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Representing a Solid by its Center of Mass

Think about center of mass 

The center of mass (or center of gravity) of a solid is similar to the centroid of solid. The centroid is the geometric center of the solid. If a solid has uniform density, then its center of mass is the same as the centroid of its shape. For simplification in physics, the solid can be represented one point to describe the trajectory of its movement. This specific point is its center of mass.

Let’s consider the center of mass of a thin plate. A thin plate can be represented by a two-dimensional region in a plane. If the region is symmetric with respect to a line, then the centroid of the region lies on the line. For example, the centroid (center of mass) of the rectangle is at its center.

For a non-symmetric region, we can calculate the coordinates of its center of mass, \[ \overline{x} = \frac{{{M_y}}}{M}\] \[ \overline{y} = \frac{{{M_x}}}{M}\] where M is the mass of the solid, \(M_y\) is the moment of the solid with respect to the y-axis, and \(M_x\) is the moment of the solid about the x-axis.

The solid to be analyzed 

We choose a shape of a thin plate which is similar to an animal’s horn or a fish’s fin. This is a common structure in nature. Many creatures use this structure to complete some functions. This perspective shows that thin plate has the value of study. The depth of the thin plate is 0.2 inch.
The blue line is determined by the function: \[x^2+y^2=16, x\ge 0, y \ge 0 \] The orange line is determined by the function: \[(x+3)^2+y^2=25, x\ge 0, y \ge 0 \]

Calculating the coordinates of the center of mass 

For the solid to be analyzed, given a value of y, the horizontal coordinates of the points on the blue line and orange line are as following: \[x_1 = \sqrt{16 - y^2 }, (0 \le y\le 4)\] \[x_2 = \sqrt{25 - y^2 } - 3, (0\le y \le 4)\]
Let the constant ρ denotes the density of the solid. First, the mass of the solid can be calculated as following: \[M=\rho \times area \times depth\] \[= 0.2 \rho \times \int_{0}^{4}{ \{ x_1(y)-x_2(y)\}dy}\] \[= 0.2 \rho \times \int_{0}^{4}{ \{ \sqrt{16 - y^2 }-\sqrt{25 - y^2 } + 3 \} dy}\] \[=1.395\rho\] Then, the moments of the solid with respect to the y axis and x axis can be calculated as following: \[M_y=\rho \times depth \times \int_{0}^{4}{ \int_{x_2(y)}^{x_1(y) } {x dx}dy}\] \[=\rho \times depth \times \int_{0}^{4}{ \int_{\sqrt{25 - y^2 } - 3}^{\sqrt{16 - y^2}}{ x dx}dy}\] \[=3.354\rho\] \[M_x=\rho \times depth \times \int_{0}^{4}{ y \times \{x_1(y) - x_2(y) \}dy}\] \[=\rho \times depth \times \int_{0}^{4}{ y \times \{ \sqrt{16 - y^2} - \sqrt{25 - y^2 } + 3 \}dy}\] \[=2.533\rho\] Finally, we can get the coordinates of the center of mass of the solid: \[ \overline{x} = \frac{{M_y}}{M} = \frac{3.354ρ}{1.395ρ}=2.404\] \[ \overline{y} = \frac{{M_x}}{M}=\frac{2.533ρ}{1.395ρ}=1.816\]
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