Chap12(3)-3
Short Description
Dynamics...
Description
Chapter 12: Kinematics of a Particle
Particle: • A particle has a mass but negligible size and shape. • A particle can undergo only translational motion. • Obects li!e roc!ets" proectile" or #ehicle may be considered as particles pro#ided motion of the body is characterized by motion of its mass center $a point% and any rotation of the body is neglected. • Chapter 12 & 1' are de#oted to dynamics of particles.
Rigid Body: A rigid body has mass" size and shape. A rigid A rigid body can undergo both translational and rotational motion. Chapters 1( ) 21 are de#oted to rigid body dynamics.
12.1: Introduction - An Overview of Mechanics echanics: +he study of ho- bodies echanics: +he react to forces acting on them.
*tatics: +he study of *tatics: +he bodies in e,uilibrium.
Dynamics: 1. ine!atics & concerned -ith the geometric aspects of motion 2. inetics ) concerned -ith the forces causing the motion
R"#$I%I&"AR I&"MA$I#' ('ection 12.2) %inear !otion $also called rectilinear motion% is motion along a straight line" and can therefore be described mathematically using only one spatial dimension.
Position of a Particle +his is step & 1 in particle analysis. +he position of the particle at any instant" relati#e to the fi/ed origin" O" is defined by the position #ector r " or the scalar s. *calar s can be positi#e or negati#e. +ypical units for r and s are meters $m% or feet $ft%.
*is+lace!ent of a Particle
+he displacement of the particle is defined as the change in its position. 4ector form:
r = r’ - r
*calar form: ∆ s s5 ) s
0isplacement is a #ector ,uantity. 0isplacement inal Position & 3nitial Position +he total distance tra#eled by the particle" s +" is a positi#e scalar that represents the total length of the path o#er -hich the particle tra#els.
A particle is mo#ing in a circular path of radius r . +he magnitude of displacement and distance after half a circle -ould be: $a% 6ero $b% 7 r $c% 2 r $d% 27 r
,"%O#I$ 4elocity is a measure of the rate of change in the position of a particle. 3t is a #ector ,uantity $it has both magnitude and direction%. athematically" # ds8dt 0irection: A positi#e #alue of #elocity indicates that the particle is mo#ing along the assumed positi#e direction of the coordinate a/is. A negati#e #alue indicates that the particle mo#es in the opposite direction.
+he magnitude of the #elocity is called speed" -ith units of m8s or ft8s. *peed is a scalar ,uantity.
A##"%"RA$IO& Acceleration is the rate of change in the #elocity of a particle. 3t is a #ector ,uantity. +ypical units are m8s2 or ft8s2. +he instantaneous acceleration is the time deri#ati#e of #elocity. 4ector form: a dv 8 dt *calar form: a d# 8 dt d 2s 8 dt2 Acceleration can be positi#e $speed increasing% or negati#e $speed decreasing" also called as decelerating%. As the te/t indicates" the deri#ati#e e,uations for #elocity and acceleration can be manipulated to get a ds # d#
'MMAR O/ I&"MA$I# R"%A$IO&': R"#$I%I&"AR MO$IO&
0ifferentiate position to get #elocity and acceleration.
# ds8dt a d#8dt a # d#8ds
9/ample: +he acceleration of a particle is a function of its displacement and is gi#en as: a ( x$1 ; kx2%< -here k is a constant. =i#en: #$ x % >.' ft8s ? #$ x .@'ft% 1'ft8s. ind k . +hin!: hich one of the follo-ing three e,uation -ill help you find k .
# ds8dt a d#8dt a ds # d#
'P"#IA% #A'": #O&'$A&$ A##"%"RA$IO& MO$IO& +he three !inematic e,uations can be integrated for the special case -hen acceleration is constant $a ac% to obtain #ery useful e,uations. A common e/ample of constant acceleration is gra#ity< i.e." a body freely falling to-ard earth. 3n this case" ac g B.1 m8s2 D2.2 ft8s2 do-n-ard. +hese e,uations are: v
t
∫ dv = ∫ a dt
vo s
c
o
v
yields
s
yields
# 2 = $#o %2 + 2a $s ) s %
o s
∫ v dv = ∫ a ds vo
#=#
o
t
∫ ds = ∫ v dt s o
yields
so
c
+at c
= s + # t + $182% a t 2 o o c
c
o
+he three !inematic e,uations are: # # ; act s s ;#t ; .' act2 #2 #2 ; 2ac $s & s% Eemember: +o use the abo#e three e,uations: 1%
+he particle must be mo#ing -ith constant acceleration.
