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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 - PowerPoint PPT Presentation
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Chapter 2 Motion in two dimensions
2.1 :An introduction to vectors
Vectors: Magnitude and directionExamples 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
2.1 :An introduction to vectors
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 A
A
This four vectors are equal because they have the same magnitude and same length
Adding vectors:
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.
Graphical method (triangle method):
Two vectors can be added using these method:1- tip to tail method.2- the parallelogram method.
1- tip to tail method.
5
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.
6
A + B = B + A(Parallelogram rule of addition)
Commutative Law of vector addition2- the parallelogram method.
Associative Law of vector addition
A+(B+C) = (A+B)+C
The order in which vectors are added together does not matter.
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)
Multiplying a vector by a scalarThe 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 ?
s
3s
s
s
s
s
s
s
Components of a vector
sin
cos
AA
AA
y
x
22yx AAA
x
y
A
A1tan
The x- and y-components of a vector:
The magnitude of a vector:
The angle between vector and x-axis:
The signs of the components Ax and Ay depend on the angle and they can be positive or negative.
Examples)
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
The unit vector notation for the vector A is:
OR in even better shorthand notation:
,x yA AA
ˆ ˆx yA i A j A
• 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
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
The components of R:Rx = Ax + Bx
Ry = Ay + By
2222 )()( yyxxyx BABARRR
The magnitude of a R:
xx
yy
x
y
BA
BA
R
RtanThe angle between vector R and x-axis:
example
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 :
The angle between where and the positive x axis is:
1. 61°2. 29°3. 151°4. 209°5. 241°
A
A x
A y
Ax 25 & Ay 45
Example :
Example :F1 = 37N 54° N of E F2 = 50N 18° N of F3 = 67 N 4° W of S
F=F1+F2+F3
W
Ex : 2 – 10 A woman walks 10 Km north, turns toward the north west , and walks 5 Km further . What is her final position?
example
Answer is d