{\mathbf{v}}=(7.0 \hat{\mathbf{i}}+6.0 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s} . \quad(a)$ Find the angular momentum
$\overrightarrow{\mathbf{L}}$ relative to the origin when the particle is at $\overrightarrow{\mathbf{r}}=(2.0 \hat{\mathbf{j}}+4.0 \hat{\mathbf{k}}) \mathrm{m} .(b)$ At position $\overrightarrow{\mathbf{r}}$ a force of $\overrightarrow{\mathbf{F}}=4.0 \mathrm{Ni} \hat{\mathbf{i}}$
is applied to the particle. Find the torque relative to the origin.

Lainey Roebuck

Numerade Educator

Problem 66

A merry-go-round with a moment of inertia equal to $1260 \mathrm{~kg} \cdot \mathrm{m}^{2}$ and a radius of $2.5 \mathrm{~m}$ rotates with negligible friction at $1.70 \mathrm{rad} / \mathrm{s}$. A child initially standing still next to the merry-go-round jumps onto the edge of the platform straight toward the axis of rotation causing the platform to slow to $1.25 \mathrm{rad} / \mathrm{s}$. What is her mass?

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 67

Why might tall narrow SUVs and buses be prone to "rollover"? Consider a vehicle rounding a curve of radius $R$ on a flat road. When just on the verge of rollover, its tires on the inside of the curve are about to leave the ground, so the friction and normal force on these two tires are zero. The total normal force on the outside tires is $F_{\mathrm{N}}$ and the total friction force is $F_{\mathrm{fr}}$. Assume that the vehicle is not skidding.
(a) Analysts define a static stability factor SSF $=w / 2 h$ where a vehicle's "track width" $w$ is the distance between tires on the same axle, and $h$ is the height of the CM above the ground. Show that the critical rollover speed is(b) Determine the ratio of highway curve radii (minimum possible) for a typical passenger car with $\mathrm{SSF}=1.40$ and an SUV with $\mathrm{SSF}=1.05$ at a speed of $90 \mathrm{~km} / \mathrm{h}$.
$$
v_{\mathrm{C}}=\sqrt{R g\left(\frac{w}{2 h}\right)}
$$

Lainey Roebuck

Numerade Educator

Problem 68

A spherical asteroid with radius $r=123 \mathrm{~m}$ and mass $M=2.25 \times 10^{10} \mathrm{~kg}$ rotates about an axis at four revolutions per day. A "tug" spaceship attaches itself to the asteroid's south pole (as defined by the axis of rotation) and fires its engine, applying a force $F$ tangentially to the asteroid's surface as shown in Fig. $11-44 .$ If $F=265 \mathrm{~N},$ how long will it take the tug to rotate the asteroid's axis of rotation through an angle of $10.0^{\circ}$ by this method?

Lainey Roebuck

Numerade Educator

Problem 69

The time-dependent position of a point object which moves counterclockwise along the circumference of a circle (radius $R$ ) in the $x y$ plane with constant speed $v$ is given by
$$
\overrightarrow{\mathbf{r}}=\hat{\mathbf{i}} R \cos \omega t+\hat{\mathbf{j}} R \sin \omega t
$$
where the constant $\omega=v / R .$ Determine the velocity $\overrightarrow{\mathbf{v}}$ and angular velocity $\overrightarrow{\boldsymbol{\omega}}$ of this object and then show that these three vectors obey the relation $\overrightarrow{\mathbf{v}}=\overrightarrow{\boldsymbol{\omega}} \times \overrightarrow{\mathbf{r}}$.

Zhaojie Xu

Numerade Educator

Problem 70

The position of a particle with mass $m$ traveling on a helical path (see Fig. $11-45$ ) is given by
$$
\overrightarrow{\mathbf{r}}=R \cos \left(\frac{2 \pi z}{d}\right) \hat{\mathbf{i}}+R \sin \left(\frac{2 \pi z}{d}\right) \hat{\mathbf{j}}+z \hat{\mathbf{k}}
$$
where $R$ and $d$ are the radius and pitch of the helix, respectively, and $z$ has time dependence $z=v_{z} t$ where $v_{z}$ is the (constant) component of velocity in the $z$ direction. Determine the time-dependent angular momentum $\overrightarrow{\mathbf{L}}$ of the particle about the origin.

Zhaojie Xu

Numerade Educator

Problem 71

A boy rolls a tire along a straight level street. The tire has mass $8.0 \mathrm{~kg}$, radius $0.32 \mathrm{~m}$ and moment of inertia about its central axis of symmetry of $0.83 \mathrm{~kg} \cdot \mathrm{m}^{2} .$ The boy pushes the tire forward away from him at a speed of $2.1 \mathrm{~m} / \mathrm{s}$ and sees that the tire leans $12^{\circ}$ to the right (Fig. $11-46$ ). ( $a$ ) How will the resultant torque affect the subsequent motion of the tire? $(b)$ Compare the change in angular momentum caused by this torque in $0.20 \mathrm{~s}$ to the original magnitude of angular momentum.

