Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 10

Rotation of a Rigid Object About a Fixed Axis - all with Video Answers

Educators

Chapter Questions

Problem 1

A wheel starts from rest and rotates with constant angular acceleration and reaches an angular speed of $12.0 \mathrm{rad} / \mathrm{s}$ in $3.00 \mathrm{~s}$. Find (a) the magnitude of the angular acceleration of the wheel and (b) the angle (in radians) through which it rotates in this time.

Keshav Singh

Keshav Singh

Numerade Educator

Problem 2

What is the angular speed in radians per second of
(a) the Earth in its orbit about the Sun and (b) the Moon in its orbit about the Earth?

Manish Jain

Numerade Educator

Problem 3

An airliner arrives at the terminal, and its engines are shut off. The rotor of one of the engines has an initial clockwise angular speed of $2000 \mathrm{rad} / \mathrm{s}$. The engine's rotation slows with an angular acceleration of magnitude $80.0 \mathrm{rad} / \mathrm{s}^{2}$. (a) Determine the angular speed after $10.0 \mathrm{~s}$. (b) How long does it take for the rotor to come to rest?

Mayukh Banik

Numerade Educator

Problem 4

(a) The positions of the hour and minute hand on a clock face coincide at 12 o'clock. Determine all other times (up to the second) at which the positions of the hands coincide. (b) If the clock also has a second hand, determine all times at which the positions of all three hands coincide, given that they all coincide at 12 o'clock.

Manish Jain

Numerade Educator

Problem 5

wAn electric motor rotating a grinding wheel at $100 \mathrm{rev} / \mathrm{min}$ is switched off. Assuming constant negative acceleration of magnitude $2.00 \mathrm{rad} / \mathrm{s}^{2},(\mathrm{a})$ how long does it take the wheel to stop? (b) Through how many radians does it turn during the time found in part (a)?

Keshav Singh

Numerade Educator

Problem 6

A centrifuge in a medical laboratory rotates at a rotational speed of 3600 rev $/ \mathrm{min}$. When switched off, it rotates $50.0$ times before coming to rest. Find the constant angular acceleration of the centrifuge.

Keshav Singh

Numerade Educator

Problem 7

The angular position of a swinging door is described by $\theta=5.00+10.0 t+2.00 t^{2} \mathrm{rad} .$ Determine the angular
position, angular speed, and angular acceleration of the door (a) at $t=0$ and (b) at $t=3.00 \mathrm{~s}$.

Keshav Singh

Numerade Educator

Problem 8

The tub of a washer goes into its spin cycle, starting from rest and gaining angular speed steadily for $8.00 \mathrm{~s}$, when it is turning at $5.00 \mathrm{rev} / \mathrm{s}$. At this point the person doing the laundry opens the lid, and a safety switch turns off the washer. The tub smoothly slows to rest in $12.0 \mathrm{~s}$. Through how many revolutions does the tub turn while it is in motion?

Keshav Singh

Numerade Educator

Problem 9

A rotating wheel requires $3.00 \mathrm{~s}$ to complete $37.0$ revolutions. Its angular speed at the end of the $3.00-\mathrm{s}$ interval is $98.0 \mathrm{rad} / \mathrm{s}$. What is the constant angular acceleration of the wheel?

Keshav Singh

Numerade Educator

Problem 10

(a) Find the angular speed of the Earth's rotation on its axis. As the Earth turns toward the east, we see the sky turning toward the west at this same rate.
(b) The rainy Pleiads wester And seek beyond the sea The head that I shall dream of That shall not dream of me.
A. E. Housman
(@ Robert E. Symons)
Cambridge, England, is at longitude $0^{\circ}$, and Saskatoon, Saskatchewan, is at longitude $107^{\circ}$ west. How much time elapses after the Pleiades set in Cambridge until these stars fall below the western horizon in Saskatoon?

Mayukh Banik

Numerade Educator

Problem 11

Make an order-of-magnitude estimate of the number of revolutions through which a typical automobile tire turns in 1 yr. State the quantities you measure or estimate and their values.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 12

The diameters of the main rotor and tail rotor of a single-engine helicopter are $7.60 \mathrm{~m}$ and $1.02 \mathrm{~m}$, respectively. The respective rotational speeds are 450 rev/min and 4138 rev/min. Calculate the speeds of the tips of both rotors. Compare these speeds with the speed of sound, $343 \mathrm{~m} / \mathrm{s}$.

Manish Jain

Numerade Educator

Problem 13

A racing car travels on a circular track with a radius of $250 \mathrm{~m}$. If the car moves with a constant linear speed of $45.0 \mathrm{~m} / \mathrm{s}$, find (a) its angular speed and (b) the magnitude and direction of its acceleration.

