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19 Specifying cable lengths to position a construction hoist (Sections 5.3 and 2.9 ). A uniform beam B is attached to a roof N by two variable-length cables (A and C). Cable A attaches to the roof at point N of N and to the beam at point B of B. Cable C attaches to the roof at point NC of N and to the beam at point BC of B. Right-handed orthogonal unit vectors nx,ny,nz and bx,by,bz are fixed in N and B, with: - nx horizontally-right from No to NC - ny vertically-downward - n2=b2 inward normal to the vertical plane containing points N0,B0,BC,NC. - b^x directed from Bo to BC. Although planar geometry and the Pythagorean theorem relate the cables' lengths to x,y,, these techniques are less effective than vector methods for complicated or 3D geometry. (a) Draw bx,by,bz, complete the previous bRn rotation table. Form NB (B's angular velocity in N). (b) Using only the picture, 4 complete the following blanks in terms of x,y,LB,LN. Result: Position from No to BoNorBo=nx+ny Position from NC to BCNCrBC=(nx+ny+bx (c) Calculate each cable's length with a dot-products. Use the following distance formulas (and the rotation matrix) to efficiently relate LA2 and LC2 to x,y,,LN,LB. Result: NrB0NrBoCrBCNCrBC=LA2=x2+y2=LC2= (d) Knowing x=2m, determine physically meaning values of y and (2 +significant digits). Result: (consider using a computer to solve the nonlinear algebraic equations). y=1.814m=27.674 (e) Implicitly differentiate LA2 and LC2 to efficiently relate LA and LC to x,y,. Result: (for this problem only, regard LA and LC as non-constant, i.e., LA(t) and LC(t) are functions of time). 2LALA=2(xx+yy)