Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 11

Rolling, Torque, and Angular Momentum - all with Video Answers

Educators

Chapter Questions

Problem 1

A car travels at $80 \mathrm{~km} / \mathrm{h}$ on a level road in the positive direction of an $x$ axis. Each tire has a diameter of $66 \mathrm{~cm}$. Relative to a woman riding in the car and in unit-vector notation, what are the velocity $\vec{v}$ at the (a) center, (b) top, and (c) bottom of the tire and the magnitude $a$ of the acceleration at the (d) center, (e) top, and (f) bottom of each tire? Relative to a hitchhiker sitting next to the road and in unit-vector notation, what are the velocity $\vec{v}$ at the $(\mathrm{g})$ center, (h) top, and (i) bottom of the tire and the magnitude $a$ of the acceleration at the (j) center, $(\mathrm{k})$ top, and (1) bottom of each tire?

Naresh Bagrecha

Naresh Bagrecha

Numerade Educator

Problem 2

An automobile traveling at $80.0 \mathrm{~km} / \mathrm{h}$ has tires of $75.0 \mathrm{~cm}$ diameter. (a) What is the angular 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 angular acceleration of the wheels? (c) How far does the car move during the braking?

Naresh Bagrecha

Naresh Bagrecha

Numerade Educator

Problem 3

A $140 \mathrm{~kg}$ hoop rolls along a horizontal floor so that the hoop's 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?

Naresh Bagrecha

Naresh Bagrecha

Numerade Educator

Problem 4

A uniform solid sphere rolls down an incline. (a) What must be the incline angle if the linear acceleration of the center of the sphere is to have a magnitude of $0.10 g ?$ (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 $0.10 \mathrm{~g}$ ? Why?

Naresh Bagrecha

Naresh Bagrecha

Numerade Educator

Problem 5

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

Naresh Bagrecha

Naresh Bagrecha

Numerade Educator

Problem 6

Figure 11-30 gives the speed $v$ versus time $t$ for a $0.500 \mathrm{~kg}$ object of radius $6.00 \mathrm{~cm}$ that rolls smoothly down a $30^{\circ}$ ramp. The scale on the velocity axis is set by $v_{s}=4.0 \mathrm{~m} / \mathrm{s}$. What is the rotational inertia of the object?

Naresh Bagrecha

Naresh Bagrecha

Numerade Educator

Problem 7

In Fig. $11-31$, a solid cylinder of radius $10 \mathrm{~cm}$ and mass $12 \mathrm{~kg}$ starts from rest and rolls without slipping a distance $L=6.0 \mathrm{~m}$ down a roof that is inclined at angle $\theta=$ $30^{\circ} .$ (a) What is the angular speed of the cylinder about its center as it leaves the roof? (b) The roof's edge is at height $H=5.0 \mathrm{~m}$. How far horizontally from the roof's edge does the cylinder hit the level ground?

Naresh Bagrecha

Naresh Bagrecha

Numerade Educator

Problem 8

Figure $11-32$ shows the potential energy $U(x)$ of a solid ball that can roll along an $x$ axis. The scale on the $U$ axis is set by $U_{s}=100 \mathrm{~J} .$ The ball is uniform, rolls smoothly, and has a mass of $0.400 \mathrm{~kg}$. It is released at $x=7.0 \mathrm{~m}$ headed in the negative direction of the $x$ axis with a mechanical energy of $75 \mathrm{~J} .$ (a) If the ball can reach $x=0 \mathrm{~m}$, what is its speed there, and if it cannot, what is its turning point? Suppose, instead, it is headed in the positive direction of the $x$ axis when it is released at $x=7.0 \mathrm{~m}$ with $75 \mathrm{~J} .(\mathrm{b})$ If the ball can reach $x=13 \mathrm{~m}$, what is its speed there, and if it cannot, what is its turning point?

Linda Winkler

Numerade Educator

Problem 9

In Fig. $11-33$, a solid ball rolls smoothly from rest (starting at height $H=6.0 \mathrm{~m}$ ) until it leaves the horizontal section at the end of the track, at height $h=2.0 \mathrm{~m}$. How far horizontally from point $A$ does the ball hit the floor?

Naresh Bagrecha

Naresh Bagrecha

Numerade Educator

Problem 10

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? When the sphere has moved $1.0 \mathrm{~m}$ up the incline from its initial position, what are (c) its total kinetic energy and (d) the speed of its center of mass?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 11

In Fig. $11-34$, a constant horizontal force $\vec{F}_{\text {app }}$ of magnitude $10 \mathrm{~N}$ is applied to a wheel of mass $10 \mathrm{~kg}$ and radius $0.30 \mathrm{~m}$. The wheel rolls smoothly on the horizontal surface, and the acceleration of its center of mass has magnitude $0.60 \mathrm{~m} / \mathrm{s}^{2}$. (a) In unit-vector notation, what is the frictional force on the wheel? (b) What is the rotational inertia of the wheel about the rotation axis through its center of mass?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 12

