diapos de vibraciones
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VECTOR MECHANICS FOR ENGINEERS:
DYNAMICSDYNAMICS
Ninth EditionNinth Edition
Ferdinand P. BeerFerdinand P. Beer
E. Russell Johnston, Jr.E. Russell Johnston, Jr.
Lecture NotesLecture Notes
J. !alt "lerJ. !alt "ler
#e$as #ech %ni&ersit'#e$as #ech %ni&ersit'
CHAPTER
2010 The McGraw-Hill Companies, Inc. All rights reserved.
12Kinetics of Particles:Newtons Second Law
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsNinth
Contents
- -
Introduction
Newtons Second Law of Motio
n
Linear Momentum of a Particle
Sstems of !nits
E"uations of Motion#namic E"uili$rium
Sam%le Pro$lem &'(&
Sam%le Pro$lem &'()
Sam%le Pro$lem &'(*
Sam%le Pro$lem &'(+
Sam%le Pro$lem &'(,
An-ular Momentum of a Particle
E"uations of Motion in Radial .
Trans/erse Com%onents
Conser/ation of An-ular Momentum
Newtons Law of 0ra/itation
Sam%le Pro$lem &'(1Sam%le Pro$lem &'(2
Tra3ector of a Particle !nder a
Central 4orce
A%%lication to S%ace Mec5anics
Sam%le Pro$lem &'(6
Ke%lers Laws of Planetar Motion
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsNinth
Introduction
- /
Newtons first and third laws are sufficient for the study of bodies at
rest (statics) or bodies in motion with no acceleration.
When a body accelerates (changes in velocity magnitude or direction),
Newtons second law is required to relate the motion of the body to the
forces acting on it.
Newtons second law:
! "article will have an acceleration "ro"ortional to the magnitude of
the resultant force acting on it and in the direction of the resultant
force.
#he resultant of the forces acting on a "article is equal to the rate of
change of linear momentum of the "article.
#he sum of the moments about Oof the forces acting on a "article is
equal to the rate of change of angular momentum of the "article
about O.
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsNinth
Newtons Second Law of Motion
- 0
Newtons Second Law: $f the resultant forceacting on a
"article is not %ero, the "article will have an acceleration
proportional to the magnitude of resultantand in the
direction of the resultant.
&onsider a "article sub'ected to constant forces,
m
a
F
a
F
a
Fmass,constant
*
* =====
When a "article of mass mis acted u"on by a force
the acceleration of the "article must satisfy
,F
amF=
!cceleration must be evaluated with res"ect to aNewtonian frame of reference, i.e., one that is not
accelerating or rotating.
$f force acting on "article is %ero, "article will not
accelerate, i.e., it will remain stationary or continue on a
straight line at constant velocity.N
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsNinth
Linear Momentum of a Particle
- 1
+e"lacing the acceleration by the derivative of the
velocity yields
( )
"articletheofmomentumlinear=
==
=
L
dt
Ldvm
dt
d
dt
vdmF
Linear Momentum Conservation Principle:
$f the resultant force on a "article is %ero, the linear
momentum of the "article remains constant in bothmagnitude and direction.
N
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsNinth
Sstems of !nits
- 2
f the units for the four "rimary dimensions (force,
mass, length, and time), three may be chosen arbitrarily.
#he fourth must be com"atible with Newtons nd -aw.
International System of nits($ /nits): base units are
the units of length (m), mass (0g), and time (second).
#he unit of force is derived,
( ) s
m0g*
s
m*0g*N*
=
=
!S! Customary nits: base units are the units of force(lb), length (m), and time (second). #he unit of mass is
derived,
ft
slb*
sft*
lb*slug*
sft.
lb*lbm*
===
N
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsNinth
E"uations of Motion
- 3
Newtons second law "rovides
amF =
olution for "article motion is facilitated by resolving
vector equation into scalar com"onent equations, e.g.,
for rectangular com"onents,
"mFymF#mF
maFmaFmaF$a%aiam$F%FiF
"y#
""yy##
"y#"y#
====== ++=++
1or tangential and normal com"onents,
v
mFdt
dvmF
maFmaF
nt
nntt
==
==
N
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsNinth
#namic E"uili$rium
- 4
!lternate e2"ression of Newtons second law,
ectorinertial vamamF
3
=
With the inclusion of the inertial vector, the system
of forces acting on the "article is equivalent to
%ero. #he "article is in dynamic e&uili'rium.
