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.

Question

Asked on August 3, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-08-03T04:21:16.040Z

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 mm Height (h_1) = 400 mm Area (A_1) = w_1 × h_1 = 200 mm × 400 mm = 80000 mm^2 Centroid x_1 = (w_1)/(2) = (200)/(2) = 100 mm Centroid y_1 = (h_1)/(2) = (400)/(2) = 200 mm Rectangle 2 (Middle): Width (w_2) = 250 mm Height (h_2) = 90 mm Area (A_2) = w_2 × h_2 = 250 mm × 90 mm = 22500 mm^2 Centroid x_2 = w_1 + (w_2)/(2) = 200 + (250)/(2) = 200 + 125 = 325 mm Centroid y_2 = (h_2)/(2) = (90)/(2) = 45 mm Rectangle 3 (Right): Width (w_3) = 300 mm Height (h_3) = 220 mm Area (A_3) = w_3 × h_3 = 300 mm × 220 mm = 66000 mm^2 Centroid x_3 = w_1 + w_2 + (w_3)/(2) = 200 + 250 + (300)/(2) = 450 + 150 = 600 mm Centroid y_3 = (h_3)/(2) = (220)/(2) = 110 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 x). A_i x_i = A_1 x_1 + A_2 x_2 + A_3 x_3 A_i x_i = (80000 mm^2 × 100 mm) + (22500 mm^2 × 325 mm) + (66000 mm^2 × 600 mm) 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 y). A_i y_i = A_1 y_1 + A_2 y_2 + A_3 y_3 A_i y_i = (80000 mm^2 × 200 mm) + (22500 mm^2 × 45 mm) + (66000 mm^2 × 110 mm) A_i y_i = 16000000 mm^3 + 1012500 mm^3 + 7260000 mm^3 = 24272500 mm^3 Step 5: Calculate the coordinates of the centroid (x, y) from point O. x = ( A_i x_i)/(A_total) = 54912500 mm^3168500 mm^2 ≈ 325.89 mm y = ( A_i y_i)/(A_total) = 24272500 mm^3168500 mm^2 ≈ 144.05 mm The position of the centroid from MO (which is the y-axis) is given by its coordinates (x, y). The position of the centroid is (325.89 mm, 144.05 mm). Send me the next one 📸

Accepted Answer · 2026-08-03T04:21:16.040Z

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 mm Height (h_1) = 400 mm Area (A_1) = w_1 × h_1 = 200 mm × 400 mm = 80000 mm^2 Centroid x_1 = (w_1)/(2) = (200)/(2) = 100 mm Centroid y_1 = (h_1)/(2) = (400)/(2) = 200 mm Rectangle 2 (Middle): Width (w_2) = 250 mm Height (h_2) = 90 mm Area (A_2) = w_2 × h_2 = 250 mm × 90 mm = 22500 mm^2 Centroid x_2 = w_1 + (w_2)/(2) = 200 + (250)/(2) = 200 + 125 = 325 mm Centroid y_2 = (h_2)/(2) = (90)/(2) = 45 mm Rectangle 3 (Right): Width (w_3) = 300 mm Height (h_3) = 220 mm Area (A_3) = w_3 × h_3 = 300 mm × 220 mm = 66000 mm^2 Centroid x_3 = w_1 + w_2 + (w_3)/(2) = 200 + 250 + (300)/(2) = 450 + 150 = 600 mm Centroid y_3 = (h_3)/(2) = (220)/(2) = 110 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 x). A_i x_i = A_1 x_1 + A_2 x_2 + A_3 x_3 A_i x_i = (80000 mm^2 × 100 mm) + (22500 mm^2 × 325 mm) + (66000 mm^2 × 600 mm) 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 y). A_i y_i = A_1 y_1 + A_2 y_2 + A_3 y_3 A_i y_i = (80000 mm^2 × 200 mm) + (22500 mm^2 × 45 mm) + (66000 mm^2 × 110 mm) A_i y_i = 16000000 mm^3 + 1012500 mm^3 + 7260000 mm^3 = 24272500 mm^3 Step 5: Calculate the coordinates of the centroid (x, y) from point O. x = ( A_i x_i)/(A_total) = 54912500 mm^3168500 mm^2 ≈ 325.89 mm y = ( A_i y_i)/(A_total) = 24272500 mm^3168500 mm^2 ≈ 144.05 mm The position of the centroid from MO (which is the y-axis) is given by its coordinates (x, y). The position of the centroid is (325.89 mm, 144.05 mm). Send me the next one 📸

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.

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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.

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