Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 10

Rotation - all with Video Answers

Educators

Chapter Questions

Problem 1

A good baseball pitcher can throw a baseball toward home plate at $85 \mathrm{mi} / \mathrm{h}$ with a spin of 1800 rev/min. How many revolutions does the baseball make on its way to home plate? For simplicity, assume that the $60 \mathrm{ft}$ path is a straight line.

Vishal Gupta

Vishal Gupta

Numerade Educator

Problem 2

What is the angular speed of (a) the second hand, (b) the minute hand, and (c) the hour hand of a smoothly running analog watch? Answer in radians per second.

Averell Hause

Carnegie Mellon University

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 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 $A t$ $t=0$, what are (a) the point's angular position and (b) its angular velocity? (c) What is its angular velocity at $t=4.0 \mathrm{~s}$ ? (d) Calculate its angular acceleration at $t=2.0 \mathrm{~s}$ (e) Is its angular acceleration constant?

Donald Albin

Numerade Educator

Problem 5

A diver makes $2.5$ revolutions on the way from a $10-\mathrm{m}$ -high platform to the water. Assuming zero initial vertical velocity, find the average angular velocity during the dive.

Donald Albin

Numerade Educator

Problem 6

The angular position of a point on the rim of a rotating wheel is given by $\theta=4.0 t-3.0 t^{2}+t^{3}$, where $\theta$ is in radians and $t$ is in seconds. What are the angular velocities at (a) $t=2.0 \mathrm{~s}$ and $(\mathrm{b}) t=4.0 \mathrm{~s} ?$
(c) What is the average angular acceleration for the time interval that begins at $t=2.0 \mathrm{~s}$ and ends at $t=4.0 \mathrm{~s}$ ? What are the instantaneous angular accelerations at (d) the beginning and (e) the end of this time interval?

Donald Albin

Numerade Educator

Problem 7

The wheel in Fig. $10-30$ has eight equally spaced spokes and a radius of $30 \mathrm{~cm}$. It is mounted on a fixed axle and is spinning at $2.5$ rey/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. Assume that the arrow and the spokes are very thin. (a) What minimum speed must the arrow have?
(b) Does it matter where between the axle and rim of the wheel you aim? If so, what is the best location?

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 $+2.0 \mathrm{rad} / \mathrm{s}$ and an angular position of $+1.0$ rad. Write expressions for (a) the angular velocity (rad/s) and
(b) the angular position (rad) as functions of time (s).

Averell Hause

Carnegie Mellon University

Problem 9

A drum rotates around its central axis at an angular velocity of $12.60 \mathrm{rad} / \mathrm{s}$. If the drum then slows at a constant rate of $4.20$ $\mathrm{rad} / \mathrm{s}^{2}$, (a) how much time does it take and (b) through what angle does it rotate in coming to rest?

Donald Albin

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 $25 \mathrm{rad}$. 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 $5.0 \mathrm{~s}$ ?

Averell Hause

Carnegie Mellon University

Problem 11

A disk, initially rotating at $120 \mathrm{rad} / \mathrm{s}$, is slowed down with a constant angular acceleration of magnitude $4.0 \mathrm{rad} / \mathrm{s}^{2} .$ (a) How much time does the disk take to stop? (b) Through what angle does the disk rotate during that time?

Salamat Ali

Numerade Educator

Problem 12

The angular speed of an automobile engine is increased at a constant rate from 1200 rev/min to 3000 rev/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 sinterval?

Donald Albin

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 rev/s. Calculate (a) the angular acceleration, (b) the time required to complete the 60 revolutions, (c) the time required to reach the $10 \mathrm{rev} / \mathrm{s}$ angular speed, and (d) the number of revolutions from rest until the time the disk reaches the 10 rev/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 $4.0$ s interval, it turns through 120 rad. How much time did it take to reach that $4.0$ sinterval?

Donald Albin

Numerade Educator

Problem 16

A merry-go-round rotates from rest with an angular acceleration of $1.50 \mathrm{rad} / \mathrm{s}^{2} .$ How long does it take to rotate through
(a) the first $2.00$ rev and (b) the next $2.00$ rev?

Averell Hause

Carnegie Mellon University

Problem 17

At $t=0$, a flywheel has an angular velocity of $4.7 \mathrm{rad} / \mathrm{s}, \mathrm{a}$ 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 $T$.

Averell Hause

Carnegie Mellon University

Problem 19

What are the magnitudes of (a) the angular velocity, (b) the radial acceleration, and (c) the tangential acceleration of a spaceship taking a circular turn of radius $3220 \mathrm{~km}$ at a speed of $29000 \mathrm{~km} / \mathrm{h}$ ?

Donald Albin

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 $\theta=0.40 e^{2 t}$, where $\theta$ is in radians and $t$ is in seconds. Consider a point on the object that is $4.0 \mathrm{~cm}$ 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?

Averell Hause

Carnegie Mellon University

Problem 21

Between 1911 and 1990 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?

