Principles of Physics

David Halliday , Robert Resnick , Jearl Walker

Chapter 10

Rotation - all with Video Answers

Educators

Chapter Questions

Problem 1

A tractor has rear wheels with a radius of $1.00 \mathrm{~m}$ and front wheels with a radius of $0.250 \mathrm{~m}$. The rear wheels are rotating at $100 \mathrm{rev} / \mathrm{min}$. Find (a) the angular speed of the front wheels in revolutions per minute and (b) the distance covered by the tractor in $10.0 \mathrm{~min}$.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 2

Just as a helicopter is landing, its blades are turning at $30.0 \mathrm{rev} / \mathrm{s}$ and slowing at a constant rate. In the $2.00 \mathrm{~min}$ required for them to stop, how many revolutions do they make?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 3

When a slice of buttered toast is accidentally pushed over the edge of a counter, it rotates as it falls. If the distance to the floor is $76 \mathrm{~cm}$ and for rotation less than $1 \mathrm{rev}$, what are the (a) smallest and (b) largest angular speeds that cause the toast to hit and then topple to be butter-side down?

Supratim Pal

Numerade Educator

Problem 4

The angular position of a point on a rotating wheel is given by $\theta=2.0+4.0 t^{2}+2.0 t^{3}$, where $\theta$ is in radians and $t$ is in seconds. At $t=0$, what are (a) the point's angular position and (b) its angular velocity? (c) What is its angular velocity at $t=3.0 \mathrm{~s} ?(\mathrm{~d})$ Calculate its angular acceleration at $t=4.0 \mathrm{~s}$. (e) Is its angular acceleration constant?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 5

At time $t=0$, a rotating bicycle wheel is thrown horizontally from a rooftop with a speed of $49 \mathrm{~m} / \mathrm{s}$. By the time its vertical speed is also $49 \mathrm{~m} / \mathrm{s}$, it has completed 40 revolutions. What has been its average angular speed to that point in the fall?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 6

A horizontal pottery wheel (a horizontal disk) with a radius of $30.0 \mathrm{~cm}$ can rotate about a vertical axis with negligible friction but is initially stationary. A horizontal rubber wheel of radius $2.00 \mathrm{~cm}$ is placed against its rim. That wheel is mounted on a motor. When the motor is switched on at time $t=0$, the rubber wheel undergoes a constant angular acceleration of $5.00 \mathrm{rad} / \mathrm{s}^{2}$. Its contact with the pottery wheel causes the pottery wheel to undergo an angular acceleration. When the pottery wheel reaches an angular speed of $5.00 \mathrm{rev} / \mathrm{s}$, the rubber wheel is pulled away from contact and thereafter the pottery wheel rotates at $5.00 \mathrm{rev} / \mathrm{s}$. From $t=0$ to $t=2.00 \mathrm{~min}$, how many full rotations does the pottery wheel make?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 7

The wheel in Fig. 10-20 has eight equally spaced spokes and a radius of $30 \mathrm{~cm}$. It is mounted on a fixed axle and is spinning at $2.5 \mathrm{rev} / \mathrm{s}$. You want to shoot a $20 \mathrm{~cm}$ long arrow parallel to this axle and through the wheel without hitting any of the spokes.
without hitting any of the spokes.

Donald Albin

Numerade Educator

Problem 8

The angular acceleration of a wheel is $\alpha=6.0 t^{4}-4.0 t^{2}$, with $\alpha$ in radians per second-squared and $t$ in seconds. At time $t=0$, the wheel has an angular velocity of $+2.5 \mathrm{rad} / \mathrm{s}$ and an angular position of $+1.5 \mathrm{rad}$. Write expressions for (a) the angular velocity ( $\mathrm{rad} / \mathrm{s}$ ) and (b) the angular position (rad) as functions of time (s).

