CHAPTER OUTLINE

• Mechanics
• Basic Concepts
• Scalars and Vectors
• Newton's Laws
• Units
• Law of Gravitation
• Accuracy, Limits, and Approximations
• Problem Solving in Statics
• Chapter Review

INTRODUCTION TO STATICS

MECHANICS

• Definition: Mechanics is the physical science which deals with the effects of forces on objects.
• Importance: Plays a crucial role in engineering analysis. Principles have wide application in fields like:
  • Vibrations
  • Stability and strength of structures and machines
  • Robotics
  • Rocket and spacecraft design
  • Automatic control
  • Engine performance
  • Fluid flow
  • Electrical machines and apparatus
  • Molecular, atomic, and sub-atomic behavior
• Historical Context:
  • Oldest of the physical sciences; early writings by Archimedes (287–212 B.C.) on levers and buoyancy.
  • Stevinus (1548–1620): Laws of vector combination of forces and principles of statics.
  • Galileo (1564–1642): First investigation of dynamics problems.
  • Newton (1642–1727): Formulated laws of motion and gravitation, introduced the infinitesimal concept in mathematics.
  • Contributions from: da Vinci, Varignon, Euler, D'Alembert, Lagrange, Laplace, among others.
• Application: The principles of mechanics are rigorously expressed through mathematics.
• Division of Mechanics:
  • Statics: Equilibrium of bodies under forces.
  • Dynamics: Motion of bodies.
  • Engineering Mechanics: Divided into Statics (Vol. 1) and Dynamics (Vol. 2).

BASIC CONCEPTS

Key Definitions:

• Space: Geometric region occupied by bodies; positions described by linear and angular measurements relative to a coordinate system.
  • 3D problems require three independent coordinates.
  • 2D problems require two coordinates.
• Time: Measure of succession of events; important in dynamics but not directly in statics.
• Mass: Measure of inertia (resistance to change in velocity) and quantity of matter in a body.
  • Affects gravitational force between bodies.
• Force: Action of one body on another; characterized by magnitude, direction, and point of application.
  • Force is a vector quantity.
• Particle: Body with negligible dimensions; effectively treated as a mass concentrated at a point.
  • Used as differential elements or when dimensions are irrelevant.
• Rigid Body: Assumed to have no change in distance between any two points; internal deformations are negligible.
  • Focus on external forces acting on rigid bodies in equilibrium for statics.

SCALARS AND VECTORS

Types of Quantities in Mechanics:

• Scalar quantities: Only magnitude; examples include time, volume, density, speed, energy, mass.
• Vector quantities: Have both magnitude and direction; examples include displacement, velocity, acceleration, force, moment, momentum.
  • Must obey the parallelogram law of addition.

Vector Classification:

• Free Vector: Not confined to a unique line in space; describes displacement of any point in a body without rotation.
• Sliding Vector: Has a unique line of action but not a unique application point (e.g., external force on a rigid body).
• Fixed Vector: Has both a unique application point and line of action; applied to deformable bodies.

Representation and Operations:

• A vector quantity V is represented as a directed line segment with a length proportional to its magnitude.
  • Boldface is used for vector quantities, lightface italic for scalar quantities (e.g., vector V).
• Direction and angle of vector V can be measured from a reference direction.
  • The negative vector -V has the same magnitude but opposite direction.
• Parallelogram Law: 2 vectors V1 and V2 can be represented as the diagonal of a parallelogram formed by the vectors.
  • Vector sum V is represented as $V = V1 + V2$ and their scalar sum is $V = V1 + V2 (scalar).$
• Difference of Vectors: Obtained by adding -V2 to V1: $V' = V1 - V2$
• Components: Vectors can be split into components, ideally into mutually perpendicular components (rectangular components).
  • For vector V: $V = V<em>x i + V</em>y j$
• Unit Vector: Defined as n = rac{V}{|V|} to indicate direction:
  • Direction cosines are $l = cos( heta<em>x), m = cos( heta</em>y), n = cos( heta<em>z)$ where $|V|^2 = V</em>x^2 + V<em>y^2 + V</em>z^2$ and $l^2 + m^2 + n^2 = 1.$