In Fig. $11-35$, a solid brass ball of mass $0.280 \mathrm{~g}$ will roll smoothly along a loop-the-loop track when released from rest along the straight section. The circular loop has radius $R=14.0 \mathrm{~cm}$, and the ball has radius $r \ll R .$ (a) What is $h$ if the ball is on the verge of leaving the track when it reaches the top of the loop? If the ball is released at height $h=6.00 R$, what are the (b) magnitude and (c) direction of the horizontal force component acting on the ball at point $Q$ ?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 13

In Fig. $11-$ 36 , a ball of mass $M$ and radius $R$ rolls smoothly from rest down a ramp and onto a circular loop of radius $0.48 \mathrm{~m}$. The initial height of the ball is $h=0.36 \mathrm{~m}$. At the loop bottom, the magnitude of the normal force on the ball is $2.00 \mathrm{Mg}$. The ball consists of an outer spherical shell (of a certain uniform density) that is glued to a central sphere (of a different uniform density). The rotational inertia of the ball can be expressed in the general form $I=\beta M R^{2}$, but $\beta$ is not $0.4$ as it is for a ball of uniform density. Determine $\beta$.

Keshav Singh

Numerade Educator

Problem 14

In Fig. 11-37, a small, solid, uniform ball is to be shot from point $P$ so that it rolls smoothly along a horizontal path, up along a ramp, and onto a plateau. Then it leaves the plateau horizontally to land on a game board, at a horizontal distance $d$ from the right edge of the plateau. The vertical heights are $h_{1}=5.00$ $\mathrm{cm}$ and $h_{2}=1.60 \mathrm{~cm} .$ With what speed must the ball be shot at point $P$ for it to land at $d=6.00 \mathrm{~cm} ?$

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 15

A bowler throws a bowling ball of radius $R=11 \mathrm{~cm}$ along a lane. The ball (Fig. $11-38$ ) slides on the lane with initial speed $v_{\mathrm{com}, 0}=8.5 \mathrm{~m} / \mathrm{s}$ and initial angular speed $\omega_{0}=0 .$ The coefficient of kinetic friction between the ball and the lane is $0.21$. The kinetic frictional force $\vec{f}_{k}$ acting on the ball causes a linear acceleration of the ball while producing a torque that causes an angular acceleration of the ball. When speed $v_{\text {com }}$ has decreased enough and angular speed $\omega$ has increased enough, the ball stops sliding and then rolls smoothly. (a) What then is $v_{\text {com }}$ in terms of $\omega$ ? During the sliding, what are the ball's (b) linear acceleration and (c) angular acceleration? (d) How long does the ball slide? (e) How far does the ball slide? (f) What is the linear speed of the ball when smooth rolling begins?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 16

In Fig. $11-39$, a cylindrical object of mass $M$ and radius $R$ rolls smoothly from rest down a ramp and onto a horizontal section. From there it rolls off the ramp and onto the floor, landing a horizontal distance $d=0.506 \mathrm{~m}$ from the end of the ramp. The initial height of the object is $H=0.90 \mathrm{~m} ;$ the end of the ramp is at height $h=0.10 \mathrm{~m}$. The object consists of an outer cylindrical shell (of a certain uniform density) that is glued to a central cylinder (of a different uniform density). The rotational inertia of the object can be expressed in the general form $I=\beta M R^{2}$, but $\beta$ is not $0.5$ as it is for a cylinder of uniform density. Determine $\beta$.

Keshav Singh

Numerade Educator

Problem 17

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 linear 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) linear speed, (d) translational kinetic energy, (e) rotational kinetic energy, and (f) angular speed?

Narayan Hari

Numerade Educator

Problem 18

In 1980 , over San Francisco Bay, a large yo-yo was released from a crane. The $116 \mathrm{~kg}$ yo-yo consisted of two uniform disks of radius $32 \mathrm{~cm}$ connected by an axle of radius $3.2 \mathrm{~cm} .$ What was the magnitude of the acceleration of the yo-yo during (a) its fall and (b) its rise? (c) What was the tension in the cord on which it rolled? (d) Was that tension near the cord's limit of $52 \mathrm{kN}$ ? Suppose you build a scaled-up version of the yo-yo (same shape and materials but larger). (e) Will the magnitude of your yo-yo's acceleration as it falls be greater than, less than, or the same as that of the San Francisco yo-yo? (f) How about the tension in the cord?

Ben Nicholson

Numerade Educator

Problem 19

In unit-vector notation, what is the net torque about the origin on a flea located at coordinates $(0,-4.0 \mathrm{~m}, 5.0 \mathrm{~m})$ when forces $\vec{F}_{1}=(3.0 \mathrm{~N}) \hat{\mathrm{k}}$ and $\vec{F}_{2}=(-2.0 \mathrm{~N}) \hat{\mathrm{j}}$ act on the flea?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 20

A plum is located at coordinates $(-2.0 \mathrm{~m}, 0,4.0 \mathrm{~m})$. In unitvector notation, what is the torque about the origin on the plum if that torque is due to a force $\vec{F}$ whose only component is (a) $F_{x}=$ $6.0 \mathrm{~N},(\mathrm{~b}) F_{x}=-6.0 \mathrm{~N},(\mathrm{c}) F_{z}=6.0 \mathrm{~N}$, and $(\mathrm{d}) F_{z}=-6.0 \mathrm{~N} ?$