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 9

In $5.00 \mathrm{~s}$, a $2.00 \mathrm{~kg}$ stone moves in a horizontal circle of radius $2.00 \mathrm{~m}$ from rest to an angular speed of $4.00 \mathrm{rev} / \mathrm{s}$. What are the stone's (a) average angular acceleration and (b) rotational inertia around the circle's center?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 10

Starting from rest, a disk rotates about its central axis with constant angular acceleration. In $5.0 \mathrm{~s}$, it rotates $20 \mathrm{rad}$. During that time, what are the magnitudes of (a) the angular acceleration and (b) the average angular velocity? (c) What is the instantaneous angular velocity of the disk at the end of the $5.0 \mathrm{~s} ?$ (d) With the angular acceleration unchanged, through what additional angle will the disk turn during the next $5.0 \mathrm{~s}$ ?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 11

Two identical disks $A$ and $B$ can spin around vertical axes. Disk $A$ is spinning with an initial angular speed of 40 rev/s when its rim touches initially stationary disk $B$, causing that disk to begin to spin. The rubbing at the contact point slows $A$ while speeding up $B$. The rate at which both disks change their angular speeds is $2.0 \mathrm{rev} / \mathrm{s}^{2}$. Find the time required for the two disks to reach the same angular speed.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 12

the angular speed of an automobile engine is increased at a constant rate from $1200 \mathrm{rev} / \mathrm{min}$ to $3200 \mathrm{rev} / \mathrm{min}$ in $12 \mathrm{~s}$. (a) What is its angular acceleration in revolutions per minute-squared? (b) How many revolutions does the engine make during this $12 \mathrm{~s}$ interval?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 13

A flywheel turns through 40 rev as it slows from an angular speed of $1.5 \mathrm{rad} / \mathrm{s}$ to a stop. (a) Assuming a constant angular acceleration, find the time for it to come to rest. (b) What is its angular acceleration? (c) How much time is required for it to complete the first 20 of the 40 revolutions?

Donald Albin

Numerade Educator

Problem 14

A disk rotates about its central axis starting from rest and accelerates with constant angular acceleration. At one time it is rotating at $10 \mathrm{rev} / \mathrm{s} ; 60$ revolutions later, its angular speed is $15 \mathrm{rev} / \mathrm{s}$. Calculate (a) the angular acceleration, (b) the time required to complete the 60 revolutions, (c) the time required to reach the 10 rev/s angular speed, and (d) the number of revolutions from rest until the time the disk reaches the $10 \mathrm{rev} / \mathrm{s}$ angular speed.

Donald Albin

Numerade Educator

Problem 15

Starting from rest, a wheel has constant $\alpha=3.0 \mathrm{rad} / \mathrm{s}^{2}$. During a certain $4.0 \mathrm{~s}$ interval, it turns through $120 \mathrm{rad}$. How much time did it take to reach that $4.0 \mathrm{~s}$ interval?

Alex Garger

Numerade Educator

Problem 16

A merry-go-round rotates from rest with an angular acceleration of $1.20 \mathrm{rad} / \mathrm{s}^{2}$. How long does it take to rotate through (a) the first $2.00$ rev and (b) the next $2.00$ rev?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 17

At $t=0$, a flywheel has an angular velocity of $4.7 \mathrm{rad} / \mathrm{s}$, a constant angular acceleration of $-0.25 \mathrm{rad} / \mathrm{s}^{2}$, and a reference line at $\theta_{0}=0$. (a) Through what maximum angle $\theta_{\max }$ will the reference line turn in the positive direction? What are the (b) first and (c) second times the reference line will be at $\theta=\frac{1}{2} \theta_{\max } ?$ At what (d) negative time and (e) positive time will the reference line be at $\theta=10.5 \mathrm{rad} ?$ (f) Graph $\theta$ versus $t$, and indicate your answers.

Salamat Ali

Numerade Educator

Problem 18

A pulsar is a rapidly rotating neutron star that emits a radio beam the way a lighthouse emits a light beam. We receive a radio pulse for each rotation of the star. The period $T$ of rotation is found by measuring the time between pulses. The pulsar in the Crab nebula has a period of rotation of $T=0.033 \mathrm{~s}$ that is increasing at the rate of $1.26 \times 10^{-5} \mathrm{~s} / \mathrm{y}$. (a) What is the pulsar's angular acceleration $\alpha ?$ (b) If $\alpha$ is constant, how many years from now will the pulsar stop rotating? (c) The pulsar originated in a supernova explosion seen in the year 1054 . Assuming constant $\alpha$, find the initial $T$.