Averell Hause

Carnegie Mellon University

Problem 73

Water drives a waterwheel (or turbine) of radius $R=3.0 \mathrm{~m}$ as shown in Fig. $11-47 .$ The water enters at a speed $v_{1}=7.0 \mathrm{~m} / \mathrm{s}$ and exits from the waterwheel at a speed $v_{2}=3.8 \mathrm{~m} / \mathrm{s} . \quad(a)$ If $85 \mathrm{~kg}$ of water passes through per second, what is the rate at which the water delivers angular momentum to the waterwheel? $(b)$ What is the torque the water applies to the waterwheel? $(c)$ If the water causes the waterwheel to make one revolution every $5.5 \mathrm{~s}$, how much power is delivered to the wheel?

Dominador Tan

Numerade Educator

Problem 74

The Moon orbits the Earth such that the same side always faces the Earth. Determine the ratio of the Moon's spin angular momentum (about its own axis) to its orbital angular momentum. (In the latter case, treat the Moon as a particle orbiting the Earth.)

Jayashree Behera

Jayashree Behera

Numerade Educator

Problem 75

A particle of mass $m$ uniformly accelerates as it moves counterclockwise along the circumference of a circle of radius $R$ :
$$
\overrightarrow{\mathbf{r}}=\hat{\mathbf{i}} R \cos \theta+\hat{\mathbf{j}} R \sin \theta
$$
with $\theta=\omega_{0} t+\frac{1}{2} \alpha t^{2},$ where the constants $\omega_{0}$ and $\alpha$ are the initial angular velocity and angular acceleration, respectively. Determine the object's tangential acceleration $\overrightarrow{\mathbf{a}}_{\tan }$ and determine the torque acting on the object using
$(a) \vec{\tau}=\overrightarrow{\mathbf{r}} \times \overrightarrow{\mathbf{F}}$
(b) $\vec{\tau}=I \vec{\alpha}$

Averell Hause

Carnegie Mellon University

Problem 76

A projectile with mass $m$ is launched from the ground and follows a trajectory given by $$ \overrightarrow{\mathbf{r}}=\left(v_{x 0} t\right) \hat{\mathbf{i}}+\left(v_{y 0} t-\frac{1}{2} g t^{2}\right) \hat{\mathbf{j}} $$ where $v_{x 0}$ and $v_{y 0}$ are the initial velocities in the $x$ and $y$ direction, respectively, and $g$ is the acceleration due to gravity. The launch position is defined to be the origin. Determine the torque acting on the projectile about the origin using $(a) \vec{\tau}=\overrightarrow{\mathbf{r}} \times \overrightarrow{\mathbf{F}},$ (b) $\overrightarrow{\boldsymbol{\tau}}=\dot{d \overrightarrow{\mathbf{L}}} / d t$

Lainey Roebuck

Numerade Educator

Problem 77

Most of our Solar System's mass is contained in the Sun, and the planets possess almost all of the Solar System's angular momentum. This observation plays a key role in theories attempting to explain the formation of our Solar System. Estimate the fraction of the Solar System's total angular momentum that is possessed by planets using a simplified model which includes only the large outer planets with the most angular momentum. The central Sun (mass $1.99 \times 10^{30} \mathrm{~kg}$, radius $6.96 \times 10^{8} \mathrm{~m}$ ) spins about its axis once every 25 days and the planets Jupiter, Saturn, Uranus, and Neptune move in nearly circular orbits around the Sun with orbital data given in the Table below. Ignore each planet's spin about its own axis.

Averell Hause

Carnegie Mellon University

Problem 78

A bicyclist traveling with speed $v=9.2 \mathrm{~m} / \mathrm{s}$ on a flat road is making a turn with a radius $r=12 \mathrm{~m}$. The forces acting on the cyclist and cycle are the normal force $\left(\overrightarrow{\mathbf{F}}_{\mathrm{N}}\right)$ and friction force $\left(\overrightarrow{\mathbf{F}}_{\mathrm{fr}}\right)$ exerted by the road on the tires and $m \overrightarrow{\mathrm{g}},$ the total weight of the cyclist and cycle. Ignore the small mass of the wheels. (a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. $11-48)$ must be given by $\tan \theta=F_{\mathrm{fr}} / F_{\mathrm{N}}$ if the cyclist is to maintain balance.
(b) Calculate $\theta$ for the values given. [Hint: Consider the "circular" translational motion of the bicycle and rider.] $(c)$ If the coefficient of static friction between tires and road is $\mu_{\mathrm{s}}=0.65,$ what is the minimum turning radius?

