figure: session.masteringengineering.com/problemAsset/1779286/7/MEng_SD_4-5_Intro.jpg
shown in the figure, the member is anchored at A and section AB lies in the x–y plane. The dimensions are x1 = 1.3 m, y1 = 1.8 m, and z1 = 1.5 m. The force applied at point C is F=[−235 i+125 j+125 k] N. Express the individual components to three significant figures, if necessary, separated by commas.
1) Calculating the moment about AB using the position vector AC. Using the position vector from A to C, calculate the moment about segment AB due to force F.
2) Calculating the moment about AB using the position vector ACUsing the position vector from A to C, calculate the moment about segment AB due to force F.
3) Calculating the moment about AB using the position vector BCUsing the position vector from B to C, calculate the moment about segment AB due to force F.
To calculate the moment about segment AB due to force F, we need to use the position vector AC and the cross product of AC and F.
1) Calculating the moment about AB using the position vector AC:
- First, write the position vector AC as AC = C - A, where C and A are the coordinates of points C and A, respectively.
- Given that the force F = [-235 i + 125 j + 125 k] N, the position vector AC = [x2 - x1, y2 - y1, z2 - z1] = [x2 - 0, y2 - 0, z2 - 0] = [x2, y2, z2].
- Substitute the given values (x1 = 1.3 m, y1 = 1.8 m, and z1 = 1.5 m) to get AC = [x2, y2, z2] = [1.3 + x2, 1.8 + y2, 1.5 + z2] (where x2, y2, and z2 represent the coordinates of point C).
- Take the cross product of AC and F by using the formula: Moment about AB (M_AB) = AC x F = [AC_y * F_z - AC_z * F_y, AC_z * F_x - AC_x * F_z, AC_x * F_y - AC_y * F_x].
- Substitute the values of AC = [1.3 + x2, 1.8 + y2, 1.5 + z2] and F = [-235, 125, 125] into the formula to calculate the moment about AB. Round the result to three significant figures if necessary.
2) Calculating the moment about AB using the position vector AC:
- This step is the same as step 1. You can follow the same procedure mentioned above to calculate the moment about AB using the position vector AC.
3) Calculating the moment about AB using the position vector BC:
- Similar to step 1, write the position vector BC as BC = C - B, where B and C are the coordinates of points B and C, respectively.
- Given that the force F = [-235 i + 125 j + 125 k] N, the position vector BC = [x2 - x3, y2 - y3, z2 - z3] = [x2 - 0, y2 - 0, z2 - 0] = [x2, y2, z2].
- Substitute the values (x3 = 0, y3 = 0, and z3 = 0) to get BC = [x2, y2, z2].
- Take the cross product of BC and F by using the formula: Moment about AB (M_AB) = BC x F = [BC_y * F_z - BC_z * F_y, BC_z * F_x - BC_x * F_z, BC_x * F_y - BC_y * F_x].
- Substitute the values of BC = [x2, y2, z2] and F = [-235, 125, 125] into the formula to calculate the moment about AB. Round the result to three significant figures if necessary.