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 21

In unit-vector notation, what is the torque about the origin on a particle located at coordinates $(0,-4.0 \mathrm{~m}, 3.0 \mathrm{~m})$ if that torque is due to (a) force $\vec{F}_{1}$ with components $F_{1 x}=2.0 \mathrm{~N}, F_{1 y}=F_{1 z}=0$, and (b) force $\vec{F}_{2}$ with components $F_{2 x}=0, F_{2 y}=2.0 \mathrm{~N}, F_{2 z}=4.0 \mathrm{~N} ?$

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 22

Aparticle moves through an $x y z$ coordinate system while a force acts on the particle. When the particle has the position vector $\vec{r}=(2.00 \mathrm{~m}) \hat{\mathrm{i}}-(3.00 \mathrm{~m}) \hat{\mathrm{j}}+(2.00 \mathrm{~m}) \hat{\mathrm{k}}$, the force is given
by $\vec{F}=F_{x} \hat{\mathrm{i}}+(7.00 \mathrm{~N}) \hat{\mathrm{j}}-(6.00 \mathrm{~N}) \hat{\mathrm{k}}$ and the corresponding torque
about the origin is $\vec{\tau}=(4.00 \mathrm{~N} \cdot \mathrm{m}) \hat{\mathrm{i}}+(2.00 \mathrm{~N} \cdot \mathrm{m}) \hat{\mathrm{j}}-(1.00 \mathrm{~N} \cdot \mathrm{m}) \hat{\mathrm{k}}$.
Determine $F_{x^{*}}$

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 23

Force $\vec{F}=(2.0 \mathrm{~N}) \hat{\mathrm{i}}-(3.0 \mathrm{~N}) \hat{\mathrm{k}}$ acts on a pebble with position vector $\vec{r}=(0.50 \mathrm{~m}) \hat{\mathrm{j}}-(2.0 \mathrm{~m}) \hat{\mathrm{k}}$ relative to the origin. In unit-vector notation, what is the resulting torque on the pebble about (a) the origin and (b) the point $(2.0 \mathrm{~m}, 0,-3.0 \mathrm{~m})$ ?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 24

In unit-vector notation, 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 $\vec{F}_{1}=(3.0 \mathrm{~N}) \hat{\mathrm{i}}-(4.0 \mathrm{~N}) \hat{\mathrm{j}}+(5.0 \mathrm{~N}) \hat{\mathrm{k}}$, (b) force $\vec{F}_{2}=(-3.0 \mathrm{~N}) \hat{\mathrm{i}}-(4.0 \mathrm{~N}) \hat{\mathrm{j}}-(5.0 \mathrm{~N}) \hat{\mathrm{k}}$, and $(\mathrm{c})$ the vector sum of $\vec{F}_{1}$ and $\vec{F}_{2} ?$ (d) Repeat part (c) for the torque about the point with coordinates $(3.0 \mathrm{~m}, 2.0 \mathrm{~m}, 4.0 \mathrm{~m})$.

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 25

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

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 26

At the instant of Fig. 11-40, a $2.0 \mathrm{~kg}$ particle $P$ has a position vector $\vec{r}$ of magnitude $3.0 \mathrm{~m}$ and angle $\theta_{1}=45^{\circ}$ and a velocity vector $\vec{v}$ of magnitude $4.0 \mathrm{~m} / \mathrm{s}$ and angle $\theta_{2}=30^{\circ}$. Force $\vec{F}$, of magnitude $2.0 \mathrm{~N}$ and angle $\theta_{3}=30^{\circ}$, acts on $P$. All three vectors lie in the $x y$ plane. About the origin, what are the (a) magnitude and (b) direction of the angular momentum of $P$ and the (c) magnitude and (d) direction of the torque acting on $P$ ?

Ben Nicholson

Numerade Educator

Problem 27

At one instant, force $\vec{F}=4.0 \hat{\mathrm{j}} \mathrm{N}$ acts on a $0.25 \mathrm{~kg}$ object that has position vector $\vec{r}=(2.0 \hat{\mathrm{i}}-2.0 \hat{\mathrm{k}}) \mathrm{m}$ and velocity vector $\vec{v}=(-5.0 \hat{\mathrm{i}}+5.0 \hat{\mathrm{k}}) \mathrm{m} / \mathrm{s}$. About the origin and in unit-vector notation, what are (a) the object's angular momentum and (b) the torque acting on the object?

Pranay Shrivastava

Pranay Shrivastava

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, in unitvector notation, what is its angular momentum relative to (a) the origin and (b) the point located at $(-2.0,-2.0) \mathrm{m} ?$

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 29

In the instant of Fig. 11-41, two particles move in an $x y$ plane. Particle $P_{1}$ has mass $6.5 \mathrm{~kg}$ and speed $v_{1}=2.2 \mathrm{~m} / \mathrm{s}$, and it is at distance $d_{1}=1.5 \mathrm{~m}$ from point $O$. Particle $P_{2}$ has mass $3.1 \mathrm{~kg}$ and speed $v_{2}=3.6 \mathrm{~m} / \mathrm{s}$, and it is at distance $d_{2}=$ $2.8 \mathrm{~m}$ from point $O$. What are the
(a) magnitude and (b) direction of the net angular momentum of the two particles about $O$ ?

