The uniform sheet of siding shown has a centrally-located doorway of width 2.0 m and height 6.0 m cut out of it. The sheet, with the doorway hole, has a mass of 264 kg. (a) What is the x coordinate of the center of mass? (b) What is the y coordinate of the center of mass? Assume the lower left hand corner of the siding is at (0, 0).

The sheet is 12 x12.

Because the shape is symmetrical on the x axis you can say that the center of mass on the x is 6. To find center of mass on the y, all you have to do it break the shape into two smaller shapes, find the center of masses of those individual shapes, and then find the center of mass of the system. You will get something like 6.2727 repeating. Sorry if this is hard to understand, and hope this helps someone!

To find the x-coordinate and y-coordinate of the center of mass, we need to calculate the individual moments of the various parts of the sheet, and then use those moments to determine the overall center of mass.

Let's divide the sheet into three separate parts: the left section, the right section, and the doorway section.

1. Left section:
The left section of the sheet is a rectangle with dimensions 6 m by 12 m. To find its moment, we need to calculate the product of its mass and the x-coordinate of its center (which is the midpoint of the section). Given that the mass of the entire sheet is 264 kg, the mass of this section can be calculated as:
mass_left = (6/12) * (12/12) * 264 kg = 132 kg

Since the center of the section is at x-coordinate 3 m, the moment of this section can be calculated as:
moment_left = mass_left * x-coordinate = 132 kg * 3 m = 396 kg·m

2. Doorway section:
The doorway section is a rectangle with dimensions 2 m by 6 m. To find the moment, we need to calculate the product of its mass and the x-coordinate of its center (which is the midpoint of the section). The mass of this section can be calculated as:
mass_doorway = (2/12) * (6/12) * 264 kg = 22 kg

Since the center of the section is at x-coordinate 6 m, the moment of this section can be calculated as:
moment_doorway = mass_doorway * x-coordinate = 22 kg * 6 m = 132 kg·m

3. Right section:
The right section is the remaining portion of the sheet after removing the doorway and the left section. Since the sheet is a square with dimensions 12 m by 12 m, the right section can be calculated as:
mass_right = mass_sheet - mass_left - mass_doorway = 264 kg - 132 kg - 22 kg = 110 kg

The center of the sheet is at x-coordinate 9 m, so the moment of the right section can be calculated as:
moment_right = mass_right * x-coordinate = 110 kg * 9 m = 990 kg·m

Now, we can determine the x-coordinate of the center of mass by summing up the moments and dividing by the total mass:
x-coordinate_center_of_mass = (moment_left + moment_doorway + moment_right) / (mass_sheet)
x-coordinate_center_of_mass = (396 kg·m + 132 kg·m + 990 kg·m) / 264 kg = 2.21 m

Therefore, the x-coordinate of the center of mass is approximately 2.21 m.

To find the y-coordinate of the center of mass, we can use the same approach, but this time we will consider the y-coordinate instead of the x-coordinate.

The y-coordinate of the center of mass is calculated similarly by dividing the sum of the individual moments in the y-direction by the total mass.

Since the sheet has a uniform density, all the sections have the same density.

The y-coordinate of the center of mass in this case will be at y = 6/2 = 3 m, which is halfway up the sheet.

Therefore, the y-coordinate of the center of mass is 3 m.

Hence, the coordinates of the center of mass of the sheet are approximately (2.21 m, 3 m).