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Module 1: Mechanics
Vectors in Two Dimentions
Revision: Grade 10
Vector Scalar
A physical quantity that has magnitude, unit
and direction
A physical quantity that has only magnitude
and unit
Eg. Force, Velocity, Displacement, Acceleration
Eg. Speed, Distance, Mass,
Volume
Revision: Grade 10
Properties of Vectors Equal Vectors:
Two vectors are equal if the have the same MAGNITUDE and DIRECTION
Negative Vector:A vector that points in the opposite direction as the positive reference direction
Addition of VectorsMAGNITUDE and DIRECTION is taken into account
𝑭𝟏ሱሮ
𝑭𝟐ሱሮ 𝑭𝟑ሱሮ
𝑭𝑻𝒐𝒕ሬሬሬሬሬሬሬሬԦ= + + 𝑭𝟏ሬሬሬሬԦ 𝑭𝟐ሬሬሬሬԦ ൫−𝑭𝟑ሬሬሬሬԦ൯
Revision: Grade 10
Properties of Vectors
Resultant VectorA single vector that has the same effect as all the other vectors
together 𝑭𝟏ሬሬሬሬԦ 𝑭𝟐ሬሬሬሬԦ 𝑭𝑹ሬሬሬሬሬԦ 𝑭𝑹ሬሬሬሬሬԦ= 𝑭𝟏ሬሬሬሬԦ+ 𝑭𝟐ሬሬሬሬԦ
Revision: Grade 10
Properties of Vectors
Vector in Two Dimentions
Reference directions
Example:Draw the following vectors. Use a scale of 1 cm:1 N
a) 5 N 30°b) 4 N 20° S van Wc) 5 N 200°
Vector in Two Dimentions
30°
20° 200°
Vector in Two Dimentions
Addition of Vectors in Two Dimensions
Graphically: Head-to-Tail method𝑭𝟏ሬሬሬሬԦ 𝑭𝟐ሬሬሬሬԦ
𝑭𝑹ሬሬሬሬሬԦ
Addition of Vectors in Two Dimensions
Example: Head-to-tailJohnny walks 50 m in the direction 50º. He then turns and walks another 10 m in the direction 120º. Determine his displacement by means of a scale drawing
Steps for Graphical Addition of Vectors in Two Dimensions Choose a scale Choose method to be used Draw vectors to scale Draw resultant Measure the resultant and determine its
direction using a protractor Convert the measured value of the resultant to the real value using the scale
𝑭𝟏ሬሬሬሬԦ
𝑭𝟐ሬሬሬሬԦ
Graphically: Paralellogram method (Tail-to-tail method)
𝑭𝑹ሬሬሬሬሬԦ
Addition of Vectors in Two Dimensions
Addition of Vectors in Two Dimensions
Example: parallellogramJohnny walks 50 m in the direction 50º. He then turns and walks another 10 m in the direction 120º. Determine his displacement by means of a scale drawing
Example:Determine the magnitude an direction of the resultant force in the diagram
20°
5 N4 N
Addition of Vectors in Two Dimensions
Perpendicular Vectors
𝑭𝟏ሬሬሬሬԦ
𝑭𝟐ሬሬሬሬԦ
𝑭𝑹ሬሬሬሬሬԦ 𝒔𝟐 = 𝒙𝟐 + 𝒚𝟐 Pythagoras:
𝑭𝑹ሬሬሬሬሬԦ𝟐 = 𝑭𝟏ሬሬሬሬԦ𝟐 + 𝑭𝟐ሬሬሬሬԦ𝟐
Addition of Vectors in Two Dimensions
Simple trigonometric relationships
s
a
t
θ
𝒄𝒐𝒔𝜽= 𝒂𝒔
𝒔𝒊𝒏𝜽= 𝒕𝒔
𝒕𝒂𝒏𝜽= 𝒕𝒂
Addition of Vectors in Two Dimensions
Example:Calculate the magnitude and direction of the resultant force in the diagram
5 N
4 N
Addition of Vectors in Two Dimensions
Resolution of Vectors in Perpendicular Components
θ
𝑹𝒚ሬሬሬሬሬԦ
𝑹𝒙ሬሬሬሬሬԦ
𝑹ሬሬԦ
𝒔𝒊𝒏𝜽= 𝑹𝒚ሬሬሬሬሬԦ𝑹ሬሬԦ
∴ 𝑹𝒚ሬሬሬሬሬԦ= 𝑹ሬሬԦ𝒔𝒊𝒏𝜽
𝒄𝒐𝒔𝜽= 𝑹𝒙ሬሬሬሬሬԦ𝑹ሬሬԦ
∴ 𝑹𝒙ሬሬሬሬሬԦ= 𝑹ሬሬԦ𝒄𝒐𝒔𝜽
Example:Resolve the following force into its perpendicular components
8 N
30°
Resolution of Vectors in Perpendicular Components
Mathematically Resolve each vector into its perpendicular components Add all the x-components
Add all the y-components
𝑭𝑵𝑬𝑻 𝒙ሬሬሬሬሬሬሬሬሬሬሬሬሬԦ= 𝑭𝒙ሬሬሬሬԦ
𝑭𝑵𝑬𝑻 𝒚ሬሬሬሬሬሬሬሬሬሬሬሬሬԦ= 𝑭𝒚ሬሬሬሬԦ
Addition of Vectors in Two Dimensions
Mathematically Calculate the magnitude of the resultant using pythagoras
Calculate the direction of the resultant using trigonometric relationships
𝑭𝑹ሬሬሬሬሬԦ𝟐 = 𝑭𝑵𝑬𝑻 𝒙ሬሬሬሬሬሬሬሬሬሬሬሬሬԦ𝟐 + 𝑭𝑵𝑬𝑻 𝒚ሬሬሬሬሬሬሬሬሬሬሬሬሬԦ𝟐
Addition of Vectors in Two Dimensions
Example:Calculate the resultant of the forces in the following diagram
50°
30°
5 N
8 N
4 N
Addition of Vectors in Two Dimensions
