Ch 4. Moments/Equivalent Systems Multimedia Engineering Statics Moment2-D Scalar Moment3-D Scalar Moment3-D Vector Couples and Equiv. System Distributed Loads, Intro
 Chapter 1. Basics 2. Vectors 3. Forces 4. Moments 5. Rigid Bodies 6. Structures 7. Centroids/Inertia 8. Internal Loads 9. Friction 10. Work & Energy Appendix Basic Math Units Sections Search eBooks Dynamics Statics Mechanics Fluids Thermodynamics Math Author(s): Kurt Gramoll ©Kurt Gramoll

 STATICS - CASE STUDY SOLUTION Position Vectors Position Vectors Coordinate systems can be attached to any point, but generally there is one or two points that are more convenient than others. For this problem, the coordiante system is placed at far right edge of the pipes. The position vector from the right edge of the pipe to joint B is      rB = rBxi + rByj + rBzk          = 45i + 0j + 15k The position vector of point Q (location of applied force F) can be determined when the wrench is rotated at an angle of α as      rQ = rQxi + rQyj + rQzk          = 0i + 30 cosαj + (60 + 30 sinα)k From vector addition, the position vector from joint B to point Q is      rB + rBQ = rQ or      rBQ = rQ - rB             = -45i + 30 cosαj + (45 + 30 sinα)k Known Information Force Vector The system of two forces can be replaced by an equivalent system consisting of a force FR and couple MR applied at point O. The force FR is found by summing the individual forces,      F = Fxi + Fyj + Fzk           = 0i - 80 sinαj + 80 cosαk Moment Vector Moment Variation with Angle α The moment of the force about joint B is given by           MB = rBQ ×> F The cross product in determinant form is       Expanding the equation gives      MB = (rBQy Fz - rBQz Fy)i + (rBQz Fx - rBQx Fz)j                         + (rBQx Fy - rBQy Fx)k Substituting the appropriate values and simplifing to determine the moment at joint B gives      MB = (2,400 + 3,600 sinα)i                  + (3,600 cosα)j + (3,600 sinα)k If the magnitude of MB is plotted as a function of the angle α, then the moment is a maximum at α = 90°, with a value of 6,997 N-cm or about 70 N-m. This is approximately 62% greater than the moment at α = 0°.

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