Draw the shear force diagram. Calculate the bending moments and draw the bending moment diagram. Calculate the position of the centroid from MO of the lamina shown in FIGURE 2 below.

Step 1: Divide the lamina into three rectangular sections and determine their dimensions, areas, and individual centroid coordinates relative to point O (the origin).

• Rectangle 1 (Left):

  • Width ($w_1$) = $200 \text{ mm}$
  • Height ($h_1$) = $400 \text{ mm}$
  • Area ($A_1$) = $w_1 \times h_1 = 200 mm \times 400 mm = 80000 mm^2$
  • Centroid $x_1 = \frac{w_1}{2} = \frac{200}{2} = 100 \text{ mm}$
  • Centroid $y_1 = \frac{h_1}{2} = \frac{400}{2} = 200 \text{ mm}$

• Rectangle 2 (Middle):

  • Width ($w_2$) = $250 \text{ mm}$
  • Height ($h_2$) = $90 \text{ mm}$
  • Area ($A_2$) = $w_2 \times h_2 = 250 mm \times 90 mm = 22500 mm^2$
  • Centroid $x_2 = w_1 + \frac{w_2}{2} = 200 + \frac{250}{2} = 200 + 125 = 325 \text{ mm}$
  • Centroid $y_2 = \frac{h_2}{2} = \frac{90}{2} = 45 \text{ mm}$

• Rectangle 3 (Right):

  • Width ($w_3$) = $300 \text{ mm}$
  • Height ($h_3$) = $220 \text{ mm}$
  • Area ($A_3$) = $w_3 \times h_3 = 300 mm \times 220 mm = 66000 mm^2$
  • Centroid $x_3 = w_1 + w_2 + \frac{w_3}{2} = 200 + 250 + \frac{300}{2} = 450 + 150 = 600 \text{ mm}$
  • Centroid $y_3 = \frac{h_3}{2} = \frac{220}{2} = 110 \text{ mm}$

Step 2: Calculate the total area of the lamina. $A_{total} = A_1 + A_2 + A_3$ $A_{total} = 80000 mm^2 + 22500 mm^2 + 66000 mm^2 = 168500 mm^2$

Step 3: Calculate the sum of the moments of the areas about the y-axis (for $\bar{x}$). $\sum A_i x_i = A_1 x_1 + A_2 x_2 + A_3 x_3$ $\sum A_i x_i = (80000 mm^2 \times 100 mm) + (22500 mm^2 \times 325 mm) + (66000 mm^2 \times 600 mm)$ $\sum A_i x_i = 8000000 mm^3 + 7312500 mm^3 + 39600000 mm^3 = 54912500 mm^3$

Step 4: Calculate the sum of the moments of the areas about the x-axis (for $\bar{y}$). $\sum A_i y_i = A_1 y_1 + A_2 y_2 + A_3 y_3$ $\sum A_i y_i = (80000 mm^2 \times 200 mm) + (22500 mm^2 \times 45 mm) + (66000 mm^2 \times 110 mm)$ $\sum A_i y_i = 16000000 mm^3 + 1012500 mm^3 + 7260000 mm^3 = 24272500 mm^3$

Step 5: Calculate the coordinates of the centroid $(\bar{x}, \bar{y})$ from point O. $\bar{x} = \frac{\sum A_i x_i}{A_{total}} = \frac{54912500 mm^3}{168500 mm^2} \approx 325.89 \text{ mm}$ $\bar{y} = \frac{\sum A_i y_i}{A_{total}} = \frac{24272500 mm^3}{168500 mm^2} \approx 144.05 \text{ mm}$

The position of the centroid from MO (which is the y-axis) is given by its coordinates $(\bar{x}, \bar{y})$. The position of the centroid is $\boxed{(*325.89 mm, 144.05 mm*)}$.