A thin rectangular plate of uniform area density σ1 = 1.01 kg/m2 has a length a = 0.580 m and a width b = 0.240 m. The lower left corner is placed at the origin, (x, y) = (0, 0). A circular hole of radius r = 0.046 m with center at (x, y) = (0.066 m, 0.066 m) is cut in the plate. The hole is plugged with a disk of the same radius that is composed of another material of uniform area density σ2 = 5.46 kg/m2. What is the distance from the origin of the resulting plate's center of mass?

To find the distance from the origin to the center of mass of the resulting plate, we can break down the problem into two steps:

Step 1: Find the center of mass of the rectangular plate without the hole.
Step 2: Account for the center of mass of the disk used to plug the hole.

Step 1: Finding the center of mass of the rectangular plate without the hole.
The center of mass of a thin rectangular plate is located at the geometric center, which is the point halfway between the upper right and lower left corners. Since the lower left corner is at the origin, the upper right corner of the rectangle would be at (a, b) = (0.580 m, 0.240 m).

To find the x-coordinate of the center of mass, we can use the formula for the x-coordinate of the center of mass of a rectangle:

x_cm_rect = (M_rect * x_rect) / (M_rect)

where:
x_cm_rect is the x-coordinate of the center of mass of the rectangle
M_rect is the mass of the rectangle
x_rect is the x-coordinate of the center of mass of the rectangle (equal to half the length of the rectangle)

The mass of the rectangular plate is given by the product of its area and the density:
M_rect = σ1 * A_rect

where:
σ1 is the area density of the rectangular plate (1.01 kg/m^2)
A_rect is the area of the rectangular plate (equal to the length times the width)

Substituting the given values, we get:
A_rect = a * b = (0.580 m)(0.240 m) = 0.1392 m^2

Thus, the mass of the rectangular plate is:
M_rect = σ1 * A_rect = (1.01 kg/m^2)(0.1392 m^2) = 0.141 kg

The x-coordinate of the center of mass of the rectangle is half its length:
x_rect = 0.5a = 0.5(0.580 m) = 0.290 m

Substituting these values into the formula, we find:
x_cm_rect = (M_rect * x_rect) / (M_rect) = (0.141 kg * 0.290 m) / (0.141 kg) = 0.290 m

So the x-coordinate of the center of mass of the rectangular plate is 0.290 m.

The y-coordinate of the center of mass of the rectangle can be found using a similar formula:

y_cm_rect = (M_rect * y_rect) / (M_rect)

where:
y_cm_rect is the y-coordinate of the center of mass of the rectangle
y_rect is the y-coordinate of the center of mass of the rectangle (equal to half the width of the rectangle)

Using the same values as before, we get:
y_rect = 0.5b = 0.5(0.240 m) = 0.120 m

Substituting into the formula, we find:
y_cm_rect = (M_rect * y_rect) / (M_rect) = (0.141 kg * 0.120 m) / (0.141 kg) = 0.120 m

So the y-coordinate of the center of mass of the rectangular plate is 0.120 m.

Therefore, the center of mass of the rectangular plate without the hole is located at (0.290 m, 0.120 m).

Step 2: Accounting for the center of mass of the disk used to plug the hole.
Since the disk used to plug the hole has the same radius as the hole, its center of mass lies at the center of the hole, which is given as (0.066 m, 0.066 m).

To account for the center of mass of the disk, we need to consider its mass and its distance from the origin.

The mass of the disk is given by the product of its area and the density:
M_disk = σ2 * A_disk

where:
σ2 is the area density of the disk (5.46 kg/m^2)
A_disk is the area of the disk (equal to the area of a circle with radius r)

The area of a circle can be calculated using the formula:
A_disk = π * r^2 = π * (0.046 m)^2

Thus, the mass of the disk is:
M_disk = σ2 * A_disk = (5.46 kg/m^2)(π * (0.046 m)^2)

To find the distance of the center of mass of the disk from the origin, we can simply use the given coordinates of the center of the hole, which is also the center of mass of the disk.

So, the distance from the origin to the center of mass of the resulting plate is the distance between the center of mass of the rectangular plate without the hole and the center of mass of the disk:

distance = sqrt((x_cm_rect - x_disk)^2 + (y_cm_rect - y_disk)^2)

Substituting the values, we have:
distance = sqrt((0.290 m - 0.066 m)^2 + (0.120 m - 0.066 m)^2)

Evaluating this equation gives the distance from the origin to the center of mass of the resulting plate.