Mayukh Banik

Numerade Educator

Problem 14

A car is traveling at $36.0 \mathrm{~km} / \mathrm{h}$ on a straight road. The radius of its tires is $25.0 \mathrm{~cm}$. Find the angular speed of one of the tires, with its axle taken as the axis of rotation.

Keshav Singh

Numerade Educator

Problem 15

A wheel $2.00 \mathrm{~m}$ in diameter lies in a vertical plane and rotates with a constant angular acceleration of $4.00 \mathrm{rad} / \mathrm{s}^{2}$. The wheel starts at rest at $t=0$, and the radius vector of point $P$ on the rim makes an angle of $57.3^{\circ}$ with the horizontal at this time. At $t=2.00 \mathrm{~s}$. find
(a) the angular speed of the wheel,
(b) the linear speed and acceleration of the point $P$, and $(c)$ the angular position of the point $P$.

Keshav Singh

Numerade Educator

Problem 16

A discus thrower accelerates a discus from rest to a speed of $25.0 \mathrm{~m} / \mathrm{s}$ by whirling it through $1.25$ rev. Assume the discus moves on the arc of a circle $1.00 \mathrm{~m}$ in
radius. (a) Calculate the final angular speed of the discus. (b) Determine the magnitude of the angular acceleration of the discus, assuming it to be constant.
(c) Calculate the acceleration time.

Keshav Singh

Numerade Educator

Problem 17

A car accelerates uniformly from rest and reaches a speed of $22.0 \mathrm{~m} / \mathrm{s}$ in $9.00 \mathrm{~s}$. If the diameter of a tire is $58.0 \mathrm{~cm}$, find (a) the number of revolutions the tire makes during this motion, assuming that no slipping $0 \mathrm{c}$ curs. (b) What is the final rotational speed of a tire in revolutions per second?

Keshav Singh

Numerade Educator

Problem 18

A $6.00$ -kg block is released from $A$ on the frictionless track shown in Figure $\mathrm{P} 10.18 .$ Determine the radial and tangential components of acceleration for the block at $P$.

Manish Jain

Numerade Educator

Problem 19

A disc $8.00 \mathrm{~cm}$ in radius rotates at a constant rate of $1200 \mathrm{rev} / \mathrm{min}$ about its central axis. Determine (a) its angular speed, (b) the linear speed at a point $3.00 \mathrm{~cm}$ from its center, (c) the radial acceleration of a point on the rim, and (d) the total distance a point on the rim moves in $2.00 \mathrm{~s}$.

Keshav Singh

Numerade Educator

Problem 20

A car traveling on a flat (unbanked) circular track accelerates uniformly from rest with a tangential acceleration of $1.70 \mathrm{~m} / \mathrm{s}^{2} .$ The car makes it one quarter of the way around the circle before it skids off the track. Deter-
mine the coefficient of static friction between the car
and track from these data.

Keshav Singh

Numerade Educator

Problem 21

A small object with mass $4.00 \mathrm{~kg}$ moves counterclockwise with constant speed $4.50 \mathrm{~m} / \mathrm{s}$ in a circle of radius $3.00 \mathrm{~m}$ centered at the origin. (a) It started at the point with cartesian coordinates $(3 \mathrm{~m}, 0) .$ When its angular displacement is $9.00 \mathrm{rad}$, what is its position vector, in cartesian unit-vector notation? (b) In what quadrant is the particle located, and what angle does its position vector make with the positive $x$ axis? (c) What is its velocity vector, in unit-vector notation? (d) In what direction is it moving? Make a sketch of the position and velocity vectors. (e) What is its acceleration, expressed in unit-vector notation? (f) What total force acts on the object? (Express your answer in unit vector notation.)

Keshav Singh

Numerade Educator

Problem 22

A standard cassette tape is placed in a standard cassette player. Each side plays for $30 \mathrm{~min}$. The two tape wheels of the cassette fit onto two spindles in the player. Suppose that a motor drives one spindle at a constant angular speed of $\sim 1 \mathrm{rad} / \mathrm{s}$ and that the other spindle is free to rotate at any angular speed. Estimate the order of magnitude of the thickness of the tape.

Manish Jain

Numerade Educator

Problem 23

Three small particles are connected by rigid rods of negligible mass lying along the $y$ axis (Fig. $\mathrm{P} 10.23) .$ If the system rotates about the $x$ axis with an angular speed of $2.00 \mathrm{rad} / \mathrm{s}$, find $(\mathrm{a})$ the moment of inertia about the $x$ axis and the total rotational kinetic energy evaluated from $\frac{1}{2} I \omega^{2}$ and (b) the linear speed of each particle and the total kinetic energy evaluated from $\sum \frac{1}{2} m_{i} v_{i}^{2}$.

