A thin rectangular plate of uniform areal density σ = 2.79 kg/m2 has length of 37.0 cm and width of 23.0 cm. The lower left hand corner is located at the origin, (x,y)= (0,0) and the length is along the x-axis.

(a)There is a circular hole of radius 8.00 cm with center at (x,y) = (12.50,9.50) cm in the plate.

(b)Calculate the mass of plate.

For part (a) I got 1.81*10^-1 kg

Now for part b I cant quite solve it.
thanks!

Calculate the distance of the plate's CM from the origin.

mass = 2.79(.23*.37-pi(.08)^2)

= 2.79(.0851-.0201) = 2.79 * .065
= .181 kg agree

moment in x (forget the 2.79 density)
.065 x = .0851*.37/2 - .0201*.125
.065 x = .0157 - .0025 = .0125
x = .192 meter = 19.2 cm
now do y of cm the same way

what is the .065??? same for y .065???

.065 is the area of the plater with the hole removed

Why did the plate go to therapy? Because it couldn't find its center of mass and it was feeling imbalanced!

To calculate the center of mass (CM) of the plate, we can break it down into two parts: the rectangular part and the circular hole.

Let's start with the rectangular part. The CM of the rectangle lies along the line connecting the midpoints of the opposite sides. Since the length is along the x-axis, the x-coordinate of the CM of the rectangular part is at x = (37.0 cm)/2 = 18.5 cm. The y-coordinate of the CM of the rectangular part is at y = (23.0 cm)/2 = 11.5 cm.

Now let's consider the circular hole. Since it's a hole, it doesn't contribute to the mass or the CM of the plate. So we can ignore it for now.

To find the overall CM of the plate, we need to consider the mass distribution of the rectangular part and the circular hole. The CM of the plate will lie along the line connecting the CMs of these two parts.

Since the circular hole is centered at (12.50, 9.50) cm, its CM will also be at these coordinates.

To find the overall CM, we need to take into account the mass distribution of the rectangular part and the circular hole. We can do this by calculating the weighted average of their respective CMs, with the weights being the masses.

Since the rectangular part has an areal density of σ = 2.79 kg/m^2, we can calculate its mass as M_rect = σ * (length * width). Convert the length and width to meters by dividing them by 100. So M_rect = 2.79 kg/m^2 * (0.37 m * 0.23 m).

The mass of the circular hole is zero, as it's a hole in the plate.

Finally, we can calculate the overall CM using the formula:

x_CM = (M_rect * x_rect + M_hole * x_hole) / (M_rect + M_hole)
y_CM = (M_rect * y_rect + M_hole * y_hole) / (M_rect + M_hole)

Plug in the values we calculated earlier and you should be able to find the coordinates of the CM of the plate.

To calculate the mass of the plate, you need to find the area of the plate first. The area of a rectangle is given by the formula:

Area = length * width

In this case, the length of the plate is 37.0 cm and the width is 23.0 cm. So, plug in these values into the formula:

Area = 37.0 cm * 23.0 cm

Now, since the density of the plate is given as σ = 2.79 kg/m^2, which means the mass per unit area, you can use this density to find the mass of the plate.

Mass = density * area

However, to calculate the mass accurately, you need to convert the cm^2 area to m^2 by dividing by 10,000 (since 1 m^2 = 10,000 cm^2). So, the area in m^2 becomes:

Area = (37.0 cm * 23.0 cm) / 10,000 m^2

Now, plug in this value into the mass formula:

Mass = 2.79 kg/m^2 * [(37.0 cm * 23.0 cm) / 10,000 m^2]

Calculating this expression will give you the mass of the plate.