Anand Jangid

Numerade Educator

Problem 19

An open jar of water moves in a vertical circle of radius $0.50 \mathrm{~m}$ with a frequency that is small enough to put the water on the verge of falling out of the jar at the top of the circle. If the same demonstration were repeated on Mars, where the gravitational acceleration is only $3.7 \mathrm{~m} / \mathrm{s}^{2}$, what is the change in the circling frequency to again put the water on the verge of falling out at the top point?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 20

An object rotates about a fixed axis, and the angular position of a reference line on the object is given by $\theta=0.40 e^{22}$, where $\theta$ is in radians and $t$ is in seconds. Consider a point on the object that is $6.0$ $\mathrm{cm}$ from the axis of rotation. At $t=0$, what are the magnitudes of the point's (a) tangential component of acceleration and (b) radial component of acceleration?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 21

Between 1911 and 1990 , the top of the leaning bell tower at Pisa, Italy, moved toward the south at an average rate of $1.2 \mathrm{~mm} / \mathrm{y}$. The tower is $55 \mathrm{~m}$ tall. In radians per second, what is the average angular speed of the tower's top about its base?

Alex Garger

Numerade Educator

Problem 22

Starting from rest at time $t=0$, a circus stunt man drives a motorbike on a horizontal circular track of radius $10.0 \mathrm{~m}$. His speed is given by $v=c t^{2}$, where $c=1.00 \mathrm{~m} / \mathrm{s}^{3}$. At $t=2.00 \mathrm{~s}$, what is the angle between his (total) acceleration vector and his radial acceleration vector?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 23

A flywheel with a diameter of $1.20 \mathrm{~m}$ is rotating at an angular speed of $200 \mathrm{rev} / \mathrm{min}$. (a) What is the angular speed of the flywheel in radians per second? (b) What is the linear speed of a point on the rim of the flywheel? (c) What constant angular acceleration (in revolutions per minute-squared) will increase the wheel's angular speed to $1000 \mathrm{rev} / \mathrm{min}$ in $60.0 \mathrm{~s}$ ? (d) How many revolutions does the wheel make during that $60.0 \mathrm{~s}$ ?

Donald Albin

Numerade Educator

Problem 24

A vinyl record is played by rotating the record so that an approximately circular groove in the vinyl slides under a stylus. Bumps in the groove run into the stylus, causing it to oscillate. The equipment converts those oscillations to electrical signals and then to sound. Suppose that a record turns at the rate of $33 \frac{1}{3} \mathrm{rev} / \mathrm{min}$, the groove being played is at a radius of $10.0 \mathrm{~cm}$, and the bumps in the groove are uniformly separated by $1.85 \mathrm{~mm}$. At what rate (hits per second) do the bumps hit the stylus?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 25

(a) What is the angular speed $\omega$ about the polar axis of a point on Earth's surface at latitude $40^{\circ} \mathrm{N} ?$ (Earth rotates about that axis.) (b) What is the linear speed $v$ of the point? What are (c) $\omega$ and (d) $v$ for a point at the equator?

Donald Albin

Numerade Educator

Problem 26

The flywheel of a steam engine runs with a constant angular velocity of $160 \mathrm{rev} / \mathrm{min}$. When steam is shut off, the friction of the bearings and of the air stops the wheel in $2.2 \mathrm{~h}$. (a) What is the constant angular acceleration, in revolutions per minute-squared, of the wheel during the slowdown? (b) How many revolutions does the wheel make before stopping? (c) At the instant the flywheel is turning at $75 \mathrm{rev} / \mathrm{min}$, what is the tangential component of the linear acceleration of a flywheel particle that is $50 \mathrm{~cm}$ from the axis of rotation? (d) What is the magnitude of the net linear acceleration of the particle in (c)?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 27

A seed is on a turntable rotating at $33 \frac{1}{3} \mathrm{rev} / \mathrm{min}, 6.0 \mathrm{~cm} \mathrm{from}$ the rotation axis. What are (a) the seed's acceleration and (b) the least coefficient of static friction to avoid slippage? (c) If the turntable had undergone constant angular acceleration from rest in $0.25 \mathrm{~s}$, what is the least coefficient to avoid slippage?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 28

In Fig. 10-21, wheel $A$ of radius $r_{A}=10 \mathrm{~cm}$ is coupled by belt $B$ to wheel $C$ of radius $r_{C}=25 \mathrm{~cm}$. The angular speed of wheel $A$ is increased from rest at a constant rate of $2.0 \mathrm{rad} / \mathrm{s}^{2}$. Find the time needed

