Understanding Physics

Karen Cummings, Priscilla W. Laws, Edward F. Redish

Chapter 12

Complex Rotations - all with Video Answers

Educators

Chapter Questions

Problem 1

An automobile traveling $80.0 \mathrm{~km} / \mathrm{h}$ has tires of $75.0 \mathrm{~cm}$ diameter. (a) What is the rotational speed of the tires about their axles? (b) If the car is brought to a stop uniformly in $30.0$ complete turns of the tires (without skidding), what is the magnitude of the rotational acceleration
of the wheels? (c) How far does the car move during the braking?

Stephen Zaffke

Stephen Zaffke

Numerade Educator

Problem 2

Consider a $66-\mathrm{cm}$ -diameter tire on a car traveling at 80 $\mathrm{km} / \mathrm{h}$ on a level road in the positive direction of an $x$ axis. Relative to a woman in the car, what are (a) the translational velocity $\vec{v}_{\text {center }}$ and
(b) the magnitude $a_{\text {center }}$ of the translational acceleration of the center of the wheel? What are (c) $\vec{v}_{\text {top }}$ and (d) $a_{\text {top }}$ for a point at the top of the tire? What are (e) $\vec{v}_{\text {bot }}$ and (f) $a_{\text {bot }}$ for a point at the bottom of the tire?
Now repeat the questions relative to a hitchhiker sitting near the road: What are $(\mathrm{g}) \vec{v}$ at the wheel's center, $(\mathrm{h}) a$ at the wheel's center, (i) $\vec{v}$ at the tire top, (j) $a$ at the tire top, $(\mathrm{k}) \vec{v}$ at the tire bottom, and (1) $a$ at the tire bottom?

Stephen Zaffke

Numerade Educator

Problem 3

A $140 \mathrm{~kg}$ hoop rolls along a horizontal floor so that its center of mass has a speed of $0.150 \mathrm{~m} / \mathrm{s}$. How much work must be done on the hoop to stop it?

Stephen Zaffke

Numerade Educator

Problem 4

A thin-walled pipe rolls along the floor. What is the ratio of its translational kinetic energy to its rotational kinetic energy about an axis parallel to its length and through its center of mass?

Stephen Zaffke

Numerade Educator

Problem 5

A $1000 \mathrm{~kg}$ car has four $10 \mathrm{~kg}$ wheels. When the car is moving, what fraction of the total kinetic energy of the car is due to rotation of the wheels about their axles? Assume that the wheels have the same rotational inertia as uniform disks of the same mass and size. Why do you not need the radius of the wheels?

Stephen Zaffke

Numerade Educator

Problem 6

A body of radius $R$ and mass $m$ is rolling smoothly with speed $v$ on a horizontal surface. It then rolls up a hill to a maximum height $h$. (a) If $h=3 v^{2} / 4 g$, what is the body's rotational inertia about the rotational axis through its center of mass?
(b) What might the body be?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 7

A uniform solid sphere rolls down an incline. (a) What must be the incline angle if the translational acceleration of the center of the sphere is to have a magnitude of $0.10 \mathrm{~g}$ ?
(b) If a frictionless block were to slide down the incline at that angle, would its acceleration magnitude be more than, less than, or equal to $0.10 g$ ? Why?

Stephen Zaffke

Numerade Educator

Problem 8

A hollow sphere of radius $0.15 \mathrm{~m}$, with rotational inertia $I=0.040 \mathrm{~kg} \cdot \mathrm{m}^{2}$ about a line through its center of mass, rolls without slipping up a surface inclined at $30^{\circ}$ to the horizontal. At a certain initial position, the sphere's total kinetic energy is $20 \mathrm{~J}$. (a) How much of this initial kinetic energy is rotational? (b) What is the speed of the center of mass of the sphere at the initial position? What are (c) the total kinetic energy of the sphere and (d)) the speed of its center of mass after it has moved $1.0 \mathrm{~m}$ up along the incline from its initial position?

Stephen Zaffke

Numerade Educator

Problem 9

A yo-yo has a rotational inertia of $950 \mathrm{~g} \cdot \mathrm{cm}^{2}$ and a mass of $120 \mathrm{~g}$. Its axle radius is $3.2 \mathrm{~mm}$, and its string is $120 \mathrm{~cm}$ long. The yo-yo rolls from rest down to the end of the string. (a) What is the magnitude of its translational acceleration? (b) How long does it take to reach the end of the string? As it reaches the end of the string, what are its (c) translational speed, (d) translational kinetic energy, (e) rotational kinetic energy, and (f) rotational speed?

