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7/25/2019 Dinamica Clases 7 y 8
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VECTOR MECHANICS FOR ENGINEERS:
DYNAMICS
Eighth Edition
Ferdinand P. Beer
E. Russell Johnston, Jr.
Leture Notes!
J. "alt #ler
$e%as $eh &ni'ersit(
CHAPTER
2007 The McGraw-Hill Companies, Inc. All rights
13Cintica de Partculas:
Metodos de Energa
and Cantidad de
Movimiento.
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ContentsIntroduction
Work of a ForcePrinciple of Work & Energy
Applications of the Principle of Work
& Energy
Power and Efficiency
Sample Problem 1!1Sample Problem 1!"
Sample Problem 1!
Sample Problem 1!#
Sample Problem 1!$
Potential Energy
%onseratie Forces
%onseration of Energy
'otion (nder a %onseratie %entral
Force
Sample Problem 1!)
Sample Problem 1!*Sample Problem 1!+
Principle of Impulse and 'omentum
Impulsie 'otion
Sample Problem 1!1,
Sample Problem 1!11
Sample Problem 1!1"
Impact
-irect %entral Impact
.bli/ue %entral Impact
Problems Inoling Energy and 'omentum
Sample Problem 1!1#
Sample Problem 1!1$
Sample Problems 1!1)
Sample Problem 0!1*
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Introduction
Preiously2 problems dealing with the motion of particles were
soled through the fundamental e/uation of motion2%urrent chapter introduces two additional methods of analysis!
!amF=
Method of work and energy3 directly relates force2 mass2
elocity and displacement!
Method of impulse and momentum3 directly relates force2
mass2 elocity2 and time!
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Work o a !orce
-ifferential ector is theparticle displacement!rd
Work of the forceis
dzFdyFdxF
dsF
rdFdU
zyx ++=
=
=
cos
Work is a scalar /uantity2 i!e!2 it has magnitude and
sign but not direction!
force!length -imensions of work are (nits are( ) ( )( ) 41!$)lb1ftm15141 ==joule
E
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Work o a !orce
Work of a force during a finite displacement2
( )
( )
++=
==
=
"
1
"
1
"
1
"
1
cos
"1
A
Azyx
s
s
t
s
s
A
A
dzFdyFdxF
dsFdsF
rdFU
Work is represented by the area under thecure ofFtplotted againsts!
E
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Work o a !orce
Work of a constant force in rectilinear motion2
( ) xFU = cos"1
Work of the force of graity2
( ) yWyyW
dyWU
dyW
dzFdyFdxFdU
y
y
zyx
==
=
=
++=
1"
"1
"
1
Work of the weightis e/ual to product of
weight Wand ertical displacement y.
Work of the weightis positie when y < ,2i!e!2 when the weight moes down!
E
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Work o a !orce 'agnitude of the force e6erted by a spring is
proportional to deflection2
( )lb7in!or57mconstantspring==
k
kxF
Work of the force exerted y spring2
"""
1"1"
1"1
"
1
kxkxdxkxU
dxkxdxFdU
x
x
==
==
Work of the force exerted y springis positie
whenx"8x
12 i!e!2 when the spring is returning to
its undeformed position!
Work of the force e6erted by the spring is e/ual to
negatie of area under cure ofFplotted against
x2 ( ) xFFU += "1"1
"1
E
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Work o a !orce
Work of a graitational force 9assume particleMoccupies fi6ed position !while particle mfollows path
shown:2
1"""1
"
"
1r
Mm"
r
Mm"dr
r
Mm"U
drr
Mm"FdrdU
r
r
==
==
E
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Work o a !orce
Forces which do notdo work #ds; , or cos = 0)3
weight of a body when its center of graity moes
hori
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Particle "inetic Energ#: Princi$le o Work % Energ#
d$m$dsF
ds
d$m$
dt
ds
ds
d$m
dt
d$
mmaF
t
tt
=
==
==
%onsider a particle of mass m acted upon by forceF
Integrating fromA%toA&2
energykineticm$'''U
m$m$d$$mdsF$
$
s
st
===
==
""1
1""1
"1"
1"""
1"
1
"
1
'he work of the force is e(ual to the change in
kinetic energy of the particle!