Salamat Ali

Numerade Educator

Problem 30

At the instant the displacement of a $2.00 \mathrm{~kg}$ object relative to the origin is $\vec{d}=(2.00 \mathrm{~m}) \hat{\mathrm{i}}+(4.00 \mathrm{~m}) \hat{\mathrm{j}}-(3.00 \mathrm{~m}) \hat{\mathrm{k}}$, its velocity is $\vec{v}=-(6.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}+(3.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}+(3.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{k}}$ and it is subject to a force $\vec{F}=(6.00 \mathrm{~N}) \hat{\mathrm{i}}-(8.00 \mathrm{~N}) \hat{\mathrm{j}}+(4.00 \mathrm{~N}) \hat{\mathrm{k}}$. Find (a) the acceleration of the object, (b) the angular momentum of the object about the origin, (c) the torque about the origin acting on the object, and (d) the angle between the velocity of the object and the force acting on the object.

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 31

In Fig. 11-42, a $0.400 \mathrm{~kg}$ ball is shot directly upward at initial speed $40.0$ $\mathrm{m} / \mathrm{s}$. What is its angular momentum about $P, 2.00 \mathrm{~m}$ horizontally from the launch point, when the ball is (a) at maximum height and (b) halfwav back to the ground? What is the torque on the ball about $P$ due to the gravitational force when the ball is (c) at maximum height and (d) halfway back to the ground?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 32

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. In unit-vector notation, find $d \vec{\ell} / d t$, where $\vec{\ell}$ is the angular momentum of the particle about the origin.

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 33

At time $t=0$, 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. About the origin, what are (a) the particle's angular momentum, (b) the torque acting on the particle, and (c) the rate at which the angular momentum is changing?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 34

A particle is to move in an $x y$ plane, clockwise around the origin as seen from the positive side of the $z$ axis. In unit-vector notation, what torque acts on the particle if the magnitude of its angular momentum about the origin is (a) $4.0 \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}$, (b) $4.0 t^{2} \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}$, (c) $4.0 \sqrt{t} \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}$, and (d) $4.0 / t^{2} \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}$ ?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 35

At time $t$, the vector $\vec{r}=4.0 t^{2} \hat{\mathrm{i}}-\left(2.0 t+6.0 t^{2}\right) \hat{\mathrm{j}}$ gives the position of a $3.0 \mathrm{~kg}$ particle relative to the origin of an $x y$ coordinate system ( $\vec{r}$ is in meters and $t$ is in seconds). (a) Find an expression for the torque acting on the particle relative to the origin. (b) Is the magnitude of the particle's angular momentum relative to the origin increasing, decreasing, or unchanging?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 36

Figure 11-43 shows three rotating, uniform disks that are coupled by belts. One belt runs around the rims of disks $A$ and $C$. Another belt runs around a central hub on disk $A$ and the rim of disk $B$. The belts move smoothly without slippage on the rims and hub. Disk $A$ has radius $R$; its hub has radius $0.5000 R$; disk $B$ has radius $0.2500 R ;$ and disk $C$ has radius $2.000 R$. Disks $B$ and $C$ have the same density (mass per unit volume) and thickness. What is the ratio of the magnitude of the angular momentum of disk $C$ to that of disk $B$ ?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 37

In Fig. $11-44$, three particles of mass $m=23 \mathrm{~g}$ are fastened to three rods of length $d=12 \mathrm{~cm}$ and negligible mass. The rigid assembly rotates around point $O$ at the angular speed $\omega=0.85 \mathrm{rad} / \mathrm{s}$. About $O$, what are (a) the rotational inertia of the assembly, (b) the magnitude of the angular momentum of the middle particle, and (c) the magnitude of the angular momentum of the asssembly?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 38

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 magnitude $16 \mathrm{~N} \cdot \mathrm{m}$ about the central axis of the disk. About that axis and with the torque applied for $33 \mathrm{~ms}$, what is the magnitude of the (a) angular momentum and (b) angular velocity of the disk?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 39

The angular 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 angular 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?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 40

A disk with a rotational inertia of $7.00 \mathrm{~kg} \cdot \mathrm{m}^{2}$ rotates like a merry-go-round while undergoing a time-dependent torque given by $\tau=(5.00+2.00 t) \mathrm{N} \cdot \mathrm{m} .$ At time $t=1.00 \mathrm{~s}$, its angular momentum is $5.00 \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}$. What is its angular momentum at $t=3.00 \mathrm{~s}$ ?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 41

Figure $11-45$ 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 angular momentum about that axis.

Keshav Singh

Numerade Educator

Problem 42

Figure $11-46$ gives the torque $\tau$ that acts on an initially stationary disk that can rotate about its center like a merry-go-round. The scale on the $\tau$ axis is set by $\tau_{s}=4.0 \mathrm{~N} \cdot \mathrm{m}$. What is the angular momentum of the disk about the rotation axis at times (a) $t=7.0 \mathrm{~s}$ and (b) $t=20 \mathrm{~s}$ ?

