Vector Mechanics for Engineers: Statics and Dynamics

Ferdinand P. Beer, E. Russell Johnston, Jr.,David F. Mazurek, Phillip J. Cornwell, Brian P. Self

Chapter 15

Kinematics of Rigid Bodies - all with Video Answers

Educators

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Section 1

Translation and Fixed Axis Rotation

Problem 1

A rectangular plate swings from arms of equal length as shown.
What is the magnitude of the angular velocity of the plate?
a. 0 rad/s
b. 1 rad/s
c. 2 rad/s
d. 3 rad/s
e. Need to know the location of the center of gravity.

Keshav Singh

Keshav Singh

Numerade Educator

Problem 1

The brake drum is attached to a larger flywheel that is not shown.
The motion of the brake drum is defined by the relation
$\theta=36 t-1.6 t^{2},$ where $\theta$ is expressed in radians and $t$ in seconds.
Determine (a) the angular velocity at $t=2 \mathrm{s},(b)$ the number of
revolutions executed by the brake drum before coming to rest.

Timothy Furey

Numerade Educator

Problem 2

Knowing that wheel $A$ rotates with a constant angular velocity and
that no slipping occurs between ring $C$ and wheel $A$ and wheel $B$,
which of the following statements concerning the angular speeds of
the three objects is true?

a. $\omega_{a}=\omega_{b}$
b. $\omega_{a}>\omega_{b}$
c. $\omega_{a}<\omega_{b}$
d. $\omega_{a}=\omega_{c}$
e. The contact points between $A$ and $C$ have the same acceleration.

Keshav Singh

Numerade Educator

Problem 2

The motion of an oscillating flywheel is defined by the relation
$\theta=\theta_{0} e^{-3 \pi} \cos 4 \pi t,$ where $\theta$ is expressed in radians and $t$ in seconds.
Knowing that $\theta_{0}=0.5$ rad, determine the angular coordinate, the
angular velocity, and the angular acceleration of the flywheel when
$(a) t=0,(b) t=0.125 \mathrm{s}$.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 3

The motion of an oscillating flywheel is defined by the relation
$\theta=\theta_{0} e^{-7 \pi v t} \sin 4 \pi t,$ where $\theta$ is expressed in radians and $t$ in
seconds. Knowing that $\theta_{0}=0.4$ rad, determine the angular coordinate,
the angular velocity, and the angular acceleration of the flywheel
when $(a) t=0.125 \mathrm{s},$ (b) $t=\infty$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 4

The rotor of a gas turbine is rotating at a speed of $6900 \mathrm{rpm}$ when
the turbine is shut down. It is observed that $4 \mathrm{min}$ is required for the
rotor to coast to rest. Assuming uniformly accelerated motion,
determine ( $a$ ) the angular acceleration, $(b)$ the number of revolutions
that the rotor executes before coming to rest.

Mihajlo Grcic

Numerade Educator

Problem 5

A small grinding wheel is attached to the shaft of an electric motor
which has a rated speed of $3600 \mathrm{rpm}$. When the power is turned on,
the unit reaches its rated speed in $5 \mathrm{s}$, and when the power is turned
off, the unit coasts to rest in $70 \mathrm{s}$. Assuming uniformly accelerated
motion, determine the number of revolutions that the motor executes
$(a)$ in reaching its rated speed, $(b)$ in coasting to rest.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 6

A connecting rod is supported by a knife-edge at point $A .$ For small
oscillations the angular acceleration of the connecting rod is govemed
by the relation $\alpha=-6 \theta$ where $\alpha$ is expressed in rad/s' and $\theta$ in
radians. Knowing that the connecting rod is released from rest when
$\theta=20^{\circ}$, determine ( $a$ ) the maximum angular velocity, (b) the angu-
lar position when $t=2 \mathrm{s}$.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 7

When studying whiplash resulting from rear-end collisions, the
rotation of the head is of primary interest. An impact test was
performed, and it was found that the angular acceleration of the head
is defined by the relation $\alpha=700 \cos \theta+70 \sin \theta,$ where $\alpha$ is
expressed in rad/s' and $\theta$ in radians. Knowing that the head is initially
at rest, determine the angular velocity of the head when $\theta=30^{\circ} .$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 8

The angular acceleration of an oscillating disk is defined by the relation $\alpha=-k \theta,$ where alpha is expressed in $\mathrm{rad} / \mathrm{s}^{2}$ and theta is expressed in radians. Determine $(a)$ the value of $k$ for which $\omega=12 \mathrm{rad} / \mathrm{s}$ when $\theta=0$ and $\theta=6$ rad when $\omega=0,(b)$ the angular velocity of the disk when $\theta=3$ rad.