Donald Albin

Numerade Educator

Problem 22

An astronaut is tested in a centrifuge with radius $10 \mathrm{~m}$ and rotating according to $\theta=0.30 t^{2}$. At $t=5.0 \mathrm{~s}$, what are the magnitudes of the (a) angular velocity, (b) linear velocity, (c) tangential acceleration, and (d) radial acceleration?

Averell Hause

Carnegie Mellon University

Problem 23

A flywheel with a diameter of $1.20 \mathrm{~m}$ is rotating at an angular speed of 200 rev/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 rev/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 $33 \frac{1}{3}$ rev/min, the groove being played is at a radius of $10.0 \mathrm{~cm}$, and the bumps in the groove are uniformly separated by $1.75 \mathrm{~mm}$. At what rate (hits per second) do the bumps hit the stylus?

Averell Hause

Carnegie Mellon University

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 $(\mathrm{d}) v$ 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 150 rev/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 revimin, 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)?

Donald Albin

Numerade Educator

Problem 27

A seed is on a turntable rotating at $33 \frac{1}{3} \mathrm{rev} / \mathrm{min}, 6.0 \mathrm{~cm}$ 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?

Salamat Ali

Numerade Educator

Problem 28

In Fig. 10-31, 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 $1.6 \mathrm{rad} / \mathrm{s}^{2}$. Find the time needed for wheel $C$ to reach an angular speed of 100 rev/min, assuming the belt does not slip. (Hint: If the belt does not slip, the linear speeds at the two rims must be equal.)

Donald Albin

Numerade Educator

Problem 29

Figure $10-32$ 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?

Salamat Ali

Numerade Educator

Problem 30

A gyroscope flywheel of radius $2.83 \mathrm{~cm}$ is accelerated from rest at $14.2 \mathrm{rad} / \mathrm{s}^{2}$ until its angular speed is 2760 rev/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?

Averell Hause

Carnegie Mellon University

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 $400 \mathrm{rad}$ 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 $30.0 \mathrm{~m}$. Its speed increases at the constant rate of $0.500 \mathrm{~m} / \mathrm{s}^{2}$.
(a) What is the magnitude of its net linear acceleration $15.0$ s later?
(b) What angle does this net acceleration vector make with the car's velocity at this time?

Averell Hause

Carnegie Mellon University

Problem 33

Calculate the rotational inertia of a wheel that has a kinetic energy of $24400 \mathrm{~J}$ when rotating at 602 rev/min.

Donald Albin

Numerade Educator

Problem 34

Figure $10-33$ 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 energy of $1.60 \mathrm{~J}$. What is its kinetic energy at $t=0$ ?

Donald Albin

Numerade Educator

Problem 35

Two uniform solid cylinders, each rotating about its central (longitudinal) axis at $235 \mathrm{rad} / \mathrm{s}$, have the same mass of $1.25 \mathrm{~kg}$ but differ in radius. What is the rotational kinetic energy of (a) the smaller cylinder, of radius $0.25 \mathrm{~m}$, and $(\mathrm{b})$ the larger cylinder, of radius $0.75 \mathrm{~m} ?$

Donald Albin

Numerade Educator

Problem 36

Figure $10-34 a$ shows a disk that can rotate about an axis at. a radial distance $h$ from the center of the disk. Figure $10-34 b$ 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?

Averell Hause

Carnegie Mellon University

Problem 37

Calculate the rotational inertia of a meter stick, with mass $0.56 \mathrm{~kg}$, about an axis perpendicular to the stick and located at the $20 \mathrm{~cm}$ mark. (Treat the stick as a thin rod.)

Donald Albin

Numerade Educator

Problem 38

Figure $10-35$ shows three $0.0100 \mathrm{~kg}$ particles that have been glued to a rod of length $L=6.00 \mathrm{~cm}$ 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 percentage does the rotational inertia of the assembly around the rotation axis decrease when that removed particle is (a) the innermost one and (b) the outermost one?

Donald Albin

Numerade Educator

Problem 39

Trucks can be run on energy stored in a rotating flywheel, with an electric motor getting the flywheel up to its top speed of $200 \pi \mathrm{rad} / \mathrm{s}$. Suppose that one such flywheel is a solid, uniform cylinder with a mass of $500 \mathrm{~kg}$ and a radius of $1.0 \mathrm{~m}$. (a) What is the kinetic energy of the flywheel after charging? (b) If the truck uses an average power of $8.0 \mathrm{~kW}$, for how many minutes can it operate between chargings?

Donald Albin

Numerade Educator

Problem 40

Figure $10-36$ 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?

Averell Hause

Carnegie Mellon University

Problem 41

In Fig. $10-37$, 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

The masses and coordinates of four particles are as follows: $50 \mathrm{~g}, x=2.0 \mathrm{~cm}, y=2.0 \mathrm{~cm} ; 25 \mathrm{~g}, x=0, y=4.0 \mathrm{~cm} ; 25 \mathrm{~g}$, $x=-3.0 \mathrm{~cm}, y=-3.0 \mathrm{~cm} ; 30 \mathrm{~g}, x=-2.0 \mathrm{~cm}, y=4.0 \mathrm{~cm} .$ What
are the rotational inertias of this collection about the (a) $x$, (b) $y$, and (c) $z$ axes? (d) Suppose that we symbolize the answers to (a) and (b) as $A$ and $B$, respectively. Then what is the answer to (c) in terms of $A$ and $B ?$

Donald Albin

Numerade Educator

Problem 43

The uniform solid block in Fig. $10-38$ 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 $0.50 \mathrm{~kg}$ each are placed at the vertices 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?

