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Chapter I Vectors and Scalars

Chapter I Vectors and Scalars. AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering. AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering.

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Chapter I Vectors and Scalars

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  1. Chapter I Vectors and Scalars AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  2. AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  3. Fundamental Principles • Preconditions to deal with problems in mechanics. • Basic concepts used in mechanics: • space, time, mass, force, particle, rigid body AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  4. Fundamental Principles Cont… • Basic concepts used in mechanics: • space, time, mass, force, particle, rigid body • coordinates - position of a point P (x, y, z) • measured from a certain point of reference AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  5. Fundamental Principles Cont… • Basic concepts used in mechanics: • space, time, mass, force, particle, rigid body • time of an event taking place, determination of velocity and acceleration AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  6. Fundamental Principles Cont… • Basic concepts used in mechanics: • space, time, mass, force, particle, rigid body • mass of a body [kg, to] • action of weight, behavior under the action of an external force AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  7. Fundamental Principles Cont… • Basic concepts used in mechanics: • space, time, mass, force, particle, rigid body • magnitude, direction, point of application • e.g. action on a rigid body, action of one body onto another AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  8. Fundamental Principles Cont… • Basic concepts used in mechanics: • space, time, mass, force, particle, rigid body • infinitesimal small piece of a body, single point in space AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  9. Fundamental Principles Cont… • Basic concepts used in mechanics: • space, time, mass, force, particle, rigid body • body consisting of a non-deformable material • (no displacement under the action of forces) AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  10. Fundamental Principles Cont… Newton’s Laws Sir Isaac Newton (1642-1727) • 1st Law: A particle remains at rest or continues to move with constant velocity if the resultant force acting on it is zero. AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  11. Fundamental Principles Cont… Newton’s Laws Sir Isaac Newton (1642-1727) • 2nd Law: The acceleration of a particle proportional to the resultant force acting on it (magnitude and direction). F = ma m = mass of particle a = acceleration AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  12. Fundamental Principles Cont… Newton’s Laws Sir Isaac Newton (1642-1727) • 3rd Law: The forces of action and reaction between bodies in contact are equal in magnitude, opposite in direction and collinear (same line of action). AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  13. Fundamental Principles Cont… Newton’s Laws • Law of Gravitation • Two particles of mass m1 and m2 are mutually attracted with equal and opposite forces F and F’ of magnitude F. G = constant of gravitation AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  14. Fundamental Principles Cont… Newton’s Laws • Law of Gravitation • Weight = Gravitational Force acting on a body • (attraction between earth and body) W = m⋅g g = acceleration of gravity = 9.81 m/s2 AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  15. Fundamental Principles Cont… Newton’s Laws • Law of Gravitation • Weight = Gravitational Force acting on a body (attraction between earth and body) W[N] = m[Kg]⋅g[m/s2] g = 9.81 m/s2 AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  16. Units • International System of Units (SI units) Mass m [to, kg] Force F [kN, N] Time t [s] Length L [m, cm, mm] AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  17. AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  18. Scalars and VectorsDefinition and properties • Scalars: quantities described by their magnitude alone e.g. time, volume, area, density, distance, energy mass AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  19. Scalars and VectorsDefinition and properties Vectors: quantities described by their magnitude and direction e.g. displacement, velocity, force, acceleration, momentum AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  20. Graphical representation of a Vector • line segment of certain length (magnitude) and orientation (θ) • arrowhead indicating direction AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  21. Symbolic representation of a Vector • magnitude, length of vector: ║V║, |V| or V e.g. in scalar equations • vectorquantities respecting the orientation: V, V e.g. mathematical vector operations AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  22. Symbolic representation of a Vector • magnitude, length of vector: ║V║, |V| or V e.g. in scalar equations • vectorquantities respecting the orientation: V, V e.g. mathematical vector operations AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  23. Representation of Vectors • Algebraically a vector is represented by its components along the three dimensions. AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  24. Representation of Vectors Cont… AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  25. Representation of Vectors Cont… AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  26. Representation of Vectors Cont… AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  27. Representation of Vectors Cont… AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  28. Representation of Vectors Cont… AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  29. Orientation of Vectors • collinear- same line of action • coplanar - located in the same plane • concurrent - passing through a common point AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  30. Classification of Vectors • Free Vector • Sliding Vector • Fixed Vector AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  31. Classification of Vectors Cont… 1. Free Vector: action in space not associated with a unique line e.g. uniform displacement of a body AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  32. Classification of Vectors Cont… 1. Free Vector: action in space not associated with a unique line e.g. uniform displacement of a body AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  33. Classification of Vectors Cont… 1. Free Vector: action in space not associated with a unique line e.g. uniform displacement of a body AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  34. Classification of Vectors Cont… 2. Sliding Vector: action in space described by a unique line e.g. action of force on rigid body AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  35. Classification of Vectors Cont… 2. Sliding Vector: action in space described by a unique line e.g. action of force on rigid body AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  36. Classification of Vectors Cont… 2. Sliding Vector: action in space described by a unique line e.g. action of force on rigid body AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  37. Classification of Vectors Cont… 3. Fixed Vector: action in space described by a unique point e.g. action of force on non rigid body AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  38. Classification of Vectors Cont… 3. Fixed Vector: action in space described by a unique point e.g. action of force on non rigid body AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  39. AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  40. AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  41. Vector Addition – graphical method The parallelogram law – resultant force • Two forces maybe replaced by a single force (resultant) obtained by drawing the diagonal of the parallelogram having sides equal to the given forces. AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  42. Vector Addition – graphical method Cont… The parallelogram law – resultant force • Two forces maybe replaced by a single force (resultant) obtained by drawing the diagonal of the parallelogram having sides equal to the given forces. AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  43. Vector Addition – graphical method Cont… The parallelogram law – resultant force • Two forces maybe replaced by a single force (resultant) obtained by drawing the diagonal of the parallelogram having sides equal to the given forces. AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  44. Vector Addition – graphical method Cont… The parallelogram law – resultant force • Two forces maybe replaced by a single force (resultant) obtained by drawing the diagonal of the parallelogram having sides equal to the given forces. AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  45. Vector Addition – graphical method Cont… The parallelogram law – resultant force • Two forces maybe replaced by a single force (resultant) obtained by drawing the diagonal of the parallelogram having sides equal to the given forces. AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  46. Vector Addition – graphical method Cont… The parallelogram law – resultant force • Two forces maybe replaced by a single force (resultant) obtained by drawing the diagonal of the parallelogram having sides equal to the given forces. AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  47. Vector Addition – graphical method Cont… The parallelogram law – resultant force • Two forces maybe replaced by a single force (resultant) obtained by drawing the diagonal of the parallelogram having sides equal to the given forces. AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  48. Vector Addition – graphical method Cont… The parallelogram law – resultant force • Two forces maybe replaced by a single force (resultant) obtained by drawing the diagonal of the parallelogram having sides equal to the given forces. AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  49. Vector Addition – graphical method Cont… The triangle rule (from parallelogram law) AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

  50. Vector Addition – Analytic Method • Trigonometric rules • applying sine and cosine rules AAIT Engineering Mechanics Statics Department of Tewodros N. Civil Engineering

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