Keshav Singh

Numerade Educator

Problem 24

The center of mass of a pitched baseball $(3.80-\mathrm{cm} \mathrm{ra}-$ dius) moves at $38.0 \mathrm{~m} / \mathrm{s} .$ The ball spins about an axis through its center of mass with an angular speed of $125 \mathrm{rad} / \mathrm{s} .$ Calculate the ratio of the rotational energy to the translational kinetic energy. Treat the ball as a uniform sphere.

Manish Jain

Numerade Educator

Problem 25

The four particles in Figure $\mathrm{P} 10.25$ are connected by rigid rods of negligible mass. The origin is at the center of the rectangle. If the system rotates in the $x y$ plane about the $z$ axis with an angular speed of $6.00 \mathrm{rad} / \mathrm{s}$, calculate (a) the moment of inertia of the system about the $z$ axis and (b) the rotational energy of the system.

Keshav Singh

Numerade Educator

Problem 26

The hour hand and the minute hand of Big Ben, the famous Parliament tower clock in London, are $2.70 \mathrm{~m}$ long and $4.50 \mathrm{~m}$ long and have masses of $60.0 \mathrm{~kg}$ and $100 \mathrm{~kg}$, respectively. Calculate the total rotational kinetic energy of the two hands about the axis of rotation. (You may model the hands as long thin rods.)

Keshav Singh

Numerade Educator

Problem 27

Two masses $M$ and $m$ are connected by a rigid rod of length $L$ and of negligible mass, as shown in Figure P10.27. For an axis perpendicular to the rod, show that the system has the minimum moment of inertia when the axis passes through the center of mass. Show that this moment of inertia is $I=\mu L^{2}$, where $\mu=$ $m M /(m+M)$.

Keshav Singh

Numerade Educator

Problem 28

Three identical thin rods, each of length $L$ and mass $m$, are welded perpendicular to each other, as shown in Figure $\mathrm{P} 10.28 .$ The entire setup is rotated about an axis that passes through the end of one rod and is parallel to another. Determine the moment of inertia of this arrangement.

Keshav Singh

Numerade Educator

Problem 29

Figure $\mathrm{P} 10.29$ shows a side view of a car tire and its radial dimensions. The rubber tire has two sidewalls of uniform thickness $0.635 \mathrm{~cm}$ and a tread wall of uniform thickness $2.50 \mathrm{~cm}$ and width $20.0 \mathrm{~cm}$. Suppose its density is uniform, with the value $1.10 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}$. Find its moment of inertia about an axis through its center perpendicular to the plane of the sidewalls.

Keshav Singh

Numerade Educator

Problem 30

Use the parallel-axis theorem and Table $10.2$ to find the moments of inertia of (a) a solid cylinder about an axis parallel to the center-of-mass axis and passing through the edge of the cylinder and (b) a solid sphere about an axis tangent to its surface.

Manish Jain

Numerade Educator

Problem 31

Attention! About face! Compute an order-of-magnitude estimate for the moment of inertia of your body as you stand tall and turn around a vertical axis passing through the top of your head and the point halfway between your ankles. In your solution state the quantities you measure or estimate and their values. $10.6 \mathrm{Tor} u \mathrm{e}$

Keshav Singh

Numerade Educator

Problem 32

Find the mass $m$ needed to balance the $1500-\mathrm{kg}$ truck on the incline shown in Figure $\mathrm{P} 10.32 .$ Assume all pulleys are frictionless and massless.

Keshav Singh

Numerade Educator

Problem 33

Find the net torque on the wheel in Figure $\mathrm{P} 10.33$ about the axle through $O$ if $a=10.0 \mathrm{~cm}$ and $b=$ $25.0 \mathrm{~cm} .$

Keshav Singh

Numerade Educator

Problem 34

The fishing pole in Figure $\mathrm{P} 10.34$ makes an angle of $20.0^{\circ}$ with the horizontal. What is the torque exerted by the fish about an axis perpendicular to the page and passing through the fisher's hand?

Mayukh Banik

Numerade Educator

Problem 35

The tires of a $1500-\mathrm{kg}$ car are $0.600 \mathrm{~m}$ in diameter, and the coefficients of friction with the road surface are $\mu_{s}=0.800$ and $\mu_{k}=0.600 .$ Assuming that the weight is evenly distributed on the four wheels, calculate the maximum torque that can be exerted by the engine on a driving wheel such that the wheel does not spin. If you wish, you may suppose that the car is at rest.