Donald Albin

Numerade Educator

Problem 45

The body in Fig. $10-39$ 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. $10-40$ is pivoted at $O .$ Three forces act on it: $F_{A}=10 \mathrm{~N}$ at point $A, 8.0$ $\mathrm{m}$ from $O ; F_{B}=16 \mathrm{~N}$ at $B, 4.0$ $\mathrm{m}$ from $\mathrm{O} ;$ and $F_{C}=19 \mathrm{~N}$ at $C$ $3.0 \mathrm{~m}$ from $O$. What is the net torque about $O$ ?

Donald Albin

Numerade Educator

Problem 47

A small ball of mass $0.75 \mathrm{~kg}$ is attached to one end of a $1.25-\mathrm{m}$ -long massless rod, and the other end of the rod is hung from a pivot. When the resulting pendulum is $30^{\circ}$ from the vertical, what is the magnitude of the gravitational torque calculated about the pivot?

Donald Albin

Numerade Educator

Problem 48

The length of a bicycle pedal arm is $0.152 \mathrm{~m}$, and a downward force of $111 \mathrm{~N}$ is applied to the pedal by the rider. What is the magnitude of the torque about the pedal arm's pivot when the arm is at angle (a) $30^{\circ}$, (b) $90^{\circ}$, and
(c) $180^{\circ}$ with the vertical?

Averell Hause

Carnegie Mellon University

Problem 49

During the launch from a board, a diver's angular speed about her center of mass changes from zero to $6.20 \mathrm{rad} / \mathrm{s}$ in 220 . $\mathrm{ms}$ Her rotational inertia about her center of mass is $12.0 \mathrm{~kg} \cdot \mathrm{m}^{2} .$ During the launch, what are the magnitudes of (a) her average angular acceleration and (b) the average external torque on her from the board?

Donald Albin

Numerade Educator

Problem 50

If a $32.0 \mathrm{~N} \cdot \mathrm{m}$ torque on a wheel causes angular acceleration $25.0 \mathrm{rad} / \mathrm{s}^{2}$, what is the wheel's rotational inertia?

Donald Albin

Numerade Educator

Problem 51

In Fig. $10-41$, 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, has radius $R=5.00 \mathrm{~cm} .$ When released from rest, block 2 falls $75.0 \mathrm{~cm}$ in $5.00 \mathrm{~s}$ without the cord slipping on the pulley. (a) What is the magnitude of the acceleration of the blocks? What are (b) tension $T_{2}$ and (c) tension $T_{1}$ ? (d) What is the magnitude of the pulley's angular acceleration? (e) What is its rotational inertia?

Donald Albin

Numerade Educator

Problem 52

In Fig. 10-42, a cylinder having a mass of $2.0 \mathrm{~kg}$ 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 $F_{4}=5.0 \mathrm{~N}$. 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.)

Donald Albin

Numerade Educator

Problem 53

Figure $10-43$ 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 $t=1.25 \mathrm{~s}$ the disk has an angular velocity of 250 $\mathrm{rad} / \mathrm{s}$ counterclockwise. Force $\bar{F}$ has a magnitude of $0.100 \mathrm{~N}$. What is magnitude $F_{2}$ ?

Donald Albin

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 $10-44$ 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 $70 \mathrm{~kg}$, and his rotational inertia about point $O$ is $65 \mathrm{~kg} \cdot \mathrm{m}^{2}$. What is the magnitude of his initial angular acceleration 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 $h=1.4 \mathrm{~m}$ ?

Donald Albin

Numerade Educator

Problem 55

In Fig. $10-45 a$, an irregularly shaped plastic plate with uniform thickness and density (mass per unit volume) is to be rotated 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. $10-45 b) .$ A string is wrapped around the edge of the disk the way a string is wrapped around a top. Then the string is pulled for $5.00 \mathrm{~s}$. As a result, the disk 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?

Donald Albin

Numerade Educator

Problem 56

Figure $10-46$ shows particles 1 and 2, each of mass $m$, fixed to the ends of a rigid massless rod of length $L_{1}+$ $L_{2}$, with $L_{1}=20 \mathrm{~cm}$ and $\overline{L_{2}}=$ $80 \mathrm{~cm} .$ 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 ?

Donald Albin

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 $F=0.50 t+0.30 r^{2}$, with Fin 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?

Donald Albin

Numerade Educator

Problem 58

(a) If $R=12 \mathrm{~cm}, M=400 \mathrm{~g}$, 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 $R=5.0 \mathrm{~cm}$.

Susan Hallstrom

Susan Hallstrom

Numerade Educator

Problem 59

An automobile crankshaft transfers energy from the engine to the axle at the rate of $100 \mathrm{hp}(=74.6 \mathrm{~kW})$ when rotating at $\mathrm{a}$ speed of 1800 rev/min. What torque (in newton-meters) does the crankshaft deliver?

