Final answer:
To find the center of mass for laminae with non-uniform densities, one must calculate the moments Mx and My through integration using the provided density functions, and then divide by the total mass. The moment calculations vary based on the object's shape and density variation.
Step-by-step explanation:
Finding Mx, My, and (x,y) for Lamina with Non-uniform Density
The task is to find the center of mass of different laminae with non-uniform densities. To find the center of mass (x, y) for such objects, we integrate to find moments Mx and My, and then divide by the total mass of the object. The moments are obtained using the density functions p(x, y), which vary depending on the shape and density variation of the object.
For a rectangular block with non-uniform density p(x, y) = Pox (where Po is constant), we would integrate over the area of the rectangle. We calculate the x-moment (Mx) by integrating the density function multiplied by the y-coordinate over the area, and the y-moment (My) by integrating the density function multiplied by the x-coordinate.
For a rod with quadratic density change p(x) = Po + (P1 − Po) (x/L)², where Po and P1 are constants, the moments would be calculated by integrating p(x) over the length of the rod, multiplied by the distance from the chosen axis.
In another example, a rectangular material with density p(x, y) = poxy, requires double integrals over its length and width to find Mx and My. All these calculations require the application of integral calculus to determine the center of mass.