Keshav Singh

Numerade Educator

Problem 36

Suppose that the car in Problem 95 has a disk brake system. Each wheel is slowed by the frictional force between a single brake pad and the disk-shaped rotor. On this particular car, the brake pad comes into contact with the rotor at an average distance of $22.0 \mathrm{~cm}$ from the axis. The coefficients of friction between the brake pad and the disk are $\mu_{s}=0.600$ and $\mu_{k}=0.500$. Calculate the normal force that must be applied to the rotor such that the car slows as quickly as possible.

Keshav Singh

Numerade Educator

Problem 37

A model airplane having a mass of $0.750 \mathrm{~kg}$ is tethered by a wire so that it flies in a circle $30.0 \mathrm{~m}$ in radius. The airplane engine provides a net thrust of $0.800 \mathrm{~N}$ perpendicular to the tethering wire. (a) Find the torque the net thrust produces about the center of the circle.
(b) Find the angular acceleration of the airplane when it is in level flight. (c) Find the linear acceleration of the airplane tangent to its flight path.

Salamat Ali

Numerade Educator

Problem 38

The combination of an applied force and a frictional force produces a constant total torque of $36.0 \mathrm{~N} \cdot \mathrm{m}$ on a wheel rotating about a fixed axis. The applied force acts for $6.00 \mathrm{~s}$; during this time the angular speed of the wheel increases from 0 to $10.0 \mathrm{rad} / \mathrm{s}$. The applied force is then removed, and the wheel comes to rest in $60.0 \mathrm{~s}$. Find (a) the moment of inertia of the wheel, (b) the magnitude of the frictional torque, and (c) the total number of revolutions of the wheel.

Keshav Singh

Numerade Educator

Problem 39

A block of mass $m_{1}=2.00 \mathrm{~kg}$ and a block of mass $m_{2}=$ $6.00 \mathrm{~kg}$ are connected by a massless string over a pulley in the shape of a disk having radius $R=0.250 \mathrm{~m}$ and mass $M=10.0 \mathrm{~kg}$. These blocks are allowed to move on a fixed block-wedge of angle $\theta=30.0^{\circ}$, as shown in Figure $\mathrm{P} 10.39 .$ The coefficient of kinetic friction for both blocks is $0.360 .$ Draw free-body diagrams of both blocks and of the pulley. Determine (a) the acceleration of the two blocks and (b) the tensions in the string on both sides of the pulley.

Keshav Singh

Numerade Educator

Problem 40

A potter's wheel -a thick stone disk with a radius of $0.500 \mathrm{~m}$ and a mass of $100 \mathrm{~kg}-$ is freely rotating at $50.0 \mathrm{rev} / \mathrm{min}$. The potter can stop the wheel in $6.00 \mathrm{~s}$ by pressing a wet rag against the rim and exerting a radially inward force of $70.0 \mathrm{~N}$. Find the effective coefficient of kinetic friction between the wheel and the rag.

Rehmat Kazmi

Numerade Educator

Problem 41

A bicycle wheel has a diameter of $64.0 \mathrm{~cm}$ and a mass of $1.80 \mathrm{~kg} .$ Assume that the wheel is a hoop with all of its mass concentrated on the outside radius. The bicycle is placed on a stationary stand on rollers, and a resistive force of $120 \mathrm{~N}$ is applied tangent to the rim of the tire.
(a) What force must be applied by a chain passing over a $9.00-\mathrm{cm}$ -diameter sprocket if the wheel is to attain an acceleration of $4.50 \mathrm{rad} / \mathrm{s}^{2} ?$ (b) What force is required if the chain shifts to a $5.60-\mathrm{cm}-\mathrm{diameter}$ sprocket?

Manish Jain

Numerade Educator

Problem 42

A cylindrical rod $24.0 \mathrm{~cm}$ long with a mass of $1.20 \mathrm{~kg}$ and a radius of $1.50 \mathrm{~cm}$ has a ball with a diameter of $8.00 \mathrm{~cm}$ and a mass of $2.00 \mathrm{~kg}$ attached to one end. The arrangement is originally vertical and stationary, with the ball at the top. The apparatus is free to pivot about the bottom end of the rod. (a) After it falls through $90^{\circ}$, what is its rotational kinetic energy? (b) What is the angular speed of the rod and ball? (c) What is the linear speed of the ball? (d) How does this compare with the speed if the ball had fallen freely through the same distance of $28 \mathrm{~cm} ?$

Keshav Singh

Numerade Educator

Problem 43

A $15.0-\mathrm{kg}$ mass and a $10.0-\mathrm{kg}$ mass are suspended by a pulley that has a radius of $10.0 \mathrm{~cm}$ and a mass of $3.00 \mathrm{~kg}$ (Fig. P10.43). The cord has a negligible mass and causes the pulley to rotate without slipping. The pulley rotates without frictoon. The masses start from rest $3.00 \mathrm{~m}$ apart. Treating the pulley as a uniform disk, determine the speeds of the two masses as they pass each other.