4ethods develo"ed for "articles in static
equilibrium may be a""lied, e.g., co"lanar forces
may be re"resented with a closed vector "olygon.
$nertia vectors are often called inertial forcesas
they measure the resistance that "articles offer tochanges in motion, i.e., changes in s"eed or
direction.
$nertial forces may be conce"tually useful but are
not li0e the contact and gravitational forces found
in statics.
N
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsNinth
Sam%le Pro$lem &'(&
- 5
! 33lb bloc0 rests on a hori%ontal
"lane. 1ind the magnitude of the force
Prequired to give the bloc0 an accelera
tion or *3 ft5s
to the right. #he coefficient of 0inetic friction between the
bloc0 and "lane is$= 3.6.
-/#$N:
+esolve the equation of motion for the
bloc0 into two rectangular com"onent
equations.
/n0nowns consist of the a""lied force
Pand the normal reaction Nfrom the"lane. #he two equations may be
solved for these un0nowns.
)N
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsNinth
Sam%le Pro$lem &'(&
- 6
N
NF
g
(m
$
6.3
ftslb*.7
sft.
lb33
)
)
==
=
==
#
y
O
-/#$N:
+esolve the equation of motion for the bloc0into two rectangular com"onent equations.
:maF#=( )( )
lb*.7
sft*3ftslb*.76.33cos
=
= NP
:3= yF3lb333sin =PN
/n0nowns consist of the a""lied force Pand
the normal reactionN
from the "lane. #he twoequations may be solved for these un0nowns.
( ) lb*.7lb333sin6.33cos
lb333sin
=++=
PP
PN
lb*6*=P
( M h i ) E i D iN
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsNinth
Sam%le Pro$lem &'()
-
#he two bloc0s shown start from rest.
#he hori%ontal "lane and the "ulleyare frictionless, and the "ulley is
assumed to be of negligible mass.
8etermine the acceleration of each
bloc0 and the tension in the cord.
-/#$N:
Write the 0inematic relationshi"s for the
de"endent motions and accelerations of
the bloc0s.
Write the equations of motion for the
bloc0s and "ulley. &ombine the 0inematic relationshi"s
with the equations of motion to solve for
the accelerations and cord tension.
)( t M h i ) E i D iN
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Sam%le Pro$lem &'()
- -
Write equations of motion for bloc0s and "ulley.
:))# amF =( ) )a* 0g*33*=
:++y amF =
( )( ) ( )( ) +
+
+++
a*
a*
am*gm
0g33N93
0g33sm;*.90g33
==
=
:3== CCy amF
3* =
**
-/#$N:
Write the 0inematic relationshi"s for the de"endentmotions and accelerations of the bloc0s.
)+)+ aa#y )*
)
* ==#
y
O
( M h i ) E i D i( t M h i ) E i D iN
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Sam%le Pro$lem &'()
- /
( )
N*7;3
N;30g*33
sm3.
sm3.;
*
*
*
====
==
=
**
a*
aa
a
)
)+
)
&ombine 0inematic relationshi"s with equations of
motion to solve for accelerations and cord tension.
)+)+ aa#y *
* ==
( ) )a* 0g*33*=
( )
( ) ( ))+
a
a*
*
0g33N93
0g33N93
=
=
( ) ( ) 30g*330g*63N93
3 *
==
)) aa
**
#
y
O
( t M h i ) E i D i( t M h i ) E i D iN
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsNinth
Sam%le Pro$lem &'(*
- 0
#he *lb bloc0+starts from rest and
slides on the 3lb wedge), which is
su""orted by a hori%ontal surface.
Neglecting friction, determine ,a-the
acceleration of the wedge, and ,'-the
acceleration of the bloc0 relative to the
wedge.
-/#$N:
#he bloc0 is constrained to slide downthe wedge. #herefore, their motions are
de"endent.
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Sam%le Pro$lem &'(*
- 1
-/#$N:
#he bloc0 is constrained to slide down thewedge. #herefore, their motions are de"endent.
)+)+ aaa +=
Write equations of motion for wedge and bloc0.
