dependent motion

8/19/2019 Dependent Motion

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Mechanics for Engineers: Dynamics, 13th SI EditionR. C. Hibbeler and Kai Beng Yap

© Pearson Eduation South !sia Pte "td#$13. !ll rights reser%ed.

ABSOLUTE DEPENDENT MOTION ANALYSISOF TWO PARTICLES

Today’s Objecties!

Students &ill be able to'1. Relate the positions,

%eloities, andaelerations o( partilesundergoing dependent

)otion.

I"#C$ass Actiities!

• Che* Ho)e&or*

• Reading +ui• !ppliations• -e(ine -ependentotion

• -e%elop Position,

/eloit0, and!elerationRelationships

• Conept +ui• roup Proble) Sol%ing• !ttention +ui


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READIN% &UI'

1. 2hen partiles are interonneted b0 a able, the

)otions o( the partiles are

!4 al&a0s independent. B4 al&a0s dependent.

C4 not al&a0s dependent. -4 5one o( the abo%e.

#. I( the )otion o( one partile is dependent on that o(another partile, eah oordinate a6is s0ste) (orthe partiles

!4 should be direted along the path o( )otion.

B4 an be direted an0&here.

C4 should ha%e the sa)e origin.

-4 5one o( the abo%e.


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APPLICATIONS

7he able and pulle0 s0ste)sho&n an be used to )odi(0 thespeed o( the )ine ar, !, relati%eto the speed o( the )otor, .

It is i)portant to establish therelationships bet&een the %arious

)otions in order to deter)ine thepo&er re8uire)ents (or the )otorand the tension in the able.

9or instane, i( the speed o( the able :P4 is *no&n beause&e *no& the )otor harateristis, ho& an &e deter)inethe speed o( the )ine ar; 2ill the slope o( the tra* ha%ean0 i)pat on the ans&er;


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APPLICATIONS :ontinued4

Rope and pulle0 arrange)entsare o(ten used to assist in li(tinghea%0 ob


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DEPENDENT MOTION (Sectio" )*+,-

In )an0 *ine)atis proble)s, the )otion o( one ob


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DEPENDENT MOTION:ontinued4

In this e6a)ple, positionoordinates s! and sB an be

de(ined (ro) (i6ed datu) linese6tending (ro) the enter o(the pulle0 along eah inline

to blo*s ! and B.

I( the ord has a (i6ed length, the position oordinates s!

and sB are related )athe)atiall0 b0 the e8uation

s! = lC- = sB > l7

Here l7 is the total ord length and lC- is the length o( ord

passing o%er the ar C- on the pulle0.


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DEPENDENT MOTION:ontinued4

7he negati%e sign indiates that as ! )o%es do&n the inline

:positi%e s! diretion4, B )o%es up the inline :negati%e sB diretion4.

!elerations an be (ound b0 di((erentiating the %eloit0e6pression. Pro%e to 0oursel( that aB > ?a! .

ds! @dt = dsB @dt > $ ⇒ %B > ?%!

7he %eloities o( blo*s ! and Ban be related b0 di((erentiating the position e8uation. 5ote thatlC- and l7 re)ain onstant, so

dlC- @dt > dl7 @dt > $


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DEPENDENT MOTION E.AMPLE

Consider a )ore o)pliatede6a)ple. Position oordinates:s! and sB4 are de(ined (ro)

(i6ed datu) lines, )easuredalong the diretion o( )otion o(eah blo*.

5ote that sB is onl0 de(ined to

the enter o( the pulle0 abo%eblo* B, sine this blo* )o%es&ith the pulle0. !lso, h is a

onstant.

7he red olored seg)ents o( the ord re)ain onstant inlength during )otion o( the blo*s.


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DEPENDENT MOTION E.AMPLE :ontinued4

7he position oordinates are related b0

the e8uation#sB = h = s! > l72here l7 is the total ord length )inus

the lengths o( the red seg)ents.

Sine l7 and h re)ain onstant

during the )otion, the %eloitiesand aelerations an be related b0t&o suessi%e ti)e deri%ati%es'

#%B > ?%! and #aB > ?a!

2hen blo* B )o%es do&n&ard :=sB4, blo* ! )o%es to the le(t

:?s!4. Re)e)ber to be onsistent &ith 0our sign on%entionA


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DEPENDENT MOTION E.AMPLE :ontinued4

7his e6a)ple an also be &or*edb0 de(ining the position oordinate(or B :sB4 (ro) the botto) pulle0

instead o( the top pulle0.

7he position, %eloit0, and

aeleration relations thenbeo)e

#:h sB4 = h = s! > l7

and #%B > %! #aB > a!

Pro%e to 0oursel( that the results are the sa)e, e%en i( the signon%entions are di((erent than the pre%ious (or)ulation.


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DEPENDENT MOTION! PROCEDURES

7hese proedures an be used to relate the dependent )otion o( partiles )o%ing along retilinear paths :onl0 the )agnitudes

o( %eloit0 and aeleration hange, not their line o( diretion4.

. -i((erentiate the position oordinate e8uation:s4 torelate %eloities and aelerations. Keep tra* o( signsA

3. I( a s0ste) ontains )ore than one ord, relate the

position o( a point on one ord to a point on anotherord. Separate e8uations are &ritten (or eah ord.