Keshav Singh

Numerade Educator

Problem 43

In Fig. 11-47, 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 of negligible mass, and the other skater grabs the other end as she passes. The skaters then rotate around the center of the pole. Assume that the friction between skates and ice is negligible. What are (a) the radius of the circle, (b) the angular speed of the skaters, and (c) the kinetic energy of the two-skater system? Next, the skaters pull along the pole until they are separated by $1.0 \mathrm{~m}$. What then are (d) their angular speed and (e) the kinetic energy of the system? (f) What provided the energy for the increased kinetic energy?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 44

A Texas cockroach of mass $0.17 \mathrm{~kg}$ runs counterclockwise around the rim of a lazy Susan (a circular disk mounted on a vertical axle) that has radius $15 \mathrm{~cm}$, rotational inertia $5.0 \times 10^{-3} \mathrm{~kg} \cdot \mathrm{m}^{2}$, and frictionless bearings. The cockroach's speed (relative to the ground) is $2.0 \mathrm{~m} / \mathrm{s}$, and the lazy Susan turns clockwise with angular speed $\omega_{0}=$ $2.8 \mathrm{rad} / \mathrm{s}$. The cockroach finds a bread crumb on the rim and, of course, stops. (a) What is the angular speed of the lazy Susan after the cockroach stops? (b) Is mechanical energy conserved as it stops?

Keshav Singh

Numerade Educator

Problem 45

A man stands on a platform that is rotating (without friction) with an angular 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 vertical axis of the platform is $6.0 \mathrm{~kg} \cdot \mathrm{m}^{2}$. If by moving the bricks the man decreases the rotational inertia of the system to $2.0$ $\mathrm{kg} \cdot \mathrm{m}^{2}$, what are (a) the resulting angular speed of the platform and
(b) the ratio of the new kinetic energy of the system to the original kinetic energy? (c) What source provided the added kinetic energy?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 46

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

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 47

A track is mounted on a large wheel that is free to turn with negligible friction about a vertical axis (Fig. $11-48$ ). A toy train of mass $m$ is placed on the track and, with the system initially at rest, the train's electrical power is turned on. The train reaches speed $0.15 \mathrm{~m} / \mathrm{s}$ with respect to the track. What is the wheel's angular speed if its mass is $1.1 \mathrm{~m}$ and its radius is $0.43 \mathrm{~m} ?$ (Treat it as a hoop, and neglect the mass of the spokes and hub.)

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 48

A Texas cockroach walks from the center of a circular disk (that rotates like a merry-go-round without external torques) out to the edge at radius $R$. The angular speed of the cockroach-disk system for the walk is given in Fig. $11-49\left(\omega_{a}=5.0 \mathrm{rad} / \mathrm{s}\right.$ and
$\omega_{b}=6.0 \mathrm{rad} / \mathrm{s}$ ). After reaching $R$ what fraction of the rotational inertia of the disk does the cockroach have?

Keshav Singh

Numerade Educator

Problem 49

Two disks are mounted (like a merry-go-round) on lowfriction bearings on the same axle and can be brought together so that they couple and rotate as one unit. The first disk, with rotational inertia $3.30 \mathrm{~kg} \cdot \mathrm{m}^{2}$ about its central axis, is set spinning counterclockwise at 450 rev/min. The second disk, with rotational inertia $6.60 \mathrm{~kg} \cdot \mathrm{m}^{2}$ about its central axis, is set spinning counterclockwise at 900 rev/min. They then couple together. (a) What is their angular speed after coupling? If instead the second disk is set spinning clockwise at $900 \mathrm{rev} / \mathrm{min}$, what are their (b) angular speed and
(c) direction of rotation after they couple together?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 50

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 along the central axis of the probe; the probe has rotational inertia $I_{p}=12 \mathrm{~kg} \cdot \mathrm{m}^{2}$ about this axis. Calculate the number of revolutions of the rotor required to turn the probe through $30^{\circ}$ about its central axis.

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 51

a wheel is rotating freely at angular 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 angular speed of the resultant combination of the shaft and two wheels? (b) What fraction of the original rotational kinetic energy is lost?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 52

A cockroach of mass $m$ lies on the rim of a uniform disk of mass $4.00 \mathrm{~m}$ that can rotate freely about its center like a merry-goround. Initially the cockroach and disk rotate together with an angular velocity of $0.260 \mathrm{rad} / \mathrm{s}$. Then the cockroach walks halfway to the center of the disk. (a) What then is the angular velocity of the cockroach-disk system? (b) What is the ratio $K / K_{0}$ of the new kinetic energy of the system to its initial kinetic energy? (c) What accounts for the change in the kinetic energy?