Keshav Singh

Numerade Educator

Problem 44

A mass $m_{1}$ and a mass $m_{2}$ are suspended by a pulley that has a radius $R$ and a mass $M$ (see Fig. $\mathrm{P} 10.43$ ). The cord has a negligible mass and causes the pulley to rotate without slipping. The pulley rotates without friction. The masses start from rest a distance $d$ apart. Treating the pulley as a uniform disk, determine the speeds of the two masses as they pass each other.

Manish Jain

Numerade Educator

Problem 45

A weight of $50.0 \mathrm{~N}$ is attached to the free end of a light string wrapped around a reel with a radius of $0.250 \mathrm{~m}$ and a mass of $3.00 \mathrm{~kg}$. The reel is a solid disk, free to $\mathrm{ro}$ tate in a vertical plane about the horizontal axis passing through its center. The weight is released $6.00 \mathrm{~m}$ above the floor. (a) Determine the tension in the string, the acceleration of the mass, and the speed with which the weight hits the floor. (b) Find the speed calculated in part (a), using the principle of conservation of energy.

Keshav Singh

Numerade Educator

Problem 46

A constant torque of $25.0 \mathrm{~N}^{*} \mathrm{~m}$ is applied to a grindstone whose moment of inertia is $0.180 \mathrm{~kg} \cdot \mathrm{m}^{2} .$ Using energy principles, find the angular speed after the grindstone has made $15.0$ revolutions. (Neglect friction.)

Manish Jain

Numerade Educator

Problem 47

This problem describes one experimental method of determining the moment of inertia of an irregularly shaped object such as the payload for a satellite. Figure P10.4 7 shows a mass $m$ suspended by a cord wound around a spool of radius $r$, forming part of a turntable supporting the object. When the mass is released from rest, it descends through a distance $h$, acquiring a speed U. Show that the moment of inertia $I$ of the equipment (including the turntable) is $m r^{2}\left(2 g h / v^{2}-1\right)$.

Keshav Singh

Numerade Educator

Problem 48

A bus is designed to draw its power from a rotating flywheel that is brought up to its maximum rate of rotation $(8000$ rev $/ \mathrm{min})$ by an electric motor. The flywheel is a solid cylinder with a mass of $1000 \mathrm{~kg}$ and a diameter of $1.00 \mathrm{~m}$. If the bus requires an average power of $10.0 \mathrm{~kW}$, how long does the flywheel rotate?

Manish Jain

Numerade Educator

Problem 49

(a) A uniform, solid disk of radius $R$ and mass $M$ is free to rotate on a frictionless pivot through a point on its rim (Fig. $\mathrm{P} 10.49$ ). If the disk is released from rest in the position shown by the blue circle, what is the speed of its center of mass when the disk reaches the position indicated by the dashed circle? (b) What is the speed of the lowest point on the disk in the dashed position?
(c) Repeat part (a), using a uniform hoop.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 50

A horizontal 800-N merry-go-round is a solid disk of radius $1.50 \mathrm{~m}$ and is started from rest by a constant horizontal force of $50.0 \mathrm{~N}$ applied tangentially to the cylinder. Find the kinetic energy of the solid cylinder after $8.00 \mathrm{~s}$.

Keshav Singh

Numerade Educator

Problem 51

Toppling chimneys often break apart in mid-fall (Fig. P10.51) because the mortar between the bricks cannot withstand much shear stress. As the chimney begins to fall, shear forces must act on the topmost sections to accelerate them tangentially so that they can keep up with the rotation of the lower part of the stack. For simplicity, let us model the chimney as a uniform rod of length $\ell$ pivoted at the lower end. The rod starts at rest in a vertical position (with the frictionless pivot at the bottom) and falls over under the influence of gravity. What fraction of the length of the rod has a tangential acceleration greater than $g \sin \theta$, where $\theta$ is the angle the chimney makes with the vertical?

