kinematics of particles relative motion with respect to translating axes
DESCRIPTION
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 ”. - PowerPoint PPT PresentationTRANSCRIPT
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 system are defined by and
.
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.
Br
Ar
Fixed axis
Translating axis
The position vector of A as measured
relative to the frame x-y is ,
where the subscript notation “A/B”
means “A relative to B” or “A with
respect to B”.
jyixr BA
/
Fixed axes
Translating axes
The position of A is, therefore, determined by the vector
BABA rrr /
Fixed axes
Translating axes
We now differentiate this vector equation
once with respect to time to obtain velocities and
twice to obtain accelerations.
Here, the velocity which we observe A to have from
our position at B attached to the moving axes x-y is
This term is the velocity of A with respect to B.
BABArrr
/
BABABABA
rrrvvv//
jyixvr BABA
//
Acceleration is obtained as
Here, the acceleration which we observe A to have from our nonrotating position on B is .
This term is the acceleration of A with respect to B.
We note that the unit vectors and have zero derivatives because their directions as well as their magnitudes remain unchanged.
i
j
BABABABABABA vvvrrraaa /// ,
jyixavr BABABA
///
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 speed of 54 km/h. When
A passes the 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, , , , and for
this instant.
r
60 m
30o
30 m A
B
r
r