F
(nits of work and kinetic energy are the same3
4m5m
s
mkg
s
mkg
"
""
"
1 ==
=
== m$'
) t M h i * E i D iE
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A$$lications o t&e Princi$le o Work and Energ#
Wish to determine elocity of pendulum bob
atA"! %onsider work & kinetic energy!
Force acts normal to path and does no
work!
)
gl$
$g
WWl
'U'
"
"
1,
"
"
"
""11
=
=+
=+
=elocity found without determining
e6pression for acceleration and integrating!
All /uantities are scalars and can be added
directly!
Forces which do no work are eliminated
from the problem!
) t M h i * E i D iE
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A$$lications o t&e Princi$le o Work and Energ#
Principle of work and energy cannot be
applied to directly determine the accelerationof the pendulum bob!
%alculating the tension in the cord re/uires
supplementing the method of work and
energy with an application of 5ewton>s
second law! As the bob passes throughA&2
Wl
gl
g
WW)
l
$
g
W
W)
amF nn
"
""
=+=
=
=
gl$ ""=
) t M h i * E i D iE
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Po'er and Eicienc#
rate at which work is done!
$F
dt
rdF
dt
dU
)ower
=
==
=
-imensions of power are work7time or force?elocity!
(nits for power are
W*#)s
lbft$$,hp1or
s
m51
s
419watt:W1 =
===
inputpower
outputpower
input work
koutput wor
efficiency
=
=
=
) t M h i * E i D iE
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(am$le Pro)lem *+.*
An automobile weighing #,,, lb is
drien down a $oincline at a speed of
), mi7h when the brakes are applied
causing a constant total breaking forceof 1$,, lb!
-etermine the distance traeled by the
automobile as it comes to a stop!
S.@(I.53
Ealuate the change in kinetic energy!
-etermine the distance re/uired for the
work to e/ual the kinetic energy change!
) t M h i * E i D iE
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(am$le Pro)lem *+.*S.@(I.53
Ealuate the change in kinetic energy!
( ) ( ) lbft#B1,,,BB"!"#,,,
sftBBs),,
h
mi
ft$"B,
h
mi),
"
"1"
1"1
1
1
===
=
=
m$'
$
( ) ,lb11$1lbft#B1,,,""11
==+
x
'U'
ft#1B=x
-etermine the distance re/uired for the work
to e/ual the kinetic energy change!
( ) ( ) ( )
( )x
xxU
lb11$1
$sinlb#,,,lb1$,,"1 =
+=
,, "" == '$
) t M h i * E i D iEi
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(am$le Pro)lem *+.,
wo blocks are Coined by an ine6tensible
cable as shown! If the system is released
from rest2 determine the elocity of block
Aafter it has moed " m! Assume that
the coefficient of friction between blockAand the plane isk; ,!"$ and that the
pulley is weightless and frictionless!
S.@(I.53
Apply the principle of work and
energy separately to blocksAand*!
When the two relations are combined2
the work of the cable forces cancel!
Sole for the elocity!
) t M h i * E i D iEi
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(am$le Pro)lem *+.,S.@(I.53
Apply the principle of work and energy separatelyto blocksAand*!
( )( )( )
( ) ( )
( ) ( ) ( ) ( ) ""1
""1
""11
"
kg",,m"5#+,m"
m"m",
3
5#+,51+)""$!,
51+)"smB1!+kg",,
$F
$mFF
'U'
W+F
W
,
AA,
AkAkA
A
=
=+=+
======
( )
( ) ( )
( ) ( ) ( ) ( ) ""1
""1
""11
"
kg,,m"5"+#,m"
m"m",
35"+#,smB1!+kg,,
$F
$mWF
'U'W
c
**c
*
=+
=+
=+==
) t M h i * E i D iEi
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(am$le Pro)lem *+., When the two relations are combined2 the work of the
cable forces cancel! Sole for the elocity!( ) ( ) ( ) ( ) "
"1 kg",,m"5#+,m" $F, =
( ) ( ) ( ) ( ) ""1 kg,,m"5"+#,m" $Fc =+
( ) ( ) ( ) ( ) ( )
( ) ""1
""1
kg$,,4#+,,
kg,,kg",,m"5#+,m"5"+#,
$
$
=
+=
sm#!#=$
)etor Mehanis *or Engineers D na+isEi
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(am$le Pro)lem *+.+
A spring is used to stop a ), kg package
which is sliding on a hori
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(am$le Pro)lem *+.+S.@(I.53
Apply principle of work and energy between initial
position and the point at which spring is fully compressed!