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 29

Figure $10-22$ shows an early method of measuring the speed of light that makes use of a rotating slotted wheel. A beam of light passes through one of the slots at the outside edge of the wheel, travels to a distant mirror, and returns to the wheel just in time to pass through the next slot in the wheel. One such slotted wheel has a radius of $5.0 \mathrm{~cm}$ and 500 slots around its edge. Measurements taken when the mirror is $L=500 \mathrm{~m}$ from the wheel indicate a speed of light of $3.0 \times 10^{5} \mathrm{~km} / \mathrm{s}$. (a) What is the (constant) angular speed of the wheel? (b) What is the linear speed of a point on the edge of the wheel?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 30

A gyroscope flywheel of radius $2.62 \mathrm{~cm}$ is accelerated from rest at $14.2 \mathrm{rad} / \mathrm{s}^{2}$ until its angular speed is $2760 \mathrm{rev} / \mathrm{min}$. (a) What is the tangential acceleration of a point on the rim of the flywheel during this spin-up process? (b) What is the radial acceleration of this point when the flywheel is spinning at full speed? (c) Through what distance does a point on the rim move during the spin-up?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 31

A disk, with a radius of $0.25 \mathrm{~m}$, is to be rotated like a merrygo-round through 800 rad, starting from rest, gaining angular speed at the constant rate $\alpha_{1}$ through the first 400 rad and then losing angular speed at the constant rate $-\alpha_{1}$ until it is again at rest. The magnitude of the centripetal acceleration of any portion of the disk is not to exceed $400 \mathrm{~m} / \mathrm{s}^{2}$. (a) What is the least time required for the rotation? (b) What is the corresponding value of $\alpha_{1} ?$

Donald Albin

Numerade Educator

Problem 32

A car starts from rest and moves around a circular track of radius $32.0 \mathrm{~m}$. Its speed increases at the constant rate of $0.600 \mathrm{~m} / \mathrm{s}^{2}$.
(a) What is the magnitude of its net linear acceleration $15.0 \mathrm{~s}$ later?
(b) What angle does this net acceleration vector make with the car's velocity at this time?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 33

a) A uniform $2.00 \mathrm{~kg}$ disk of radius $0.300 \mathrm{~m}$ can rotate around its central axis like a merry-go-round. Beginning from resi at time $t=0$, it undergoes a constant angular acceleration of $30.0 \mathrm{rad} / \mathrm{s}^{2}$. When is the rotational kinetic energy equal to $2000 \mathrm{~J}$
(b) Repeat the calculation but substitute a ring of the same $\omega(\mathrm{rad} / \mathrm{s})$
substitute a ring of the same
mass and radius and assume that
the spokes have negligible mass.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 34

Figure $10-23$ gives angular speed versus time for a thin rod that rotates around one end. The scale on the $\omega$ axis is set by $\omega_{s}=6.0 \mathrm{rad} / \mathrm{s}$. (a) What is the magnitude of the rod's angular acceleration? (b) At $t=4.0 \mathrm{~s}$, the rod has a rotational kinetic the rod has a rotational kinetic energy of $1.60 \mathrm{~J}$. What is its kinetic energy at $t=0$ ?

Donald Albin

Numerade Educator

Problem 35

A meter stick of negligible mass can rotate about a vertical axis through a point at distance $x$ from the point marked " $0 " .$ A small block of mass $0.100 \mathrm{~kg}$ is glued at the mark of " 0 " and a small block of mass $0.500 \mathrm{~kg}$ is glued at the opposite end, at the mark of "1." The stick and blocks are to rotate with an angular speed of $5.00 \mathrm{rad} / \mathrm{min}$. (a) For what choice of $x$ is the associated rotational kinetic energy least and (b) what is that least energy?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 36

Figure $10-24 a$ shows a disk that can rotate about an axis at a radial distance $h$ from the center of the disk. Figure $10-24 b$ gives the rotational inertia $I$ of the disk about the axis as a function of that distance $h$, from the center out to the edge of the disk. The scale on the $I$ axis is set by $I_{A}=0.050 \mathrm{~kg} \cdot \mathrm{m}^{2}$ and $I_{B}=0.150 \mathrm{~kg} \cdot \mathrm{m}^{2}$. What is the mass of the disk?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 37