Stephen Zaffke

Numerade Educator

Problem 10

Suppose that the yo-yo in Problem 9, instead of rolling from rest, is thrown so that its initial speed down the string is $1.3 \mathrm{~m} / \mathrm{s}$. (a) How long does the yo-yo take to reach the end of the string? As it reaches the end of the string, what are its
(b) total kinetic energy, (c) translational speed,
(d) translational kinetic energy, (e) rotational speed, and (f) rotational kinetic energy?

Stephen Zaffke

Numerade Educator

Problem 11

Show that the area of the triangle contained between $\vec{a}$ and $\vec{b}$ and the solid line connecting their tips in Fig. $12-22$ is $\frac{1}{2}|\vec{a} \times \vec{b}|$.

Stephen Zaffke

Numerade Educator

Problem 12

The Product In the product $\vec{F}=q \vec{v} \times \vec{B}$, take $q=2$,
$$
\vec{v}=2.0 \hat{\hat{i}}+4.0 \hat{\mathrm{j}}+6.0 \hat{\mathrm{k}}
$$
and
$$
\vec{F}=4.0 \hat{\mathrm{i}}-20 \hat{\mathrm{j}}+12 \hat{\mathrm{k}}
$$
What then is $\vec{B}$ in unit-vector notation if $B_{x}=B_{y} ?$

Derek Walkama

Numerade Educator

Problem 13

(A) Show that $\vec{a} \cdot(\vec{b} \times \vec{a})$ is zero for all vectors $\vec{a}$ and $\vec{b}$. (b) What is the magnitude of $\vec{a} \times(\vec{b} \times \vec{a})$ if there is an angle $\phi$ between the directions of $\vec{a}$ and $\vec{b}$ ?

Stephen Zaffke

Numerade Educator

Problem 14

For the following three vectors, what is $3 \vec{C}$.
$(2 \vec{A} \times \vec{B}) ?$
$$
\begin{array}{l}
\vec{A}=2.00 \hat{\mathrm{i}}+3.00 \hat{\mathrm{j}}-4.00 \hat{\mathrm{k}} \\
\vec{B}=-3.00 \hat{\mathrm{i}}+4.00 \hat{\mathrm{j}}+2.00 \hat{\mathrm{k}} \\
\overrightarrow{\boldsymbol{C}}=7.00 \hat{\mathrm{i}}-8.00 \hat{\mathrm{j}}
\end{array}
$$

Derek Walkama

Numerade Educator

Problem 15

Show that, if $\vec{r}$ and $\vec{F}$ lie in a given plane, the torque $\vec{\tau}=\vec{r} \times \vec{F}$ has no component in that plane.

Supratim Pal

Numerade Educator

Problem 16

What are the magnitude and direction of the torque about the origin on a plum located at coordinates $(-2.0,0.0,4.0) \mathrm{m}$ due to force $\vec{F}$ whose only component is (a) $F_{x}=6.0 \mathrm{~N}$, (b) $F_{x}=$ $-6.0 \mathrm{~N},(\mathrm{c}) F_{z}=6.0 \mathrm{~N}$, and $(\mathrm{d}) F_{z}=-6.0 \mathrm{~N} ?$

Stephen Zaffke

Numerade Educator

Problem 17

What are the magnitude and direction of the torque about the origin on a particle located at coordinates $(0.0,-4.0,3.0) \mathrm{m}$ due to (a) force $\vec{F}_{A}$ with components $F_{A x}=2.0 \mathrm{~N}$ and $F_{A y}=F_{A z}=0$, and $(\mathrm{b})$ force $\vec{F}_{B}$ with components $F_{B x}=0$, $F_{B v}=2.0 \mathrm{~N}$, and $F_{B z}=4.0 \mathrm{~N} ?$

Stephen Zaffke

Numerade Educator

Problem 18

Force $\vec{F}=(2.0 \mathrm{~N}) \hat{\mathrm{i}}-(3.0 \mathrm{~N}) \hat{\mathrm{k}}$ acts on a pebble with
position vector $\vec{r}=(0.50 \mathrm{~m}) \hat{j}-(2.0 \mathrm{~m}) \hat{\mathrm{k}}$, relative to the origin. What is the resulting torque acting on the pebble about (a) the origin and (b) a point with coordinates $(2.0,0.0,-3.0) \mathrm{m}$ ?

