The plate is a thin square of thickness t and sides d. The left half of the plate has (uniform) mass density p1 and the right half (uniform) mass density p2. Place the XY-axis with every axis parallel to the sides of the square and origin at the geometrical center of the plate. Then answer the following questions:
A. Show that the total mass of the place is given by 4t(d/2)^2*paver where paver=(p1+p2)/2
B. Because every half of the plate is uniformly mass-distributed and because it is symmetrical, you can find the location of their center of mass and use a composite approach to discover the center of mass of the whole plate. Show that this total center of mass is located at (d/4) pdiff/paver on the X-axis (here pdiff=(p2-p1)/2)
C. Using the volume integral approach outlined in the lecture, show that the center of mass of the whole plate (again) is given by d/4 pdiff/paver on the X-axis
Note: Use volume integrals in the Cartesian system to calculate the CM of a thin plate of varying mass density.