Post on 07-Aug-2018
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Kinematics
Absolute Dependant Motion Analysisof Two Particles
&Relative Motion
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Overview
• Absolute Dependant Motion Analysis of TwoParticles
• Relative Motion
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KinematicsAbsolute Dependant Motion Analysis
of Two Particles
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Absolute Dependent Motion•
In some cases the motion of one particledepends on the motion of another.
• Consider two objects physically
interconnected by inextensible chords of apulley system.• sA and sB are the position
coordinates from a datum(fixed point or fixed line)with positive sense“down” the inclines
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Absolute Dependent Motion
• Total chord length,l T = sA + l CD + sB
where l CD is the chord length over arc CD of
the pulleyl CD and l T remain constant
• Differentiating l T with respect
to time
or 0
dt
ds
dt
ds B A
A B vv
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Absolute Dependent Motion
• The negative sign confirms that “upward”velocity of A causes “downward” velocity of Band vice versa.
•
Differentiating velocity yields acceleration A B
aa
Ship elevator, Three Gorges Dam, China
Aircraft elevator, USS Kitty Hawk
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Absolute Dependent Motion•
Let us consider a more complex system• The red colored segments of the chord will
remain constant and can be omitted•
Consider positive position coordinates asshown by arrowheads
• For total length of the chord, l
in this set up h is constant,therefore time derivatives are A
B
s A
sB
Datum
Datum
hl sh s A B 2
A B
vv
2
A B aa
2
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Absolute Dependent Motion•
We could have defined the position of objectB from the small pulley at the bottom of theassembly
•
total length of the chord, l
therefore time derivatives are
l sh sh A B
)(2
A B
vv 2
A B
aa 2
A
B
s A
sB
Datum
Datum
h
New Datum
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Conclusion
• Examples
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Kinematics
Relative Motion
Albert Einstein
Apollo-Soyuz Rendezvous
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Relative Motion-Translating Axes
• So far we have considered absolute motion ofa particle within a fixed frame of reference
• In some cases it is easier to analyze themotion using two or more frames of reference
• In this chapter we shall set up another frameof reference by translating frames of referenceto change our point of view of the object(s) inmotion.
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Relative Motion•
Consider two joggers (A and B) on the beachrunning in a straight path and beingmonitored by a stationary observer at a fixedorigin O.
• At an instant, with this frame of reference, wecan consider the positions of the joggers fromthe origin as their absolute position
o
A B
r A
r B
r B/A
Stationaryobserver
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Relative Position
• However, if we were to change ourperspective and consider the position of B asobserved by A, we would call this the position
of B relative to A , or the relative position of B from A, and denote it by r B/A• It follows that
orand by differentiating, velocity
A B A B
r r r /
A B A B
r r r /
A B A B
vvv/
o
A B
r Ar B
r B/A
Stationaryobserver
Translatingobserver
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Translating Axes
• We can extend this concept to vectors in orderto analyze motion along curvilinear paths thatalso may not coincide.
•
Let r A, r B, denote the corresponding absolute position vectors , and r B/A denote the relative position vector
Then:
o
A
B
r A
r B
r B/A x y
z Translatingobserver
Stationaryobserver
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Relative Motion-Translating Axes• Position:
r B = r A + r B/A• The time derivative of position yields the
absolute and relative velocity ;v B = v A + v B/A
v B/A is the velocity of jogger B as observed by jogger A who is also in motion with absolutevelocity v A
• The time derivative of velocity yields theacceleration;
a B = a A + a B/A
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Conclusion
• Examples