Keshav Singh

Numerade Educator

Problem 52

A mixing beater consists of three thin rods: Each is $10.0 \mathrm{~cm}$ long, diverges from a central hub, and is separated from the others by $120^{\circ}$. All turn in the same plane. A ball is attached to the end of each rod. Each ball has a cross-sectional area of $4.00 \mathrm{~cm}^{2}$ and is shaped so that it has a drag coefficient of $0.600 .$ Calculate the power input required to spin the beater at $1000 \mathrm{rev} / \mathrm{min}(\mathrm{a}) \mathrm{in}$ air and $(\mathrm{b})$ in water.

Keshav Singh

Numerade Educator

Problem 53

A grinding wheel is in the form of a uniform solid disk having a radius of $7.00 \mathrm{~cm}$ and a mass of $2.00 \mathrm{~kg} . \mathrm{It}$ starts from rest and accelerates uniformly under the $\mathrm{ac}-$ tion of the constant torque of $0.600 \mathrm{~N} \cdot \mathrm{m}$ that the motor exerts on the wheel. (a) How long does the wheel take to reach its final rotational speed of 1200 rev/min?
(b) Through how many revolutions does it turn while accelerating?

Keshav Singh

Numerade Educator

Problem 54

The density of the Earth, at any distance $r$ from its center, is approximately
$$
\rho=[14.2-11.6 \mathrm{r} / \mathrm{R}] \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}
$$
where $R$ is the radius of the Earth. Show that this density leads to a moment of inertia $I=0.930 M R^{2}$ about an axis through the center, where $M$ is the mass of the Earth.

Keshav Singh

Numerade Educator

Problem 55

A $4.00-\mathrm{m}$ length of light nylon cord is wound around a uniform cylindrical spool of radius $0.500 \mathrm{~m}$ and mass $1.00 \mathrm{~kg} .$ The spool is mounted on a frictionless axle and is initially at rest. The cord is pulled from the spool with a constant acceleration of magnitude $2.50 \mathrm{~m} / \mathrm{s}^{2}$.
(a) How much work has been done on the spool when it reaches an angular speed of $8.00 \mathrm{rad} / \mathrm{s} ?$ (b) Assuming that there is enough cord on the spool, how long does it take the spool to reach this angular speed? (c) Is there enough cord on the spool?

Mayukh Banik

Numerade Educator

Problem 56

A flywheel in the form of a heavy circular disk of diameter $0.600 \mathrm{~m}$ and mass $200 \mathrm{~kg}$ is mounted on a frictionless bearing. A motor connected to the flywheel accelerates it from rest to 1000 rev/min. (a) What is the moment of inertia of the flywheel? (b) How much work is done on it during this acceleration? (c) When the angular speed reaches 1000 rev/min, the motor is disengaged. A friction brake is used to slow the rotational rate to $500 \mathrm{rev} / \mathrm{min}$. How much energy is dissipated as internal energy in the friction brake?

Manish Jain

Numerade Educator

Problem 57

A shaft is turning at $65.0 \mathrm{rad} / \mathrm{s}$ at time zero. Thereafter, its angular acceleration is given by
$$
\alpha=-10 \mathrm{rad} / \mathrm{s}^{2}-5 t \mathrm{rad} / \mathrm{s}^{3}
$$
where $t$ is the elapsed time. (a) Find its angular speed at $t=3.00 \mathrm{~s}$. (b) How far does it turn in these 3 s?

Keshav Singh

Numerade Educator

Problem 58

For any given rotational axis, the radius of gyration $K$ of a rigid body is defined by the expression $K^{2}=I / M$, where $M$ is the total mass of the body and $I$ is its moment of inertia. Thus, the radius of gyration is equal to the distance between an imaginary point mass $M$ and the axis of rotation such that $I$ for the point mass about that axis is the same as that for the rigid body. Find the radius of gyration of (a) a solid disk of radius $R,(\mathrm{~b})$ a uniform rod of length $L$, and (c) a solid sphere of radius $R$, all three of which are rotating about a central axis.

Manish Jain

Numerade Educator

Problem 59

A long, uniform rod of length $L$ and mass $M$ is pivoted about a horizontal, frictionless pin passing through one end. The rod is released from rest in a vertical position, as shown in Figure $\mathrm{P} 10.59 .$ At the instant the rod is horizontal, find (a) its angular speed, (b) the magnitude of its angular acceleration, (c) the $x$ and $y$ components of the acceleration of its center of mass, and (d) the components of the reaction force at the pivot.

