b) Determine the coordinates of the centroid for the shape given below

To determine the coordinates of the centroid (x, y) for the given composite shape, we will decompose it into simpler geometric figures: a large rectangle, a triangle, and a semi-circle (which is a cutout, so its area will be subtracted). Step 1: Decompose the composite shape and identify properties of each component. The overall shape can be considered as: • Shape 1: A rectangle with width 250 mm and height 100 mm. • Shape 3: A right-angled triangle on top of the rectangle, with base 250 mm and height 150 mm - 100 mm = 50 mm. Its vertices are (0, 100), (250, 100), and (250, 150). • Shape 2: A semi-circle removed from the bottom right, with a diameter of 100 mm (radius r = 50 mm). Its diameter lies on the x-axis, centered at x = 150 mm + 50 mm = 200 mm. Step 2: Calculate the area (A_i) and centroid coordinates (x_i, y_i) for each individual shape. For Shape 1 (Rectangle): A_1 = 250 mm × 100 mm = 25000 mm^2 x_1 = 250 mm2 = 125 mm y_1 = 100 mm2 = 50 mm For Shape 3 (Triangle): A_3 = (1)/(2) × base × height = (1)/(2) × 250 mm × 50 mm = 6250 mm^2 The centroid of a triangle is the average of its vertices' coordinates. x_3 = (0 + 250 + 250)/(3) = (500)/(3) mm ≈ 166.67 mm y_3 = (100 + 100 + 150)/(3) = (350)/(3) mm ≈ 116.67 mm For Shape 2 (Semi-circle, removed): A_2 = -(1)/(2) r^2 = -(1)/(2) (50 mm)^2 = -1250 mm^2 The centroid of a semi-circle is located at its axis of symmetry. The x-coordinate is the center of its diameter. The y-coordinate is (4r)/(3) from the diameter. x_2 = 200 mm y_2 = 4 × 50 mm3 = (200)/(3) mm ≈ 21.22 mm Step 3: Calculate the first moment of area (A_i x_i and A_i y_i) for each shape. For Shape 1: A_1 x_1 = 25000 mm^2 × 125 mm = 3125000 mm^3 A_1 y_1 = 25000 mm^2 × 50 mm = 1250000 mm^3 For Shape 3: A_3 x_3 = 6250 mm^2 × (500)/(3) mm = (3125000)/(3) mm^3 A_3 y_3 = 6250 mm^2 × (350)/(3) mm = (2187500)/(3) mm^3 For Shape 2: A_2 x_2 = -1250 mm^2 × 200 mm = -250000 mm^3 A_2 y_2 = -1250 mm^2 × (200)/(3) mm = -(250000)/(3) mm^3 Step 4: Sum the areas and the first moments of area. Total Area: A_i = A_1 + A_3 + A_2 = 25000 + 6250 - 1250 = (31250 - 1250) mm^2 A_i ≈ 31250 - 1250(3.14159) ≈ 31250 - 3926.99 = 27323.01 mm^2 Sum of moments about y-axis: A_i x_i = 3125000 + (3125000)/(3) - 250000 = ((12500000)/(3) - 250000) mm^3 A_i x_i ≈ 4166666.67 - 785398.16 = 3381268.51 mm^3 Sum of moments about x-axis: A_i y_i = 1250000 + (2187500)/(3) - (250000)/(3) = (1250000 + (1937500)/(3)) mm^3 A_i y_i = (3750000 + 1937500)/(3) = (5687500)/(3) mm^3 A_i y_i ≈ 1895833.33 mm^3 Step 5: Calculate the centroid coordinates (x, y). x = ( A_i x_i)/( A_i) = 3381268.51 mm^327323.01 mm^2 ≈ 123.75 mm y = ( A_i y_i)/( A_i) = 1895833.33 mm^327323.01 mm^2 ≈ 69.39 mm The coordinates of the centroid are: (x, y) = (123.75 mm, 69.39 mm) Send me the next one 📸