Physics (vector and kinematics)

• Vectors


  A physical quantity described by both magnitude and direction
• Scalars


  A physical quantity described by a single number
• Scalars


  • Length, Mass, Time, Pressure, Volume, Area, etc.
• Vectors


  • Can be represented by boldface italic symbols with an arrow above them
  • Drawn as a line with an arrowhead at its tip
• Five characteristics of vectors

  • Length – represents the vector's magnitude/modulus
  • Direction – represented by the vector's arrowhead
  • Sense (e.g. direction of torque)
  • Line of action (or carrier)
  • Origin (point of application on carrier)
• 1. Dimensional Vectors


  One dimensional Vectors can use a "+" or "–" sign to show direction
• 1. Dimensional Vectors


  • 𝑨 = (𝐴!, 𝐴")
  • 𝐴! = 𝑥# − 𝑥$ = 𝐴 cos 𝜃 (𝒙-component)
  • 𝐴" = 𝑦# − 𝑦$ = 𝐴 sin 𝜃 (𝒚-component)
• Magnitude (Modulus) of a Vector


  𝑨 = 𝐴 = √𝐴!^2 + 𝐴"^2
• Direction (Argument) of a Vector


  𝜃 = tan^-1(𝐴"/𝐴!)
• Unit Vector of a 2-D Vector


  • The normalized vector of a vector that has a magnitude of 1 and indicates only the direction of a vector
  • 𝑨 = 𝐴/𝐴
• Unit Vectors in Cartesian Plane


  • î (1,0)
  • ĵ (0,1)
• Types of Vectors


  • Equal Vectors
  • Parallel Vectors (Like)
  • Antiparallel Vectors (Unlike)
  • Negative Vectors (Opposite)
• Properties of Vectors


  • Commutative
  • Associative
  • Multiplication by a Scalar k
  • Negative of a Vector
  • Null/Zero Vector, φ
• Vector Addition


  1. Resultant, R

  2. Equilibrant, E

  3. Null vector or Zero Vector, φ
• 1. D Vector Addition


  • If the Vectors are in the same direction, add their magnitudes and retain their direction
  • If the Vectors are in opposite directions, subtract their magnitudes
• Graphical Methods for Vector Addition

  • Head-to-Tail Method
  • Parallelogram Method
• Adding Multiple Vectors


  Graphical Method
• Vector Subtraction (Graphical)


  • 𝑹 = 𝑨 − 𝑩
  • 𝑹 = 𝑨 + (−𝑩)
• Analytical Method for Vector Addition


  • Uses the laws and theorems of mathematics
  • Requires working diagram
• Triangle Method: Pythagorean Theorem


  1. 𝑹 = √𝐴^2 + 𝐵^2

  2. 𝜃 = tan^-1(𝐵/𝐴)
• Triangle Method: Laws of Sines and Cosines


  1. 𝑅^2 = 𝐴^2 + 𝐵^2 - 2𝐴𝐵 cos(180° - 𝛾)

  2. sin 𝜃 = 𝐵/𝑅
• Force Table


  Used to analyze forces acting on an object in equilibrium
• Lami's Theorem


  If the system is in equilibrium, Lami's Theorem may be applied: 𝐹_𝛼/sin 𝛼 = 𝐹_𝛽/sin 𝛽 = 𝐹_𝛾/sin 𝛾
• Vectors


  Quantities that have both magnitude and direction
• Magnitude


  The length or size of a vector
• Direction


  The orientation of a vector
• Addition of two vectors with opposite directions results in 𝑪 = 𝑨 + −𝑩
• Lami's Theorem


  𝐹%sin 𝛼 = 𝐹&sin 𝛽 = 𝐹'sin 𝛾
• Vectors in 3-Dimensions


  • Components: 𝑨 = 𝐴!, 𝐴", 𝐴%
  • Magnitude: 𝑨 = 𝐴!𝟐 + 𝐴"𝟐 + 𝐴%𝟐
  • Unit vectors: ̂=, ̂>, ?𝒌
  • Component form: 𝑨 = 𝐴! ̂= + 𝐴" ̂> + 𝐴% ?𝒌
• Unit Vectors in 3D Space


  • ̂2 = 1,0,0
  • ̂3 = 0,1,0
  • H𝒌 = 0,0,1
• Dot (Scalar or Inner) Product


  𝑨 G 𝑩 = 𝐴||𝐵 = 𝐴𝐵 cos 𝜃
• Dot Product of Cartesian Vectors


  𝑨 G 𝑩 = 𝐴!𝐵! + 𝐴"𝐵"
• Dot Product Activity


  • Given 𝑨 = 4 ̂2 + 3 ̂3 − 5H𝒌 and 𝑩 = 4 ̂2 − 2 ̂3 + 3H𝒌, determine (a) 𝑨 G 𝑩, (b) 𝑨 = 𝐴 and 𝑩 = 𝐵, (c) The angle, 𝜃 between 𝑨 and 𝑩
• Cross Product (Vector Product)


  𝑪 = 𝑨×𝑩 = 𝑨 𝑩 sin 𝜃 P𝒏
• Cross Product Activity


  • Given 𝑨 = 4 ̂2 + 3 ̂3 − 5H𝒌 and 𝑩 = 4 ̂2 − 2 ̂3 + 3H𝒌, evaluate 𝑨×𝑩 using the determinant and levi-civita symbol
• Classical Mechanics


  The study of the relationships among force, matter, and motion
• Kinematics


  • Describes motion
• Dynamics


  • Understanding why objects move in different ways
• Classical Mechanics continues to give a lot of applications successfully and gives an approximate description of how the world behaves
• Motion


  The phenomena in which a particle or an object changes its position