Stephen Zaffke

Numerade Educator

Problem 19

Force $\vec{F}=(-8.0 \mathrm{~N}) \hat{\mathrm{i}}+(6.0 \mathrm{~N}) \hat{\mathrm{j}}$ acts on a
particle with position vector $\vec{r}=(3.0 \mathrm{~m}) \hat{\mathrm{i}}+(4.0 \mathrm{~m}) \mathrm{j} .$ What are (a) the torque on the particle about the origin and (b) the angle between the directions of $\vec{r}$ and $\vec{F} ?$

Stephen Zaffke

Numerade Educator

Problem 20

What is the torque about the origin on a jar of jalapeño peppers located at coordinates $(3.0 \mathrm{~m},-2.0 \mathrm{~m}, 4.0 \mathrm{~m})$ due to
(a) force $\vec{F}_{A}=(3.0 \mathrm{~N}) \hat{\mathrm{i}}-(4.0 \mathrm{~N}) \hat{\mathrm{j}}+(5.0 \mathrm{~N}) \hat{\mathrm{k}},(\mathrm{b})$ force $\vec{F}_{B}=$
$(-3.0 \mathrm{~N}) \hat{\mathrm{i}}-(4.0 \mathrm{~N}) \hat{\mathrm{j}}-(5.0 \mathrm{~N}) \hat{\mathrm{k}}$, and $(\mathrm{c})$ the vector sum of $\vec{F}_{A}$ and
$\vec{F}_{B} ?$ (d) Repeat part (c) about a point with coordinates $(3.0 \mathrm{~m},$, $2.0 \mathrm{~m}, 4.0 \mathrm{~m}$ ) instead of about the origin.

Stephen Zaffke

Numerade Educator

Problem 21

A $3.0 \mathrm{~kg}$ particle with velocity $\vec{v}=$ $(5.0 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}-(6.0 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}$ is at $x=3.0 \mathrm{~m}, y=8.0 \mathrm{~m} .$ It is pulled by a
$7.0 \mathrm{~N}$ force in the negative $x$ direction. (a) What is the rotational momentum of the particle about the origin? (b) What torque about the origin acts on the particle? (c) At what rate is the rotational momentum of the particle changing with time?

Stephen Zaffke

Numerade Educator

Problem 22

A particle is acted on by two torques about the origin: $\vec{\tau}_{1}$ has a magnitude of $2.0 \mathrm{~N} \cdot \mathrm{m}$ and is directed in the positive direction of the $x$ axis, and $\vec{\tau}_{2}$ has a magnitude of
$4.0 \mathrm{~N} \cdot \mathrm{m}$ and is directed in the negative direction of the $y$ axis. What are the magnitude and direction of $d \vec{\ell} / d t$, where $\vec{\ell}$ is the rotational momentum of the particle about the origin?

Stephen Zaffke

Numerade Educator

Problem 23

What torque about the origin acts on a particle moving in the $x y$ plane, clockwise about the origin, if the particle has the following magnitudes of rotational momentum about the origin:
(a) $4.0 \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}$
(b) $\left(4.0 \frac{1}{s}\right) t^{2} \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}$
(c) $\left(4.0 \frac{1}{s^{1}}\right) \sqrt{t} \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}$
(d) $\left(4.0 \mathrm{~s}^{2}\right) / t^{2} \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}$ ?

Stephen Zaffke

Numerade Educator

Problem 24

At time $t=0$, a $2.0 \mathrm{~kg}$ particle has position vector $\vec{r}=(4.0 \mathrm{~m}) \hat{\mathrm{i}}-(2.0 \mathrm{~m}) \hat{\mathrm{j}}$ relative to the origin. Its velocity just then
is given by $\vec{v}=\left(-6.0 \mathrm{~m} / \mathrm{s}^{3}\right) t^{2} \hat{\mathrm{i}}$. About the origin and for $t>0$, what are (a) the particle's rotational momentum and (b) the torque acting on the particle? (c) Repeat (a) and (b) about a point with coordinates $(-2.0,-3.0,0.0) \mathrm{m}$ instead of about the origin.

Stephen Zaffke

Numerade Educator

Problem 25

Two objects are moving as shown in Fig. $12-23$. What is their total rotational momentum about point $O$ ?

Stephen Zaffke

Numerade Educator

Problem 26

A Particle $P$ In Fig. $12-24$, a particle $P$ with mass $2.0 \mathrm{~kg}$ has position vector $\vec{r}$ of magnitude $3.0 \mathrm{~m}$ and velocity $\vec{v}$ of magnitude $4.0 \mathrm{~m} / \mathrm{s}$. A force $\vec{F}$ of magnitude 2.0 $\mathrm{N}$ acts on the particle. All three vectors lie in the $x y$ plane oriented as shown. About the origin, what are (a) the rotational momentum of the particle and (b) the torque acting on the particle?

Stephen Zaffke

Numerade Educator

Problem 27

At a certain time, a $0.25 \mathrm{~kg}$ object has a position vector $\vec{r}=(2.0 \mathrm{~m}) \hat{\mathrm{i}}+$
$(-2.0 \mathrm{~m}) \hat{\mathrm{y}}$ in meters. At that instant, its velocity in meters per second is $\vec{v}=(-5.0 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}+(5.0 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}$
and the force in newtons acting on it is $\vec{F}=(4.0 \mathrm{~N}) \hat{\mathrm{j}} .$ (a) What is the rotational momentum of the object about the origin? (b) What torque acts on it?