Salamat Ali

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 $4.0 \mathrm{rad} / \mathrm{s}$. 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.

Averell Hause

Carnegie Mellon University

Problem 61

A $32.0 \mathrm{~kg}$ wheel, essentially a thin hoop with radius $1.20 \mathrm{~m}$, is rotating at 280 rev/min. It must be brought to a stop in $15.0 \mathrm{~s}$.
(a) How much work must be done to stop it?
(b) What is the required average power?

Salamat Ali

Numerade Educator

Problem 62

In Fig. $10-35$, three $0.0100 \mathrm{~kg}$ particles have been glued to a 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
$60.0 \mathrm{rad} / \mathrm{s} ?$ (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)?

Donald Albin

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 $10 \mathrm{~cm}$ and mass $20 \mathrm{~kg}$ 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?

Donald Albin

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. $10-47$ ). 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 $10-48$ 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's angular speed about the rotation axis when it passes through the upside-down (inverted) orientation?

Salamat Ali

Numerade Educator

Problem 68

Two uniform solid spheres have the same mass of $1.65 \mathrm{~kg}$, but one has a radius of $0.226 \mathrm{~m}$ and the other has a radius of $0.854 \mathrm{~m}$. Each can rotate about an axis through its center. (a) What is the magnitude $\tau$ of the torque required to bring the smaller sphere from rest to an angular speed of $317 \mathrm{rad} / \mathrm{s}$ in $15.5 \mathrm{~s}$ ? (b) What is the magnitude $F$ of the force that must be applied tangentially at the sphere's equator to give that torque? What are the corresponding values of (c) $\tau$ and (d) $F$ for the larger sphere?

Donald Albin

Numerade Educator

Problem 69

In Fig. $10-49$, a small disk of radius $r=2.00 \mathrm{~cm}$ has been glued to the edge of a larger disk of radius $R=4.00 \mathrm{~cm}$ so that the disks lie in the same plane. The disks can be rotated around a perpendicular axis through point $O$ at the center of the larger disk. The disks both have a uniform density (mass per unit volume) of $1.40 \times$ $10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ and a uniform thickness of $5.00 \mathrm{~mm}$. What is the rotational inertia of the two-disk assembly about the rotation axis through $O$ ?

Donald Albin

Numerade Educator

Problem 70

A wheel, starting from rest, rotates with a constant angular acceleration of $2.00 \mathrm{rad} / \mathrm{s}^{2}$. During a certain $3.00 \mathrm{~s}$ interval, it turns through $90.0$ rad. (a) What is the angular velocity of the wheel at the start of the $3.00 \mathrm{~s}$ interval? (b) How long has the wheel been turning before the start of the $3.00$ s interval?

Donald Albin

Numerade Educator

Problem 71

In Fig. $10-50$, two $6.20 \mathrm{~kg}$ blocks are connected by a massless string over a pulley of radius $2.40 \mathrm{~cm}$ and rotational inertia $7.40 \times 10^{-4}$ $\mathrm{kg} \cdot \mathrm{m}^{2} .$ The string does not slip on the pulley; it is not known whether there is friction between the table and the sliding block; the pulley's axis is frictionless. When this system is released from rest, the pulley turns through $0.130$ rad in $91.0 \mathrm{~ms}$ and the acceleration of the blocks is constant. What are (a) the magnitude of the pulley's angular acceleration, (b) the magnitude of either block's acceleration, (c) string tension $T_{1}$, and (d) string tension $T_{2}$ ?

Donald Albin

Numerade Educator

Problem 72

Attached to each end of a thin steel rod of length $1.20 \mathrm{~m}$ and mass $6.40 \mathrm{~kg}$ is a small ball of mass $1.06 \mathrm{~kg} .$ The rod is constrained to rotate in a horizontal plane about a vertical axis through its midpoint. At a certain instant, it is rotating at $39.0 \mathrm{rev} / \mathrm{s}$. Because of friction, it slows to a stop in $32.0$ s Assuming a constant retarding torque due to friction, compute (a) the angular acceleration, (b) the retarding torque, (c) the total energy transferred from mechanical energy to thermal energy by friction, and (d) the number of revolutions rotated during the $32.0 \mathrm{~s}$ (e) Now suppose that the retarding torque is known not to be constant. If any of the quantities (a), (b), (c), and (d) can still be computed without additional information, give its value.

Donald Albin

Numerade Educator

Problem 73

A uniform helicopter rotor blade is $7.80 \mathrm{~m}$ long, has a mass of $110 \mathrm{~kg}$, and is attached to the rotor axle by a single bolt. (a) What is the magnitude of the force on the bolt from the axle when the rotor is turning at 320 rev/min? (Hint: For this calculation the blade can be considered to be a point mass at its center of mass. Why?)
(b) Calculate the torque that must be applied to the rotor to bring it to full speed from rest in $6.70 \mathrm{~s}$. Ignore air resistance. (The blade cannot be considered to be a point mass for this calculation. Why not? Assume the mass distribution of a uniform thin rod.) (c) How much work does the torque do on the blade in order for the blade to reach a speed of 320 rev/min?