2%
Fou must define a fi/ed coordinate a/is and specify positi#e8 negati#e directions before you use these e,ns.
D%
+he magnitudes and signs of s" # and ac" used in the abo#e three e,uations are determined -.r.t to the chosen coordinate a/is.
9/ample: An airplane begins its ta!e)off run at A -ith zero #elocity and a constant acceleration a. Kno-ing that it becomes airborne D s later at B and that the distance AB is 2> ft" determine $a% the acceleration a" $b% the ta!e)off #elocity G B.
9/ample: Automobiles A and B are tra#eling in adacent high-ay lanes and at t ha#e the positions and speeds sho-n. Kno-ing that automobile A has a constant acceleration of .( m8s 2 and that B has a constant deceleration of .@ m8s 2" determine $a% -hen and -here A -ill o#erta!e B" $b% the speed of each automobile at that time.
12.0 MO$IO& O/ A PRO"#$I%"
MO$IO& O/ A PRO"#$I%"
('ection 12.0)
3f -e ignore the air drag8 air)friction on the particle" then the only force acting on a proectile is its -eight
Horizontal − Component
Vertical − Component
→ ∑ F x = Max = : ⇒ a x = :
↑ ∑ F y = Ma y = − Mg
⇒
dv x
=:
dt ⇒ v x = Cons tan t
⇒ a y = − g ⇒ a y = Cons tan t Constant acceleration motion along y a/is
Proectile motion can be treated as t-o rectilinear motions" one in the horizontal direction e/periencing zero acceleration and the other in the #ertical direction e/periencing constant acceleration $i.e." from gra#ity%.
I&"MA$I# "A$IO&': 3ORI4O&$A% MO$IO&
# # ; act s s ;#t ; .' a ct2
#2 #2 ; 2ac $s & s% *ince a/ " the #elocity in the horizontal direction remains constant $#/ #o/% and the position in the / direction can be determined by: / /o ; $#o/% t
I&"MA$I# "A$IO&': ,"R$I#A% MO$IO&
*ince the positi#e y)a/is is directed up-ard" ay & g. Application of the constant acceleration e,uations yields: #y #oy & g t
# # ; act s s ;#t ; .' a ct
2
#2 #2 ; 2ac $s & s%
y yo ; $#oy% t & H g t2
#y2 #oy2 & 2 g $y & y o%
IMPOR$A&$ "A$IO&' /OR PRO"#$I%" MO$IO&
or Iorizontal otion: / / ; $#/% t
or 4ertical otion: #y #y & g t y y ; $#y% t & H g t2 #y2 #y2 & 2 g $y & y % 3mportant: +ime" Jt is the only common term that appears in the horizontal and #ertical motion e,uations.
9/ample: A s!i umper starts -ith a horizontal ta!e)off #elocity of 2' m8s and lands on a straight landing hill inclined at DL. 0etermine $a% the time bet-een ta!e)off and landing" $b% the length d of the ump.
9/ample: A golf ball is struc! -ith a #elocity of ft8s as sho-n. 0etermine the distance" d " to -here it -ill land.
9/ample: A pump is located near the edge of the horizontal platform sho-n. +he nozzle at A discharges -ater -ith an initial #elocity of 2' ft8s at an angle of ''M -ith the #ertical. 0etermine the range of #alues of the height h for -hich the -ater enters the opening BC .
12.5: 6"&RA% #R,I%I&"AR MO$IO& - Introduction
Cur#ilinear motion occurs -hen the particle mo#es along a cur#ed path.
A roller coaster car +he path of motion of a plane
Cur#ed road
#R,I%I&"AR MO$IO&
*ince the path is often described in D)0" #ector analysis -ill be used to formulate the particle5s position" #elocity and acceleration. A particle mo#es along a cur#e defined by the path function" s.
+he position of the particle at any instant is designated by the #ector r r $t%. Noth the magnitude and direction of r may #ary -ith time.
