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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