#
y
:))#
amF =
( ) ))
))
ag(N
amN
==
*
*
6.3
3sin
:3cos )+)+#+# aamamF ==
( )+=
=3sin3cos
3cos3sin
gaa
aag((
))+
)+)++
( ) :3sin == )+y+y amamF( ) = 3sin3cos
* )++ ag((N
( t M h i ) E i D i( t M h i ) E i D iN
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Sam%le Pro$lem &'(*
- 2
( ) )) ag(N =*6.3
olve for the accelerations.
( )
( ) ( )
( )( )( ) ( ) +
=
+
=
==
3sinlb*lb3
3coslb*sft.
3sin
3cos
3sin3cos
3sin3cos
*
)
+)
+
)
)++))
)++
a
((
g(a
ag((ag(
ag((N
sft3=.6=)a
( ) ( ) +=+=
3sinsft.3cossft3=.6
3sin3cos
)+
))+
a
gaa
sft6.)3=)+a
( t M h i ) E i D i( t M h i ) E i D iNi
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsNinth
Sam%le Pro$lem &'(+
- 3
#he bob of a m "endulum describes
an arc of a circle in a vertical "lane. $f
the tension in the cord is .6 times the
weight of the bob for the "ositionshown, find the velocity and accel
eration of the bob in that "osition.
-/#$N:
+esolve the equation of motion for thebob into tangential and normal
com"onents.
olve the com"onent equations for the
normal and tangential accelerations.
olve for the velocity in terms of the
normal acceleration.
( t M h i ) E i D i( t M h i ) E i D iNi
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsinth
Sam%le Pro$lem &'(+
- 4
-/#$N:
+esolve the equation of motion for the bob intotangential and normal com"onents.
olve the com"onent equations for the normal and
tangential accelerations.
:tt maF =
=
=
3sin
3sin
ga
mamg
t
t
sm9.=ta
:nn maF =( )=
=
3cos6.
3cos6.
ga
mamgmg
n
n
sm3.*7=na
olve for velocity in terms of normal acceleration.
( )( )
sm3.*7m=== nn avv
a
sm77.6=v
( t M h i ) E i D i( t M h i ) E i D iNi
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsinth
Sam%le Pro$lem &'(,
- 5
8etermine the rated s"eed of a
highway curve of radius > 33 ft
ban0ed through an angle > *;o. #he
rated s"eed of a ban0ed highway curveis the s"eed at which a car should
travel if no lateral friction force is to
be e2erted at its wheels.
-/#$N:
#he car travels in a hori%ontal circular"ath with a normal com"onent of
acceleration directed toward the center
of the "ath.#he forces acting on the car
are its weight and a normal reaction
from the road surface.
+esolve the equation of motion for
the car into vertical and normal
com"onents.
olve for the vehicle s"eed.
( t M h i ) E i D i(ector Mechanics )or En*ineers D'na+icsNi
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsnth
Sam%le Pro$lem &'(,
- -6
-/#$N:
#he car travels in a hori%ontal circular
"ath with a normal com"onent of
acceleration directed toward the centerof the "ath.#he forces acting on the
car are its weight and a normal
reaction from the road surface.
+esolve the equation of motion for
the car into vertical and normal
com"onents.
:3= yF
cos
3cos
(.
(.
=
=
:nn maF =
sincos
sin
v
g
((
ag(. n
=
=
olve for the vehicle s"eed.
( )( ) ==
*;tanft33sft.
tan
)
) gv
hmi*.sft=.7 ==v
( t M h i ) E i D i(ector Mechanics )or En*ineers D'na+icsNi
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsnth
An-ular Momentum of a Particle
- -
moment of momentumor the angular
momentumof the "article about O.
== /mr0O
8erivative of angular momentum with res"ect to time,
=
=
+=+=
O
O
M
Fr
amr/m//mr/mr0
$t follows from Newtons second law that the sum of
the moments about Oof the forces acting on the
"article is equal to the rate of change of the angular
momentum of the "article about O!
