A uniform, square metal plate with side L = 3.50 cm and mass 0.087 kg is located with its lower left corner at (x, y) = (0, 0) as shown in the figure. Two squares with side length L/4 are removed from the plate.

(a) What is the x-coordinate of the center of mass?
= L/2

(b) What is the y-coordinate of the center of mass?

Do this as two problems:

find the cg with the large plate, no corner cut, then the cg of the corner that is gone.

newCg*mass=oldCG*mass-cornercg*mass

where cg has to be figured in x, y coordinates.

To find the y-coordinate of the center of mass, we need to consider the distribution of mass along the y-axis.

First, we need to determine the mass and center of mass of the remaining portion of the plate after the two squares are removed.

The original plate has a mass of 0.087 kg and dimensions of L × L. After removing the two squares with side length L/4, we are left with a plate with dimensions 3L/4 × 3L/4.

The mass of the remaining portion can be calculated as follows:

Mass of remaining portion = mass of original plate - mass of two removed squares
= 0.087 kg - (2 × (L/4)^2)
= 0.087 kg - (2 × (3.50 cm / 4)^2)

Now, let's calculate the y-coordinate of the center of mass:

y-coordinate of center of mass = (mass1 × y1 + mass2 × y2) / (mass1 + mass2)

Where:
mass1 = mass of the remaining portion
mass2 = mass of each removed square
y1 = y-coordinate of the center of mass of the remaining portion
y2 = y-coordinate of the center of mass of each removed square

The y-coordinate of the center of mass of the remaining portion (y1) is 3L/8, as the remaining portion is symmetric along the y-axis.

The y-coordinate of the center of mass of each removed square (y2) is L/8, as the removed squares are symmetric along the y-axis.

Substituting the values:

y-coordinate of center of mass = (mass1 × (3L/8) + mass2 × (L/8)) / (mass1 + mass2)
= (mass1 × (3 × L/8) + mass2 × (L/8)) / (mass1 + mass2)
= ((0.087 kg - (2 × (3.50 cm / 4)^2)) × (3 × L/8) + (2 × (3.50 cm / 4)^2) × (L/8)) / (0.087 kg - (2 × (3.50 cm / 4)^2) + (2 × (3.50 cm / 4)^2))

By substituting the values and simplifying the equation, we can find the y-coordinate of the center of mass.

To find the y-coordinate of the center of mass, we need to consider the masses and their respective distances from the y-axis.

In this case, we have a uniform, square metal plate with side L = 3.50 cm and mass 0.087 kg. Two squares with side length L/4 are removed from the plate. The remaining plate forms a rectangular shape.

Let's label the length and width of the rectangular plate as L_r and W_r, respectively. Since two squares with side length L/4 are removed, L_r = L - (L/4) - (L/4) = L/2.

The center of mass of a uniform rectangular plate lies at its geometric center. Thus, the y-coordinate of the center of mass is given by the formula:

y_cm = (W_r/2)

In this case, since the metal plate is a square, W_r = L_r = L/2. Therefore, the y-coordinate of the center of mass is:

y_cm = (L/2)/2 = L/4 = 3.50 cm / 4 = 0.875 cm