Keshav Singh

Numerade Educator

Problem 60

A bicycle is turned upside down while its owner repairs a flat tire. A friend spins the other wheel, of radius $0.381 \mathrm{~m}$, and observes that drops of water fly off tangentially. She measures the height reached by drops moving vertically (Fig. P10.60). A drop that breaks loose from the tire on one turn rises $h=54.0 \mathrm{~cm}$ above the tangent point. A drop that breaks loose on the next turn rises $51.0 \mathrm{~cm}$ above the tangent point. The height to which the drops rise decreases because the angular speed of the wheel decreases. From this information, determine the magnitude of the average angular acceleration of the wheel.

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 61

A bicycle is turned upside down while its owner repairs a flat tire. $A$ friend spins the other wheel of radius $R$ and observes that drops of water fly off tangentially. She measures the height reached by drops moving vertically (see Fig. $\mathrm{P} 10.60$ ). A drop that breaks loose from the tire on one turn rises a distance $h_{1}$ above the tangent point.
A drop that breaks loose on the next turn rises a distance $h_{2}<h_{1}$ above the tangent point. The height to which the drops rise decreases because the angular speed of the wheel decreases. From this information, determine the magnitude of the average angular acceleration of the wheel.

Keshav Singh

Numerade Educator

Problem 62

The top shown in Figure $\mathrm{P} 10.62$ has a moment of inertia of $4.00 \times 10^{-4} \mathrm{~kg}^{*} \mathrm{~m}^{2}$ and is initially at rest. It is free to rotate about the stationary axis $A A^{\prime} .$ A string, wrapped around a peg along the axis of the top, is pulled in such a manner that a constant tension of $5.57 \mathrm{~N}$ is maintained. If the string does not slip while it is unwound from the peg, what is the angular speed of the top after $80.0 \mathrm{~cm}$ of string has been pulled off the peg?

Keshav Singh

Numerade Educator

Problem 63

A cord is wrapped around a pulley of mass $m$ and of radius $r .$ The free end of the cord is connected to a block of mass $M$. The block starts from rest and then slides down an incline that makes an angle $\theta$ with the horizontal. The coefficient of kinetic friction between block and incline is $\mu .$ (a) Use energy methods to show that the block's speed as a function of displacement $d$ down the incline is
$$
v=\left[4 g d M(m+2 M)^{-1}(\sin \theta-\mu \cos \theta)\right]^{1 / 2}
$$
(b) Find the magnitude of the acceleration of the block in terms of $\mu, m, M, g$, and $\theta$.

Keshav Singh

Numerade Educator

Problem 64

(a) What is the rotational energy of the Earth about its spin axis? The radius of the Earth is $6870 \mathrm{~km}$, and its mass is $5.98 \times 10^{24} \mathrm{~kg} .$ Treat the Earth as a sphere of moment of inertia $\frac{2}{5} M R^{2}$. (b) The rotational energy of the Earth is decreasing steadily because of tidal friction. Estimate the change in one day, given that the rotational period increases by about $10 \mu$ search year.

Keshav Singh

Numerade Educator

Problem 65

The speed of a moving bullet can be determined by allowing the bullet to pass through two rotating paper disks mounted a distance $d$ apart on the same axle (Fig. P10.65). From the angular displacement $\Delta \theta$ of the two bullet holes in the disks and the rotational speed of the disks, we can determine the speed $v$ of the bullet. Find the bullet speed for the following data: $d=80 \mathrm{~cm}$, $\omega=900 \mathrm{rev} / \mathrm{min}$, and $\Delta \theta=31.0^{\circ} .$

Mayukh Banik

Numerade Educator

Problem 66

A wheel is formed from a hoop and $n$ equally spaced spokes extending from the center of the hoop to its rim. The mass of the hoop is $M$, and the radius of the hoop (and hence the length of each spoke) is $R$. The mass of each spoke is $m$. Determine (a) the moment of inertia of the wheel about an axis through its center and perpendicular to the plane of the wheel and
(b) the moment of inertia of the wheel about an axis
through its rim and perpendicular to the plane of the wheel.

Manish Jain

Numerade Educator

Problem 67

A uniform, thin, solid door has a height of $2.20 \mathrm{~m}, \mathrm{a}$ width of $0.870 \mathrm{~m}$, and a mass of $28.0 \mathrm{~kg}$. Find its moment of inertia for rotation on its hinges. Are any of the data unnecessary?