"y#
O
mvmvmv
"y#
$%i
0
=
is "er"endicular to "lane containingO0 /mr and
)
sin
mr
vrm
rm/0O
=
==
(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsNin
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsnth
E"s of Motion in Radial . Trans/erse Com%onents
- --
( )( )
rrmmaF
rrmmaF rr
)
+==
==
&onsider "article at rand , in "olar coordinates,
( )
( )( )
rrmF
rrrm
mrdt
dFr
mr0O
)
)
+=
+=
=
=
#his result may also be derived from conservation
of angular momentum,
(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsNin
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsnth
Conser/ation of An-ular Momentum
- -/
When only force acting on "article is directed
toward or away from a fi2ed "oint O, the "article
is said to be moving under a central force.
ince the line of action of the central force
"asses through O, and3 == OO 0M
constant== O
0/mr
?osition vector and motion of "article are in a
"lane "er"endicular to .O0
4agnitude of angular momentum,
333 sin
constantsin
/mr
/rm0O
===
massunit
momentumangular
constant
)
)
===
==
hr
m
0
mr0
O
O
or
(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsNin
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsnth
Conser/ation of An-ular Momentum
- -0
+adius vector OPswee"s infinitesimal area
drd) ))
*=
8efine ===
*
* r
dt
dr
dt
d)areal velocity
+ecall, for a body moving under a central force,
constant) == rh
When a "article moves under a central force, its
areal velocity is constant.
(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsNin
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsnth
Newtons Law of 0ra/itation
- -1
@ravitational force e2erted by the sun on a "lanet or by
the earth on a satellite is an im"ortant e2am"le ofgravitational force.
Newtons law of universal gravitation two "articles of
massMand mattract each other with equal and o""osite
force directed along the line connecting the "articles,
9
*
slb
ft*3.
s0g
m*3=.77
ngravitatioofconstant
=
=
=
=
1
r
Mm1F
1or "article of mass mon the earths surface,
s
ft.
s
m;*.9 ==== gmg
.
M1m(
(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsNin
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsnth
Sam%le Pro$lem &'(1
- -2
! bloc0+of mass mcan slide freely on
a frictionless arm O)which rotates in a
hori%ontal "lane at a constant rate .3
a) the com"onent vrof the velocity of+
along O), and
b) the magnitude of the hori%ontal force
e2erted on+by the arm O).
Anowing that+is released at a distance
r3from O2 e2"ress as a function of r
-/#$N:
Write the radial and transverseequations of motion for the bloc0.
$ntegrate the radial equation to find an
e2"ression for the radial velocity.
ubstitute 0nown information into thetransverse equation to find an
e2"ression for the force on the bloc0.
(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsNin
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsnth
Sam%le Pro$lem &'(1
- -3
-/#$N:
Write the radial and transverse
equations of motion for the bloc0.
:
:
amF
amFrr
== ( )
rrmF
rrm
3
+= =
$ntegrate the radial equation to find an
e2"ression for the radial velocity.
=
==
====
r
r
v
rr
rr
rr
rrr
drrdvv
drrdrrdvv
dr
dvv
dt
dr
dr
dv
dt
dvvr
r
3
3
3
3
dr
dvv
dt
dr
dr
dv
dt
dvvr rr
rrr ====
3
3
rrvr =
ubstitute 0nown information into the
transverse equation to find an e2"ression
for the force on the bloc0.
( ) *3
3 rrmF =
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsnth
Sam%le Pro$lem &'(2
- -4
! satellite is launched in a direction
"arallel to the surface of the earth
with a velocity of *;;3 mi5h from
an altitude of 3 mi. 8etermine the
velocity of the satellite as it reaches itma2imum altitude of 3 mi. #he
radius of the earth is 973 mi.
-/#$N:
ince the satellite is moving under a
central force, its angular momentum is
constant.
-
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsnth
Sam%le Pro$lem &'(2
- -5
-/#$N:
ince the satellite is moving under a
central force, its angular momentum is
constant.
-
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsth
Tra3ector of a Particle !nder a Central 4orce
- /6
1or "article moving under central force directed towards force center,
( ) 3 ==+== FrrmFFrrm r
econd e2"ression is equivalent to from which,,constant)
==hr
==
rd
d
r
hr
r
h *and
!fter substituting into the radial equation of motion and sim"lifying,
ru
umh
Fu
d
ud *where
==+
$fFis a 0nown function of ror u2then "article tra'ectory may be
found by integrating for u 3 f,-, with constants of integration
determined from initial conditions.