Keshav Singh

Numerade Educator

Problem 53

In Fig. 11-50 (an overhead view), a uniform thin rod of length $0.500 \mathrm{~m}$ and mass $4.00 \mathrm{~kg}$ can rotate in a horizontal plane about a vertical axis through its center. The rod is at rest when a $3.00 \mathrm{~g}$ bullet traveling in the rotation plane is fired into one end of the rod. In the view from above, the bullet's path makes angle $\theta=60.0^{\circ}$ with the rod (Fig. 1150 ). If the bullet lodges in the rod and the angular velocity of the rod is $10 \mathrm{rad} / \mathrm{s}$ immediately after the collision, what is the bullet's speed just before impact?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 54

Figure 11-51 shows an overhead view of a ring that can rotate about its center like a merrygo-round. Its outer radius $R_{2}$ is $0.800 \mathrm{~m}$, its inner radius $R_{1}$ is $R_{2} / 2.00$, its mass $M$ is $8.00 \mathrm{~kg}$, and the mass of the crossbars at its center is negligible. It initially rotates at an angular speed of $8.00 \mathrm{rad} / \mathrm{s}$ with a cat of mass $m=M / 4.00$ on its outer edge, at radius $R_{2}$. By how much does the cat increase the kinetic energy of the cat-ring system if the cat crawls to the inner edge, at radius $R_{1}$ ?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 55

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 an angular speed of $4.7 \mathrm{rad} / \mathrm{s}$ and a rotational inertia of $5.0 \times 10^{-4} \mathrm{~kg} \cdot \mathrm{m}^{2}$
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 angular speed of the record immediately afterwards?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 56

In a long jump, an athlete leaves the ground with an initial angular momentum that tends to rotate her body forward, threatening to ruin her landing. To counter this tendency, she rotates her outstretched arms to "take up" the angular momentum (Fig. 11- 18). In $0.700 \mathrm{~s}$, one arm sweeps through $0.500$ rev and the other arm sweeps through $1.000$ rev. Treat each arm as a thin rod of mass $4.0 \mathrm{~kg}$ and length $0.60 \mathrm{~m}$, rotating around one end. In the athlete's reference frame, what is the magnitude of the total angular momentum of the arms around the common rotation axis through the shoulders?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 57

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 an angular 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 angular velocity about the center of the larger disk? (b) What is the ratio $K / K_{0}$ of the new kinetic energy of the two-disk system to the system's initial kinetic energy?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 58

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 angular speed of the system is $1.5 \mathrm{rad} / \mathrm{s}$ when the student starts at the rim, what is the angular speed when she is $0.50 \mathrm{~m}$ from the center?

Nishant Kumar

Numerade Educator

Problem 59

Figure $11-52$ is an overhead view of a thin uniform rod of length $0.800 \mathrm{~m}$ and mass $M$ rotating horizontally at angular speed $20.0 \mathrm{rad} / \mathrm{s}$ about an axis through its center. A particle of mass $M / 3.00$ initially attached to one end is ejected from the rod and travels along a path that is perpendicular to the rod at the instant of ejection. If the particle's speed $v_{p}$ is $6.00 \mathrm{~m} / \mathrm{s}$ greater than the speed of the rod end just after ejection, what is the value of $v_{p}$ ?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 60

In Fig. 11-53, a $1.0 \mathrm{~g}$ bullet is fired into a $0.50 \mathrm{~kg}$ block attached to the end of a $0.60 \mathrm{~m}$ nonuniform rod of mass $0.50 \mathrm{~kg}$. The block-rod-bullet system then rotates in the plane of the figure, about a fixed axis at $A$. The rotational inertia of the rod alone about that axis at $A$ is $0.060 \mathrm{~kg} \cdot \mathrm{m}^{2}$. Treat the block as a particle. (a) What then is the rotational inertia of the block-rod-bullet system about point $A ?$ (b) If the angular speed of the system about $A$ just after impact is $4.5 \mathrm{rad} / \mathrm{s}$, what is the bullet's speed just before impact?

Keshav Singh

Numerade Educator

Problem 61

The uniform rod (length $0.60 \mathrm{~m}$, mass $1.0 \mathrm{~kg}$ ) in Fig. $11-54$ rotates in the plane of the figure 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, it collides with a $0.20 \mathrm{~kg}$ putty wad that sticks to the end of the rod. If the rod's angular speed just before collision is $2.4 \mathrm{rad} / \mathrm{s}$, what is the angular speed of the rod-putty system immediately after collision?

Keshav Singh

Numerade Educator

Problem 62

During a jump to his partner, an aerialist is to make a quadruple somersault lasting a time $t=1.87 \mathrm{~s}$. For the first and last quarter-revolution, he is in the extended orientation shown in Fig. $11-55$, with rotational inertia $I_{1}=19.9 \mathrm{~kg} \cdot \mathrm{m}^{2}$ around his center of mass (the dot). During the rest of the flight he is in a tight tuck, with rotational inertia $I_{2}=3.93 \mathrm{~kg} \cdot \mathrm{m}^{2} .$ What must be his angular speed $\omega_{2}$ around his center of mass during the tuck?

Keshav Singh

Numerade Educator

Problem 63

In Fig. $11-56$, a $30 \mathrm{~kg}$ child stands on the edge of a stationary merry-go-round of radius $2.0 \mathrm{~m}$. The rotational inertia of the merrygo-round about its rotation axis 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 $\vec{v}$ of magnitude $12 \mathrm{~m} / \mathrm{s}$, at angle $\phi=37^{\circ}$ with a line tangent to the outer edge of the merry-go-round, as shown. What is the angular speed of the merry-go-round just after the ball is caught?