A $0.50 \mathrm{~kg}$ meter stick can rotate around an axis perpendicular to the stick. Find the difference in the stick's rotational inertia about the rotation axis if that axis is initially at the point marked "40 $\mathrm{cm}$ " and then at the point marked " $10 \mathrm{~cm} . "$

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 38

Figure $10-25$ shows three $0.0100 \mathrm{~kg}$ particles that have been glued to a rod of length $L=8.00 \mathrm{~cm}$ and negligible mass. The assembly can rotate around a perpendicular axis through point $O$ at the left end. If we remove one particle (that is, $33 \%$ of the mass), by what (that is, $33 \%$ of the mass), by what percentage does the rotational iner- $\quad|-d \rightarrow|-d \rightarrow \mid-d \rightarrow 1$ tia of the assembly around the rotaFigure 10-25 Problems tion axis decrease when that re38 and 62 . moved particle is (a) the innermost one and (b) the outermost one?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 39

A wheel with a rotational inertia of $0.50 \mathrm{~kg} \cdot \mathrm{m}^{2}$ about its central axis is initially rotating at an angular speed of $15 \mathrm{rad} / \mathrm{s}$. At time $t=0$, a man begins to slow it at a uniform rate until it stops at $t=$ $5.0 \mathrm{~s}$. (a) By time $t=3.0 \mathrm{~s}$, how much work had the man done?
(b) For the full $5.0 \mathrm{~s}$, at what average rate did the man do work?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 40

Figure $10-26$ shows an arrangement of 15 identical disks that have been glued together in a rod-like shape of length $L=1.0000 \mathrm{~m}$ and (total) mass $M=100.0 \mathrm{mg}$. The disks are uniform, and the disk arrangement can rotate about a perpendicular axis through its central disk at point $O$. (a) What is the rotational inertia of the arrangement about that axis? (b) If we approximated the arrangement as being a uniform rod of mass $M$ and length $L$, what percentage error would we make in using the formula in Table $10-2 e$ to calculate the rotational inertia?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 41

In Fig. 10-27, two particles, each with mass $m=0.85 \mathrm{~kg}$, are fastened to each other, and to a rotation axis at $O$, by two thin rods, each with length $d=5.6 \mathrm{~cm}$ and mass $M=1.2 \mathrm{~kg}$. The combination rotates around the rotation axis with the angular speed $\omega=0.30 \mathrm{rad} / \mathrm{s}$. Measured about $O$, what are the combination's (a) rotational inertia and (b) kinetic energy?

Mukesh Devi

Numerade Educator

Problem 42

\begin{aligned}
&42 . \text { Figure } 10-28 \text { is an overhead } \\
&\text { view of a rod of length } 1.0 \mathrm{~m} \text { and } \\
&\text { mass } 1.0 \mathrm{~kg} \text { that is lying station- } \\
&\text { ary on a frictionless surface } \\
&\text { when three bullets hit it simulta- } \\
&\text { neously. The bullets move along } \\
&\text { paths that are in the plane of the } \\
&\text { rod and perpendicular to the } \\
&\text { rod. Bullet } 1 \text { has mass } 10 \mathrm{~g} \text { and } \\
&\text { speed } 2.0 \mathrm{~m} / \mathrm{s} \text {. Bullet } 2 \text { has mass } 20 \mathrm{~g} \text { and speed } 3.0 \mathrm{~m} / \mathrm{s} \text {. Bullet } 3 \\
&\text { has mass } 30 \mathrm{~g} \text { and speed } 5.0 \mathrm{~m} / \mathrm{s} \text {. The labelled distances are } a= \\
&10 \mathrm{~cm}, b=60 \mathrm{~cm}, \text { and } c=80 \mathrm{~cm} \text {. As a result of the impacts, the } \\
&\text { rod-bullets system rotates around its center of mass while the } \\
&\text { center of mass moves in a straight line over the frictionless sur- } \\
&\text { face. (a) What is the linear speed of the system's center of mass? } \\
&\text { (b) What is the distance between the rod's center and the sys- } \\
&\text { tem's center of mass? (c) What is the rotational inertia of the } \\
&\text { system about the system's center of mass? }
\end{aligned}

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 43

The uniform solid block in Fig. 10-29 has mass $0.172 \mathrm{~kg}$ and edge lengths $a=3.5 \mathrm{~cm}, b=8.4 \mathrm{~cm}$, and $c=1.4 \mathrm{~cm}$. Calculate its rotational inertia about an axis through one corner and perpendicular to the large faces.