CM

Corinne Montell

Numerade Educator

Problem 9

The angular acceleration of a shaft is defined by the relation
$\alpha=-0.5 \omega,$ where $\alpha$ is expressed in rad/s and $\omega$ in rad/s. Knowing
that at $t=0$ the angular velocity of the shaft is 30 rad/s, determine
$(a)$ the number of revolutions the shaft will execute before coming
to rest, (b) the time required for the shaft to come to rest, ( $c$ ) the
time required for the angular velocity of the shaft to reduce to
2 percent of its initial value.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 10

The bent rod $A B C D E$ rotates about a line joining points $A$ and $E$
with a constant angular velocity of 9 rad/s. Knowing that the rotation
is clockwise as viewed from $E$, determine the velocity and acceleration of corner $C .$

Mihajlo Grcic

Numerade Educator

Problem 11

In Prob. 15.10 , determine the velocity and acceleration of corner $B$,
assuming that the angular velocity is 9 rad/s and increases at the
rate of $45 \mathrm{rad} / \mathrm{s}^{2}$.

Mihajlo Grcic

Numerade Educator

Problem 12

The rectangular block shown rotates about the diagonal $O A$ with a
constant angular velocity of 6.76 rad/s. Knowing that the rotation is
counterclockwise as viewed from $A$, determine the velocity and
acceleration of point $B$ at the instant shown.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 13

The rectangular block shown rotates about the diagonal $O A$ with an
angular velocity of 3.38 rad/s that is decreasing at the rate of 5.07 radis?
Knowing that the rotation is counterclockwise as viewed from $A$,
determine the velocity and acceleration of point $B$ at the instant shown.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 14

A circular plate of $120-\mathrm{mm}$ radius is supported by two bearings $A$
and $B$ as shown. The plate rotates about the rod joining $A$ and $B$ with
a constant angular velocity of $26 \mathrm{rad} / \mathrm{s}$. Knowing that, at the instant
considered, the velocity of point $C$ is directed to the right, determine
the velocity and acceleration of point $E .$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 15

In Prob. 15.14 , determine the velocity and acceleration of point $E,$
assuming that the angular velocity is 26 rad/s and increases at the
rate of $65 \mathrm{rad} / \mathrm{s}^{2}$.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 16

The earth makes one complete revolution around the sun in
365.24 days. Assuming that the orbit of the earth is circular and has
a radius of $93,000,000 \mathrm{mi}$, determine the velocity and acceleration of
the earth.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 17

The earth makes one complete revolution on its axis in $23 \mathrm{h} 56 \mathrm{min}$.
Knowing that the mean radius of the earth is $3960 \mathrm{mi}$, determine the
linear velocity and acceleration of a point on the surface of the earth
$(a)$ at the equator, $(b)$ at Philadelphia, latitude $40^{\circ}$ north, (c) at the
North Pole.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 18

A series of small machine components being moved by a conveyor
belt pass over a $120-\mathrm{mmm}$ -radius idler pulley. At the instant shown,
the velocity of point $A$ is $300 \mathrm{mm} / \mathrm{s}$ to the left and its acceleration is
$180 \mathrm{mm} / \mathrm{s}^{2}$ to the right. Determine (a) the angular velocity and
angular acceleration of the idler pulley, ( $b$ ) the total acceleration
of the machine component at $B .$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 19

A series of small machine components being moved by a conveyor
belt pass over a 120 -mm-radius idler pulley. At the instant shown, the
angular velocity of the idler pulley is 4 rad/s clockwise. Determine
the angular acceleration of the pulley for which the magnitude of the
total acceleration of the machine component at $B$ is $2400 \mathrm{mm} / \mathrm{s}^{2}$.

