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

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 = 4.70 ft and b = 2.90 ft. HINTS: Think of the board as consisting of two rectangular pieces: one piece that is everything to the left of "a", and one rectangle that is everything to the right of "a". Find the x,y location of the centers of gravity of these two rectangles. Then, find the final center of gravity of two "point masses" at those locations. This will be the center of gravity of the board.

Explanation / Answer

Since the mass of the plywood sheet is uniformly distributed, we can represent the mass of each section as the area of the section:

X = [(4 x (4x8)) - (6.45 x (3.1x1.5))] / [(4x8) - (3.1x1.5)]

= (128 - 29.9925) / (32 - 4.65) = 3.583 feet

Y = [(2 x (4x8)) - (3.25 x (3.1x1.5))] / [(4x8) - (3.1x1.5)]

= (64 - 15.1125) / (32 - 4.65) = 1.787 feet

Hence, the center of mass of the plywood is at (3.583, 1.787) feet from the lower left hand corner of the plywood.

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