In order to understand motion, the first concept to be introduced is the
concept of frame of reference.
Frame of Reference
If a cast away, in the middle of the ocean, has been rowing
for several hours, how does he know if he has been moving? The trouble in
answering this question is associated with the ocean not having reference
points. The most likely answer to this question is that, after rowing for several
hours, the cast away does not know about a possible change of location. On the
other side, for a person lost in a city the response will be different. In the
city case, even when the person may not be able to conclude about progress
toward reaching the final destination, the person will certainty know about
being on the same or different place with respect to the starting point. The
difference in result is because a city has different land marks that
allow people to identified different locations. In mechanics, the city land
marks are equivalent to frames of references.
The figure on the left represents the most common
one-dimension frame of reference. The one-dimensional frame of reference
is characterized by a single straight line with equally spaced divisions
proportional to the unit of length used. At the center, usually the so
called origin of coordinates is located.
In this case, the figure on the left is a
two-dimensions coordinate system. Here, two axis are use to identify
every point on the plane described by the coordinates. The
identification of points is equivalent to the one used in elementary
mathematics with the first coordinate representing the value of the
point with reference to the x-axis and the second coordinate
representing the y-axis; thus, any point in the plane will be
represented by the ordinate pair (x,y). The
two dimensional coordinate system is represented by two orthogonal
(perpendicular) axis that intercept at a point called the origin of
coordinates, (0, 0). The segment of the axis to the right of the origin
is the positive side of the x-axis and the segment of the axis to
the left of the origin is the negative segment of the x-axis.
Similarly, the top part of the y-axis is the positive side while
the lower part of the y-axis is the negative side.
The picture below shows a possible use of a two dimensional coordinate
system to represent position of objects on the floor of the three
dimensional room . For example, if the motion of an object such as a toy
car is always on the floor of the room, the motion of the toy car can be
completely study with a two dimensional coordinates system represented
by the x and y axis.
The figure at the left represents a
three-dimensional coordinate system generated from the vertex of a room.
The abstract representation of the same coordinate system is also presented
on the figure
. The negative
portion of the axis are not
represented in the picture. The negative z-axis is the projection
down of the picturedpositivez-axis.
Tree-dimensional coordinates can represent points on the space by a set
of three numbers, the x coordinate, y coordinate, and the
z coordinate. The notation for representing a point in the plane
is ( x, y, z). In these notes, most of the problems
will be reduced to a single plane. Thus, the most use coordinate system
will be the two dimensional system.
The applet on the left can help to understand
coordinate systems in two dimensions. Click on the graph, the
coordinates of the point will be displayed. The number of points that
can be display at a single time is limited. Press the Clear button to
clean the points from the graph.
What are the sign of the coordinate numbers for
points on the left-top quadrant (II quadrant) of the graph?
In which quadrant are the two coordinate numbers
negative? and, positive?