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 sheet of plywood, follow these steps:
1. Divide the sheet of plywood into two rectangles: the main rectangular piece and the quadrant cutout.
2. Calculate the area of each rectangle to determine its mass. Given the dimensions of the main rectangular piece (4.00 ft by 8.00 ft), its area is 4.00 ft * 8.00 ft = 32.00 square ft.
3. The area of the quadrant cutout can be calculated by subtracting its rectangular area from the area of the larger rectangle. The dimensions of the cutout are given as a = 4.50 ft and b = 1.70 ft. Therefore, its area is (4.50 ft * 1.70 ft) = 7.65 square ft.
4. Assuming the mass density of the plywood is constant and equal to 1.0 kg/m², convert the areas of the rectangles to square meters by multiplying by 0.092903 to obtain masses in kg. So, the mass of the main rectangular piece is 32.00 ft² * 0.092903 m²/ft² = 2.98256 kg. The mass of the cutout is 7.65 ft² * 0.092903 m²/ft² = 0.712752 kg.
5. Find the x- and y-coordinates of the centers of gravity for each rectangle. The center of gravity for each rectangle is located at the midpoint of its sides. For the main rectangular piece, the x-coordinate of its COG is halfway between 0 and 4.00 ft, so x = 2.00 ft. The y-coordinate of its COG is halfway between 0 and 8.00 ft, so y = 4.00 ft.
6. For the quadrant cutout, the x-coordinate of its COG is a little more complicated to determine. It should be located at the midpoint of the bottom side of the cutout, which would be 2.25 ft. However, since we are only considering the upper right quadrant, we need to disregard the bottom left quadrant and shift the x-coordinate accordingly. This means we subtract b/2 from the x-coordinate, giving us x = 2.25 ft - 1.70 ft/2 = 1.90 ft. The y-coordinate of the COG is straightforward, located at b/2, so y = 1.70 ft/2 = 0.85 ft.
7. Determine the overall x- and y-coordinates of the sheet of plywood's COG using the "textbook" equations for Xcg and Ycg.
Xcg = (m1 * x1 + m2 * x2) / (m1 + m2)
Ycg = (m1 * y1 + m2 * y2) / (m1 + m2)
Substituting the values we have:
Xcg = (2.98256 kg * 2.00 ft + 0.712752 kg * 1.90 ft) / (2.98256 kg + 0.712752 kg)
= (5.96512 kg*ft + 1.3542296 kg*ft) / 3.695312 kg
= 7.3193496 kg*ft / 3.695312 kg
= 1.980911 ft
Ycg = (2.98256 kg * 4.00 ft + 0.712752 kg * 0.85 ft) / (2.98256 kg + 0.712752 kg)
= (11.93024 kg*ft + 0.6053448 kg*ft) / 3.695312 kg
= 12.5355848 kg*ft / 3.695312 kg
= 3.390053 ft
Therefore, the x-coordinate of the center of gravity of the plywood sheet is approximately 1.980911 ft, and the y-coordinate is approximately 3.390053 ft.