general physics i, lec 2 by: t.a.eleyan 1 lecture 2 coordinate systems & vectors

General physics I, lec 2 By: T.A.Eleyan

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Lecture 2

Coordinate Systems & Vectors


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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


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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,)


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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,)


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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


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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


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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


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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


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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.


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Adding Vectors

When adding vectors, their directions must be taken into account and units must be the same

First: Graphical Methods

Second: Algebraic Methods


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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


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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


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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


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Vector Subtraction

Special case of vector addition

If A – B, then use A+(-B)

Continue with standard vector addition procedure

general physics i, lec 2 by: t.a.eleyan 1 lecture 2 coordinate systems & vectors

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