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Position Vectors

Position Vectors. The position vector r is defined as a fixed vector which locates a point in space relative to another point. r = x i + y j + z k. z. zk. P(x,y,z). r AB A = from point B = to point r OA =r A. r. yj. y. O. xi. x. Position Vectors. r A + r = r B

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Position Vectors

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  1. Position Vectors The position vector r is defined as a fixed vector which locates a point in space relative to another point. r = xi + yj + zk z zk P(x,y,z) rAB A = from point B = to point rOA=rA r yj y O xi x

  2. Position Vectors rA+ r = rB r= rB- rA=(xBi+yBj+zBk)-(xAi+yAj+zAk) r= (xB-xA)i+(yB-yA)j+(zB-zA)k z B(xB,yB,zB) r A(xA,yA,zA) rB y rA O x

  3. Moment of a force about a point A measure of the tendency of the force to cause a body to rotate about the point or axis. The moment of a force about a point O; MO = r x F MO r O A F

  4. Moment of a force about a point MO = rFsinθ = F (rsinθ) = Fd MO d r θ r O A F Mx My Mz

  5. Moment of a force about a line In order to find the projected component of the moment about an axis. a Mo Ma F r a’

  6. Moment of a force about a line as a Cartesian vector;

  7. Example The magnitude of the force shown in the figure is 140 N. Determine; • The moment of F about point D. • The moment of F about a line joining D and A. • The angle btw. BG & DG. z G(2,0,3) A(0,6,0) y D(4,0,2) E(2,0,2) F C B(4,6,0) x

  8. Moment of a Couple A couple consists of two forces of equal magnitude having parallel lines of action but opposite directions. F d F Moment of couple

  9. Example Add couples whose forces act along diagonal of the rectangular prism. z 10 N 5 N 5 N y 5 10 10 N x

  10. Equivalent Force Systems • Translation of a force to a parallel position • Resultant of a force system • Distributed force systems

  11. Translation of a force to a parallel position A force can be moved to any parallel position provided that a couple moment of the correct orientation and size is simultaneously provided. z F=6i+3j+6k Replace this force by an equivalent force system at point P? A(2,1,10) P(6,10,12) y x

  12. Resultant of a force system Any force system can be replaced by any point by a single force and a single couple moment. MC:couple moments

  13. Example F2 • If θ=60°, F1=15N, F2=80N, F3=50N compute the equivalent resultant force. • Determine the intercepts of this resultant force with x and y axes. • Find the magnitude and direction of force F3 such that the simplest resultant is only A COUPLE MOMENT at point O F1=15N, F2=80N. • If F1=15N, F2=80N, F3=50N find the direction of force F3 such that the simplest resultant is only a force at point O. • Determine the magnitude of force F1 and the magnitude and direction of F3 such that the equivalent force and the couple moment at point O are ZERO. F3 2m θ 3m F1 4m x O

  14. Distributed Force Systems • Volume Distribution If the force is distributed over the volume of a body. (e.g. Force of gravitational attraction. N/m3) • Area Distribution If the force is distributed over an area. (e.g. Hydraulic pressure of fluid. N/m2) • Line Distribution If the force is distributed along a line. (e.g. Vertical load supported by a suspended cable. N/m)

  15. Center of a Geometry • Center of Mass • Center of Gravity • Centroid

  16. Line Force Distribution w: load intensity (N/m) y w(x) x FR x

  17. Example y w(x) FR=? 200 N/m x 9m

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