#. Relate the position oordinates to the ord length.Seg)ents o( ord that do not hange in length duringthe )otion )a0 be le(t out.

1. -e(ine position oordinates (ro) (i6ed datu) lines,along the path o( eah partile. -i((erent datu) linesan be used (or eah partile.


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E.AMPLE

%ie"!In the (igure on the le(t,the ord at ! is pulleddo&n &ith a speed o( #)@s.

Fi"d! 7he speed o( blo* B.

P$a"! 7here are t&o ords in%ol%edin the )otion in thise6a)ple. 7here &ill be t&oposition e8uations :one (oreah ord4. 2rite these t&o

e8uations, o)bine the),and then di((erentiate the).


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E.AMPLE :ontinued4

So$/tio"!

• -e(ine the datu) line through the top

pulle0 :&hih has a (i6ed position4.• s! an be de(ined to the point !.

• sB an be de(ined to the enter o( the

pulle0 abo%e B.

• sC is de(ined to the enter o( pulle0 C.

•

!ll oordinates are de(ined as positi%edo&n and along the diretion o()otion o( eah point@ob


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E.AMPLE :ontinued4

34 Eli)inating sC bet&een the t&o

e8uations, &e gets! = sB > l1 = #l#

#4 2rite position@length e8uations

(or eah ord. -e(ine l1 as thelength o( the (irst ord, )inusan0 seg)ents o( onstant length.-e(ine l# in a si)ilar )anner (or

the seond ord'

4 Relate %eloities b0 di((erentiating this e6pression. 5otethat l1 and l# are onstant lengths.

%! = %B > $ ⇒ %B > $.#D%! > $.#D:#4 > $.D )@s

7he %eloit0 o( blo* B is $.D )@s up :negati%e sB diretion4.

Cord 1' s! = #sC > l1Cord #' sB = :sB sC4 > l#


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CONCEPT &UI'

#. 7&o blo*s are interonneted b0 aable. 2hih o( the (ollo&ing isorret;

!4 :%64!> ? :%64B B4 %!> ? %B

C4 :%04!> ? :%04B -4 !ll o( the abo%e.0

y

1. -eter)ine the speed o( blo* B.

!4 1 )@s B4 # )@s

C4 )@s -4 5one o( the abo%e.


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%ROUP PROBLEM SOL1IN% I

%ie"!7he rope is dra&n to&ardsthe )otor, , at a speed o(:Dt3@#4 )@s, &here t is inseonds.

Fi"d! 7he speed o( blo* ! &hen

t > 1 s.

P$a"!

7here is onl0 one ord in%ol%ed in the )otion, soone position@length e8uation &ill be re8uired.-e(ine position oordinates (or blo* ! and theable, &rite the position relation and thendi((erentiate it to (ind the relationship bet&een thet&o %eloities.


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%ROUP PROBLEM SOL1IN% I:ontinued4So$/tio"!

#4 -e(ining s! and s as sho&n, the

position relation an be &ritten'3 s

! = s

> l

34 7a*ing the ti)e deri%ati%e o( theabo%e e8uation to relate%eloities'

3 %! = % > $ := ↓4

14 ! datu) line an be dra&n through the upper, (i6ed,

pulle0s. 7&o oordinates )ust be de(ined' one (or blo* !:s!4, one (or the able:s4, is dra&n to&ards the )otor.

Sine the rope is dra&n to&ards the)otor at a speed o( :Dt3@#4 )@s,

% > :Dt3@#4 ⇒ % > D )@s at t > 1s.

3 %! = D > $ ⇒ %! > −1.F )@s > 1.F )@s ↑

sA

sM


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%ROUP PROBLEM SOL1IN% II

%ie"!In this pulle0 s0ste), blo* ! is)o%ing do&n&ard &ith a speed

o( )@s &hile blo* C is )o%ingup at # )@s.

Fi"d! 7he speed o( blo* B.

P$a"!

!ll blo*s are onneted to a single able, so onl0 oneposition@length e8uation &ill be re8uired. -e(ineposition oordinates (or eah blo*, &rite out theposition relation, and then di((erentiate it to relate the%eloities.


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%ROUP PROBLEM SOL1IN% II:ontinued4So$/tio"!

#4 -e(ining s!, sB, and sC as sho&n, the

position relation an be &ritten's! = #sB = sC > l

34 -i((erentiate to relate %eloities'%! = #%B = %C > $

⇒ = #%B

= :?#4 >$

⇒ %B > ?1 )@s

14 ! datu) line an be dra&n through the upper, (i6ed, pulle0s

and position oordinates de(ined (ro) this line to eah blo*:or the pulle0 abo%e the blo*4.

7he %eloit0 o( blo* B is 1 )@s up :negati%e sB diretion4.


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ATTENTION &UI'

1. -eter)ine the speed o( blo* B &hen

blo* ! is )o%ing do&n at )@s &hileblo* C is )o%ing do&n at 1G )@s.

!4 # )@s B4 3 )@s

C4 1# )@s -4 )@s

A23 45s C2)6 45s

#. -eter)ine the %eloit0 %etor o(blo* ! &hen blo* B is )o%ingdo&n&ard &ith a speed o( 1$ )@s.

!4 :Gi = j 4 )@s B4 :i = 3 j 4 )@s

C4 :?Gi ? j 4 )@s -4 :3i = j 4 )@sB2)7 45s

j

i

B2)7 45s

dependent motion

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