Moment of a Force-Vector Formulation Learning Goal: To use the vector cross product to calculate the moment produced by a force, or forces, about a specified point on a member. The moment of a force F about the moment axis passing through O and perpendicular to the plane containing O and F can be expressed using the vector cross product, Morx F. In a properly constructed Cartesian coordinate system, the vector cross product can be calculated using a matrix determinant: i j k M=rxF T 1% ry F. Fy F Notice that the order of the elements of the matrix determinant is important; switching rows 2 and 3 of the determinant would change the sign of the moment from positive to negative (or vice versa.) Part A - Moment due to a force specified by magnitude and endpoints As shown, a member is fixed at the origin, point O, and has an applied force F. the tension in the rope, applied at the free end, point B. The force has magnitude F = 180 N and is directed as shown. The dimensions are 21 = 0.450 m, x2 = 1.70 m, y₁ = 2.30 m, and z₁ = 1.10 m. What is the moment about the origin due to the applied force F? Express the individual components of the Cartesian vector to three significant figures, separated by commas. B You did not open hints for this part. ANSWER: Mo =[ i,j,k] N. m Part B - Moment due to a force specified as a Cartesian vector As shown, a member is fixed at the origin, point O, and has an applied force F, the tension in the rope, applied at the free end, point B. The force is given by F = 120 Ni - 125 Nj + 80 N k. The dimensions are x1 = 1.35 m, 91 =1.90 m, and 21 = 1.15 m. What is the moment about the origin due to the applied force F? Express the individual components of the Cartesian vector to three significant figures, separated by commas. You did not open hints for this part. ANSWER: Mo=[ i,j,k] N.m Part C Moment due to two forces As shown, a member is fixed at the origin, point O, and has two applied forces, F1 and F2, applied at the free end, point B. The forces are given by F₁ = 105 Ni- 110 Nj +55 N k and F₂ has magnitude 175 N and direction angles a = 149.0°, 8= 70.0°, and y = 67.4°. The dimensions are 21=1.30 m, y₁ = 1.85 m, and z₁ = 1.10 m. What is the moment about the origin due to the applied forces? Express the individual components of the Cartesian vector to three significant figures, separated by commas. You did not open hints for this part. ANSWER: Mo -| i,j,k] N.m x B