Donald Albin

Numerade Educator

Problem 74

Racing disks. Figure $10-51$ shows two disks that can rotate about their centers like a merry-go-round. At time $t=0$, the reference lines of the two disks have the same orientation. Disk $A$ is already rotating, with a constant angular velocity of $9.5 \mathrm{rad} / \mathrm{s}$.
Disk $B$ has been stationary but now begins to rotate at a constant angular acceleration of $2.2 \mathrm{rad} / \mathrm{s}^{2} .(\mathrm{a})$ At what time $t$ will the reference lines of the two disks momentarily have the same angular displacement $\theta$ ? (b) Will that time $t$ be the first time since $t=0$ that the reference lines are momentarily aligned?

Donald Albin

Numerade Educator

Problem 75

A high-wire walker always attempts to keep his center of mass over the wire (or rope). He normally carries a long, heavy pole to help: If he leans, say, to his right (his com moves to the right) and is in danger of rotating around the wire, he moves the pole to his left (its com moves to the left) to slow the rotation and allow himself time to adjust his balance. Assume that the walker has a mass of $70.0 \mathrm{~kg}$ and a rotational inertia of $15.0 \mathrm{~kg} \cdot \mathrm{m}^{2}$ about the wire. What is the magnitude of his angular acceleration about the wire if his com is $5.0 \mathrm{~cm}$ to the right of the wire and (a) he carries no pole and (b) the $14.0 \mathrm{~kg}$ pole he carries has its $\operatorname{com} 10 \mathrm{~cm}$ to the left of the wire?

Salamat Ali

Numerade Educator

Problem 76

Starting from rest at $t=0$, a wheel undergoes a constant angular acceleration. When $t=2.0 \mathrm{~s}$, the angular velocity of the wheel is $5.0 \mathrm{rad} / \mathrm{s}$. The acceleration continues until $t=20 \mathrm{~s}$, when it abruptly ceases. Through what angle does the wheel rotate in the interval $t=0$ to $t=40 \mathrm{~s}$ ?

Donald Albin

Numerade Educator

Problem 77

A record turntable rotating at $33 \frac{1}{3}$ rev/min slows down and stops in $30 \mathrm{~s}$ after the motor is turned off. (a) Find its (constant) angular acceleration in revolutions per minute-squared.
(b) How many revolutions does it make in this time?

Donald Albin

Numerade Educator

Problem 78

A rigid body is made of three identical thin rods, each with length $L=0.600 \mathrm{~m}$, fastened together in the form of a letter $\mathbf{H}$ (Fig. $10-52$ ). The body is free to rotate about a horizontal axis that runs along the length of one of the legs of the $\mathbf{H}$. The body is allowed to fall from rest from a position in which the plane of the $\mathbf{H}$ is horizontal. What is the angular speed of the body when the plane of the $\mathbf{H}$ is vertical?

Donald Albin

Numerade Educator

Problem 79

(a) Show that the rotational inertia of a solid cylinder of mass $M$ and radius $R$ about its central axis is equal to the rotational inertia of a thin hoop of mass $M$ and radius $R / \sqrt{2}$ about its central axis. (b) Show that the rotational inertia $I$ of any given body of mass $M$ about any given axis is equal to the rotational inertia of an equivalent hoop about that axis, if the hoop has the same mass $M$ and a radius $k$ given by $$
k=\sqrt{\frac{I}{M}}
$$
The radius $k$ of the equivalent hoop is called the radius of gyration of the given body.

Donald Albin

Numerade Educator

Problem 80

A disk rotates at constant angular acceleration, from angular position $\theta_{1}=10.0$ rad to angular position $\theta_{2}=70.0$ rad in $6.00 \mathrm{~s}$. Its angular velocity at $\theta_{2}$ is $15.0 \mathrm{rad} / \mathrm{s}$. (a) What was its angular velocity at $\theta_{1} ?$ (b) What is the angular acceleration? (c) At what angular position was the disk initially at rest? (d) Graph $\theta$ versus time $t$ and angular speed $\omega$ versus $t$ for the disk, from the beginning of the motion (let $t=0$ then $)$.

Averell Hause

Carnegie Mellon University

Problem 81

The thin uniform rod in Fig. $10-53$ has length $2.0 \mathrm{~m}$ and can pivot about a horizontal, frictionless pin through one end. It is released from rest at angle $\theta=40^{\circ}$ above the horizontal. Use the principle of conservation of energy to determine the angular speed of the rod as it passes through the horizontal position.

Salamat Ali

Numerade Educator

Problem 82

George Washington Gale Ferris, Jr., a civil engineering graduate from Rensselaer Polytechnic Institute, built the original Ferris wheel for the 1893 World's Columbian Exposition in Chicago. The wheel, an astounding engineering construction at the time, carried 36 wooden cars, each holding up to 60 passengers, around a circle $76 \mathrm{~m}$ in diameter. The cars were loaded 6 at a time, and once all 36 cars were full, the wheel made a complete rotation at constant angular speed in about $2 \mathrm{~min}$. Estimate the amount of work that was required of the machinery to rotate the passengers alone.

