Engineering Mechanics Electrical Engineering Department Engr. Abdul Qadir channa

1. Cartesian Vectors Right-Handed Coordinate System A rectangular or Cartesian coordinate system is said to be right-handed provided: - Thumb of right hand points in the direction of the positive z axis when the right-hand fingers are curled about this axis and directed from the positive x towards the positive y axis

Rectangular Components of a Vector - A vector A may have one, two or three rectangular components along the x, y and z axes, depending on orientation - By two successive application of the parallelogram law A = A’ + A z A’ = A x + A y - Combing the equations, A can be expressed as A = A x + A y + A z 1. Cartesian Vectors

Unit Vector - Direction of A can be specified using a unit vector - Unit vector has a magnitude of 1 - If A is a vector having a magnitude of A ≠ 0, unit vector having the same direction as A is expressed by u A = A / A So that A = A u A 1. Cartesian Vectors

Unit Vector - Since A is of a certain type, like force vector, a proper set of units are used for the description - Magnitude A has the same sets of units, hence unit vector is dimensionless - A ( a positive scalar) defines magnitude of A - u A defines the direction and sense of A 1. Cartesian Vectors

Cartesian Unit Vectors - Cartesian unit vectors, i, j and k are used to designate the directions of z, y and z axes - Sense (or arrowhead) of these vectors are described by a plus or minus sign (depending on pointing towards the positive or negative axes) 1. Cartesian Vectors

Cartesian Vector Representations - Three components of A act in the positive i, j and k directions A = A x i + A y j + A Z k *Note the magnitude and direction of each components are separated, easing vector algebraic operations. 1. Cartesian Vectors

Magnitude of a Cartesian Vector - From the colored triangle, - From the shaded triangle, - Combining the equations gives magnitude of A 1. Cartesian Vectors

Direction of a Cartesian Vector - Orientation of A is defined as the coordinate direction angles α, β and γ measured between the tail of A and the positive x, y and z axes - 0° ≤ α, β and γ ≤ 180 ° 1. Cartesian Vectors

Direction of a Cartesian Vector - For angles α, β and γ (blue colored triangles), we calculate the direction cosines of A 1. Cartesian Vectors

Direction of a Cartesian Vector - For angles α, β and γ (blue colored triangles), we calculate the direction cosines of A 1. Cartesian Vectors

Direction of a Cartesian Vector - For angles α, β and γ (blue colored triangles), we calculate the direction cosines of A 1. Cartesian Vectors

Direction of a Cartesian Vector - Angles α, β and γ can be determined by the inverse cosines - Given A = A x i + A y j + A Z k Then, u A = A /A = (A x /A)i + (A y /A)j + (A Z /A)k where 1. Cartesian Vectors

Direction of a Cartesian Vector - u A can also be expressed as u A = cosαi + cosβj + cosγk - Since and magnitude of u A = 1, -A as expressed in Cartesian vector form A = Au A = Acosαi + Acosβj + Acosγk = A x i + A y j + A Z k 1. Cartesian Vectors

1.1. Addition and Subtraction of Cartesian Vectors Example Given: A = A x i + A y j + A Z k and B = B x i + B y j + B Z k Vector Addition Resultant R = A + B = (A x + B x )i + (A y + B y )j + (A Z + B Z ) k Vector Substraction Resultant R = A - B = (A x - B x )i + (A y - B y )j + (A Z - B Z ) k

Concurrent Force Systems - Force resultant is the vector sum of all the forces in the system F R = ∑F = ∑F x i + ∑F y j + ∑F z k where ∑F x, ∑F y and ∑F z represent the algebraic sums of the x, y and z or i, j or k components of each force in the system 1.1. Addition and Subtraction of Cartesian Vectors

Force, F that the tie down rope exerts on the ground support at O is directed along the rope Angles α, β and γ can be solved with axes x, y and z 1.1. Addition and Subtraction of Cartesian Vectors

1.1 Addition and Subtraction of Cartesian Vectors Cosines of their values forms a unit vector u that acts in the direction of the rope Force F has a magnitude of F F = Fu = Fcosαi + Fcosβj + Fcosγk

Example 3 Express the force F as Cartesian vector 1.1 Addition and Subtraction of Cartesian Vectors

Solution Since two angles are specified, the third angle is found by Two possibilities exit, namely or 1.1 Addition and Subtraction of Cartesian Vectors

Solution By inspection, α = 60° since F x is in the +x direction Given F = 200N F = Fcosαi + Fcosβj + Fcosγk = (200cos60°N)i + (200cos60°N)j + (200cos45°N)k = {100.0i j k}N Checking: 1.1 Addition and Subtraction of Cartesian Vectors

Example Addition and Subtraction of Cartesian Vectors

GENERAL DEFINITION A coordinate system is a system designed to establish positions with respect to given reference points. The coordinate system consists of one or more reference points, the styles of measurement (linear or angular) from those reference points, and the directions (or axes) in which those measurements will be taken. In satellite navigation various coordinate (reference) systems are used to precisely define the satellite and user locations.

POLAR COORDINATE SYSTEM Polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a fixed point and an angle from a fixed direction.

