University:
Michigan State University
Course:
ERG 160 | Success in Science, Technology, Engineering, and Mathematics
Academic year:

2021
Views:

101

Pages:

2
Author:

Neeleworvorhx4

COSMOS: Complete Online Solutions Manual Organization System Chapter 3, Solution 139. First, choose a coordinate system so that the xy plane coincides with the given plane. Also, position the coordinate system so that the line of action of the wrench passes through the origin as shown in Figure a. Since the orientation of the plane and the components (R, M) of the wrench are known, it follows that the scalar components of R and M are known relative to the shown coordinate system. A force system to be shown as equivalent is illustrated in Figure b. Let A be the force passing through the given point P and B be the force that lies in the given plane. Let b be the x-axis intercept of B. The known components of the wrench can be expressed as R = Rxi + Ry j + Rzk M = M xi + M y j + M zk and while the unknown forces A and B can be expressed as A = Axi + Ay j + Azk and B = Bxi + Bzk Since the position vector of point P is given, it follows that the scalar components (x, y, z) of the position vector rP are also known. Then, for equivalence of the two systems ΣFx : Rx = Ax + Bx ΣFy : Ry = Ay (1) (2) ΣFz : Rz = Az + Bz (3) ΣM x : M x = yAz − zAy (4) ΣM y : M y = zAx − xAz − bBz (5) ΣM z : M z = xAy − yAx (6) continued . COSMOS: Complete Online Solutions Manual Organization System ( ) Based on the above six independent equations for the six unknowns Ax , Ay , Az , Bx , Bz , b , there exists a unique solution for A and B. Ay = Ry From Equation (2) . Equation (6) 1 Ax =   xRy − M z  y Equation (1) 1 Bx = Rx −   xRy − M z  y Equation (4) 1 Az =   M x + zRy  y Equation (3) 1 Bz = Rz −   M x + zRy  y Equation (5) b= ( ) ( ( ) ) ( ( xM x + yM y + zM z ) ( M x − yRz + zRy ) )