ENSC 11

• Scalar Quantity


  Physical quantity characterized solely by magnitude (e.g. distance, speed, and mass), described by a real number with units of measurement, and can be positive or negative
• Vector Quantity


  Physical quantity characterized by magnitude, direction, and sense (e.g. displacement, velocity, and weight)
• Parts of a Vector

  • Length of Arrow – Magnitude
  • Line of Action – Direction
  • Arrowhead – Sense
  • Tail – Point of Application of the Force
• Resultant of Vectors

  1. Simplest force system

  2. Single equivalent force to a given force system or the sum of two or more vectors
• Parallelogram Method or Triangular Law

  1. If two vectors A and B are added, the resultant R is equal to the diagonal of the parallelogram drawn using vectors A and B as sides through their common point, magnitude, and direction

  2. If three or more vectors are present, parallelogram method can be used successively

  3. Summation of forces is commutative
• Polygon Method or Tip-to-Tail Method


  Extension of triangle law where vectors are connected tip-to-tail and represent a side of the polygon while the resultant represents the side that closes the polygon
• Rectangular Components Method

  1. Most accurate method in analyzing vectors

  2. Each vector is resolved into its components along the x- and y- axes

  3. Resultant is determined using Pythagorean Theorem while direction is determined using inverse trigonometry
• Vector Resolution

  1. One force vector is broken down into two or more component vectors

  2. There are infinite number of possible sets of forces when resolving a single force
• Moment of a Force


  Tendency of the force to cause an object to rotate about a point or an axis
• Couple


  Two parallel forces that have the same magnitude but opposite in direction, cancel each other out in summation of forces but cause a couple moment in summation of moments
• Equivalency


  Two systems of forces are said to be equivalent if they have the same effect on the body
• Important Notes about Moment

  • Force F causing rotation or moment if its line of action does not pass through a reference point
  • Force F causing zero rotation or moment if its line of action passes through a reference point
  • Moment is a vector quantity with magnitude and direction (clockwise CW or counterclockwise CCW)
  • Moment is always Force multiplied by the perpendicular distance
• Varignon’s Theorem


  “The moment of a force about a certain point is equivalent to the sum of the moments of the force’s components about the point”
• Theorem: 'A point is equivalent to the sum of the moments of the force’s components about the point'
• Theorem


  Basis for resolving forces into x- and y-components when determining perpendicular distances
• Couple Moment or Moment of a Couple

  Couple forces do not produce a net force but produce a couple moment
• Couple Moment


  Has its own rotation and does not depend on a reference point. Magnitude and direction remain the same regardless of location
• Equivalent Systems

  Concurrent force system can be reduced to a single force at the point of concurrency. Coplanar force system can be reduced to a single force and a moment or a single force acting at a point
• Sum of moments for both sides

  If point O is used as a reference point, the sum of moments for both the Left-Hand side and Right-Hand side will be zero
• Coplanar force systems are used in the sample problems and problem sets for a better foundation on equivalent systems
• Conditions for Static Equilibrium: The particle should be at rest (acceleration is zero)
• Newton’s First Law of Motion states that when the resultant of all forces acting on the particle is zero, the particle is in equilibrium
• ∑ FX = 0 (The net force along x is zero meaning no net movement to the left or right)
• ∑ FY = 0 (The net force along y is zero meaning no net movement up or down)
• We are not considering the summation of moment in the Particle Model of Analysis in this module
• Free Body Diagram

  Diagram of a body with all external forces acting on it, excluding internal forces
• Forces acting on a Body

  Applied Forces, Structural Members (Cable Force, Bar Force, Spring Force), Supports
• Springs

  Assumed to be linear elastic with force along the length, F=ks
• Bars


  Assumed to have negligible weight and force along the length
• Cables


  Assumed to be inextensible, flexible, and have tension only
• Mechanics can be defined as the branch of physical science concerned with the state of rest or motion of bodies that are subjected to the action of forces
• Mechanics explains the relationship between a force and its effect on a body
• Effects of forces when applied on bodies

  • Development of forces at points or surfaces of contact
  • Development of forces within the body
  • Deformation of the body
  • Change in state of motion
• Development of forces at points or surfaces of contact: Whenever two bodies come into contact, contact forces such as friction and normal forces are developed. These forces occur at the points where the bodies are in contact with each other
• Development of forces within the body: When an external force is applied, internal forces are developed to keep the particles inside the body intact
• Deformation of the body: When internal forces are not able to keep the particles in the body together, changes in shape and size or even breakage of the body may occur
• Change in state of motion: A body initially at rest may begin to move as force is applied. It could also be that upon application of force on a moving body, the body would come to a stop or change the direction of motion
• Divisions of the study of mechanics

  • Mechanics of rigid bodies
  • Mechanics of deformable bodies
  • Mechanics of fluids
• Rigid bodies are assumed to have no significant changes in their form after force is applied. While in real-life situations bodies deform when forces are applied and are never absolutely rigid, the deformation in rigid bodies is considered to have insignificant effects
• Mechanics of rigid bodies can be divided into

  • Statics
  • Dynamics