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A%%lication to S%ace Mec5anics
- /
constant
*where
==+
====+
h
1Mu
d
ud
1Mmur
1MmF
ru
umh
Fu
d
ud
&onsider earth satellites sub'ected to only gravitational "ull
of the earth,
olution is equation of conic section,
( ) tyeccentricicos**
==+==
1M
hC
h
1M
ru
rigin, located at earths center, is a focus of the conic section.
#ra'ectory may be elli"se, "arabola, or hy"erbola de"ending
on value of eccentricity.
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A%%lication to S%ace Mec5anics
- /-
( ) tyeccentricicos**
==+=
1MhC
h1M
r
#ra'ectory of earth satellite is defined by
hyper'ola, 4 * or C 4 1M5h. #he radius vector
becomes infinite for
=
==+
**** cos*cos3cos*
hC1M
para'ola, > * or C 3 1M5h. #he radius vector
becomes infinite for
==+ *;33cos*
ellipse, B * or C 6 1M5h. #he radius vector is finite
for and is constant, i.e., a circle, for B 3.
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A%%lication to S%ace Mec5anics
- //
$ntegration constant Cis determined by
conditions at beginning of free flight, >3, r 3 r72
( ) 333
3
3
**
3cos**
vr
1M
rh
1M
rC
1M
Ch
h
1M
r
==
+=
( )
3
3
33
or*
r
1Mvv
vr1Mh1MC
esc ==
=
atellite esca"es earth orbit for
#ra'ectory is elli"tic for v76 vescand becomes
circular for > 3 or C> 3,
3r
1Mvcirc =
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A%%lication to S%ace Mec5anics
- /0
+ecall that for a "article moving under a central
force, the areal velocityis constant, i.e.,
constant*
* === hr
dt
d)
Periodic timeor time required for a satellite to
com"lete an orbit is equal to area within the orbit
divided by areal velocity,
h
a'
h
a'
==
where ( )
*3
*3*
rr'
rra
=
+=
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Sam%le Pro$lem &'(6
- /1
8etermine:
a) the ma2imum altitude reached by
the satellite, and
b) the "eriodic time of the satellite.
! satellite is launched in a direction
"arallel to the surface of the earth
with a velocity of 7,933 0m5h at an
altitude of 633 0m.
-/#$N:
#ra'ectory of the satellite is described by
cos*
C
h
1M
r+=
3.
8etermine the ma2imum altitude by
finding rat > *;3o.
With the altitudes at the "erigee and
a"ogee 0nown, the "eriodic time canbe evaluated.
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Sam%le Pro$lem &'(6
- /2
-/#$N:
#ra'ectory of the satellite is described by
cos*
C
h
1M
r+=
3.
( )
( )( )
( )( )*
7
9
733
3
73
sm*39;
m*3=.7sm;*.9
sm*3.=3
sm*36.*3m*37.;=
sm*36.*3
s5h733
m50m*333
h
0m7933
m*37.;=
0m6337=3
=
==
=
==
=
=
=
+=
g.1M
vrh
v
r
( )*9
*
7
3
m*3.76
sm.=3
sm*39;
m*3;=.7
*
*
=
=
= h1M
rC
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Sam%le Pro$lem &'(6
- /3
8etermine the ma2imum altitude by finding r*
at > *;3o.
( )0m77=33m*3=.77
m
**3.76
sm.=3
sm*39;*
7*
9
*
*
==
==
r
Ch
1M
r
( ) 0m73330m7=377=33altitudema2 ==
With the altitudes at the "erigee and a"ogee 0nown,
the "eriodic time can be evaluated.
( ) ( )
( )( )sm*3=3.
m*3*.m*37.;
h
m*3*.m*3=.77;=.7
m*37.;m*3=.77;=.7
9
77
77*3
77
*
*3*
==
===
=+=+=
a'
rr'
rra
min*h*9s*3.=3 ==
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(ector Mechanics )or En*ineers D'na+ics(ector Mechanics )or En*ineers D'na+icsh
Ke%lers Laws of Planetar Motion
+esults obtained for tra'ectories of satellites around earth may also be
a""lied to tra'ectories of "lanets around the sun.
?ro"erties of "lanetary orbits around the sun were determined
astronomical observations by Cohann Ae"ler (*6=**73) before
Newton had develo"ed his fundamental theory.
*)
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