Donald Albin

Numerade Educator

Problem 44

Four identical particles of mass
Figure 10-29 Problem 43.
$0.75 \mathrm{~kg}$ each are placed at the ver-
tices of a $2.0 \mathrm{~m} \times 2.0 \mathrm{~m}$ square and
held there by four massless rods, which form the sides of the square. What is the rotational inertia of this rigid body about an axis that (a) passes through the midpoints of opposite sides and lies in the plane of the square, (b) passes through the midpoint of one of the sides and is perpendicular to the plane of the square, and (c) lies in the plane of the square and passes through two diagonally opposite particles?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 45

The body in Fig. $10-30$ is pivoted at $O$, and two forces act on it as shown. If $r_{1}=1.30 \mathrm{~m}$, $r_{2}=2.15 \mathrm{~m}, F_{1}=4.20 \mathrm{~N}, F_{2}=$ $4.90 \mathrm{~N}, \theta_{1}=75.0^{\circ}$, and $\theta_{2}=$ $60.0^{\circ}$, what is the net torque about the pivot?

Donald Albin

Numerade Educator

Problem 46

The body in Fig. 10-31 is pivoted at $O$. Three forces act on it: $F_{A}=12 \mathrm{~N}$ at point $A, 8.0 \mathrm{~m}$ from $O ; F_{B}=14 \mathrm{~N}$ at $B, 4.0 \mathrm{~m}$ from $O$; and $F_{c}=$ $23 \mathrm{~N}$ at $C, 3.0 \mathrm{~m}$ from $O .$ What is the net torque about $O ?$

Supratim Pal

Numerade Educator

Problem 47

quad \mathrm{~A} 60 \mathrm{~kg}$ father and $20 \mathrm{~kg}$ child sit on opposite ends of a
Figure 10-31 Problem $46 .$ seesaw consisting of a board of length $4.0 \mathrm{~m}$ and negligible mass. The pivot can be placed anywhere between the father and the child. At what distance from the child should it be placed so that the seesaw is balanced when the father and child are stationary?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 48

A $100 \mathrm{~kg}$ cubical box lies on a floor. A child pushes horizontally at a top edge. What force magnitude puts the box on the verge of tipping over if there is sufficient friction between it and the floor to prevent sliding?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 49

A cord with negligible mass is wrapped around a pulley that is a uniform disk of mass $5.00 \mathrm{~kg}$ and radius $0.300 \mathrm{~m}$ and that can rotate without friction about its central axis. A $1.0 \mathrm{~kg}$ bucket is attached at the free end of the cord hanging down from the pulley and then released at time $t=0$. The cord begins to unwrap from the pulley as the bucket descends. At $t=5.00 \mathrm{~s}$, through how many rotations has the pulley turned (the bucket is still descending)?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 50

If a $42.0 \mathrm{~N} \cdot \mathrm{m}$ torque on a wheel causes angular acceleration $25.0 \mathrm{rad} / \mathrm{s}^{2}$, what is the wheel's rotational inertia?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 51

In Fig. 10-32, block 1 has mass $m_{1}=460 \mathrm{~g}$, block 2 has mass $m_{2}=500 \mathrm{~g}$, and the pulley, which is mounted on a horizontal axle with negligible friction,

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 52

In Fig. 10-33, a cylinder having a mass of $3.0 \mathrm{~kg}$ can rotate about its central axis through point $O$. Forces are applied as shown: $F_{1}=6.0 \mathrm{~N}, F_{2}=4.0 \mathrm{~N}, F_{3}=2.0 \mathrm{~N}$, and $F_{4}=5.0 \mathrm{~N} . \mathrm{Also}$, $r=5.0 \mathrm{~cm}$ and $R=12 \mathrm{~cm}$. Find the (a) magnitude and (b) direction of the angular acceleration of the cylinder. (During the rotation, the forces maintain their same angles relative to the cylinder.)