Lainey Roebuck

Numerade Educator

Problem 79

Competitive ice skaters commonly perform single, double, and triple axel jumps in which they rotate $1 \frac{1}{2}, 2 \frac{1}{2},$ and $3 \frac{1}{2}$ revolutions, respectively, about a vertical axis while airborne. For all these jumps, a typical skater remains airborne for about 0.70 s. Suppose a skater leaves the ground in an "open" position (e.g., arms outstretched) with moment of inertia $I_{0}$ and rotational frequency $f_{0}=1.2 \mathrm{rev} / \mathrm{s},$ maintaining this position for $0.10 \mathrm{~s}$. The skater then assumes a "closed" position (arms brought closer) with moment of inertia I, acquiring a rotational frequency $f,$ which is maintained for 0.50 s. Finally, the skater immediately returns to the "open" position for 0.10 s until landing (see Fig. $11-49$ ).
(a) Why is angular momentum conserved during the skater's jump? Neglect air resistance. (b) Determine the minimum rotational frequency $f$ during the flight's middle section for the skater to successfully complete a single and a triple axel.
(c) Show that, according to this model, a skater must be able to reduce his or her moment of inertia in midflight by a factor of about 2 and 5 in order to complete a single and triple axel, respectively.

Lainey Roebuck

Numerade Educator

Problem 80

A radio transmission tower has a mass of $80 \mathrm{~kg}$ and is $12 \mathrm{~m}$ high. The tower is anchored to the ground by a flexible joint at its base, but it is secured by three cables $120^{\circ}$ apart (Fig. $11-50)$. In an analysis of a potential failure, a mechanical engineer needs to determine the behavior of the tower if one of the cables broke. The tower would fall away from the broken cable, rotating about its base. Determine the speed of the top of the tower as a function of the rotation angle $\theta$. Start your analysis with the rotational dynamics equation of motion $d \overrightarrow{\mathbf{L}} / d t=\overrightarrow{\boldsymbol{\tau}}_{\text {net }}$. Approximate the tower as a tall thin rod.

Lainey Roebuck

Numerade Educator

Problem 81

Suppose a star the size of our Sun, but with mass 8.0 times as great, were rotating at a speed of 1.0 revolution every 9.0 days. If it were to undergo gravitational collapse to a neutron star of radius $12 \mathrm{~km}$, losing $\frac{3}{4}$ of its mass in the process, what would its rotation speed be? Assume the star is a uniform sphere at all times. Assume also that the thrown-off mass carries off either $(a)$ no angular momentum, or $(b)$ its proportional share $\left(\frac{3}{4}\right)$ of the initial angular momentum.

Averell Hause

Carnegie Mellon University

Problem 82

A baseball bat has a "sweet spot" where a ball can be hit with almost effortless transmission of energy. A careful analysis of baseball dynamics shows that this special spot is located at the point where an applied force would result in pure rotation of the bat about the handle grip. Determine the location of the sweet spot of the bat shown in Fig. $11-51 .$ The linear mass density of the bat is given roughly by $\left(0.61+3.3 x^{2}\right) \mathrm{kg} / \mathrm{m},$ where $x$ is in meters measured from the end of the handle. The entire bat is $0.84 \mathrm{~m}$ long. The desired rotation point should be $5.0 \mathrm{~cm}$ from the end where the bat is held. [Hint: Where is the CM of the bat?]

Lainey Roebuck

Numerade Educator

Problem 83

A uniform stick $1.00 \mathrm{~m}$ long with a total mass of $330 \mathrm{~g}$ is pivoted at its center. A 3.0 -g bullet is shot through the stick a distance $x$ from the pivot. The bullet approaches at $250 \mathrm{~m} / \mathrm{s}$ and leaves at $140 \mathrm{~m} / \mathrm{s}$ (Fig. $11-36$ ). (a) Determine a formula for the angular speed of the spinning stick after the collision as a function of $x$. (b) Graph the angular speed as a function of $x,$ from $x=0$ to $x=0.50 \mathrm{~m}$

Lainey Roebuck

Numerade Educator

Problem 84

Figure $11-39$ shows a thin rod of mass $M$ and length $\ell$ resting on a frictionless table. The rod is struck at a distance $x$ from its CM by a clay ball of mass $m$ moving at speed $v$. The ball sticks to the rod. (a) Determine a formula for the rotational motion of the system after the collision. (b) Graph the rotational motion of the system as a function of $x,$ from $x=0$ to $x=\ell / 2,$ with values of $M=450 \mathrm{~g}, \quad m=15 \mathrm{~g}$
$\ell=1.20 \mathrm{~m},$ and $v=12 \mathrm{~m} / \mathrm{s} .$ (c) Does the translational
motion depend on $x ?$ Explain.

Lainey Roebuck

Numerade Educator