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
X=2.75
Y=1.75
To find the x- and y-coordinates of the center of gravity (COG) of the plywood sheet, we can follow these steps:
1. Divide the plywood sheet into two rectangles, one representing the bottom right quadrant (with dimensions 8.00 ft by 4.00 ft) and the other representing the cutout (with dimensions 1.70 ft by 4.50 ft).
2. Calculate the area of each rectangle:
Area of bottom right quadrant = length x width = 8.00 ft x 4.00 ft = 32.00 ft²
Area of cutout = length x width = 1.70 ft x 4.50 ft = 7.65 ft²
3. Determine the mass of each rectangle using the given mass density of the plywood (1.0 kg/m²).
Mass of bottom right quadrant = Area of bottom right quadrant x mass density = 32.00 ft² x 1.0 kg/m² = 32.00 kg
Mass of cutout = Area of cutout x mass density = 7.65 ft² x 1.0 kg/m² = 7.65 kg
4. Find the x- and y-coordinates of the centers of gravity of each rectangle using the formulas:
Xcg = (sum of (mass × x-coordinate)) / total mass
Ycg = (sum of (mass × y-coordinate)) / total mass
For the bottom right quadrant:
Xcg₁ = (32.00 kg × (8.00 ft / 2)) / 32.00 kg = 4.00 ft
Ycg₁ = (32.00 kg × (4.00 ft / 2)) / 32.00 kg = 2.00 ft
For the cutout:
Xcg₂ = (7.65 kg × (8.00 ft + 1.70 ft / 2)) / 7.65 kg = 8.81 ft
Ycg₂ = (7.65 kg × (4.00 ft + 4.50 ft / 2)) / 7.65 kg = 4.76 ft
5. Calculate the total mass of the plywood sheet:
Total mass = mass of bottom right quadrant + mass of cutout = 32.00 kg + 7.65 kg = 39.65 kg
6. Finally, find the x- and y-coordinates of the center of gravity of the entire plywood sheet using the formula:
Xcg = (sum of (mass × x-coordinate)) / total mass
Ycg = (sum of (mass × y-coordinate)) / total mass
Xcg = (32.00 kg × 4.00 ft + 7.65 kg × 8.81 ft) / 39.65 kg ≈ 4.03 ft
Ycg = (32.00 kg × 2.00 ft + 7.65 kg × 4.76 ft) / 39.65 kg ≈ 2.64 ft
So, the x-coordinate of the center of gravity of the plywood sheet is approximately 4.03 ft, and the y-coordinate is approximately 2.64 ft.
To find the x- and y-coordinates of the center of gravity (COG) of the plywood sheet, we can break it down into two rectangular pieces: the larger rectangle and the cut-out smaller rectangle.
1. Find the center of gravity of the larger rectangle:
- The dimensions of the larger rectangle are 4.00 ft x 8.00 ft.
- The center of gravity of a rectangle is located at the midpoint of its width and height.
- For the larger rectangle, the midpoint of the width is 4.00 ft / 2 = 2.00 ft, and the midpoint of the height is 8.00 ft / 2 = 4.00 ft.
- So the center of gravity of the larger rectangle is at (x1, y1) = (2.00 ft, 4.00 ft).
2. Find the center of gravity of the smaller rectangle (cut-out):
- The dimensions of the cut-out rectangle are given as a = 4.50 ft (width) and b = 1.70 ft (height).
- The center of gravity of a rectangle is located at the midpoint of its width and height.
- For the smaller rectangle, the midpoint of the width is 4.50 ft / 2 = 2.25 ft, and the midpoint of the height is 1.70 ft / 2 = 0.85 ft.
- So the center of gravity of the smaller rectangle is at (x2, y2) = (2.25 ft, 0.85 ft).
3. Calculate the mass of each rectangle:
- The mass of each rectangle is proportional to its area.
- Let's assume the mass density of the plywood is 1.0 kg/m^2 (you could use other values if given).
- The area of the larger rectangle is A1 = 4.00 ft * 8.00 ft = 32.00 ft^2.
- The mass of the larger rectangle is m1 = 32.00 ft^2 * 1.0 kg/ft^2 = 32.00 kg (approximation).
- The area of the smaller rectangle is A2 = 4.50 ft * 1.70 ft = 7.65 ft^2.
- The mass of the smaller rectangle is m2 = 7.65 ft^2 * 1.0 kg/ft^2 = 7.65 kg (approximation).
4. Predict the approximate location of the center of gravity of the entire board:
- Since the mass of the larger rectangle is greater than the mass of the smaller rectangle, we expect the COG to be closer to the center of the larger rectangle.
- Based on our predictions, we can approximate the COG to be slightly higher and to the right of (x1, y1), but we'll need to calculate it exactly.
5. Calculate the coordinates of the center of gravity of the entire board:
- The COG coordinates can be calculated using the following formulas:
Xcg = (m1 * x1 + m2 * x2) / (m1 + m2)
Ycg = (m1 * y1 + m2 * y2) / (m1 + m2)
- Substitute the values into the formulas:
Xcg = (32.00 kg * 2.00 ft + 7.65 kg * 2.25 ft) / (32.00 kg + 7.65 kg) ≈ 1.998 ft
Ycg = (32.00 kg * 4.00 ft + 7.65 kg * 0.85 ft) / (32.00 kg + 7.65 kg) ≈ 3.975 ft
Therefore, the x-coordinate of the center of gravity is approximately 1.998 ft, and the y-coordinate is approximately 3.975 ft.