Stephen Zaffke

Numerade Educator

Problem 28

A $2.0 \mathrm{~kg}$ particle-like object moves in a plane with velocity components $v_{x}=30 \mathrm{~m} / \mathrm{s}$ and $v_{y}=60 \mathrm{~m} / \mathrm{s}$ as it passes through the point with $(x, y)$ coordinates of $(3.0,-4.0) \mathrm{m}$. Just then, what is its rotational momentum relative to (a) the origin and (b) the point $(-2.0,-2.0) \mathrm{m}$ ?

Stephen Zaffke

Numerade Educator

Problem 29

Two particles, each of mass $m$ and speed $v$, travel in opposite directions along parallel lines separated by a distance $d$. (a) In terms of $m, v$, and $d$, find an expression for the magnitude $L$ of the rotational momentum of the two-particle system around a point midway between the two lines. (b) Does the expression change if the point about which $L$ is calculated is not midway between the lines? (c) Now reverse the direction of travel for one of the particles and repeat (a) and (b).

Stephen Zaffke

Numerade Educator

Problem 30

A $4.0 \mathrm{~kg}$ particle moves in an $x y$ plane. At the instant when the particle's position and velocity are $\vec{r}=$ $(2.0 \mathrm{~m}) \hat{\mathrm{i}}+(4.0 \mathrm{~m}) \hat{\mathrm{j}}$ and $\vec{v}=(-4.0 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}$, the force on the particle is
$\vec{F}=(-3.0 \mathrm{~N}) \hat{\mathrm{i}}$. At this instant, determine (a) the particle's rotational momentum about the origin, (b) the particle's rotational momentum about the point $x=0, y=4.0 \mathrm{~m},(\mathrm{c})$ the torque acting on the particle about the origin, and (d) the torque acting on the particle about the point $x=0.0 \mathrm{~m}, y=4.0 \mathrm{~m}$.

Stephen Zaffke

Numerade Educator

Problem 31

The rotational momentum of a flywheel having a rotational inertia of $0.140 \mathrm{~kg} \cdot \mathrm{m}^{2}$ about its central axis decreases from $3.00$ to $0.800 \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}$ in $1.50 \mathrm{~s}$. (a) What is the magnitude of the average torque acting on the flywheel about its central axis during this period? (b) Assuming a constant rotational acceleration, through what angle does the flywheel turn? (c) How much work is done on the wheel? (d) What is the average power of the flywheel?

Stephen Zaffke

Numerade Educator

Problem 32

A sanding disk with rotational inertia $1.2 \times$ $10^{-3} \mathrm{~kg} \cdot \mathrm{m}^{2}$ is attached to an electric drill whose motor delivers a torque of $16 \mathrm{~N} \cdot \mathrm{m}$. Find (a) the rotational momentum of the disk about its central axis and (b) the rotational speed of the disk $33 \mathrm{~ms}$ after the motor is turned on.

Stephen Zaffke

Numerade Educator

Problem 33

Three particles, each of mass $m$, are fastened to each other and to a rotation axis at $O$ by three massless strings, each with length $d$ as shown in Fig. $12-25 .$ The combination rotates around the rotational axis with rotational velocity $\omega$ in such a way that the particles remain in a straight line.
In terms of $m, d$, and $\omega$, and relative to point $O$, what are $($ a) the rotational inertia of the combination, (b) the rotational momentum of the middle particle, and (c) the total rotational momentum of the three particles?

Stephen Zaffke

Numerade Educator

Problem 34

An impulsive force $\vec{F}(t)=F_{x}(t) \hat{\mathrm{i}}$ acts for a short time $\Delta t$ on a rotating rigid body constrained to rotate about the $z$ axis with rotational inertia $I$. Show that
$$
\left(\int \tau_{z} d t\right) \hat{\mathrm{k}}=(\mid\langle\vec{F}\rangle R \Delta t) \hat{\mathrm{k}}=I\left(\omega_{2 z}-\omega_{1}\right) \hat{\mathrm{k}}
$$
where $\tau_{z} \hat{k}$ is the torque due to the force, $R$ is the moment arm of the force, $\langle\vec{F}\rangle$ is the average value of the force during the time it acts on the body, and $\omega_{1}, \hat{k}$ and $\omega_{2} z \hat{k}$ are the rotational velocities of the body just before and just after the force acts. (The quantity $\left(\int \tau_{z} d t\right) \hat{\mathrm{k}}=(|\langle\vec{F}\rangle| R \Delta t) \hat{\mathrm{k}}$ is called the rotational impulse, in analogy
with $\langle\vec{F} / \Delta t$, the translational impulse.)

