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Statics-61110

An-Najah National University College of Engineering. Statics-61110. Chapter [2]. Dr. Sameer Shadeed. Force Systems. Chapter Objectives. Students will be able to: Resolve a 2D and 3D vector into components Work with 2D and 3D vectors using Cartesian vector notations

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Statics-61110

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  1. An-Najah National University College of Engineering Statics-61110 Chapter [2] Dr. Sameer Shadeed Force Systems Dr. Sameer Shadeed

  2. Chapter Objectives • Students will be able to: • Resolve a 2D and 3D vector into components • Work with 2D and 3D vectors using Cartesian vector notations • Estimate the resultant of forces and couples in 2D and 3D Dr. Sameer Shadeed

  3. Applications • There are four concurrent cable forces acting on the bracket • How do you determine the resultant force acting on the bracket ? Dr. Sameer Shadeed

  4. Applications • Given the forces in the cables, how will you determine the resultant force acting at D, the top of the tower? Dr. Sameer Shadeed

  5. Scalars and Vectors • Scalars: A mathematical quantity possessing magnitude only (e.g. area, volume, mass, energy) • Vectors: A mathematical quantity possessing magnitude and direction (e.g. forces, velocity, displacement) Dr. Sameer Shadeed

  6. Types of Vectors • A Free vector:is a vector whose action is not confined to or associated with a unique line in space. • A Sliding vector:is a vector which has a unique line of action in space but not a unique point of application. • A Fixed vector: is a vector for which a unique point of application is specified and thus cannot be moved without modifying the conditions of the problem. Dr. Sameer Shadeed

  7. V V V -V Working with Vectors • For two vectors to be equal they must have the same: 1. Magnitude 2. Direction • But they do not need to have the same point of application • A negative vectorof a given vectorhas same magnitude butopposite direction • V and -V are equal and opposite V + (-V) = 0 Dr. Sameer Shadeed

  8. Vector Operations • Product of a scalar and a vector • V + V + V = 3V (the number 3 is a scalar) • This is a vector in the same direction as V but 3 times as long • (+n)V= vector same direction as V, n times as long • (-n)V= vector opposite direction as V, n times as long Dr. Sameer Shadeed

  9. V1 R1 R V2 V3 Vector Addition • The sum of two vectors can be obtained by attaching the two vectors to the same point and constructing a parallelogram (Parallelogram law) • As vector:V = V1+ V2 • As scalar:V ≠ V1+ V2 • Addition of vectors is commutative: V1+ V2 = V2 + V1 • The sum of three vector (Parallelogram law) • R1 = V1 + V2 • R = R1 + V3 = V1 + V2 + V3 • Vector addition is associative: V1 + V2 + V3 = (V1 + V2 ) + V3 = V1 + (V2 + V3 ) Dr. Sameer Shadeed

  10. Vector Subtraction • Vector Subtraction:is the addition of the corresponding negative vector Dr. Sameer Shadeed

  11. Two-Dimensional Force System • Rectangular Components:is the most common two-dimensional resolution of a force vector • where Fxand Fy are the vector components of F in the x- and y-directions • In terms of the unit vectors i and j, Fx = Fxi and Fy = Fyj • where the scalers Fxand Fy are the x and y scaler componentsof the vector F Dr. Sameer Shadeed

  12. Two-Dimensional Force System • The scalar components can be positive or negative, depending on the quadrantinto which F points • The magnitude and direction of F is expressed by: Dr. Sameer Shadeed

  13. Determining the Components of a Force • Dimensions are not always given in horizontal and vertical directions • Angles need not be measured counterclockwise from the x-axis • The origin of coordinates need not be on the line of action of a force • Therefore, it is essential that we be able to determine the correct components of a force no matter how the axes are originated or how angles are measured Dr. Sameer Shadeed

  14. Resultant of Two Concurrent Forces or • From which we canconclude that: • The term means “the algebric sum of x-scalar components” Dr. Sameer Shadeed