Keshav Singh

Numerade Educator

Problem 68

A uniform, hollow, cylindrical spool has inside radius $R / 2$, outside radius $R$, and mass $M$ (Fig. P10.68). It is mounted so that it rotates on a massless horizontal axle. A mass $m$ is connected to the end of a string wound around the spool. The mass $m$ falls from rest through a distance $y$ in time $t$. Show that the torque due to the frictional forces between spool and axle is
$$
\tau_{f}=R\left[m\left(g-2 y / t^{2}\right)-M\left(5 y / 4 t^{2}\right)\right]
$$

Keshav Singh

Numerade Educator

Problem 69

An electric motor can accelerate a Ferris wheel of moment of inertia $I=20000 \mathrm{~kg} \cdot \mathrm{m}^{2}$ from rest to $10.0$ rev/min in $12.0 \mathrm{~s}$. When the motor is turned off, friction causes the wheel to slow down from $10.0$ to $8.00 \mathrm{rev} / \mathrm{min}$ in $10.0 \mathrm{~s}$. Determine (a) the torque generated by the motor to bring the wheel to $10.0 \mathrm{rev} / \mathrm{min}$ and (b) the power that would be needed to maintain this rotational speed.

Manish Jain

Numerade Educator

Problem 70

The pulley shown in Figure $\mathrm{P} 10.70$ has radius $R$ and moment of inertia $I$. One end of the mass $m$ is connected to a spring of force constant $k$, and the other end is fastened to a cord wrapped around the pulley. The pulley axle and the incline are frictionless. If the pulley is wound counterclockwise so that the spring is stretched a distance $d$ from its unstretched position and is then released from rest, find (a) the angular speed of the pulley when the spring is again unstretched and
(b) a numerical value for the angular speed at this point if $I=1.00 \mathrm{~kg} \cdot \mathrm{m}^{2}, R=0.300 \mathrm{~m}, k=50.0 \mathrm{~N} / \mathrm{m}$
$m=0.500 \mathrm{~kg}, d=0.200 \mathrm{~m}$, and $\theta=37.0^{\circ} .$

Keshav Singh

Numerade Educator

Problem 71

Two blocks, as shown in Figure $\mathrm{P} 10.71$, are connected by a string of negligible mass passing over a pulley of radius $0.250 \mathrm{~m}$ and moment of inertia $I .$ The block on the frictionless incline is moving upward with a constant acceleration of $2.00 \mathrm{~m} / \mathrm{s}^{2} .$ (a) Determine $T_{1}$ and $T_{2}$, the tensions in the two parts of the string. (b) Find the moment of inertia of the pulley.

Keshav Singh

Numerade Educator

Problem 72

A common demonstration, illustrated in Figure $\mathrm{P} 10.72$, consists of a ball resting at one end of a uniform board of length $\boldsymbol{\ell}$, hinged at the other end, and elevated at an angle $\theta .$ A light cup is attached to the board at $r_{c}$ so that it will catch the ball when the support stick is suddenly removed. (a) Show that the ball will lag behind the falling board when $\theta$ is less than $35.8^{\circ} ;$ and that (b) the ball will fall into the cup when the board is supported at this limiting angle and the cup is placed at
$$
r_{e}=\frac{2 \ell}{3 \cos \theta}
$$
(c) If a ball is at the end of a $1.00-\mathrm{m}$ stick at this critical angle, show that the cup must be $18.4 \mathrm{~cm}$ from the moving end.

Manish Jain

Numerade Educator

Problem 73

As a result of friction, the angular speed of a wheel changes with time according to the relationship
$$
d \theta / d t=\omega_{0} e^{-\sigma t}
$$
where $\omega_{0}$ and $\sigma$ are constants. The angular speed changes from $3.50 \mathrm{rad} / \mathrm{s}$ at $t=0$ to $2.00 \mathrm{rad} / \mathrm{s}$ at $t=9.30 \mathrm{~s}$. Use this information to determine $\sigma$ and $\omega_{0}$. Then, determine (a) the magnitude of the angular acceleration at $t=3.00 \mathrm{~s},(\mathrm{~b})$ the number of revolutions the wheel makes in the first $2.50 \mathrm{~s}$, and $(\mathrm{c})$ the number of revolutions it makes before coming to rest.

Keshav Singh

Numerade Educator

Problem 74

The hour hand and the minute hand of Big Ben, the famous Parliament tower clock in London, are $2.70 \mathrm{~m}$ long and $4.50 \mathrm{~m}$ long and have masses of $60.0 \mathrm{~kg}$ and $100 \mathrm{~kg}$, respectively (see Fig. $\mathrm{P} 10.26$ ). (a) Determine the total torque due to the weight of these hands about the axis of rotation when the time reads (i) $8=00$,
(ii) $5: 15$, (iii) $6: 00$, (iv) $8: 20$, and
(v) 9:45. (You may model the hands as long thin rods.) (b) Determine all times at which the total torque about the axis of rotation is zero. Determine the times to the nearest second, solving a transcendental equation numerically.

Manish Jain

Numerade Educator