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# Chapter 2 - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about ' Chapter 2' - oleg-mclaughlin

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Presentation Transcript

Motion in two dimensions

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

• 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

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.

Resultant vector

R=A+B+C+D

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

2- the parallelogram method.

A + B = B + A

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

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

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

A

-A

Subtracting vectors:

A - B = A + (-B)

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 ?

The x- and y-components of a vector:

The magnitude of a vector:

The angle q between vector and x-axis:

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

(Examples)

• 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:

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:

Example 1.

• 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

The angle between where and the positive x axis is:

• 61°

• 29°

• 151°

• 209°

• 241°

W 1.

Example :

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

F=F1+F2+F3

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

example 1.