  15. Example 1 Dr. Sameer Shadeed

  16. Example 1 (Solution) Dr. Sameer Shadeed

  17. Example 1 (Solution) Dr. Sameer Shadeed

  18. Example 2 Dr. Sameer Shadeed

  19. Example 2 (Solution) Dr. Sameer Shadeed

  20. Example 2 (Solution) Dr. Sameer Shadeed

  21. Example 3 Dr. Sameer Shadeed

  22. Example 3 (Solution) Dr. Sameer Shadeed

  23. Example 4 Dr. Sameer Shadeed

  24. Example 4 (Solution) Dr. Sameer Shadeed

  25. Example 4 (Solution) Dr. Sameer Shadeed

  26. Example 4 (Solution) Dr. Sameer Shadeed

  27. Moment (Moment about a Point) • Moment:is the measure of the tendency of the force to make a rigid body rotate about a point or fixed axis • The magnitudeof the momentM is proportional both to the magnitude of the forceF and the moment arm d • The basic units of the moment in SIunits are newton-meters (N.m), and in the U.S. Custamary system are pound-feet (Ib.ft) Dr. Sameer Shadeed

  28. Moment (Moment about a Point) • The moment is a vectorMperpendicular to the plane of the body • The sense ofMdepends on the direction in which Ftends to rotate the body • The right hand rule is used to identify this sense • Moment direction:a plus sign (+) for counterclockwise moments and a minus sign (−) for clockwise moments, or vice versa Dr. Sameer Shadeed

  29. Moment (The Cross Product) • The moment of F about point A may be represented by the cross-product expression • where r is a posision vector which runs from the momentreference point A to any point on the line of action of F • The order of r × F of the vectors must be maintainedbecause F × r would produce a vector with a sense opposite to that of M F × r = -M Dr. Sameer Shadeed

  30. Moment (The Cross Product) • The magnitude of the moment M is gevin by: M = F (r sin α) • From the shown diagram: r sin α = d M = Fd Dr. Sameer Shadeed

  31. Moment (Varignon’s Theorem) • The moment of a force about any point is equal to the sum of the moments of the components of the force about the same point Mo = r × R R = P + Q Mo = r × P + r × Q Mo = Rd = Pp − Qq Dr. Sameer Shadeed

  32. Example 5 Dr. Sameer Shadeed

  33. Example 5 (Solution) Dr. Sameer Shadeed

  34. Example 5 (Solution) Dr. Sameer Shadeed

  35. Example 5 (Solution) Dr. Sameer Shadeed

  36. Example 6 • Given: • Find: • Moment of the 100 N force about O • Magnitude of a horizontal forceapplied at A which create the same moment about O • The smallest forceapplied at A which creates the same moment about O Dr. Sameer Shadeed

  37. Example 6 (Solution) 1. Moment of the 100 N force about O Mo = Fd = 100(5 cos60o) = 250 N.m 2. Magnitude of a horizontal force applied at A which create the same moment about O Mo = 250 = PL L = 5 sin 60o = 4.33 m P = 250/4.33 = 57.7 N Dr. Sameer Shadeed

  38. Example 6 (Solution) 3. The smallest force applied at A which creates the same moment about O • Pis smallest when d in M = Fd is a maximum • This occurs when P is perpendicular to the lever Mo = Pd = P × 5 = 250 P = 250/5 = 50 N Dr. Sameer Shadeed

  39. Couple • Couple:is the moment produced by two equal, opposite, and non-collinear forces • The combined moment of the two forcesabout axis normal to their plane and passing through any point such as O in their plane is the coupleM M = F(a + d) - Fa M = Fd Dr. Sameer Shadeed

  40. Couple (Vector Algebra Method) • With the cross-product notation, the combined moment about point O of the forces forming the couple is M = rA × F + rB × (–F) = (rA – rB) × F • where rA and rB are position vectorswhich run from point O to arbitrary points A and B on the lines of action of F and –F • Because rA – rB = r, M can be express as: M = r × F Dr. Sameer Shadeed

  41. The Sense of a Couple Vector • The sense of a couple vectorM can represent as clockwise or counterclockwise by one of the shown conventions Dr. Sameer Shadeed

  42. Equivalent Couple • Changing the values of F and ddoes not change a given couple as long the productFd remains the same • A couple is not affectedif the forces act ina different but parallel plane Dr. Sameer Shadeed

  43. Force – Couple System • The combination of the force and couplein the shown figure is refered to as a force – couple system Dr. Sameer Shadeed

  44. Example 7 Dr. Sameer Shadeed

  45. Example 7 (Solution) Dr. Sameer Shadeed

  46. Example 8 Dr. Sameer Shadeed

  47. Example 8 (Solution) Dr. Sameer Shadeed

  48. Resultants • The Resultantof a system of forces is thesimplest force combinationwhich can replace the original forces withoutaltering the external effect on the rigid body to which theforces are applied Dr. Sameer Shadeed

  49. Resultants • Algebraic Method Dr. Sameer Shadeed

  50. Resultants • Algebraic Method Dr. Sameer Shadeed

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