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2 Force Systems

2 Force Systems. Force, Moment, Couple and Resultants. Force Definition. Force is an action that tends to cause acceleration of an object. [Dynamics] The SI unit of force magnitude is the newton ( N ).

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2 Force Systems

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  1. 2 Force Systems Force, Moment, Couple and Resultants

  2. Force Definition Force is an action that tends to cause acceleration of an object. [Dynamics] The SI unit of force magnitude is the newton(N). One Newton is equivalent to one kilogram-meter per second squared (kg·m/s2 or kg·m·s – 2) • Force is a vector quantity (why?) • Force is the action of one body on another. [Statics] Examples of mechanical force include the thrust of a rocket engine, the impetus that causes a car to speed up when you step on the accelerator, and the pull of gravity on your body. • Force can result from the action of electric fields, magnetic fields, and various other phenomena. 3

  3. FORCE SYSTEMS Force is a vector Line of action is a straight line colinear with the force Force System: concurrentif the lines of action intersect at a point parallel if the lines of action are parallel y coplanar if the lines of action lie on the same plane x

  4. Writing Convention same symbol In this course, you have to write in this convention. Recommended Style

  5. FORCE SYSTEMS Vector (2D&3D) Basic Concept 3-D Force Systems 2-D Force Systems Moment Couple Resultants Moment Couple Resultants

  6. Magnitude: orV : Direction Vector : or V Free Vectors:associated with “Magnitude” and “Direction” Representation parallelogram triangle +

  7. OperationAddition #5 Vector Commutative

  8. OperationAddition #6 Vector Associative

  9. OperationScalar Multiplication #2 wrt = with respect to associative distributive wrt scalar addition distributive wrt vector addition

  10. ComponentResolution of a Vector Vector A vector may be resolved into two components.

  11. Basic relations of Triangle (C/6, law of cosine, sine) Law of cosine a Law of sine c b

  12. (Law of sine) (Law of sine) 2 Hint a 1 b b q c b Given V,  and , find Law of cosine Law of sine

  13. Vector Component and Projection = b : vector components of (along axis a and b) a : projections of (onto axis a and b) b special case: projection vectors are orthogonal to each other a : orthogonal projections & vector components

  14. Rectangular Components vector component = vector projection • Most commonly used y q x

  15. Fx=? Fy=? y F p-b x minus (b>90) b x  y b-q y x b   y x

  16. y T  x EXAMPLE 2-1 Given the magnitude of the tension in the cable, T = 9 kN, express T in terms of unit vector i and j 3 S.F. Correct? ANS kN

  17. P = 90 N P = 90 N We are using robot arm to put the cylindrical part into a hole. Determine the components of the force which the cylindrical part exerts on the robot along axes (a) parallel and perpendicular to arm AB (b) parallel and perpendicular to arm BC par per Defining direction per par arm AB ANS arm BC ANS

  18. R T=600 N (6cm) P R T 2/2 Combine the two forces P and T, which act on the fixed structure at B, into a single equivalent force R P=800 N (8cm) Graphics Geometric Vector Component (Algebraic) Correct? Point of application is B

  19. Example Hibbeler Ex 2-1 #1 Determine the magnitude and direction of the resultant force. Two forces is not acting at the same point. Geometric

  20. Vector Component (Algebraic) Geometric Good? (get full score?) - more explanation - mark answer - 5S.F. Then 3S.F.

  21. Good Answer Sheet Geometric O a

  22. Point of Application

  23. Example Hibbeler Ex 2-6 #1 Vector

  24. Example Hibbeler Ex 2-6 #2 Vector

  25. y F2y F1y Ry o x F1x F2x Rx • Reference axis (very very important) • Many problems do not come with ref. axis. • Assignment based on convenience/experience Originally pass through O • Vector summation (addition) • Three ways to be mastered 1. Graphically 2. Geometrically 3. Vector component (algebraically) The calculations do not reveal the point of application of the resultant force. In case where forces do not apply at the same point of application, you have to find it too!

  26. Recommended Problem 2/9, H2-17, 2/12, 2/26, H2-28

  27. Three Dimensional Coordinate System y Real-life Coordinate System is 3D. Introduce rule for defining the 3rd axis - “right-hand rule”: x-y-z - for consistency in math calculation (cross vector) z x How does 2D differs from 3D? y 2D z x

  28. Rectangular Components (3D) - cos(x), cos(y), cos(z) : “directional cosines” of is a unit vector in the direction of & component projection z y x (maybe +/-) - cos2(x)+cos2(y)+cos2(z) = 1 - If you known the magnitude and all directional cosines, you can write force in the form of directional cosine Method

  29. ExampleHibbeler Ex 2-8 Find Cartesian components of F z x y

  30. z B y A x Given the cable tension T = 2 kN. Write the vector expression of 1) directional cosine method Real directional cosine directionl cosine = -0.92 B A

  31. B z z B B y y A A A x x ANS Thus, B A

  32. Directional Cosines by Graphics cos2(x)+cos2(y)+cos2(z) = 1

  33. - Usually, the direction of force is not given using the directional cosines. Need some calculation. - Two examples (a) Two points on the line of action of force is given (F also given). z B (x2, y2, z2) Position vector Two-Point Method A (x1, y1, z1) y x

  34. z 0.5 2) 2-point construction y 0.4 B A 0.3 1.2 x kN Ans

  35. where = unit vector from B to A Thus Write vector expression of . Also determine angle x, y, z, of T with respect to positive x, y and z axes Consider: T as force of tension acting on the bar ANS

  36. Example Hibbeler Ex 2-9 #1 Vector Determine the magnitude and the coordinate direction angles of the resultant force acting on the ring

  37. Example Hibbeler Ex 2-9 #2 Vector

  38. Example Hibbeler Ex 2-11 #1 Vector Specify the coordinate direction angles of F2 so that the resultant FR acts along the positive y axis and has a magnitude of 800 N.

  39. Example Hibbeler Ex 2-11 #2 Vector

  40. Example Hibbeler Ex 2-11 #3 Vector

  41. Example Hibbeler Ex 2-15 #1 Force The roof is supported by cables as shown. If the cables exert forces FAB = 100 N and FAC = 120 N on the wall hook at A as shown, determine the magnitude of the resultant force acting at A.

  42. Example Hibbeler Ex 2-15 #2 Force

  43. Example Hibbeler Ex 2-15 #3 Force

  44. Example Hibbeler Ex 2-15 #4 Force

  45. (b) Two Anglesorienting the line of action of forceare given (, ) Othorgonal projection Method Resolve into two components at a time z y Fz= F sin() Fxy = F cos()   Fx=Fxy cos() = F cos() cos() Fy= Fxy sin() = F cos() sin() x

  46. z Fz F 50o Fy Fx 65o y x Fxy Ans

  47. B z TAB 15o C A T y x x TZ Ans

  48. 2/110 A force F is applied to the surface of the sphere as shown. The 2 angles (zeta, phi) locate Point P, and point M is the midpoint of ON. Express F in vector form, using the given x-,y- z-coordinates.

  49. Recommended Problems • 3D Rectangular Component: • 2/99 2/100 2/107 2/110

  50. Operation Products Vector • Dot Products • Cross Products • Mixed Triple Products

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