Manish Jain

Numerade Educator

Problem 35

Two cylinders having radii $R_{A}$ and $R_{B}$ and rotational inertias $I_{A}$ and $I_{B}$ about their central axes are supported by axles perpendicular to the plane of Fig. 12-26. The large cylinder is initially rotating clockwise with rotational velocity $\vec{\omega}_{1}$.
The small cylinder is moved to the right until it touches the large cylinder and is caused to rotate by the frictional force between the two. Eventually, slipping ceases, and the two cylinders rotate at constant rates in opposite directions. Find the final rotational velocity $\vec{\omega}_{2}$ of the small cylinder in terms of $I_{A}, I_{B}, R_{A}, R_{B}$, and $\vec{\omega}_{1}$. (Hint: Neither rotational momentum nor kinetic energy is conserved. Apply the rotational impulse equation of Problem 34.)

Manish Jain

Numerade Educator

Problem 36

Figure $12-27$ shows a rigid structure consisting of a circular hoop of radius $R$ and mass $m$, and a square made of four thin bars, each of length $R$ and mass $m$. The rigid structure rotates at a constant speed about a vertical axis, with a period of rotation of $2.5$ s. Assuming $R=0.50 \mathrm{~m}$ and $m$
$=2.0 \mathrm{~kg}$, calculate (a) the structure's rotational inertia about the axis of rotation and (b) its rotational momentum about that axis.

Stephen Zaffke

Numerade Educator

Problem 37

A man stands on a platform that is rotating (without friction) with a rotational speed of $1.2 \mathrm{rev} / \mathrm{s} ;$ his arms are outstretched and he holds a brick in each hand. The rotational inertia of the system consisting of the man, bricks, and platform about the central axis is $6.0 \mathrm{~kg} \cdot \mathrm{m}^{2} .$ If by moving the bricks the man decreases the rotational inertia of the system to $2.0 \mathrm{~kg} \cdot \mathrm{m}^{2}$, (a) what is the resulting rotational speed of the platform and (b) what is the ratio of the new kinetic energy of the system to the original kinetic energy? (c) What provided the added kinetic energy?

Stephen Zaffke

Numerade Educator

Problem 38

The rotor of an electric motor has rotational inertia $I_{m}=2.0 \times 10^{-3} \mathrm{~kg} \cdot \mathrm{m}^{2}$ about its central axis. The motor is used to change the orientation of the space probe in which it is mounted. The motor axis is mounted parallel to the axis of the probe, which has rotational inertia $I_{p}=12 \mathrm{~kg} \cdot \mathrm{m}^{2}$ about its axis. Calculate the number of revolutions of the rotor required to turn the probe through $30^{\circ}$ about its axis.

Stephen Zaffke

Numerade Educator

Problem 39

A wheel is rotating freely at rotational speed 800 rev/min on a shaft whose rotational inertia is negligible. A second wheel, initially at rest and with twice the rotational inertia of the first, is suddenly coupled to the same shaft. (a) What is the rotational speed of the resultant combination of the shaft and two wheels? (b) What fraction of the original rotational kinetic energy is lost?

Stephen Zaffke

Numerade Educator

Problem 40

Two disks are mounted on low-friction bearings on the same axle and can be brought together so that they couple and rotate as one unit. (a) The first disk, with rotational inertia $3.3 \mathrm{~kg} \cdot \mathrm{m}^{2}$ about its central axis, is set spinning at 450 rev/min. The second disk, with rotational inertia $6.6 \mathrm{~kg} \cdot \mathrm{m}^{2}$ about its central axis, is set spinning at 900 rev/min in the same direction as the first. They then couple together. What is their rotational speed after coupling?
(b) If instead the second disk is set spinning at 900 rev/min in the direction opposite the first disk's rotation, what is their rotational speed and direction of rotation after coupling?

Stephen Zaffke

Numerade Educator

Problem 41

In a playground, there is a small merry-go-round of radius $1.20 \mathrm{~m}$ and mass $180 \mathrm{~kg} .$ Its radius of gyration (see Problem 43 of Chapter 11 ) is $91.0 \mathrm{~cm}$. A child of mass $44.0 \mathrm{~kg}$ runs at a speed of $3.00 \mathrm{~m} / \mathrm{s}$ along a path that is tangent to the rim of the initially stationary merry-go-round and then jumps on. Neglect friction between the bearings and the shaft of the merry-go-round. Calculate (a) the rotational inertia of the merry-go-round about its axis of rotation, (b) the magnitude of the rotational momentum of the running child about the axis of rotation of the merry-go-round, and
(c) the rotational speed of the merry-go-round and child after the child has jumped on.

