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Coordinate system enables us to specify a point in its defined volumetric space. We must recognize that a point is a concept without dimensions; whereas the objects or bodies under motion themselves are not points. The real bodies, however, approximates a point in translational motion, when paths followed by the particles, composing the body are parallel to each other (See Figure). As we are concerned with the geometry of the path of motion in kinematics, it is, therefore, reasonable to treat real bodies as “point like” mass for description of translational motion.
Translational motion
We conceptualize a particle in order to facilitate the geometric description of motion. A particle is considered to be dimensionless, but having a mass. This hypothetical construct provides the basis for the logical correspondence of point with the position occupied by a particle.
Without any loss of purpose, we can designate motion to begin at A or A’ or A’’ corresponding to final positions B or B’ or B’’ respectively as shown in the figure above.
For the reasons as outlined above, we shall freely use the terms “body” or “object” or “particle” in one and the same way as far as description of translational motion is concerned. Here, pure translation conveys the meaning that the object is under motion without rotation, like sliding of a block on a smooth inclined plane.
Position of a point object
The position of a point like object, in three dimensional coordinate space, is defined by three values of coordinates i.e. x, y and z in Cartesian coordinate system as shown in the figure above.
It is evident that the relative position of a point with respect to a fixed point such as the origin of the system “O” has directional property. The position of the object, for example, can lie either to the left or to the right of the origin or at a certain angle from the positive x - direction. As such the position of an object is associated with directional attribute with respect to a frame of reference (coordinate system).
Problem : The length of the second’s hand of a round wall clock is ‘r’ meters. Specify the coordinates of the tip of the second’s hand corresponding to the markings 3,6,9 and 12 (Consider the center of the clock as the origin of the coordinate system.).
Coordinates of the tip of the second’s hand
Solution : The coordinates of the tip of the second’s hand is given by the coordinates :
3 : r, 0, 0
6 : 0, -r, 0
9 : -r, 0, 0
12 : 0, r, 0
What would be the coordinates of the markings 3,6,9 and 12 in the earlier example, if the origin coincides with the marking 6 on the clock ?
Coordinates of the tip of the second’s hand
The coordinates of the tip of the second’s hand is given by the coordinates :
3 : r, r, 0
6 : 0, 0, 0
9 : -r, r, 0
12 : 0, 2r, 0
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