Transcript Slide 1

KINEMATICS OF
PARTICLES
RELATIVE MOTION WITH
RESPECT TO
TRANSLATING AXES
In the previous articles, we have described particle
motion using coordinates with respect to fixed
reference axes. The displacements, velocities and
accelerations so determined are termed “absolute”.
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
made with respect to a moving reference system. These
measurements, 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”.
In this article, we will confine our attention to moving
reference systems which translate but do not rotate.
Now let’s consider two particles A and B
which may have separate curvilinear
motions in a given plane or in parallel
planes; the positions of the particles at any
time with respect to fixed OXY reference
Translating axis


system are defined by rA and rB .
Fixed axis
Let’s 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.
The position vector of A as measured
 

relative to the frame x-y is rA / B  xi  yj ,
where the subscript notation “A/B” means
“A relative to B” or “A with respect to B”.
Translating axes
Fixed axes
The position of A is,
therefore, determined by the
vector
  
rA  rB  rA / B
Translating axes
Fixed axes
  
We now differentiate this vector equation rA  rB  rA / B
once with respect to time to obtain velocities and twice to
obtain accelerations.

 
v A  vB  v A / B
  
rA  rB  rA / B 
Here, the velocity which we observe A to have from our
position at B attached to the moving axes x-y is
 


rA/ B  vA/ B  xi  yj
This term is the velocity of A with respect to B.
Acceleration is obtained as



rA  rB  rA / B  ,
a A  aB  a A / B
vA  vB  vA / B 
Here, the acceleration which we observe A to have from our
nonrotating position on B is
 

 .

rA/ B  vA/ B  aA/ B  xi  yj
This term is the acceleration of A with respect to B.


We note that the unit vectors i and j have zero derivatives
because their directions as well as their magnitudes remain
unchanged.
We can express the relative motion terms in whatever
coordinate system is convenient – rectangular, normal and
tangential or polar, and use their relevant expressions.
1. The car A 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 observer B, who rides in a nonrotating chair on the Ferris wheel.
The angular rate  = 3 rev/min of the Ferris wheel is constant.
4. Car A is traveling along a circular
curve of 60 m radius at a constant
30
speed of 54 km/h. When A passes the
B
o
60 m
position shown, car B is 30 m from
the intersection, traveling with a
speed of 72 km/h and accelerating at
the rate of 1.5 m/s2. Determine the
velocity and acceleration which A
appears to have when observed by an
occupant of B at this instant. Also
determine r, q,
instant.
r , q , r and q for this
r
30 m
q
A