Stephen Zaffke

Numerade Educator

Problem 42

The rotational inertia of a collapsing spinning star changes to $\frac{1}{3}$ its initial value. What is the ratio of the new rotational kinetic energy to the initial rotational kinetic energy?

Stephen Zaffke

Numerade Educator

Problem 43

A track is mounted on a large wheel that is free to turn with negligible friction about a vertical axis (Fig. $12-28) \cdot \mathrm{A}$ toy train of mass $m$ is placed on the track and, with the system initially at rest, the electrical power is turned on. The train reaches a steady speed $v$ with respect to the track. What is the rotational speed of the wheel if its mass is $M$ and its radius is $R ?$ (Treat the wheel as a hoop, and neglect the mass of the spokes and hub.)

Stephen Zaffke

Numerade Educator

Problem 44

In Fig. $12-29$, two skaters, each of mass $50 \mathrm{~kg}$, approach each other along parallel paths separated by $3.0 \mathrm{~m}$. They have opposite velocities of $1.4 \mathrm{~m} / \mathrm{s}$ each. One skater carries one end of a long pole with negligible mass, and the other skater grabs the other end of it as she passes. Assume frictionless ice. (a) Describe quantitatively the motion of the skaters after they have become connected by the pole. (b) What is the kinetic energy of the two-skater system?
Next, the skaters each pull along the pole so as to reduce their separation to $1.0 \mathrm{~m}$. What then are (c) their rotational speed and
(d) the kinetic energy of the system?
(e) Explain the source of the increased kinetic energy.

Stephen Zaffke

Numerade Educator

Problem 45

A cockroach of mass $m$ runs counterclockwise around the rim of a lazy Susan (a circular dish mounted on a vertical axle) of radius $R$ and rotational inertia $I$ and having frictionless bearings. The cockroach's speed (relative to the ground) is $v$, whereas the lazy Susan turns clockwise with rotational speed $\omega_{1}$. The cockroach finds a bread crumb on the rim and, of course, stops.
(a) What is the rotational speed of the lazy Susan after the cockroach stops? (b) Is mechanical energy conserved?

Stephen Zaffke

Numerade Educator

Problem 46

A girl of mass $M$ stands on the rim of a frictionless merry-go-round of radius $R$ and rotational inertia $I$ that is not moving. She throws a rock of mass $m$ horizontally in a direction that is tangent to the outer edge of the merry-go-round. The speed of the rock, relative to the ground, is $v .$ Afterward, what are
(a) the rotational speed of the merry-go-round and (b) the translational speed of the girl?

Stephen Zaffke

Numerade Educator

Problem 47

A horizontal vinyl record of mass $0.10 \mathrm{~kg}$ and radius $0.10 \mathrm{~m}$ rotates freely about a vertical axis through its center
with a rotational speed of $4.7 \mathrm{rad} / \mathrm{s}$. The rotational inertia of the record about its axis of rotation is $5.0 \times 10^{-4} \mathrm{~kg} \cdot \mathrm{m}^{2} .$ A wad of wet putty of mass $0.020 \mathrm{~kg}$ drops vertically onto the record from above and sticks to the edge of the record. What is the rotational speed of the record immediately after the putty sticks to it?

Stephen Zaffke

Numerade Educator

Problem 48

A uniform thin rod of length $0.50 \mathrm{~m}$ and mass $4.0 \mathrm{~kg}$ can rotate in a horizontal plane about a vertical axis through its center. The rod is at rest when a $3.0 \mathrm{~g}$ bullet traveling in the horizontal plane of the rod is fired into one end of the rod. As viewed from above, the direction of the
bullet's velocity makes an angle of $60^{\circ}$ with the rod (Fig. $12-30$ ). If the bullet lodges in the rod and the rotational velocity of the rod is $10 \mathrm{rad} / \mathrm{s}$ immediately after the collision, what is the bullet's speed just before impact?

Stephen Zaffke

Numerade Educator

Problem 49

Two $2.00 \mathrm{~kg}$ balls are attached to the ends of a thin rod of negligible mass, $50.0 \mathrm{~cm}$ long. The rod is free to rotate in a vertical plane without friction about a horizontal axis through its. center. With the rod initially horizontal (Fig. $12-31$ ), a $50.0 \mathrm{~g}$ wad of wet putty drops onto one of the balls, hitting it with a speed of $3.00 \mathrm{~m} / \mathrm{s}$ and then sticking to it. (a) What is the rotational speed of the system just after the putty wad hits? (b) What is the ratio of the kinetic energy of the entire system after the collision to that of the putty wad just before? (c) Through what angle will the system rotate until it momentarily stops?

