Find the x and y coordinates of the center of gravity of a 4.00 ft by 8.00 ft un

Question

Find the x and y coordinates of the center of gravity of a 4.00 ft by 8.00 ft uniform sheet of plywood with the upper right quadrant removed as shown in the figure below. The dimensions of the cutout are a = 3.10 ft and b = 2.50 ft. Find the x, y location of the centers of gravity of these two rectangles. Figure out how much mass each of the rectangles has. The mass will be proportional to the area of the rectangle. If you like, take the mass density of the plywood to be 1.0 kg/m2 for each square meter of board. With this information, predict approximately where the center of gravity of the board should be. Then use these masses (and their locations) to find the center of gravity of the entire board using the "textbook" equations for Xcg and Ycg. Check that these coordinates make sense compared with your prediction.

Explanation / Answer

Consider the remaining plywood to consist of two
parts: A1 is a 4.00-ft by 4.00-ft section with center of
gravity located at (2.00 ft,2.00ft), while A2  is a
2.00-ft by 4.00-ft section with center of gravity at
(6.00 ft,1.00ft). Since the plywood is uniform, its
mass per area is constant and the mass of a section having area A is m= A. the center of gravity of the remaining plywood has coordinates given by:

Xcg=mixi/mi=A1X1+A2X2/A1+A2=(16*2+8*6)/(16+8)=3.3 ft

Ycg=miyi/mi=(A1y1+A2y2)/A1+A2=(16*2+8*1)/16+8=1.67 ft

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