( ) ( ) ,4$!1B*sm$!"kg), ""
"1"
1"1
1 ==== 'm$'
( )
( ) ( )( ) ( ) kk
kfxWU
4**m)#,!,smB1!+kg),"
"1
==
=
( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( ) 4,!11"m,#,!,5",,5"#,,
5",,m1),!,mk5",
5"#,,m1",!,mk5",
"1
ma6min"1
"1
,ma6
,min
=+=
+=
==+====
x))U
xxk)
kx)
e
( ) ( ) ( ) 411"4**"1"1"1 =+= kef UUU
( ) ,411"4**D4$!1B*
3""11
=
=+
k
'U'
",!,
=k
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(am$le Pro)lem *+.+ Apply the principle of work and energy for the rebound
of the package!
( ) ""1"
"1
" kg),, $m$'' ===
( ) ( ) ( )
4)!$
411"4**"""
+=
+=+= kef UUU
( ) ""1
""
kg),4$!),
3
$
'U'
=+
=+
sm1,!1=$
)etor Mehanis *or Engineers! D(na+isEig
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- //
(am$le Pro)lem *+.-
A ",,, lb car starts from rest at point 1
and moes without friction down the
track shown!
-etermine3
a: the force e6erted by the track on
the car at point "2 and
b: the minimum safe alue of the
radius of curature at point !
S.@(I.53
Apply principle of work and energy to
determine elocity at point "!
Apply 5ewton>s second law to find
normal force by the track at point "!
Apply principle of work and energy todetermine elocity at point !
Apply 5ewton>s second law to find
minimum radius of curature at point
such that a positie normal force is
e6erted by the track!
)etor Mehanis *or Engineers! D(na+isEig
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(am$le Pro)lem *+.-S.@(I.53
Apply principle of work and energy to determine
elocity at point "!
( )
( )
( ) ( ) ( ) sftB!$,ft"!"ft#,"ft#,""
1ft#,,3
ft#,
"
1,
"""
"
""""11
"1
""
"""
1"1
===
=+=+
+=
===
$sg$
$g
WW'U'
WU
$g
Wm$''
Apply 5ewton>s second law to find normal force by
the track at point "!
3nn amF =+ ( )
W+
g
g
W$
g
Wam+W n
$
ft",
ft#,"
"
""
=
===+
lb1,,,,=+
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(am$le Pro)lem *+.- Apply principle of work and energy to determine
elocity at point !
( )
( ) ( ) ( ) sft1!#,sft"!"ft"$"ft"$"
"
1ft"$,
"
"11
===
=+=+
$g$
$g
WW'U'
Apply 5ewton>s second law to find minimum radius ofcurature at point such that a positie normal force is
e6erted by the track!
3nn amF =+
( )
" ft"$"
g
g
W$
g
W
amWn
===
ft$,=
)etor Mehanis *or Engineers! D(na+isEig
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- /1
(am$le Pro)lem *+.
he dumbwaiterand its load hae a
combined weight of ),, lb2 while the
counterweight , weighs B,, lb!
-etermine the power deliered by theelectric motorMwhen the dumbwaiter
#a-is moing up at a constant speed of
B ft7s and #-has an instantaneous
elocity of B ft7s and an acceleration of
"!$ ft7s"2 both directed upwards!
S.@(I.53
Force e6erted by the motorcable has same direction as
the dumbwaiter elocity!
Power deliered by motor is
e/ual to F$/ $0 B ft7s!
In the first case2 bodies are in uniform
motion! -etermine force e6erted by
motor cable from conditions for static
e/uilibrium!
In the second case2 both bodies are
accelerating! Apply 5ewton>s
second law to each body to
determine the re/uired motor cable
force!
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(am$le Pro)lem *+. In the first case2 bodies are in uniform motion!
-etermine force e6erted by motor cable from
conditions for static e/uilibrium!