Stephen Zaffke

Numerade Educator

Problem 50

A cockroach of mass $m$ lies on the rim of a uniform disk of mass $10.0 m$ that can rotate freely about its center like a merry-go-round. Initially the cockroach and disk rotate together with a rotational velocity of $\omega_{1}$. Then the cockroach walks halfway to the center of the disk. (a) What is the change $\Delta \omega$ in the rotational velocity of the cockroach-disk system? (b) What is the ratio $K_{2} / K_{1}$ of the new kinetic energy of the system to its initial kinetic energy? (c) What accounts for the change in the kinetic energy?

Stephen Zaffke

Numerade Educator

Problem 51

If Earth's polar ice caps fully melted and the water returned to the oceans, the oceans would be deeper by about $30 \mathrm{~m}$. What effect would this have on Earth's rotation? Make an estimate of the resulting change in the length of the day. (Concern has been expressed that warming of the atmosphere resulting from industrial pollution could cause the ice caps to melt.)

Manish Jain

Numerade Educator

Problem 52

A horizontal platform in the shape of a circular disk rotates on a frictionless bearing about a vertical axle through the center of the disk. The platform has a mass of $150 \mathrm{~kg}$, a radius of $2.0 \mathrm{~m}$, and a rotational inertia of $300 \mathrm{~kg} \cdot \mathrm{m}^{2}$ about the axis of rotation. A $60 \mathrm{~kg}$ student walks slowly from the rim of the platform toward the center. If the rotational speed of the system is $1.5 \mathrm{rad} / \mathrm{s}$ when the student starts at the rim, what is the rotational speed when she is $0.50 \mathrm{~m}$ from the center?

Stephen Zaffke

Numerade Educator

Problem 53

A uniform disk of mass $10 \mathrm{~m}$ and radius $3.0 r$ can rotate freely about its fixed center like a merry-go-round. A smaller uniform disk of mass $m$ and radius $r$ lies on top of the larger disk, concentric with it. Initially the two disks rotate together with a rota-
tional velocity of $20 \mathrm{rad} / \mathrm{s}$. Then a slight disturbance causes the smaller disk to slide outward across the larger disk, until the outer edge of the smaller disk catches on the outer edge of the larger disk. Afterward, the two disks again rotate together (without further sliding). (a) What then is their rotational velocity about the center of the larger disk? (b) What is the ratio $K_{2} / K_{1}$ of the new kinetic energy of the two-disk system to the system's initial kinetic energy?

Stephen Zaffke

Numerade Educator

Problem 54

A $30 \mathrm{~kg}$ child stands on the edge of a stationary merry-go-round of mass $100 \mathrm{~kg}$
and radius $2.0 \mathrm{~m}$. The rotational inertia of the merry-go-round about its axis of rotation is $150 \mathrm{~kg} \cdot \mathrm{m}^{2}$. The child catches a ball of mass $1.0$ $\mathrm{kg}$ thrown by a friend. Just before the ball is caught, it has a horizontal velocity of $12 \mathrm{~m} / \mathrm{s}$ that makes an angle of $37^{\circ}$ with a line tangent to the outer edge of the merry-goround, as shown in the overhead view of Fig. $12-32 .$ What is the rotational speed of the merry-go-round just after the ball is caught?

Stephen Zaffke

Numerade Educator

Problem 55

In Fig. 12-33, a $1.0 \mathrm{~g}$ bullet is fired into a $0.50 \mathrm{~kg}$ block that is mounted on the end of ? $0.60 \mathrm{~m}$ nonuniform rod of mass $0.50 \mathrm{~kg}$. The block-rod-bullet system then rotates about a fixed axis at point $A$. The rotational inertia of the rod alone about $A$ is $0.060$ $\mathrm{kg} \cdot \mathrm{m}^{2}$. Assume the block is small enough to treat as a particle on the end of the rod. (a) What is the rotational inertia of the block-rod-bullet system about point $A ?$ (b) If the rotational speed of the system about $A$ just after the bullet's impact is $4.5$ $\mathrm{rad} / \mathrm{s}$, what is the speed of the bullet just before the impact?

Stephen Zaffke

Numerade Educator

Problem 56

56. Uniform Rod In Fig. $12-34$, a uniform rod (length $=0.60 \mathrm{~m}$, mass $1.0 \mathrm{~kg}$ ) rotates about an axis through one end, with a rotational inertia of $0.12 \mathrm{~kg} \cdot \mathrm{m}^{2}$. As the rod swings
through its lowest position, the end of the rod collides with a small $0.20$ $\mathrm{kg}$ putty wad that sticks to the end of the rod. If the rotational speed of the rod just before the collision is $2.4 \mathrm{rad} / \mathrm{s}$, what is the rotational speed of the rod-putty system immediately after the collision?

