The particle M in Figure 1 is suspended by using the two (soft) ropes MA and MB, which are made of the same material and have equal diameters. The x and y coordinates of points A and B, measured in metres, are (-1,2) and (1,2), respectively. In the design, point M is required to be located at a grid point (blue circle) on the grid that is uniform and defined over the region 1.2 m x 1.2 m, 0.5 m y 1.5 m with the elements being squares of side lengthx=y=0.1m. LetF1andF2be the internal force vector in MA and MB, respectively. Your tasks are to find possible grid points for particle M to be suspended, and which grid points the ropes can have a minimum diameter. It is noted that one can adapt the lengths of the ropes to new locations of point M, the ropes are soft (i.e. subjected to only tensile forces), and the lower the internal force magnitude the smaller the diameter of the rope will be.

For each location of point M, the conditions for point M to be in equilibrium (Figure 2) are

• the total force vector acting on point M is zero

F1+F2+W=0 equation (1)

whereW=(0,-W) is the weight (W is the magnitude of W).

• F1=(F1x,F1y)is a tensile force lying on MA.
• F2=(F2x,F2y)is a tensile force lying on MB

Solve the directions ofF1andF2are chosen from M to A and from M to B, respectively (i.e. tensile forces), and the vector equation (1) is converted into the following algebraic equation set:

F1x+F2x=0 equation (2)

F1y+F2yW=0 equation (3)

or

MAxAxMF1+MBxBxMF2=0 equation (4)

MAyAyMF1+yBMByMF2=W equation (5)

where

MA=(xAxM)2+(yAyM)2&MB=(xBxM)2+(yByM)2

are the lengths of the ropes MA and MB, respectively. The unknowns to be found in equations (4) & (5) are the two force magnitudesF1andF2(2 equations and 2 unknowns). If the negative force magnitude is obtained, the corresponding rope is not in tension and thus the equilibrium conditions are not satisfied.

W= 5

Perform hand calculations to solve:

1. Establish the algebraic equation system based on (4)-(5) for each following case: M(1, 0.5) m and M(0, 0.5) m.

2. Express the two systems of algebraic equations in Requirement 1 in matrix form.

3. Find the internal forces for both cases in Requirement 1. Report the answers rounded to three significant figures.