( ) ( )
slbft1),,
sftBlb",,
=== F$)ower
( ) hp+1!"slbft$$,
hp1slbft1),, =
=)ower
FreeDbody %3
3,=+ yF lb#,,,lbB,," == ''
FreeDbody -3
3,=+ yFlb",,lb#,,lb),,lb),,
,lb),,
====+
'F
'F
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(am$le Pro)lem *+. In the second case2 both bodies are accelerating! Apply
5ewton>s second law to each body to determine the re/uired
motor cable force!
=== ""1" sft"$!1sft$!" , aaa
FreeDbody %3
3,,y amF =+ ( ) lb$!B#"$!1"!"
B,,"B,, == ''
FreeDbody -3
3y amF =+ ( )
lb1!")")!#)),,$!B#
$!""!"
),,),,
==+
=+
FF
'F
( ) ( ) slbft",+*sftBlb1!")" === F$)ower
( ) hpB1!
slbft$$,
hp1slbft",+* =
=)ower
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Potential Energ#
"1"1 yWyWU =
Work of the force of graity 2W
Work is independent of path followed depends
only on the initial and final alues of Wy.
=
=Wy1g
potential energyof the body with respect
toforce of gra$ity!
"1"1 gg 11U =
(nits of work and potential energy are the same3
4m5 === Wy1g
%hoice of datum from which the eleationyismeasured is arbitrary!
)etor Mehanis *or Engineers! D(na+isEig
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Potential Energ#
Preious e6pression for potential energy of a body
with respect to graity is only alid when theweight of the body can be assumed constant!
For a space ehicle2 the ariation of the force of
graity with distance from the center of the earth
should be considered!
Work of a graitational force2
1""1
r
"Mm
r
"MmU =
Potential energy 1gwhen the ariation in the
force of graity can not be neglected2
r
W2
r
"Mm1g
"
==
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Potential Energ#
Work of the force e6erted by a spring depends
only on the initial and final deflections of thespring2
"
""
1"
1"
1"1 kxkxU =
he potential energy of the body with respect
to the elastic force2
( ) ( )"1"1
""1
ee
e
11U
kx1
=
=
5ote that the preceding e6pression for 1eis
alid only if the deflection of the spring is
measured from its undeformed position!
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Conservative !orces %oncept of potential energy can be applied if the
work of the force is independent of the path
followed by its point of application!
( ) ( )"""111"1 2222 zyx1zyx1U =Such forces are described as conser$ati$e forces!
For any conseratie force applied on a closed path2
,= rdF Elementary work corresponding to displacement
between two neighboring points2
( ) ( )
( )zyxd1
dzzdyydxx1zyx1dU
22
2222
=
+++=
1z
1
y
1
x
1F
dzz
1dy
y
1dx
x
1dzFdyFdxF zyx
grad=
+
+
=
+
+
=++
)etor Mehanis *or Engineers! D(na+isEigh
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- -/
Conservation o Energ# Work of a conseratie force2
"1"1
11U =
%oncept of work and energy2
1""1 ''U =
Follows that
constant
""11
=+=+=+
1'3
1'1'
When a particle moes under the action of
conseratie forces2 the total mechanical
energy is constant!
W1'
W1'
=+==
11
11 ,
( )
W1'
1WggWm$'
=+
====
""
""""1" ,"
"1 Friction forces are not conseratie! otal
mechanical energy of a system inoling
friction decreases!
'echanical energy is dissipated by friction
into thermal energy! otal energy is constant!
)etor Mehanis *or Engineers! D(na+isEigh
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- --
Motion /nder a Conservative Central !orce When a particle moes under a conseratie central
force2 both the principle of conseration of angular
momentum
and the principle of conseration of energy
may be applied!
sinsin ,,, rm$m$r =
r
"Mmm$
r
"Mmm$
1'1'
=
+=+
""1
,
","
1
,,
ien r2 the e/uations may be soled for $and .
At minimum and ma6imum r2 = +,o! ien the
launch conditions2 the e/uations may be soled for
rmin/ rmax/ $min2 and $max!
