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General physics I, lec 2 By: T.A.Eleyan
2
Coordinate Systems and Frames of Reference
The location of a point on a line can be described by one coordinate; a point on a plane can be described by two coordinates; a point in a three dimensional volume can be described by three coordinates. In general, the number of coordinates equals the number of dimensions. A coordinate system consists of:
1 .a fixed reference point (origin)
2 .a set of axes with specified directions and scales
3 .instructions that specify how to label a point in space relative to the origin and axes
General physics I, lec 2 By: T.A.Eleyan
3
Coordinate Systems
In 1 dimension, only 1 kind of system, Linear Coordinates (x) +/-
In 2 dimensions there are two commonly used systems, Cartesian Coordinates (x,y) Polar Coordinates (r,)
In 3 dimensions there are three commonly used systems, Cartesian Coordinates (x,y,z) Cylindrical Coordinates (r,,z) Spherical Coordinates (r,)
General physics I, lec 2 By: T.A.Eleyan
4
Cartesian coordinate system also called rectangular
coordinate system x and y axes points are labeled (x,y)
Plane polar coordinate system
origin and reference line are noted
point is distance r from the origin in the direction of angle
points are labeled (r,)
General physics I, lec 2 By: T.A.Eleyan
5
The relation between coordinates
x rcos sinry
22 yxr
x
ytan
Furthermore, it follows that
Problem: A point is located in polar coordinate system by the coordinate and.
Find the x and y coordinates of this point, assuming the two coordinate systems have the same origin .
5.2r 35
General physics I, lec 2 By: T.A.Eleyan
6
Example: The Cartesian coordinates of a point are given by
)x,y-) =(3.5,-2.5 (meter. Find the polar coordinate of this point .
Solution:
21636180
714.05.3
5.2
x
ytan
m3.4)5.2()5.3(yxr 2222
Note that you must use the signs of x and y to find that is in the third quadrant of coordinate system. That is not 36
216
General physics I, lec 2 By: T.A.Eleyan
7
Scalars and Vectors Scalars have magnitude only. Length, time, mass, speed and volume are examples of scalars.
Vectors have magnitude and direction. The magnitude of is written Position, displacement, velocity, acceleration and force are examples of vector quantities.
v
v
General physics I, lec 2 By: T.A.Eleyan
8
Properties of Vectors
Equality of Two Vectors
Two vectors are equal if they have the same magnitude and the same direction
Movement of vectors in a diagram
Any vector can be moved parallel to itself without being affected
General physics I, lec 2 By: T.A.Eleyan
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Negative Vectors
Two vectors are negative if they have the same magnitude
but are 180° apart (opposite directions)
Multiplication or division of a vector by a scalar results in a vector for which
)a (only the magnitude changes if the scalar is positive )b (the magnitude changes and the direction
is reversed if the scalar is negative.
General physics I, lec 2 By: T.A.Eleyan
10
Adding Vectors
When adding vectors, their directions must be taken into account and units must be the same
First: Graphical Methods
Second: Algebraic Methods
General physics I, lec 2 By: T.A.Eleyan
11
Adding Vectors Graphically (Triangle Method)
Continue drawing the vectors “tip-to-tail”
The resultant is drawn from the origin of A to the end of the last vector
Measure the length of R and its angle
General physics I, lec 2 By: T.A.Eleyan
12
When you have many vectors, just keep repeating the process until all are included
The resultant is still drawn from the origin of the first vector to the end of the last vector
General physics I, lec 2 By: T.A.Eleyan
13
Alternative Graphical Method (Parallelogram Method)
When you have only two vectors, you may use the Parallelogram Method
All vectors, including the resultant, are drawn from a common origin
The remaining sides of the parallelogram are sketched to determine the diagonal, R