Relative Motion Wrt Translating Axes [Uyumluluk Modu]

7/28/2019 Relative Motion Wrt Translating Axes [Uyumluluk Modu]

1/14

KINEMATICS OF

PARTICLES

RELATIVE MOTION WITHRESPECT TO

TRANSLATING AXES


2/14

In the previous articles, we have described particlemotion using coordinates with respect to fixed

reference xes. The dis lacements, velocities nd

accelerations so determined are termed absolute.


3/14

It is not always possible or convenient to use a fixed set

of axes to describe or to measure motion. In addition,

there are many engineering problems for which the

analysis of motion is simplified by using measurements

ma e w re pec o mov ng re erence y em. e emeasurements, when combined with the absolute motion

of the moving coordinate system, enable us to determine

the absolute motion in question.

This approach is called the relative motion analysis.


4/14

In this article, we will confine our attention to moving

reference systems which translate but do not rotate.

Motion measured in rotating systems will be discussed in

,

important application.

We will also confine our attention here to relative motion

analysis for plane motion.


5/14

Now lets consider two particlesA andB

which may have separate curvilinear

motions in a given plane or in parallelplanes; the positions of the particles at any

time with respect to fixed OXYreferenceTranslating axis

system are defined by and .

Lets attach the origin of a set of

translating (nonrotating) axes to particle B

and observe the motion of A from our

moving position on B.

Brv

Arv

Fixed axis


6/14

The position vector ofA as measured

relative to the framex-y is ,

where the subscript notation A/B means

jyixr BAvvv +=

/

Translating axes.

Fixed axes


7/14

The position ofA

is, therefore,determined bythe vector

BABA rrr /vvv

+=

Fixed axes

Translating axes


8/14

We now differentiate this vector equationonce with respect to time to

obtain velocities and twice to obtain

accelerations.

BABArrr

/

vvv+=

( )BABABABA

rrrvvv//

&v

&v

&vvvv

+=+=

Here, the velocity which we observeA tohave from our position atB attached to themoving axesx-y is .

This term is the velocity ofA with respecttoB.

jyixvr BABAv&

v&

v&v

+==//


9/14

Acceleration is obtained as

Here, the acceleration which we observe A tohave from our nonrotating position on B is

( ) ( )BBBBBB vvvrrraaa /// &v

&v

&v

&&v

&&v

&&vvvv

+=+=+= ,

v&&

v&&

v&v

&&v .

This term is the acceleration of A with respectto B.

We note that the unit vectors and havezero derivatives because their directions aswell as their magnitudes remain unchanged.

iv jv

xavr BABABA===

///


10/14

The velocity and acceleration equations state that the

absolute velocity (or acceleration) equals the absolute

velocity (or acceleration) ofB plus, vectorially, thevelocity (or acceleration) ofA relative toB. The relative

an observer attached to the moving coordinate systemx-y

would make. We can express the relative motion terms in

whatever coordinate system is convenient rectangular,

normal and tangential or polar, and use their relevant

expressions.


11/14

1. The carA has a forward speed of 18 km/h and is accelerating at

3 m/s2. Determine the velocity and acceleration of the car relative

to observerB, who rides in a nonrotating chair on the Ferris wheel.

The angular rate = 3 rev/min of the Ferris wheel is constant.


12/14

2. A batter hits the baseballA with an initial velocity of v0=30 m/s directly

toward fielderB at an angle of 30 to the horizontal; the initial position of the

ball is 0.9 m above ground level. FielderB requires s to judge where the ball

should be caught and begins moving to that position with constant speed. Because

of great experience, fielderB chooses his running speed so that he arrives at the

catch position simultaneously with the ball. The catch position is the field

location at which the ball altitude is 2.1 m. Determine the velocity of the ballrelative to the fielder at the instant the catch is made.


13/14

3. AirplaneA is flying horizontally with a constant speed of 200

km/h and is towing the gliderB, which is gaining altitude. If the

tow cable has a length r= 60 m and is increasing at the

constant rate of 5 degrees per second, determine the

magnitudes of the velocity and acceleration of the glidervv a

v

or t e instant w en

= .


14/14

4. Car A is traveling along a circular

curve of 60 m radius at a constant

speed of 54 km/h. When A passes theposition shown, car B is 30 m from

the intersection, traveling with a

60 m

30

o

30 mA

B

r

speed of 72 km/h and accelerating atthe rate of 1.5 m/s2. Determine the

velocity and acceleration which A

appears to have when observed by anoccupant of B at this instant. Also

determine r, , , , and for this

instant.

r& & r&& &&

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Relative Motion Wrt Translating Axes [Uyumluluk Modu]