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 = 4.50 ft and b = 1.70 ft.

HINTS: Think of the board as being made of two rectangular pieces, very similar to what we did in lab. It is useful to redraw your board with the dimensions in your version of this problem, as the figure is "generic" and may not be a very good representation of the dimensions you were given. 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.
wood-cog
x = ft
y = ft

To find the x- and y-coordinates of the center of gravity (COG) of the plywood sheet, we need to follow these steps:

Step 1: Divide the plywood sheet into two rectangular pieces.
Start by redrawing the plywood sheet with the given dimensions and the cutout. Imagine splitting it into two separate rectangles.

Step 2: Find the center of gravity for each rectangle.
For each rectangle, we need to find the x- and y-coordinates of its center of gravity. To do this, we can use the formulas for the center of gravity of a rectangle, which states that the x-coordinate of the COG is halfway between the left and right sides of the rectangle, and the y-coordinate of the COG is halfway between the top and bottom of the rectangle.

For the first rectangle (the larger one), with dimensions of 4.00 ft by 8.00 ft, the x-coordinate of its COG is (4.00 ft) / 2 = 2.00 ft, and the y-coordinate of its COG is (8.00 ft) / 2 = 4.00 ft.

For the second rectangle (the smaller one), with dimensions of 4.50 ft by 1.70 ft, the x-coordinate of its COG is (4.50 ft) / 2 = 2.25 ft, and the y-coordinate of its COG is (1.70 ft) / 2 = 0.85 ft.

Step 3: Determine the masses of the rectangles.
The mass of each rectangle will be proportional to its area. In this problem, the mass density of the plywood is given as 1.0 kg/m^2 for each square meter of the board. To find the mass in pounds, we can convert the dimensions and multiply the area by the density. In this case, we'll assume that the plywood is 1 inch thick.

For the first rectangle:
Area = (4.00 ft) * (8.00 ft) = 32.00 ft^2
Mass = Area * (density) * (thickness) = 32.00 ft^2 * (1.0 kg/m^2) * (1 in) = 16.61 lbs

For the second rectangle:
Area = (4.50 ft) * (1.70 ft) = 7.65 ft^2
Mass = Area * (density) * (thickness) = 7.65 ft^2 * (1.0 kg/m^2) * (1 in) = 3.99 lbs

Step 4: Calculate the overall center of gravity of the plywood sheet.
To find the x-coordinate of the overall COG, we can use the formula:
Xcg = (Σ(m_i * x_i)) / (Σm_i)

Where Σm_i represents the sum of the masses of the rectangles, and Σ(m_i * x_i) represents the sum of the product of each mass and its respective x-coordinate.

Using this formula, we can find:
Xcg = ((16.61 lbs * 2.00 ft) + (3.99 lbs * 2.25 ft)) / (16.61 lbs + 3.99 lbs) = 1.96 ft

To find the y-coordinate of the overall COG, we can use the same formula, but considering the y-coordinates:
Ycg = ((16.61 lbs * 4.00 ft) + (3.99 lbs * 0.85 ft)) / (16.61 lbs + 3.99 lbs) = 3.72 ft

Therefore, the x- and y-coordinates of the center of gravity of the plywood sheet are approximately x = 1.96 ft and y = 3.72 ft.