Rashmi Sinha

Numerade Educator

Problem 20

The belt sander shown is initially at rest. If the driving drum $B$ has
a constant angular acceleration of $120 \mathrm{rad} / \mathrm{s}^{2}$ counterclockwise,
determine the magnitude of the acceleration of the belt at point $C$
when $(a) t=0.5 \mathrm{s},(b) t=2 \mathrm{s}$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 21

The rated speed of drum $B$ of the belt sander shown is $2400 \mathrm{rpm}$.
When the power is turned off, it is observed that the sander coasts
from its rated speed to rest in $10 \mathrm{s}$. Assuming uniformly decelerated
motion, determine the velocity and acceleration of point $C$ of the
belt, (a) immediately before the power is turned off, $(b) 9$ s s later.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 22

The two pulleys shown may be operated with the $V$ belt in any of
three positions. If the angular acceleration of shaft $A$ is 6 radis' and
if the system is initially at rest, determine the time required for
shaft $B$ to reach a speed of 400 rpm with the belt in each of the three
positions.

Jay Yoshimi

Numerade Educator

Problem 23

Three belts move over two pulleys without slipping in the speed
reduction system shown. At the instant shown, the velocity of point
$A$ on the input belt is $2 \mathrm{ft} / \mathrm{s}$ to the right, decreasing at the rate of
$6 \mathrm{ft} / \mathrm{s}^{2}$. Determine, at this instant, $(a)$ the velocity and acceleration
of point $C$ on the output belt, (b) the acceleration of point $B$ on the
output pulley.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 24

A gear reduction system consists of three gears $A, B,$ and $C .$ Knowing
that gear $A$ rotates clockwise with a constant angular velocity
$\omega_{A}=600 \mathrm{rpm}$, determine $(a)$ the angular velocities of gears $B$ and
$C,(b)$ the accelerations of the points on gears $B$ and $C$ which are
in contact.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 25

A belt is pulled to the right between cylinders $A$ and $B$. Knowing
that the speed of the belt is a constant $5 \mathrm{ft} / \mathrm{s}$ and no slippage occurs,
determine $(a)$ the angular velocities of $A$ and $B,(b)$ the accelerations
of the points which are in contact with the belt.

Narayan Hari

Numerade Educator

Problem 26

Ring $C$ has an inside radius of $55 \mathrm{mm}$ and an outside radius of
$60 \mathrm{mm}$ and is positioned between two wheels $A$ and $B$, each of $24 \mathrm{mm}$
outside radius. Knowing that wheel $A$ rotates with a constant angular
velocity of $300 \mathrm{rpm}$ and that no slipping occurs, determine (a) the
angular velocity of ring $C$ and of wheel $B,(b)$ the acceleration of the
points on $A$ and $B$ that are in contact with $C .$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 27

Ring $B$ has an inside radius $r_{2}$ and hangs from the horizontal shaft
$A$ as shown. Shaft $A$ rotates with a constant angular velocity of
25 rad/s and no slipping occurs. Knowing that $r_{1}=12 \mathrm{mm}, r_{2}=$
$30 \mathrm{mm},$ and $r_{3}=40 \mathrm{mm}$, determine (a) the angular velocity of ring
$B,(b)$ the accelerations of the points of shaft $A$ and ring $B$ which
are in contact, (c) the magnitude of the acceleration of point $D .$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 28

A plastic film moves over two drums. During a 4 -s interval the speed
of the tape is increased uniformly from $v_{0}=2 \mathrm{ft} / \mathrm{s}$ to $v_{1}=4 \mathrm{ft} / \mathrm{s}$.
Knowing that the tape does not slip on the drums, determine (a) the
angular acceleration of drum $B,(b)$ the number of revolutions executed
by drum $B$ during the 4 - interval.