Keshav Singh

Numerade Educator

Problem 64

A ballerina begins a tour jeté (Fig. $11-19 a$ ) with angular speed $\omega_{i}$ and a rotational inertia consisting of two parts:
$I_{\text {leg }}=1.44 \mathrm{~kg} \cdot \mathrm{m}^{2}$ for her leg extended outward at angle $\theta=90.0^{\circ}$ to her body and $I_{\text {trunk }}=0.660 \mathrm{~kg} \cdot \mathrm{m}^{2}$ for the rest of her body (primarily her trunk). Near her maximum height she holds both legs at angle $\theta=30.0^{\circ}$ to her body and has angular speed $\omega_{f}$ (Fig. $11-19 b$ ). Assuming that $I_{\text {trunk }}$ has not changed, what is the ratio $\omega_{f} / \omega_{i}$ ?

Keshav Singh

Numerade Educator

Problem 65

Two $2.00 \mathrm{~kg}$ balls are attached to the ends of a thin rod of length $50.0 \mathrm{~cm}$ and negligible mass. 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. 11-57), 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 angular speed of the system just after the putty wad hits? (b) What is the ratio of the kinetic energy of the system after the collision to that of the putty wad just before? (c) Through what angle will the system rotate before it momentarily stops?

Linda Winkler

Numerade Educator

Problem 66

In Fig. 11-58, a small $50 \mathrm{~g}$ block slides down a frictionless surface through height $h=20 \mathrm{~cm}$ and then sticks to a uniform rod of mass $100 \mathrm{~g}$ and length $40 \mathrm{~cm}$. The rod pivots about point $O$ through angle $\theta$ before momentarily stopping. Find $\theta$.

Keshav Singh

Numerade Educator

Problem 67

Figure $11-59$ is an overhead view of a thin uniform rod of length $0.600 \mathrm{~m}$ and mass $M$ rotating horizontally at $80.0 \mathrm{rad} / \mathrm{s}$ counterclockwise about an axis through its center. A particle of mass $M / 3.00$ and traveling horizontally at speed $40.0 \mathrm{~m} / \mathrm{s}$ hits the rod and sticks. The particle's path is perpendicular to the rod at the instant of the hit, at a distance $d$ from the rod's center. (a) At what value of $d$ are rod and particle stationary after the hit? (b) In which direction do rod and particle rotate if $d$ is greater than this value?

Keshav Singh

Numerade Educator

Problem 68

A top spins at 30 rev/s about an axis that makes an angle of $30^{\circ}$ with the vertical. The mass of the top is $0.50 \mathrm{~kg}$, its rotational inertia about its central axis is $5.0 \times 10^{-4} \mathrm{~kg} \cdot \mathrm{m}^{2}$, and its center of mass is $4.0 \mathrm{~cm}$ from the pivot point. If the spin is clockwise from an overhead view, what are the (a) precession rate and (b) direction of the precession as viewed from overhead?

Keshav Singh

Numerade Educator

Problem 69

A certain gyroscope consists of a uniform disk with a $50 \mathrm{~cm}$ radius mounted at the center of an axle that is $11 \mathrm{~cm}$ long and of negligible mass. The axle is horizontal and supported at one end. If the spin rate is 1000 rev $/ \mathrm{min}$, what is the precession rate?

Salamat Ali

Numerade Educator

Problem 70

A uniform solid ball rolls smoothly along a floor, then up a ramp inclined at $15.0^{\circ} .$ It momentarily stops when it has rolled $1.50 \mathrm{~m}$ along the ramp. What was its initial speed?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 71

In Fig. 11-60, a constant horizontal force $\vec{F}_{\text {app }}$ of magnitude 12 $\mathrm{N}$ is applied to a uniform solid cylinder by fishing line wrapped around the cylinder. The mass of the cylinder is $10 \mathrm{~kg}$, its radius is $0.10 \mathrm{~m}$, and the cylinder rolls smoothly on the horizontal surface. (a) What is the magnitude of the acceleration of the center of mass of the cylinder? (b) What is the magnitude of the angular acceleration of the cylinder about the center of mass? (c) In unit-vector notation, what is the frictional force acting on the cylinder?

Keshav Singh

Numerade Educator

Problem 72

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

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 73

$A 3.0 \mathrm{~kg}$ toy car moves along an $x$ axis with a velocity given by $\vec{v}=-2.0 t^{3} \mathrm{i} \mathrm{m} / \mathrm{s}$, with $t$ in seconds. For $t>0$, what are (a) the angular momentum $\vec{L}$ of the car and (b) the torque $\vec{\tau}$ on the car, both calculated about the origin? What are (c) $\vec{L}$ and $(\mathrm{d}) \bar{\tau}$ about the point $(2.0 \mathrm{~m}, 5.0 \mathrm{~m}, 0) ?$ What are (e) $\vec{L}$ and (f) $\vec{\tau}$ about the point $(2.0 \mathrm{~m},-5.0 \mathrm{~m}, 0)$ ?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 74

A wheel rotates clockwise about its central axis with an angular momentum of $600 \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}$. At time $t=0$, a torque of magnitude $50 \mathrm{~N} \cdot \mathrm{m}$ is applied to the wheel to reverse the rotation. At what time $t$ is the angular speed zero?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 75

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 79 of Chapter 10 ) 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 angular momentum of the running child about the axis of rotation of the merry-go-round, and (c) the angular speed of the merry-go-round and child after the child has jumped onto the merry-go-round.