Averell Hause

Carnegie Mellon University

Problem 83

In Fig. $10-41$, two blocks, of mass $m_{1}=400 \mathrm{~g}$ and $m_{2}=600 \mathrm{~g}$, are connected by a massless cord that is wrapped around a uniform disk of mass $M=500 \mathrm{~g}$ and radius $R=12.0 \mathrm{~cm}$. The disk can rotate without friction about a fixed horizontal axis through its center; the cord cannot slip on the disk. The system is released from rest. Find (a) the magnitude of the acceleration of the blocks, (b) the tension $T_{1}$ in the cord at the left, and (c) the tension $T_{2}$ in the cord at the right.

Salamat Ali

Numerade Educator

Problem 84

At $7: 14$ A.M. on June 30,1908 , a huge explosion occurred above remote central Siberia, at latitude $61^{\circ} \mathrm{N}$ and longitude $102^{\circ} \mathrm{E} ;$ the fireball thus created was the brightest flash seen by anyone before nuclear weapons. The Tunguska Event, which according to one chance witness "covered an enormous part of the sky," was probably the explosion of a stony asteroid about 140 $\mathrm{m}$ wide. (a) Considering only Earth's rotation, determine how much later the asteroid would have had to arrive to put the explosion above Helsinki at longitude $25^{\circ} \mathrm{E}$. This would have obliterated the city. (b) If the asteroid had, instead, been a metallic asteroid, it could have reached Earth's surface. How much later would such an asteroid have had to arrive to put the impact in the Atlantic Ocean at longitude $20^{\circ} \mathrm{W} ?$ (The resulting tsunamis would have wiped out coastal civilization on both sides of the Atlantic.)

Averell Hause

Carnegie Mellon University

Problem 85

A golf ball is launched at an angle of $20^{\circ}$ to the horizontal, with a speed of $60 \mathrm{~m} / \mathrm{s}$ and a rotation rate of $90 \mathrm{rad} / \mathrm{s}$. Neglecting air drag, determine the number of revolutions the ball makes by the time it reaches maximum height.

Salamat Ali

Numerade Educator

Problem 86

Figure $10-54$ shows a flat construction of two circular rings that have a common center and are held together by three rods of negligible mass. The construction, which is initially at rest, can rotate around the common center (like a merrygo-round), where another rod of negligible mass lies. The mass, inner radius, and outer radius of the rings are given in the following table. A tangential force of magnitude $12.0 \mathrm{~N}$ is applied to the outer edge of the outer ring for $0.300 \mathrm{~s}$. What is the change in the angular speed of the construction during the time interval?
$$
\begin{array}{cccc}
\hline \text { Ring } & \text { Mass }(\mathrm{kg}) & \text { Inner Radius }(\mathrm{m}) & \text { Outer Radius }(\mathrm{m}) \\
\hline 1 & 0.120 & 0.0160 & 0.0450 \\
2 & 0.240 & 0.0900 & 0.1400 \\
\hline
\end{array}
$$

Donald Albin

Numerade Educator

Problem 87

In Fig. $10-55$, a wheel of radius $0.20 \mathrm{~m}$ is mounted on a frictionless horizontal axle. A massless cord is wrapped around the wheel and attached to a $2.0 \mathrm{~kg}$ box that slides on a frictionless surface inclined at angle $\theta=20^{\circ}$ with the horizontal. The box accelerates down the surface at $2.0 \mathrm{~m} / \mathrm{s}^{2} .$ What is the rotational inertia of the wheel about the axle?

Salamat Ali

Numerade Educator

Problem 88

A thin spherical shell has a radius of $1.90 \mathrm{~m}$. An applied torque of $960 \mathrm{~N} \cdot \mathrm{m}$ gives the shell an angular acceleration of $6.20 \mathrm{rad} / \mathrm{s}^{2}$ about an axis through the center of the shell. What are (a) the rotational inertia of the shell about that axis and (b) the mass of the shell?

Averell Hause

Carnegie Mellon University

Problem 89

A bicyclist of mass $70 \mathrm{~kg}$ puts all his mass on each downwardmoving pedal as he pedals up a steep road. Take the diameter of the circle in which the pedals rotate to be $0.40 \mathrm{~m}$, and determine the magnitude of the maximum torque he exerts about the rotation axis of the pedals.

Donald Albin

Numerade Educator

Problem 90

The flywheel of an engine is rotating at $25.0 \mathrm{rad} / \mathrm{s}$. When the engine is turned off, the flywheel slows at a constant rate and stops in $20.0 \mathrm{~s}$. Calculate (a) the angular acceleration of the flywheel,
(b) the angle through which the flywheel rotates in stopping, and
(c) the number of revolutions made by the flywheel in stopping.

