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ABSOLUTE DEPENDENT MOTION ANALYSIS
OF TWO PARTICLES (Section 12.9)
O!ecti"e#$
To relate the positions, velocities,and accelerations of particles
undergoing dependent motion.
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APPLICATIONS
The cable and pulley modify the speed
of B relative to the speed of the motor.
It is important to relate the various
motions to determine the powerrequirements for the motor and the
tension in the cable.
If the speed of the A is known, how can
we determine the speed of block B?
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APPLICATIONS (continued
!ope and pulley arran"ements are
used to assist in liftin" heavy
ob#ects. The total liftin" force
required from the truck depends onthe acceleration of the cabinet A.
$ow can we determine theacceleration and velocity of A if the
acceleration of B is known?
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DEPENDENT MOTION
COMPATABILITY EQUATIONS
In many kinematics problems, the motion of one ob#ect will
depend on the motion of another.
The blocks are connected by an
ine%tensible cord wrapped arounda pulley. If block A moves
downward, block B will move up.
s A and sB define motion of blocks. &ach starts from a fi%ed
point, positive in the direction of motion of block.
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DEPENDENT MOTION (continued
s A and sB are defined from
the center of the pulley to
blocks A and B.
If the cord has a fi%ed len"th, then' s A l)* sB + lT
lT is total cord len"th and l)* is the len"th of cord
passin" over arc )* on the pulley.
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DEPENDENT MOTION (continued
The ne"ative si"n indicates that as A moves down (positive
s A, B moves up (ne"ative sB direction.
Accelerations can be found by differentiatin" the velocity
e%pression. rove' aB + -a A .
elocities can be found by
differentiatin" the position
equation. /ince l)* and lT remain
constant, so dl)*0dt + dlT0dt + 1
dsA/dt + dsB/dt = 0 => vB = -vA
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DEPENDENT MOTION EXAMPLE
s A and sB are defined from fi%ed
datum lines, measured alon" the
direction of motion of each block.
sB is defined to the center of the
pulley above block B, since this
block moves with the pulley.
The red colored se"ments of
the cord and h remain constant
in len"th
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DEPENDENT MOTION
EXAMPLE (continued
The position coordinates are related
by 2sB h s A + l
3here l is the total cord len"th minus
the len"ths of the red se"ments.
elocities and accelerations can
be related by two successive
time derivatives'2vB + -v A and 2aB + -a A
3hen block B moves downward (sB, block A moves tothe left -s .
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DEPENDENT MOTION
EXAMPLE (continued
The e%ample can also be worked
by definin" the sB from the bottom
pulley instead of the top pulley.
The position, velocity, and
acceleration relations become
2(h 4 sB h s A + l
and 2vB + v A 2aB + a A
rove that the results are the same, even if the si"nconventions are different than the previous formulation.
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EXAMPLE PROBLEM
5iven'In the fi"ure the cord at
A is pulled down with a
speed of 6 m0s.
7ind' The speed of block B.
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EXAMPLE (/olution
*efine the position coordinates one for point A (s A, one for
block B (sB, and one relatin" positions on the two
cords (pulley ).
:
)oordinates are defined as ve
down and alon" the direction of
motion of each ob#ect.
sAs sB
!AT"#
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EXAMPLE (continued
&liminatin" s), 2s A 8sB + l9 2l2
If l9+len"th of the first cord, minus
any se"ments of constant
len"th and l2 for the second'
elocities are found by differentiatin"' (l9 and l2 constants'
2v A 8vB + 1 +: vB + - 1.;v A + - 1.;(6 + - 8 m0s
sAs sB
!AT"#
)ord 9' 2s A 2s) + l9)ord 2' s
B (sB 4 s) + l2
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GROUP PROBLEM SOLVING
5iven'In this system, block A is
movin" downward with
a speed of 8 m0s while
block ) is movin" up at2 m0s.
7ind' The speed of block B.
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%ROUP PROBLEM SOL&IN%
$%olution&
*efine s A, sB, and s)
A datum line is drawn throu"h the upper, fi%ed, pulleys.
*ifferentiate to relate
velocities'
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