RK

Ryan Keefe

Numerade Educator

Problem 29

Cylinder $A$ is moving downward with a velocity of $3 \mathrm{m} / \mathrm{s}$ when the
brake is suddenly applied to the drum. Knowing that the cylinder
moves $6 \mathrm{m}$ downward before coming to rest and assuming uniformly
accelerated motion, determine $(a)$ the angular acceleration of the
drum, $(b)$ the time required for the cylinder to come to rest.

Prashant Bana

Numerade Educator

Problem 30

The system shown is held at rest by the brake-and-drum system
shewn. After the brake is partially released at $t=0$ it is obsterved
that the cylinder moves $5 \mathrm{m}$ in $4.5 \mathrm{s}$. Assuming uniformly accelerated
motion, determine $(a)$ the angular acceleration of the drum, $(b)$ the
angular velocity of the drum at $t=3.5 \mathrm{s}$.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 31

A load is to be raised $20 \mathrm{ft}$ by the hoisting system shown. Assuming
gear $A$ is initially at rest, accelerates uniformly to a speed of $120 \mathrm{rpm}$
in $5 \mathrm{s}$, and then maintains a constant speed of $120 \mathrm{rpm}$, determine
$(a)$ the number of revolutions executed by gear $A$ in raising the load,
$(b)$ the time required to raise the load.

PR

Paul Ridder

Numerade Educator

Problem 32

A simple friction drive consists of two disks $A$ and $B$. Initially,
disk $B$ has a clockwise angular velocity of $500 \mathrm{m}$, and disk $A$ is at
rest. It is known that disk $B$ will coast to rest in $60 \mathrm{s}$. However, rather
than waiting until both disks are at rest to bring them together, disk
$A$ is given a constant angular acceleration of $3 \mathrm{rad} / \mathrm{s}^{2}$ counterclockwise.
Determine $(a)$ at what time the disks can be brought together if they
are not to slip, ( $b)$ the angular velocity of each disk as contact is made.

Ajay Singhal

Numerade Educator

Problem 33

Two friction wheels $A$ and $B$ are both rotating freely at $300 \mathrm{rpm}$
counterclockwise when they are brought into contact. After $12 \mathrm{s}$ of
slippage, during which time each wheel has a constant angular accel-
eration, wheel $B$ reaches a final angular velocity of $75 \mathrm{rpm}$ counter-
clockwise. Determine (a) the angular acceleration of each wheel
during the period of slippage, ( $b)$ the time at which the angular
velocity of wheel $A$ is equal to zero.

Prashant Bana

Numerade Educator

Problem 34

Two friction disks $A$ and $B$ are to be brought into contact without
slipping when the angular velocity of disk $A$ is 240 rpm counter-
clockwise. Disk $A$ starts from rest at time $t=0$ and is given a
constant angular acceleration with a magnitude $\alpha .$ Disk $B$ starts from
rest at time $t=2 \mathrm{s}$ and is given a constant clockwise angular accel-
eration, also with a magnitude $\alpha .$ Determine $(a)$ the required angular
acceleration magnitude $\alpha,(b)$ the time at which the contact occurs.

Ajay Singhal

Numerade Educator

Problem 35

Two friction disks $A$ and $B$ are brought into contact when the angular
velocity of disk $A$ is 240 rpm counterclockwise and disk $B$ is at rest.
A period of slipping follows and disk $B$ makes two revolutions
before reaching its final angular velocity. Assuming that the angular
acceleration of each disk is constant and inversely proportional to
the cube of its radius, determine ( $a$ ) the angular acceleration of each
disk, ( $b$ ) the time during which the disks slip.

Ajay Singhal

Numerade Educator

Problem 36

Steel tape is being wound onto a spool that rotates with a constant
angular velocity $\omega_{0}$. Denoting by $r$ the radius of the spool and tape
at any given time and by the thickness of the tape, derive an
expression for the acceleration of the tape as it approaches the spool.

Jared Schuler

Numerade Educator

Problem 37

In a continuous printing process, paper is drawn into the presses at
a constant speed $v$. Denoting by $r$ the radius of the paper roll at any
given time and by $b$ the thickness of the paper, derive an expression
for the angular acceleration of the paper roll.

Ajay Singhal

Numerade Educator