Chapter I Vectors and Scalars

AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

Fundamental Principles• Preconditions to deal with

problems in mechanics.• Basic concepts used in

mechanics:• space, time, mass, force, particle,

rigid body

Fundamental Principles Cont…

• Basic concepts used in mechanics:

• space, time, mass, force, particle, rigid body

• coordinates - position of a point P (x, y, z)

• measured from a certain point of reference

• time of an event taking place,determination of velocity and acceleration

• mass of a body [kg, to]• action of weight, behavior under the actionof an external force

• magnitude, direction, point of application• e.g. action on a rigid body, action of one bodyonto another

• infinitesimal small piece of a body, single point in space

• body consisting of a non-deformable material• (no displacement under the action of forces)

Newton’s LawsSir Isaac Newton (1642-1727)• 1st Law: A particle remains at rest or

continues to move with constant velocity if the resultant force acting on it is zero.

Newton’s LawsSir Isaac Newton (1642-1727)• 2nd Law: The acceleration of a particle

proportional to the resultant force acting on it (magnitude and direction). F = ma m = mass of particlea = acceleration

Newton’s LawsSir Isaac Newton (1642-1727)• 3rd Law: The forces of action and reaction

between bodies in contact are equal in magnitude, opposite in direction and collinear (same line of action).

Newton’s Laws• Law of Gravitation• Two particles of mass m1 and m2 are

mutually attracted with equal and opposite forces F and F’ of magnitude F.

G = constant of gravitation

Newton’s Laws• Law of Gravitation• Weight = Gravitational Force acting on a body• (attraction between earth and body)

W = m g ⋅g = acceleration of gravity = 9.81 m/s2

Newton’s Laws• Law of Gravitation• Weight = Gravitational Force acting on a body

(attraction between earth and body)W[N] = m[Kg] g[m/s⋅ 2]

g = 9.81 m/s2

Units• International System of Units (SI units)

Mass m [to, kg]Force F [kN, N]Time t [s]Length L [m, cm, mm]

Scalars and VectorsDefinition and properties

• Scalars: quantities described by their magnitude alonee.g. time, volume, area, density,

distance, energy mass

Vectors: quantities described by their magnitude and direction

e.g. displacement, velocity, force, acceleration, momentum

Scalars and VectorsDefinition and properties

Graphical representation of a Vector

• line segment of certain length (magnitude) and orientation (θ)

• arrowhead indicating direction

Symbolic representation of a Vector

• magnitude, length of vector: V , |V| or ║ ║ Ve.g. in scalar equations

• vector quantities respecting the orientation: V, V

e.g. mathematical vector operations

Symbolic representation of a Vector

• magnitude, length of vector: V , |V| or ║ ║ Ve.g. in scalar equations

• vector quantities respecting the orientation: V, V

e.g. mathematical vector operations

Representation of Vectors

• Algebraically a vector is represented by its components along the three dimensions.

Representation of Vectors Cont…

Orientation of Vectors

• collinear - same line of action• coplanar - located in the same plane• concurrent - passing through a common point

Classification of Vectors

•Free Vector•Sliding Vector•Fixed Vector

1. Free Vector: action in space not associated with a unique line

e.g. uniform displacement of a body

Classification of Vectors Cont…

2. Sliding Vector: action in space described by a unique line

e.g. action of force on rigid body

3. Fixed Vector: action in space described by a unique point

e.g. action of force on non rigid body

3. Fixed Vector: action in space described by a unique point

e.g. action of force on non rigid body

Vector Addition – graphical method

The parallelogram law – resultant force• Two forces maybe replaced by a single force

(resultant) obtained by drawing the diagonal of the parallelogram having sides equal to the given forces.

Vector Addition – graphical method Cont…

The triangle rule (from parallelogram law)

Vector Addition – Analytic Method

• Trigonometric rules• applying sine and cosine rules

Vector Addition – Analytic Method Cont…

Decomposition of Vectors

• Components,• perpendicular

horizontal component of V

Decomposition of Vectors Cont…

• Components,

horizontal component of V

• Components,

horizontal component of Vvertical component of V

• Components,

Multiplication• Multiplication of vectors by scalars

Multiplication Cont…• Multiplication of vectors by vectors

- dot product (scalar product)- cross product (vector product)

Dot Product (scalar product)

• Vectors A and B are θ inclined from each other

Dot Product (scalar product) Cont…

• Vectors A and B are θ inclined from each other• Result : Vector of determined magnitude and

direction perpendicular to the plane formed by A and B

Cross Product (vector product)

• Determination of resulting vector by three by three matrix

Cross Product (vector product) Cont…

• Moment of a vector V about any point 0

Thank You!

Chapter I Vectors and Scalars

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