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Course Introduction

  • Instructor: Dr. Uzair Khaleeq uz Zaman

  • Course Code: DMTS - College of E&ME

  • Course Title: ME 113 – ENGINEERING MECHANICS I: STATICS

  • Week: 02

Chapter 2: Force Vectors

  • Author: Dr. Uzair Khaleeq uz Zaman

Outline

  • Scalars & Vectors

  • Vector Operations

  • Vector Addition of Forces

  • Addition of a System of Coplanar Forces

  • Cartesian Vectors

  • Addition of Cartesian Vectors

  • Position Vectors

  • Force Vector Directed along a Line

  • Dot Product

Vectors

Types of Vectors

  1. Free Vector:

    • Action not confined to a line; defined by magnitude and direction.

    • Example: Movement of a body without rotation.

  2. Sliding Vector:

    • Unique line of action, but no specific point of application; described by magnitude, direction, and line of action.

    • Example: External force on a rigid body.

  3. Fixed Vector:

    • Unique point of application; described by magnitude, direction, and point of application.

Vector Operations

Basic Operations

  • Multiplication & Division by a Scalar:

    • Positive scalar: increases magnitude.

    • Negative scalar: decreases magnitude and reverses direction.

Vector Addition

  1. Parallelogram Law:

    • Vectors A and B combined to form resultant R = A + B.

    • Steps:

      • Join tails of A and B.

      • Draw lines from heads of A and B parallel to opposite vectors.

      • Intersection point defines resultant vector R.

  2. Triangle Rule:

    • Vectors A and B added in head-to-tail configuration.

    • Resultant R extends from tail of A to head of B.

    • Commutative property: R = A + B = B + A.

  3. Collinear Case:

    • If A and B are collinear, thence R = A + B becomes scalar addition.

Vector Subtraction

  • Defined as R' = A – B = A + (-B); follows addition rules.

Zero Vector

  • Resultant of a vector subtracted from itself: P - P = 0 (Null Vector).

Vector Addition of Forces

Finding Resultant Forces

  1. Finding Resultant Force:

    • Two force components F1 and F2 lead to resultant force FR = F1 + F2.

  2. Applying Laws:

    • Use parallelogram construction or the triangle rule with laws of cosines or sines.

Resolving Forces

  • Force resolved into two components with respect to specific axes (u and v).

Addition of Several Forces

  • Use successive applications of the parallelogram law for multiple forces:

    FR = (F1 + F2) + F3.

Important Points

  • Scalars: Positive or negative numbers.

  • Vectors: Quantities with magnitude, direction, and sense.

  • Proper analysis uses x-y coordinate systems for resolving forces.

Practice & Example Problems

  • Example 2-1: Calculate resultant force generated by two forces at a specified angle using the parallelogram law.

  • Example 2-3: Component forces determination involving vector operations.

Addition of a System of Coplanar Forces

Types of Forces

  1. Coplanar Forces: Lines of action reside in the same plane.

  2. Collinear Forces: Lines of action reside on the same line.

  3. Concurrent Forces: Meet at a point (not necessarily collinear).

  4. Coplanar Concurrent Forces: Meet at one point on the same plane.

  5. Coplanar Non-concurrent Forces: Do not meet at a point, but in the same plane.

  6. Non-coplanar Concurrent Forces: Meet at one point not in the same plane.

  7. Non-coplanar Non-concurrent Forces: Do not meet and are not on the same plane.

Notation Approaches

  • Scalar Notation:

    • F = Fx + Fy

    • Fx = F Cos θ ; Fy = F Sin θ.

  • Cartesian Vector Notation:

    • F = Fx i + Fy j, where i and j are unit vectors.

Coplanar Force Resultants

  • Resolving forces into x and y components and adding them.

  • Resultant force calculated using law of cosines or using component addition techniques.

Important Synopsis

  • Forces resolved in an established x, y coordinate system expedite resultant determinations.

  • Positive directions designated by Cartesian unit vectors.

  • Magnitude calculated using the Pythagorean theorem and direction from trigonometric relations.

Conclusion

  • A systematized approach to vectors in statics enhances understanding of force interactions.

Thank You

  • Instructor: Dr. Uzair Khaleeq uz Zaman

  • Affiliation: College of E&ME