NEWTON'S LAWS

Newton's Laws:

• Law I: A particle remains at rest or continues to move with uniform velocity if there are no unbalanced forces acting on it.
• Law II: The acceleration of a particle is proportional to the vector sum of forces acting upon it, expressed as: $F = ma$
  • Here, F is the vector sum of forces, m is mass, and a is the acceleration vector.
• Law III: Every action has an equal and opposite reaction; forces exist in action-reaction pairs.
• Significance: These laws are fundamental for understanding motion, analysis in dynamics, and the principle of equilibrium in statics.
• Correct Force Consideration: It's crucial to identify which force of the action-reaction pair applies to the subject in analysis.

UNITS

Fundamental Quantities and Systems of Units:

• Fundamental Quantities: Length, mass, force, and time.
• Two primary systems:
  • SI Units:
  • Length: meter (m)
  • Mass: kilogram (kg)
  • Time: second (s)
  • Force: newton (N)
  • $1 N = 1 kg imes 1 m/s^2$
  • U.S. Customary Units (FPS System):
  • Length: feet (ft)
  • Mass: slug
  • Time: seconds (sec)
  • Force: pounds (lb)
  • $1 lb = 1 slug imes 1 ft/s^2$
• Gravitational Attraction of Earth: Affects the force on bodies; depends on mass m and gravitational acceleration g near Earth:
  • Weight W expressed as: $W = mg$ where approximate values for g are 9.81 m/s² (SI) and 32.1740 ft/s² (U.S.)
• Unit Conversions and Standards: Accurate conversions between SI and U.S. units are crucial for engineering calculations.

LAW OF GRAVITATION

Law of Gravitation:

• Formulated by Newton, expressed as: F = G rac{m1 m2}{r^2}
  • G is the gravitational constant ( $G = 6.673 imes 10^{-11} m^3/(kg imes s^2)$ ).
• Gravitational forces exist between masses, notably Earth's attraction.
• Earth’s gravitational force is typically the only relevant force in practical engineering applications.
• Weight: The gravitational attraction on a body is measured in newtons (N)
  • Weight varies based on gravitational forces, commonly calculated using $W = mg$ where m is in kg (SI) or slugs (FPS).

ACCURACY, LIMITS, AND APPROXIMATIONS

• Significant Figures: Results should match the accuracy of given data; expressed with appropriate significant figures.
• Differentials: Higher-order differentials may be neglected in calculations; focus is on primary (lower-order) differentials.
• Small-Angle Approximations: Utilized when angles are small; examples include:
  • $sin( heta) hickapprox heta$, $tan( heta) hickapprox heta$, $cos( heta) hickapprox 1$ when angles are in radians.

PROBLEM SOLVING IN STATICS

Problem-Solving Methodology

• Analyze forces acting on structures in equilibrium using mathematical descriptions.
• Assumptions: Recognize the idealization of physical problems, often involving mathematical/physical approximations.
• Graphics in Analysis:
  1. Represent physical systems through sketches or diagrams for improved understanding.
  2. Utilize graphical solutions when viable, as they can simplify analysis.
  3. Employ charts or graphs to display results succinctly.
• Formulating Problems:
  • Steps:
  1. State data, desired result, assumptions.
  2. Gather necessary diagrams.
  3. Apply governing principles and perform calculations.
• Free-Body Diagrams: Essential for isolating bodies and accurately depicting forces; a core analytical tool in mechanics.

Solution Methods:

• Use various methods, including:
  1. Hand calculations (algebraic or numeric)
  2. Graphical methods
  3. Computational solutions (preferred for larger equations or parameter variations)

CHAPTER REVIEW

Proficiencies After Completion:

1. Express vectors in unit and perpendicular components, perform vector operations.
2. State and understand Newton’s laws of motion.
3. Carry out calculations using both SI and U.S. units with correct accuracy.
4. Express and apply the law of gravitation to compute weight.
5. Apply simplifications based on differentials and small angles.
6. Understand the methodology for formulating and solving statics problems.