,"%O#I$
4elocity represents the rate of change in the position of a particle. +he instantaneous #elocity is the time)deri#ati#e of position v dr 8dt .
+he #elocity #ector" v" is al-ays tangent to the path of motion.
A##"%"RA$IO&
Acceleration represents the rate of change in the #elocity of a particle.
+he instantaneous acceleration is the time)deri#ati#e of #elocity:
a dv8dt d2r 8dt2
A&A%'I' O/ #R,I%I&"AR MO$IO& 1% sing Eectangular $/" y" z% Components 2% sing ormal and +angential Components $n)t % D% Cylindrical Coordinates $r" Q" z%
#R,I%I&"AR MO$IO&: R"#$A&6%AR #OMPO&"&$' ('ection 12.7)
3t is often con#enient to describe the motion of a particle in terms of its /" y" z or rectangular components" relati#e to a fi/ed frame of reference. +he position of the particle can be defined at any instant by the position #ector r x i ; y j ; z k . +he /" y" z components may all be functions of time" i.e." / /$t%" y y$t%" and z z$t% . +he magnitude of the position #ector is: r $/ 2 ; y2 ; z2%.'
R"#$A&6%AR #OMPO&"&$': ,"%O#I$
+he #elocity #ector is the time deri#ati#e of the position #ector: v dr 8dt d$/i %8dt ; d$y j %8dt ; d$zk %8dt
*ince the unit #ectors i " j " k are constant in magnitude and direction" this e,uation reduces to # # / i ; #y j ; #z k •
•
•
-here #/ / d/8dt" #y y dy8dt" #z z dz8dt +he magnitude of the #elocity #ector is # R$#/%2 ; $#y%2 ; $#z%2S.' +he direction of v is tangent to the path of motion.
R"#$A&6%AR #OMPO&"&$': A##"%"RA$IO&
+he acceleration #ector is the time deri#ati#e of the #elocity #ector $second deri#ati#e of the position #ector%: a dv8dt d2r 8dt2 a/ i ; ay j ; az k •
-here •
••
•
••
a/ #/ / d#/ 8dt" ay #y y d#y 8dt" ••
az #z z d#z 8dt +he magnitude of the acceleration #ector is a R$a/%2 ; $ay%2 ; $az%2 S.' +he direction of a is usually not tangent to the path of the particle.
"8a!+le: +he bo/ slides do-n the slope described by the e,uation y $.' x2% m" -here x is in meters. #/ )D m8s" a x )1.' m8s2 at x ' m. ind the y
components of the #elocity and the acceleration of the bo/ at x ' m.
12.9 #R,I%I&"AR MO$IO& &ORMA% A&* $A&6"&$IA% #OMPO&"&$' O;ective: 0etermine the #elocity and acceleration of a particle tra#eling along a cur#ed path using normal and tangential components.
&ORMA% A&* $A&6"&$IA% #OMPO&"&$' $*ection 12.>% hen a particle mo#es along a cur#ed path" it is sometimes con#enient to describe its motion using coordinates other than Cartesian. hen the path of motion is !no-n" normal $n% and tangential $t % coordinates are often used. 3n the n-t coordinate system" the origin" O" is located on the particle $the origin mo#es -ith the particle%.
+he t )a/is is tangent to the path $cur#e% at the instant considered" positi#e in the direction of the particle5s motion. +he n)a/is is perpendicular to the t )a/is -ith the positi#e direction to-ard the center of cur#ature of the cur#e.
&ORMA% A&* $A&6"&$IA% #OMPO&"&$' $continued%
+he positi#e n and t directions are defined by the unit #ectors un and ut" respecti#ely.
+he center of cur#ature" OT" al-ays lies on the conca#e side of the cur#e. +he radius of cur#ature" ρ" is defined as the perpendicular distance from the cur#e to the center of cur#ature at that point.
,"%O#I$ I& $3" n-t #OOR*I&A$" ''$"M
+he #elocity #ector is al-ays tangent to the path of motion $t )direction%. v # ut
A##"%"RA$IO& I& $3" n-t #OOR*I&A$" ''$"M
Acceleration is the time rate of change of #elocity : . . a dv8dt d$#ut%8dt #ut ; #ut
After mathematical manipulation" the acceleration #ector can be e/pressed as: . a v ut ; $#28ρ% un at ut ; an un.