CILINDRICAL COORDINATE SYSTEM A cylindrical coordinate system is a three-dimensional coordinate system, where each point is specified by the two polar coordinates of its perpendicular projection onto some fixed plane, and by its (signed) distance from that plane

SPHERICAL COORDINATE SYSTEM A spherical coordinate system is a coordinate system for three- dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its elevation angle measured from a fixed plane, and the azimuth angle of its orthogonal projection on that plane.

CARTESIAN COORDINATE SYSTEM Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to two fixed perpendicular directed lines, measured in the same unit of length.

The geometric significance of the scalar triple product can be seen by considering the parallelepiped determined by the vectors a, b, and c. Scalar Triple Product

The area of the base parallelogram is: A = |b x c| Scalar Triple Product

If θ is the angle between a and b x c, then the height h of the parallelepiped is: h = |a||cos θ| Scalar Triple Product

Hence, the volume of the parallelepiped is: V = Ah = |a · (b x c)| Scalar Triple Product

Let a = b = c =

Scalar Triple Product a · (b x c) = (a x b) · c

Curl and Divergence Here, we define two operations that: Can be performed on vector fields. Play a basic role in the applications of vector calculus to fluid flow, electricity, and magnetism.

Curl and Divergence The motion of a wind or fluid can be described by a vector field. The concept of a force field plays an important role in mechanics, electricity, and magnetism.

Curl and Divergence

Curl Suppose: F = P i + Q j + R k is a vector field. The partial derivatives of P, Q, and R all exist

Curl Then, the curl of F is the vector field on defined by:

Curl As a memory aid, let’s rewrite Equation 1 using operator notation. We introduce the vector differential operator (“del”) as:

Curl It has meaning when it operates on a scalar function to produce the gradient of f :

Curl If we think of as a vector with components ∂/∂x, ∂/∂y, and ∂/∂z, we can also consider the formal cross product of with the vector field F as follows.

Curl

Curl

DIVERGENCE If F = P i + Q j + R k is a vector field and ∂P/∂x, ∂Q/∂y, and ∂R/∂z exist, the divergence of F is the function of three variables defined by:

DIVERGENCE In terms of the gradient operator the divergence of F can be written symbolically as the dot product of and F:

FORCE An agent which produces or tends to produce, destroy or tends to destroy motion of a body is called force. Unit of force is Newton. It is a vector quantity.

CHARACHTERISTIC OF A FORCE MAGNITUDE: Magnitude of force may be 10kn etc. DIRECTION: It has certain fixed direction. NATURE: It may be tensile or compressive. POINT OF APPLICATION: Point at which force acts.

SYSTEM OF FORCES When two or more forces act on a body, they are called to form a system of forces.  Coplanar Forces  Concurrent Forces  Collinear Forces  Coplanar Concurrent Forces  Coplanar non Concurrent Forces  Non coplanar Concurrent Forces  Non coplanar Non concurrent Forces  Like parallel Forces  Unlike parallel Forces  Spatial Forces

COPLANAR FORCES Forces whose line of action lie on the same plane, are known as coplanar forces.

CONCURRENT FORCES The forces which meet at one point, are known as concurrent forces.

CO-LINEAR FORCES The forces whose line of action lie on the same line, are known as collinear force.

COPLANAR CONCURRENT FORCES Forces which meet at one point & lines of action also lie on the same plane are known as coplanar concurrent forces.

NON-COPLANAR CONCURRENT FORCES Forces whose line of action do not lie on the same plane, but they meet at one point.

COPLANAR NON-CONCURRENT FORCES The forces whose line of action lie on the same plane but they do not meet at one point are known as coplanar non-concurrent forces.

NON-COPLANAR NON-CONCURRENT FORCES Forces whose line of action do not lie on the same plane & they do not meet at any point.

LIKE PARALLEL FORCES Forces whose line of action are parallel to each other & all of them act in same direction.

UNLIKE PARALLEL FORCES Forces whose line of action are parallel to each other but all of them do not act in the same direction.

SPATIAL FORCES Forces acting in the space are known as spatial forces. Forces acting in space but meeting at one point are known as spatial concurrent forces.

RESOLVED FORCES Spliting of forces into their component unit is called resolution of forces. This is the reverse process which consist of expressing a single force in terms of their components.

Class test. 01 Q.02 If A= (2i-3j+4k) and B= (4i+2j+5k) then find the value of vector A X B Q.01 Q.03 Draw a cartesian vector, write the equation for resultant of a cartesian vector and direction cosines.

Resultant of force Graphical Approach Triangle method Parallelogram method Polygon method (good if more than one force)

Parallelogram method.

Laws of force Parallelogram law Two “component” forces F1 and F2 add according to the parallelogram law, yielding a resultant force FR that forms the diagonal of the parallelogram. Label all the known and unknown force magnitudes and the angles on the sketch and identify the two unknowns as the magnitude and direction of FR, or the magnitudes of its components. Trigonometry. Redraw a half portion of the parallelogram to illustrate the triangular head-to-tail addition of the components. From this triangle, the magnitude of the resultant force can be determined using the law of cosines, and its direction is determined from the law of sines. The magnitudes of two force components are determined from the law of sines.

Example. 1 Hint: Apply Cosine Law

Example Hint: Apply Sine Law

Solution