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 76

A uniform block of granite in the shape of a book has face dimensions of $20 \mathrm{~cm}$ and $15 \mathrm{~cm}$ and a thickness of $1.2 \mathrm{~cm}$. The density (mass per unit volume) of granite is $2.64 \mathrm{~g} / \mathrm{cm}^{3}$. The block rotates around an axis that is perpendicular to its face and halfway between its center and a corner. Its angular momentum about that axis is $0.104 \mathrm{~kg} \cdot \mathrm{m}^{2} / \mathrm{s}$. What is its rotational kinetic energy about that axis?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 77

Two particles, each of mass $2.90 \times 10^{-4} \mathrm{~kg}$ and speed $5.46 \mathrm{~m} / \mathrm{s}$, travel in opposite directions along parallel lines separated by $4.20 \mathrm{~cm} .$ (a) What is the magnitude $L$ of the angular momentum of the two-particle system around a point midway between the two lines? (b) Is the value different for a different location of the point? If the direction of either particle is reversed, what are the answers for (c) part (a) and (d) part (b)?

Linda Winkler

Numerade Educator

Problem 78

A wheel of radius $0.250 \mathrm{~m}$, moving initially at $43.0 \mathrm{~m} / \mathrm{s}$, rolls to a stop in $225 \mathrm{~m}$. Calculate the magnitudes of its (a) linear acceleration and (b) angular acceleration. (c) Its rotational inertia is $0.155$ $\mathrm{kg} \cdot \mathrm{m}^{2}$ about its central axis. Find the magnitude of the torque about the central axis due to friction on the wheel.

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 79

Wheels $A$ and $B$ in Fig. 11-61 are connected by a belt that does not slip. The radius of $B$ is $3.00$ times the radius of $A$. What would be the ratio of the rotational inertias $I_{A} / I_{B}$ if the two wheels had (a) the same angular momentum about their central axes and (b) the same rotational kinetic energy?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 80

A $2.50 \mathrm{~kg}$ particle that is moving horizontally over a floor with velocity $(-3.00 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}$ undergoes a completely inelastic collision with a $4.00 \mathrm{~kg}$ particle that is moving horizontally over the floor with velocity $(4.50 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{i}}$. The collision occurs at $x y$ coordinates $(-0.500 \mathrm{~m},-0.100 \mathrm{~m})$. After the collision and in unit-vector notation, what is the angular momentum of the stuck-together particles with respect to the origin?

Keshav Singh

Numerade Educator

Problem 81

A uniform wheel of mass $10.0 \mathrm{~kg}$ and radius $0.400 \mathrm{~m}$ is mounted rigidly on a massless axle through its center (Fig. $11-62$ ). The radius of the axle is $0.200$ $\mathrm{m}$, and the rotational inertia of the wheel-axle combination about its central axis is $0.600 \mathrm{~kg} \cdot \mathrm{m}^{2}$. The wheel is initially at rest at the top of a surface that is inclined at angle $\theta=$ $30.0^{\circ}$ with the horizontal; the axle rests on the surface while the wheel extends into a groove in the surface without touching the surface. Once released, the axle rolls down along the surface smoothly and without slipping. When the wheel-axle combination has moved down the surface by $2.00 \mathrm{~m}$, what are (a) its rotational kinetic energy and (b) its translational kinetic energy?

Keshav Singh

Numerade Educator

Problem 82

A uniform rod rotates in a horizontal plane about a vertical axis through one end. The rod is $6.00 \mathrm{~m}$ long, weighs $10.0 \mathrm{~N}$, and rotates at 240 rev/min. Calculate (a) its rotational inertia about the axis of rotation and (b) the magnitude of its angular momentum about that axis.

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 83

A solid sphere of weight $36.0 \mathrm{~N}$ rolls up an incline at an angle of $30.0^{\circ}$. At the bottom of the incline the center of mass of the sphere has a translational speed of $4.90 \mathrm{~m} / \mathrm{s}$. (a) What is the kinetic energy of the sphere at the bottom of the incline? (b) How far does the sphere travel up along the incline?
(c) Does the answer to
(b) depend on the sphere's mass?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 84

Suppose that the yo-yo in Problem 17, 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) linear speed, (d) translational kinetic energy, (e) angular speed, and (f) rotational kinetic energy?

Ajay Singhal

Numerade Educator

Problem 85

A girl of mass $M$ stands on the rim of a frictionless merrygo-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 angular speed of the merry-go-round and (b) the linear speed of the girl?

Pranay Shrivastava

Pranay Shrivastava

Numerade Educator

Problem 86

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