A##"%"RA$IO& I& $3" n-t #OOR*I&A$" ''$"M $continued%
*o" there are t-o components to the acceleration #ector: a at ut ; an un
• +he tangential component is tangent to the cur#e and in the direction of increasing or decreasing #elocity. . at # • +he normal or centripetal component is al-ays directed to-ard the center of cur#ature of the cur#e. a n #28ρ • +he magnitude of the acceleration #ector is a R$a t%2 ; $an%2S.'
3POE+A+ 9UA+3O*
+angential Acceleration" at d#8dt ormal Acceleration" an #28V
+he magnitude of the acceleration #ector is a R$at %2 ; $an%2S.'
'P"#IA% #A'"' O/ MO$IO& +here are some special cases of motion to consider. 1% +he particle mo#es along a straight line. . 2 ρ ∞ W an # 8ρ = 0 => a at # +he tangential component represents the time rate of change in the magnitude of the #elocity. 2% +he particle mo#es along a cur#e at constant speed. . at # W a an #28ρ
'P"#IA% #A'"' O/ MO$IO& $continued%
D% *ometimes" the e,uation of the cur#e path follo-ed by the particle may be gi#en" y $ x%. +he radius of cur#ature" V" at any point on the path can be calculated from the follo-ing e,uation: D
dy 1 + ÷ dx 2
ρ =
d2y dx
2
2
"8a!+le: A oat travels around a circular +ath at a s+eed that increases with ti!e< v = (>.>027 t 2) !?s. /ind the !agnitudes of the oat@s velocity and acceleration at the instant t = 1> s.
"8a!+le: $he !otorcyclist travels along the curve at a constant s+eed of > ft?s. *eter!ine his acceleration when he is located at +oint A. 3int: $reat the !otorcycle and rider as a +article.
"8a!+le: $he auto!oile has a s+eed of > ft?s at +oint A and an acceleration< a< having a !agnitude of 1> ft?s2< acting in the direction shown. *eter!ine the radius of curvature of the +ath at +oint A and the tangential co!+onent of acceleration.
R"%A$I,"-MO$IO& A&A%'I' O/ $CO PAR$I#%"'
All motion is relati#e to some frame of reference
2 trains approaching each other $along a line% at B' !m8h each. Obser#ers on either train see the other coming at B' ; B' 1B !m8h. Obser#er on ground sees ± B' !m8h. Velocity depends on reference frame!!
R"%A$ R"%A $I," PO'I$IO& $*ection 12.1%
+he absolute position of position of t-o particles A and N -ith respect to the fi/ed /" y" z reference frame are gi#en by r A and r B. +he position +he position of N relati#e to A is A is represented by r BA r A r B $#ector addition% r BA r B & r A
ote: r B!A means position of B of B as as obser#ed from A from A r A!B means position of A of A as as obser#ed from B from B +herefore" if
r B $1 i ; ; 2 j % m
and
r A $@ i ; ; ' j % m"
then
& D j % m. r BA $( i &
R"%A$I," ,"%O#I$
+o determine the relati#e #elocity of #elocity of N -ith respect to A" the time deri#ati#e of of the relati#e position e,uation is ta!en. v BA v B & v A or v B v A ; v BA 3n these e,uations" v B and v A are called absolute #elocities and v BA is the relati#e #elocity of #elocity of N -ith respect to A. ote that v BA ) v AB .
R"%A$I," R"%A$ I," A##"%"RA$ A##"%"RA$IO& IO&
+he time deri#ati#e of the relati#e #elocity e,uation yields a similar #ector relationship bet-een the absolute and absolute and relati#e accelerations of particles A and N. +hese deri#ati#es yield: a BA a B & a A or a B a A ; a BA
'O%,I&6 PROB%"M' *ince the relati#e motion e,uations are #ector e,uations" problems in#ol#ing them re,uires resol#ing #elocity $or acceleration% #ector along x and y and then performing the #ector operation: v B v A ; v BA a B a A ; a BA
"DAMP%": 'hown in figure are two air+lanes flying at sa!e altitude with their res+ective velocities and direction shown. /ind vB?A.