Averell Hause

Carnegie Mellon University

Problem 91

In Fig. $10-19 a$, a wheel of radius $0.20 \mathrm{~m}$ is mounted on a frictionless horizontal axis The rotational inertia of the wheel about the axis is $0.40 \mathrm{~kg} \cdot \mathrm{m}^{2}$. A massless cord wrapped around the wheel's circumference is attached to a $6.0 \mathrm{~kg}$ box. The system is released from rest. When the box has a kinetic energy of $6.0 \mathrm{~J}$, what are (a) the wheel's rotational kinetic energy and (b) the distance the box has fallen?

Donald Albin

Numerade Educator

Problem 92

Our Sun is $2.3 \times 10^{4}$ ly (light-years) from the center of our Milky Way galaxy and is moving in a circle around that center at a speed of $250 \mathrm{~km} / \mathrm{s}$. (a) How long does it take the Sun to make one revolution about the galactic center? (b) How many revolutions has the Sun completed since it was formed about $4.5 \times 10^{9}$ years ago?

Averell Hause

Carnegie Mellon University

Problem 93

A wheel of radius $0.20 \mathrm{~m}$ is mounted on a frictionless horizontal axis. The rotational inertia of the wheel about the axis is $0.050 \mathrm{~kg} \cdot \mathrm{m}^{2}$. A massless cord wrapped around the wheel is attached to a $2.0 \mathrm{~kg}$ block that slides on a horizontal frictionless surface. If a horizontal force of magnitude $P=3.0 \mathrm{~N}$ is applied to the block as shown in Fig. $10-56$, what is the magnitude of the angular acceleration of the wheel? Assume the cord does not slip on the wheel.

Donald Albin

Numerade Educator

Problem 94

If an airplane propeller rotates at 2000 rev/min while the airplane flies at a speed of $480 \mathrm{~km} / \mathrm{h}$ relative to the ground, what is the linear speed of a point on the tip of the propeller, at radius $1.5 \mathrm{~m}$, as seen by (a) the pilot and (b) an observer on the ground? The plane's velocity is parallel to the propeller's axis of rotation.

Prabhu Ramji

Numerade Educator

Problem 95

The rigid body shown in Fig. $10-57$ consists of three particles connected by massless rods. It is to be rotated about an axis perpendicular to its plane through point $P .$ If $M=$ $0.40 \mathrm{~kg}, a=30 \mathrm{~cm}$, and $b=50 \mathrm{~cm}$
how much work is required to take the body from rest to an angular speed of $5.0 \mathrm{rad} / \mathrm{s} ?$

Salamat Ali

Numerade Educator

Problem 96

The pull tab was a major advance in the engineering design of beverage containers. The tab pivots on a central bolt in the can's top. When you pull upward on one end of the tab, the other end presses downward on a portion of the can's top that has been scored. If you pull upward with a $10 \mathrm{~N}$ force, what force magnitude acts on the scored section? (You will need to examine a can with a pull tab.)

Averell Hause

Carnegie Mellon University

Problem 97

Figure $10-58$ shows a propeller blade that rotates at 2000 rev/min about a perpendicular axis at point $B .$ Point $A$ is at the outer tip of the blade, at radial distance $1.50 \mathrm{~m}$. (a) What is the difference in the magnitudes $a$ of the centripetal acceleration of point $A$ and of a point at radial distance $0.150 \mathrm{~m} ?$ (b) Find the slope of a plot of $a$ versus radial distance along the blade.

Salamat Ali

Numerade Educator

Problem 98

A yo-yo-shaped device mounted on a horizontal frictionless axis is used to lift a $30 \mathrm{~kg}$ box as shown in Fig. $10-59 .$ The outer radius $R$ of the device is $0.50 \mathrm{~m}$, and the radius $r$ of the hub is $0.20 \mathrm{~m}$. When a constant horizontal force $\vec{F}_{\text {app }}$ of magnitude $140 \mathrm{~N}$ is applied to a rope wrapped around the outside of the device, the box, which is suspended from a rope wrapped around the hub, has an upward acceleration of magnitude $0.80$ $\mathrm{m} / \mathrm{s}^{2}$. What is the rotational inertia of the device about its axis of rotation?

Donald Albin

Numerade Educator

Problem 99

A small ball with mass $1.30 \mathrm{~kg}$ is mounted on one end of a rod $0.780 \mathrm{~m}$ long and of negligible mass. The system rotates in a horizontal circle about the other end of the rod at 5010 rev/min. (a) Calculate the rotational inertia of the system about the axis of rotation. (b) There is an air drag of $2.30 \times 10^{-2} \mathrm{~N}$ on the ball, directed opposite its motion. What torque must be applied to the system to keep it rotating at constant speed?

Salamat Ali

Numerade Educator

Problem 100

Two thin rods (each of mass $0.20$ $\mathrm{kg}$ ) are joined together to form a rigid body as shown in Fig. $10-60 .$ One of the rods has length $L_{1}=0.40 \mathrm{~m}$, and the other has length $L_{2}=0.50 \mathrm{~m} .$ What is the rotational inertia of this rigid body about (a) an axis that is perpendicular to the plane of the paper and passes through the center of the shorter rod and (b) an axis that is perpendicular to the plane of the paper and passes through the center of the longer rod?