)etor Mehanis *or Engineers! D(na+isEigh
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)etor Mehanis *or Engineers! D(na+ishth
- -0
(am$le Pro)lem *+.0
A ", lb collar slides without friction
along a ertical rod as shown! he
spring attached to the collar has an
undeflected length of # in! and aconstant of lb7in!
If the collar is released from rest at
position 12 determine its elocity after
it has moed ) in! to position "!
S.@(I.53
Apply the principle of conseration of
energy between positions 1 and "!
he elastic and graitational potential
energies at 1 and " are ealuated from
the gien information! he initialkinetic energy is
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)etor Mehanis *or Engineers! D(na+ishth
- -1
(am$le Pro)lem *+.0S.@(I.53
Apply the principle of conseration of energy between
positions 1 and "!
Position 13 ( ) ( )
,
lbft",lbin!"#
lbin!"#in!#in!Bin!lb
1
1
"
"1"
1"1
=
=+=+=
===
'
111
kx1
ge
e
Position "3 ( ) ( )
( ) ( )
""
""
""
"
1"
"
""1"
""1
11!,
"!"
",
"
1
lbft$!$lbin!))1",$#
lbin!1",in!)lb",
lbin!$#in!#in!,1in!lb
$$m$'
111
Wy1
kx1
ge
g
e
===
===+=
===
===
%onseration of Energy3
lbft$!$,!11lbft", ""
""11
=+
+=+
$
1'1'
= sft+1!#"$
)etor Mehanis *or Engineers! D(na+isEigh
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)etor Mehanis *or Engineers! D(na+ishth
- -2
(am$le Pro)lem *+.1
he ,!$ lb pellet is pushed against the
spring and released from rest atA!
5eglecting friction2 determine thesmallest deflection of the spring for
which the pellet will trael around the
loop and remain in contact with the
loop at all times!
S.@(I.53
Since the pellet must remain in contactwith the loop2 the force e6erted on the
pellet must be greater than or e/ual to
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)etor Mehanis *or Engineers! D(na+ishth
- -3
(am$le Pro)lem *+.1S.@(I.53
Setting the force e6erted by the loop to
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)etor Mehanis *or Engineers! D(na+ishth
- -4
(am$le Pro)lem *+.2
A satellite is launched in a direction
parallel to the surface of the earth
with a elocity of )+,, km7h from
an altitude of $,, km!
-etermine #a-the ma6imum altitudereached by the satellite2 and #-the
ma6imum allowable error in the
direction of launching if the satellite
is to come no closer than ",, km to
the surface of the earth
S.@(I.53
For motion under a conseratie centralforce2 the principles of conseration of
energy and conseration of angular
momentum may be applied simultaneously!
Apply the principles to the points of
minimum and ma6imum altitude todetermine the ma6imum altitude!
Apply the principles to the orbit insertion
point and the point of minimum altitude to
determine ma6imum allowable orbit
insertion angle error!
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)etor Mehanis *or Engineers! D(na+ishth
- -5
(am$le Pro)lem *+.2 Apply the principles of conseration of energy and
conseration of angular momentum to the points of minimum
and ma6imum altitude to determine the ma6imum altitude!
%onseration of energy3
1
"1"
1
,
","
1
r
"Mmm$
r
"Mmm$1'1' AAAA =+=+
%onseration of angular momentum3
1,,111,,r
r$$m$rm$r ==
%ombining2
",,1
,
1
,
,"
1
","
,"1 "111
$r
"M
r
r
r
r
r
"M
r
r$ =+
=
( )( ) "1"")""
),
,
sm1,+Bm1,*!)smB1!+
sm1,"$!1,hkm)+,,
km)B*,km$,,km)*,
===
===+=
g2"M
$
r
km),#,,m1,#!), )
1
==r
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)etor Mehanis *or Engineers! D(na+isth
(am$le Pro)lem *+.2 Apply the principles to the orbit insertion point and the point
of minimum altitude to determine ma6imum allowable orbit
insertion angle error!
%onseration of energy3
min
"ma6"
1
,
","
1,,
r
"Mmm$
r
"Mmm$1'1' AA =+=+
%onseration of angular momentum3
min
,,,ma6ma6min,,, sinsin
rr$$m$rm$r ==
%ombining and soling for sin02
=
=
$!11+,
+B,1!,sin
,
,
= $!11errorallowable