Stephen Zaffke

Numerade Educator

Problem 57

The particle of mass $m$ in Fig. $12-35$ slides down the frictionless surface through height $h$ and collides with the uniform vertical rod (of mass $M$ and length $d$ ), sticking to it. The rod pivots about point $O$ through the angle $\theta$ before momentarily stopping. Find $\theta$

Keshav Singh

Numerade Educator

Problem 58

As part of an examination a few years ago, a student went through the algebraic manipulations on an exam shown in Fig. $12-36 .$ At this point you don't know what the symbols mean, but given the information about the dimensions associated with each symbol, decide the following:
(a) Is it possible that the final equation in Fig. $12-36$ is correct? Justify your answer.
(b) If the final equation is not correct, does that mean that the starting equation is necessarily wrong? Explain.
(c) If the final equation is not correct and the starting equation is not wrong, can you find the error using dimensional analysis? If so, do so. If not, explain why.
$[M]=\mathrm{M} \quad M g h=\frac{1}{2} M v^{2}+\frac{1}{2} I \omega^{2}$
$\begin{array}{ll}{[g]=\mathbf{L} / \mathrm{T}^{2}} & M g h=\frac{1}{2} M v^{2}+\frac{1}{2}\left(M R^{2}\right) \omega^{2}\end{array}$
$[h]=\mathrm{L}$
$[\omega]=1 / \mathrm{T} \quad M g h=\frac{1}{2} M v^{2}+\frac{1}{2}\left(M R^{2}\right)\left(\frac{v^{2}}{R}\right)^{2}$
$\begin{array}{l}{[v]=\mathrm{L} / \mathrm{T}} \\ {[R]=\mathrm{L}}\end{array} \quad g h=\frac{1}{2} v^{2}+\frac{1}{2} v^{4}$
$[I]=\mathrm{ML}^{2}$
Note: M stands for a mass unit, $\mathrm{L}$ is for length unit, and $\mathrm{T}$ is for a time unit.

Brandy Heflin

Numerade Educator

Problem 59

The four objects in Fig. $12-37$ are moving as indicated by the arrows. A curved arrow indicates rolling without slipping in the direction. For object $(a)$, use the coordinates shown. For the others, take the origin at the center of the circle. Use the directions associated with the coordinate axes shown for object $(a) .$ Construct a table with the values of the magnitudes total translational momentum, total rotational momentum, and total energy of motion at the instant shown for each case. Express your answers in terms of $m, v$, and $R$. (Include an indicator of the direction where appropriate.) Which system has the largest and smallest of each of the quantities? Explain your reasoning.

Averell Hause

Carnegie Mellon University

Problem 60

In testing a design for a yo-yo, an engineer begins by constructing a simple prototype-a string wound about the rim of a wooden disk. She puts an axle riding on nearly frictionless ball bearings through the axis of the wooden disk and fixes the ends of the axle. See Fig. $12-38$. In order to measure the moment of inertia of the disk, she attaches a weight of mass $m$ to the string and measures how long it takes to fall a given distance. (a) Assuming the rotational inertia of the disk is given by $I$, and the radius of the disk is $R$, find the time for the mass to fall a distance $h$ starting from rest. (b) The engineer doesn't have a very accurate stopwatch but wants to get a measurement good to a few percent. She decides that a fall time of 2 seconds would work. How big a mass should she use? Imagine you were setting up this experiment, and make reasonable estimates of the parameters you need.

Sarah Mccrumb

Numerade Educator

Problem 61

Figure $12-39$ shows an Atwood's machine with two unequal masses attached by a massless string. The pulley has a mass of $20 \mathrm{~g}$ and a radius of $2 \mathrm{~cm} .$ (a) State three approximations that you can make to simplify your calculation of the motion of the blocks. ("Making an approximation" is the process of ignoring a physical effect because you expect it to be small and have little effect on your result if you only care about a few significant figures. If you want more significant figures, you may have to include those effects.)
(b) Using your approximations, find the accel-
eration of block $A$. (c) What happens to your result if the two masses are equal? Is the result what you expect? Explain. (d) If you have ignored the rotational inertia of the pulley in your calculation in part
(a) of this problem, set up the equations that would allow you to solve for the acceleration when it is included (but don't solve them).

Tarandeep Singh

Tarandeep Singh

Numerade Educator

Problem 62

A refrigerator has separate shelves on the door for storing bottles. Thin plastic straps keep the bottles from falling off the door. Someone in the house slams the door with a bit too much vigor and a heavy bottle breaks the strap. Do you think the bottle would be more likely to break the plastic strap if it is close to the hinge? Close to the handle? Or doesn't it matter? Explain your answer in terms of the physics we have learned.

Sheh Lit Chang

University of Washington