Averell Hause

Carnegie Mellon University

Problem 101

In Fig. $10-61$, four pulleys are connected by two belts. Pulley $A$ (radius $15 \mathrm{~cm}$ ) is the drive pulley, and it rotates at $10 \mathrm{rad} / \mathrm{s}$. Pulley $B$ (radius $10 \mathrm{~cm}$ ) is connected by belt 1 to pulley $A$. Pulley $B^{\prime}$ (radius $5 \mathrm{~cm}$ ) is concentric with pulley $B$ and is rigidly attached to it. Pulley $C$ (radius $25 \mathrm{~cm}$ ) is connected by belt 2 to pulley $\underline{B}^{\prime}$. Calculate (a) the linear speed of a point on belt $1,($ b) the angular speed of pulley $B,(\mathrm{c})$ the angular speed of pulley $B^{\prime},(\mathrm{d})$ the linear speed of a point on belt 2, and (e) the angular speed of pulley C. (Hint:
If the belt between two pulleys does not slip, the linear speeds at the rims of the two pulleys must be equal.)

Salamat Ali

Numerade Educator

Problem 102

The rigid object shown in Fig. $10-62$ consists of three balls and three connecting rods, with $M=1.6 \mathrm{~kg}, L=0.60 \mathrm{~m}$, and $\theta=30^{\circ}$. The balls may be treated as particles, and the connecting rods have negligible mass. Determine the rotational kinetic energy of the object if it has an angular speed of $1.2 \mathrm{rad} / \mathrm{s}$ about (a) an axis that passes through point $P$ and is perpendicular to the plane of the figure and (b) an axis that passes through point $P$, is perpendicular to the rod of length $2 L$. and lies in the plane of the figure.

Averell Hause

Carnegie Mellon University

Problem 103

In Fig. $10-63$, a thin uniform rod (mass $3.0 \mathrm{~kg}$, length $4.0 \mathrm{~m}$ ) rotates freely about a horizontal axis $A$ that is perpendicular to the rod and passes through a point at distance $d=1.0 \mathrm{~m}$ from the end of the rod. The kinetic energy of the rod as it passes through the vertical position is $20 \mathrm{~J}$. (a) What is the rotational inertia of the rod about axis $A ?$ (b) What is the (linear) speed of the end $B$ of the rod as the rod passes through the vertical position?
(c) At what angle $\theta$ will the rod momentarily stop in its upward swing?

Salamat Ali

Numerade Educator

Problem 104

Four particles, each of mass, $0.20 \mathrm{~kg}$, are placed at the vertices of a square with sides of length $0.50 \mathrm{~m}$. The particles are connected by rods of negligible mass. This rigid body can rotate in a vertical plane about a horizontal axis $A$ that passes through one of the particles. The body is released from rest with rod $A B$ horizontal (Fig. $10-64$ ).
(a) What is the rotational inertia of the body about axis $A ?$ (b) What is the angular speed of the body about axis $A$ when $\operatorname{rod} A B$ swings through the vertical position?

Averell Hause

Carnegie Mellon University

Problem 105

Cheetahs running at top speed have been reported at an astounding $114 \mathrm{~km} / \mathrm{h}$ (about $71 \mathrm{mi} / \mathrm{h})$ by observers driving alongside the animals. Imagine trying to measure a cheetah's speed by keeping your vehicle abreast of the animal while also glancing at your speedometer, which is registering $114 \mathrm{~km} / \mathrm{h}$. You keep the vehicle a constant $8.0 \mathrm{~m}$ from the cheetah, but the noise of the vehicle causes the cheetah to continuously veer away from you along a circular path of radius $92 \mathrm{~m}$. Thus, you travel along a circular path of radius $100 \mathrm{~m} .$ (a) What is the angular speed of you and the cheetah around the circular paths? (b) What is the linear speed of the cheetah along its path? (If you did not account for the circular motion, you would conclude erroneously that the cheetah's speed is $114 \mathrm{~km} / \mathrm{h}$, and that type of error was apparently made in the published reports)

Donald Albin

Numerade Educator

Problem 106

A point on the rim of a $0.75-\mathrm{m}$ -diameter grinding wheel changes speed at a constant rate from $12 \mathrm{~m} / \mathrm{s}$ to $25 \mathrm{~m} / \mathrm{s}$ in $6.2 \mathrm{~s}$. What is the average angular acceleration of the wheel?

Averell Hause

Carnegie Mellon University

Problem 107

A pulley wheel that is $8.0 \mathrm{~cm}$ in diameter has a $5.6$ -m-long cord wrapped around its periphery. Starting from rest, the wheel is given a constant angular acceleration of $1.5 \mathrm{rad} / \mathrm{s}^{2}$. (a) Through what angle must the wheel turn for the cord to unwind completely? (b) How long will this take?

Salamat Ali

Numerade Educator

Problem 108

A vinyl record on a turntable rotates at $33 \frac{1}{3}$ revimin.
(a) What is its angular speed in radians per second? What is the linear speed of a point on the record (b) $15 \mathrm{~cm}$ and (c) $7.4 \mathrm{~cm}$ from the turntable axis?

Donald Albin

Numerade Educator