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Chapter 2. Motion in two dimensions. 2.1: An introduction to vectors. 2.1: An introduction to vectors. Vectors : Magnitude and direction Examples for Vectors: force – acceleration- displacement Scalars : Only Magnitude

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Presentation Transcript
slide1

Chapter 2

Motion in two dimensions

2.1: An introduction to vectors

slide2

2.1: An introduction to vectors

  • Vectors: Magnitude and direction
  • Examples for Vectors: force – acceleration- displacement
  • Scalars: Only Magnitude
  • A scalar quantity has a single value with an appropriate unit and has no direction.
    • Examples for Scalars: mass- speed- work-Distance- Energy-Work-Pressure

Motion of a particle from A to B along an arbitrary path (dotted line). Displacement is a vector

slide3

Vectors:

  • Represented by arrows (example displacement).
  • Tip points away from the starting point.
  • Length of the arrow represents the magnitude
  • In text: a vector is often represented in bold face (A) or by an arrow over the letter.
  • In text: Magnitude is written as A or

This four vectors are equal because they have the same magnitude and same length

slide4

Adding vectors:

Two vectors can be added using these method:

1- tip to tail method.

2- the parallelogram method.

1- tip to tail method.

Graphical method (triangle method):

Draw vector A. Draw vector B starting at the tip of vector A.

The resultant vector R = A + B is drawn from the tail of A to the tip of B.

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Adding several vectors together.

Resultant vector

R=A+B+C+D

is drawn from the tail of the first vector to the tip of the last vector.

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Commutative Law of vector addition

2- the parallelogram method.

A + B = B + A

(Parallelogram rule of addition)

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Associative Law of vector addition

A+(B+C) = (A+B)+C

The order in which vectors are added together does not matter.

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Negative of a vector.

The vectors A and –A have the same magnitude but opposite directions. A + (-A) = 0

A

-A

Subtracting vectors:

A - B = A + (-B)

slide9

Multiplying a vector by a scalar

The product mA is a vector that has the same direction as A and magnitude mA.

The product –mA is a vector that has the opposite direction of A and magnitude mA.

Examples: 5A; -1/3A

  • Given , what is ?
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Components of a vector

The x- and y-components of a vector:

The magnitude of a vector:

The angle q between vector and x-axis:

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The signs of the components Ax and Ay depend on the angle q and they can be positive or negative.

(Examples)

slide13

Unit vectors

  • A unit vector is a dimensionless vector having a magnitude 1.
  • Unit vectors are used to indicate a direction.
  • i, j, k represent unit vectors along the x-, y- and z- direction
  • i, j, k form a right-handed coordinate system
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A unit vector is a dimensionless vector having a magnitude 1.

  • Unit vectors are used to indicate a direction.
  • i, j, k represent unit vectors along the x-, y- and z- direction
  • i, j, k form a right-handed coordinate system

The unit vector notation for the vector A is:

OR in even better shorthand notation:

slide15

Adding Vectors by Components

We want to calculate: R = A + B

From diagram: R = (Axi + Ayj) + (Bxi + Byj)

R = (Ax + Bx)i + (Ay + By)j

Rx = Ax+ Bx

Ry = Ay+ By

The components of R:

The magnitude of a R:

The angle q between vector R and x-axis:

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Example

  • A force of 800 N is exerted on a bolt A as show in Figure (a). Determine the horizontal and vertical components of the force.

The vector components of F are thus,

and we can write F in the form

example
Example :

The angle between where and the positive x axis is:

  • 61°
  • 29°
  • 151°
  • 209°
  • 241°
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W

Example :

F1 = 37N 54° N of E F2 = 50N 18° N of F3 = 67 N 4° W of S

F=F1+F2+F3

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Ex : 2 – 10

A woman walks 10 Km north, turns toward the north west , and walks 5 Km further . What is her final position?

slide24

example

Answer is d

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