A uniform thin metal plate of mass $$10 \text{ kg}$$ with dimensions is shown in the figure below. The ratio of $$x$$ and $$y$$ coordinates of center of mass of the plate is $$\frac{n}{9}$$. The value of $$n$$ is ________.

Since the plate is uniform, we can find the COM by treating the object as a large rectangle with a smaller section removed from its top.

We consider the full $$3 \times 2$$ rectangle and subtract the $$1 \times 1$$ empty square region in the upper middle.

Total Rectangle ($$A_1$$): The area is $$3 \times 2 = 6$$, with its geometric center at $$(1.5, 1)$$.

Removed Square ($$A_2$$): The area is $$1 \times 1 = 1$$, with its geometric center at $$(1.5, 1.5)$$.

Coordinates of the Center of Mass

Because the figure is symmetric about the line $$x = 1.5$$, the $$x$$-coordinate is:

$$x_{cm} = \frac{A_1x_1 - A_2x_2}{A_1 - A_2} = \frac{6(1.5) - 1(1.5)}{5} = 1.5$$

The $$y$$-coordinate is determined by the distribution of the remaining area along the vertical axis:

$$y_{cm} = \frac{A_1y_1 - A_2y_2}{A_1 - A_2} = \frac{6(1) - 1(1.5)}{6 - 1} = \frac{4.5}{5} = 0.9$$

Taking the ratio of the $$x$$ and $$y$$ coordinates:

$$\text{Ratio} = \frac{x_{cm}}{y_{cm}} = \frac{1.5}{0.9}$$

$$\frac{15}{9} = \frac{n}{9}$$

The value of $$n$$ is 15.