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 53

Figure $10-34$ shows a uniform disk that can rotate around its center like a merry-go-round. The disk has a radius of $2.00 \mathrm{~cm}$ and a mass of $20.0$ grams and is initially at rest. Starting at time $t=0$, two forces are to be applied tangentially to the rim as indicated, so that at time $t=1.25 \mathrm{~s}$ Problem 53 the disk has an angular velocity of 250 rad/s counterclockwise. Force $\vec{F}_{1}$ has a magnitude of $0.100 \mathrm{~N}$. What is magnitude $F_{2}$ ?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 54

In a judo foot-sweep move, you sweep your opponent's left foot out from under him while pulling on his gi (uniform) toward that side. As a result, your opponent rotates around his right foot and onto the mat. Figure $10-35$ shows a simplified diagram of your opponent as you face him, with his left foot swept out. The rotational axis is through point $O$. The gravitational force $\vec{F}_{g}$ on him effectively acts at his center of mass, which is a horizontal distance $d=28 \mathrm{~cm}$ from point $O$. His mass is $75 \mathrm{~kg}$, and his rotational inertia about point $O$ is $65 \mathrm{~kg} \cdot \mathrm{m}^{2}$. What is the magnitude of his initial angular acceleration - Figure 10-35 his initial angular acceleration about point $O$ if your pull $\vec{F}_{a}$ on his gi is (a) negligible and (b) horizontal with a magnitude of $300 \mathrm{~N}$ and applied at height $h=1.4 \mathrm{~m}$ ?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 55

In Fig. 10-36a, an irregularly shaped plastic plate with uniform thickness and density (mass per unit volume) is to be rotated Axle around an axle that is perpendicular to the plate face and through point $O$. The rotational inertia of the plate about that axle is measured with the following method. A circular disk of mass $0.500 \mathrm{~kg}$ and radius $2.00 \mathrm{~cm}$ is glued to the plate, with its center aligned with point $O$ (Fig. $10-36 b$ ). A string is aligned with point $O$ (Fig. 10-36b). A string is String wrapped around the edge of the disk the way
(b) a string is wrapped around a top. Then the string is pulled for $5.00 \mathrm{~s}$. As a result, the disk
Figure 10-36 and plate are rotated by a constant force of $0.400 \mathrm{~N}$ that is applied by the string tangentially to the edge of the disk. The resulting angular speed is $114 \mathrm{rad} / \mathrm{s}$. What is the rotational inertia of the plate about the axle?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 56

Figure $10-37$ shows
particles 1 and 2 , each of particles 1 and 2 , each of mass $m$, fixed to the ends of mass $m$, fixed to the ends of a rigid massiess rod of 1 $\triangle$ length $L_{1}+L_{2}$, with $L_{1}=$ Figure 10-37 Problem $56 .$ $20 \mathrm{~cm}$ and $L_{2}=80 \mathrm{~cm}$. The rod is held horizontally on the fulcrum and then released. What are the magnitudes of the initial accelerations of (a) particle 1 and (b) particle 2 ?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 57

A pulley, with a rotational inertia of $1.0 \times 10^{-3} \mathrm{~kg} \cdot \mathrm{m}^{2}$ about its axle and a radius of $10 \mathrm{~cm}$, is acted on by a force applied tangentially at its rim. The force magnitude varies in time as $F=0.50 t+0.30 t^{2}$, with $F$ in newtons and $t$ in seconds. The pulley is initially at rest. At $t=3.0 \mathrm{~s}$ what are its (a) angular acceleration and (b) angular speed?

Salamat Ali

Numerade Educator

Problem 58

(a) If $R=15 \mathrm{~cm}, M=350 \mathrm{~g}$, and $m=50 \mathrm{~g}$ in Fig. $10-19$, find the speed of the block after it has descended $50 \mathrm{~cm}$ starting from rest. Solve the problem using energy conservation principles. (b) Repeat (a) with $R=5.0 \mathrm{~cm}$.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 59

A uniform metal pole of height $30.0 \mathrm{~m}$ and mass $100 \mathrm{~kg}$ is ini tially standing upright but then falls over to one side without it: lower end sliding or losing contact with the ground. What is the lin ear speed of the pole's upper end just before impact?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 60

A thin rod of length $0.75 \mathrm{~m}$ and mass $0.42 \mathrm{~kg}$ is suspended freely from one end. It is pulled to one side and then allowed to swing like a pendulum, passing through its lowest position with angular speed $3.5$ rad/s. Neglecting friction and air resistance, find (a) the rod's kinetic energy at its lowest position and (b) how far above that position the center of mass rises.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 61

