A uniform square plate is placed on an xy -plane with its vertices at (0,0) , (6,0) , (6,6) , and (0,6) . A triangular portion with vertices at (0,0) , (6,0) , and (6,3) is removed from it. If the coordinates of the center of mass of the remaining plate are (x_c, y_c) , the value of 9 x_c y_c is ________.

Step-by-step solution

Let the mass density of the uniform plate be . The area of the complete square plate is A₁ = 6 6 = 36 . The center of mass of the complete square plate is at (x₁, y₁) = (3, 3) . The removed portion is a triangle with vertices (0,0) , (6,0) , and (6,3) . The area of this triangle is A₂ = 1 2 base height = 1 2 6 3 = 9 . The center of mass of this triangle is at its centroid: x₂ = 0 + 6 + 6 3 = 4 y₂ = 0 + 0 + 3 3 = 1 Using the principle of superposition (negative mass method), the coordinates of the center of mass of