6iven:
#A (' !m8h #N !m8h
/ind:
v BA
"DAMP%" (continued% 'olution:
v A $(' i % !m8h v B & cos ( i & sin ( j
$ &@ i & (B2. j % !m8h v BA v B & v A $&1' i & (B2. j % !m8h
# B 8 A =
(−1050)2 +(−692.8)2 = 1258 !m8h
θ tan)1$
692.8 % DD.@° 1050
θ
9/ample: At the instant sho-n" race car A is passing race car B -ith a relati#e #elocity $# A8N% of 1 m8s. Kno-ing that the speeds of both cars are constant and that the relati#e acceleration of car A -ith respect to car B $aA8N% is .2' m8s 2 directed to-ard the center of cur#ature" determine the speed of cars A and B at the instant sho-n.
"8a!+le: At a given instant in an air+lane race< air+lane A is flying horiEontally in a straight line< and its s+eed is eing increased at a rate of 0 !?s2. Air+lane B is flying at the sa!e altitude as air+lane A and is following a circular +ath of 2>>-! radius. nowing that at the given instant the s+eed of B is eing decreased at the rate of 2 !?s 2< deter!ine< for the +ositions shown< the acceleration of B relative to A.
AB'O%$" *"P"&*"&$ MO$IO& A&A%'I' O/ $CO PAR$I#%"' ('ection-12.F)
3n many !inematics problems" the motion of one obect -ill depend on the motion of another obect.
• 3f bloc! A mo#es do-n-ard along the inclined plane" bloc! N -ill mo#e up the other incline. • +his dependency commonly occurs if the particles are interconnected by ine/tensible8 inelastic cords -hich are -rapped around pulley$s%.
PRO#"*R" /OR A&A%'I'- *"P"&*"&$ MO$IO&
ote: +he length of the cord connecting the t-o particles -ill al-ays remain same8 constant" irrespecti#e of the position8 location of the t-o particles.
+he motion of each bloc! can be related mathematically by defining position coordinates" sA and sN. 9ach coordinate a/is is defined from a fi/ed point or datum line" measured positi#e along each plane in the direction of motion of each bloc!.
*"P"&*"&$ MO$IO& $continued%
3n this e/ample" position coordinates sA and sN can be defined from fi/ed datum lines e/tending from the center of the pulley along each incline to bloc!s A and N. *ince" the cord has a fi/ed length" the position coordinates sA and sN are related mathematically by the e,uation sA ; lC0 ; sN l+ Iere l+ is the total cord length and l C0 is the length of cord passing o#er the arc C0 on the pulley.
*"P"&*"&$ MO$IO& $continued%
+he #elocities of bloc!s A and N can be related by differentiating the position e,uation. ote that lC0 and l+ remain constant" so dlC08dt dl+8dt dsA8dt ; dsN8dt
W
#N )#A
+he negati#e sign indicates that as A mo#es do-n the incline $positi#e sA direction%" N mo#es up the incline $negati#e s N direction%. Accelerations can be found by differentiating the #elocity e/pression. aN )aA .
*"P"&*"&$ MO$IO&: PRO#"*R"'
+hese procedures can be used to relate the dependent motion of particles mo#ing along rectilinear paths $only the magnitudes of #elocity and acceleration change" not their line of direction%. 1. 0efine position coordinates from fi/ed datum lines" along the path of each particle. 0ifferent datum lines can be used for each particle. 2. Eelate the position coordinates to the cord length. *egments of cord that do not change in length during the motion may be left out. D. 3f a system contains more than one cord" relate the position of a point on one cord to a point on another cord. *eparate e,uations are -ritten for each cord. @. 0ifferentiate the position coordinate e,uation$s% to relate #elocities and accelerations. Keep trac! of signsX
"DAMP%"-1
=i#en: 3n the figure on the left" the cord at A is pulled do-n -ith a speed of 2 m8s. ind: +he speed of bloc! N.
"8a!+le - 2: 'lider locG B !oves to the right with a constant velocity of >> !!?s. *eter!ine (a) the velocity of slider locG A< () the velocity of +oint " on the cale< (c) the relative velocity of " w.r.t. A # v"A $.
View more...
Comments