Disks $A$ and $B$ each have a rotational inertia of $0.300 \mathrm{~kg} \cdot \mathrm{m}^{2}$ about the central axis and a radius of $20.0 \mathrm{~cm}$ and are free to rotate on a central rod through both of them. To set them spinning around the rod in the same direction, each is wrapped with a string that is then pulled for $10.0 \mathrm{~s}$ (the string detaches at the end). The magnitudes of the forces pulling the strings are $30.0 \mathrm{~N}$ for disk $A$ and $20.0 \mathrm{~N}$ for disk $B$. After the strings detach, the disks happen to collide and the frictional force between them brings them to the same final angular speed in $6.00 \mathrm{~s}$. What are (a) magnitude of the average frictional torque that brings them to the final angular speed and (b) the loss in kinetic energy as that torque acts on them? (c) Where did the "lost energy" go?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 62

62 In Fig. $10-25$, three $0.0100 \mathrm{~kg}$ particles have been glued to rod of length $L=6.00 \mathrm{~cm}$ and negligible mass and can rotate around a perpendicular axis through point $O$ at one end. How much work is required to change the rotational rate (a) from 0 to $20.0 \mathrm{rad} / \mathrm{s}$, (b) from $20.0 \mathrm{rad} / \mathrm{s}$ to $40.0 \mathrm{rad} / \mathrm{s}$, and (c) from $40.0 \mathrm{rad} / \mathrm{s}$ to $60.0 \mathrm{rad} / \mathrm{s} ?(\mathrm{~d})$ What is the slope of a plot of the assembly's kinetic energy (in joules) versus the square of its rotation rate (in radianssquared per second-squared)?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 63

A meter stick is held vertically with one end on the floor and is then allowed to fall. Find the speed of the other end just before it hits the floor, assuming that the end on the floor does not slip. (Hint: Consider the stick to be a thin rod and use the conservation of energy principle.)

Donald Albin

Numerade Educator

Problem 64

A uniform cylinder of radius $12 \mathrm{~cm}$ and mass $25 \mathrm{~kg}$ is mounted so as to rotate freely about a horizontal axis that is parallel to and $5.0 \mathrm{~cm}$ from the central longitudinal axis of the cylinder. (a) What is the rotational inertia of the cylinder about the axis of rotation? (b) If the cylinder is released from rest with its central longitudinal axis at the same height as the axis about which the cylinder rotates, what is the angular speed of the cylinder as it passes through its lowest position?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 65

A tall, cylindrical chimney falls over when its base is ruptured. Treat the chimney as a thin rod of length $55.0 \mathrm{~m}$. At the instant it makes an angle of $35.0^{\circ}$ with the vertical as it falls, what are (a) the radial acceleration of the top, and (b) the tangential acceleration of the top. (Hint: Use energy considerations, not a torque.) (c) At what angle $\theta$ is the tangential acceleration equal to $g$ ?

Salamat Ali

Numerade Educator

Problem 66

A uniform spherical shell of mass $M=4.5 \mathrm{~kg}$ and radius $R=8.5 \mathrm{~cm}$ can rotate about a vertical axis on frictionless bearings (Fig. 10-38). A massless cord passes around the equator of the shell, over a pulley of rotational inertia $I=3.0 \times 10^{-3} \mathrm{~kg} \cdot \mathrm{m}^{2}$ and radius $r=5.0 \mathrm{~cm}$, and is attached to a small object of mass $m=0.60 \mathrm{~kg}$. There is no friction on the pulley's axle; the cord does not slip on the pulley. What is the speed of the object when it has fallen $82 \mathrm{~cm}$ after being released from rest? Use energy considerations.

Averell Hause

Carnegie Mellon University

Problem 67

Figure $10-39$ shows a rigid assembly of a thin hoop (of mass $m$ and radius $R=0.150 \mathrm{~m}$ ) and a thin radial rod (of mass $m$ and length $L=2.00 R$ ). The assembly is upright, but if we give it a slight nudge, it will rotate around a horizontal axis in the plane of the rod and hoop, through the lower end of the rod. Assuming that the energy given to the assembly in such a nudge is negligible, what would be the assembly rotation axis when it passes through th orientation?

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator