Skip to main content

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: \[ \overline{x} = \frac{{M_y}}{M} = \frac{3.354ρ}{1.395ρ}=2.404\] \[ \overline{y} = \frac{{M_x}}{M}=\frac{2.533ρ}{1.395ρ}=1.816\]
Labels:

Comments

Post a Comment

Popular posts from this blog

Knot 9-31

Knot 9-31 A knot is mathematics is defined as a closed, non-self-intersecting curve that is placed in three dimensions and cannot be the "unknot". The main difference between a knot in the real world and a known in mathematics is that a knot in mathematics does not contain any extra strands. The example following will help visualize this. Today, I have specifically chosen knot 9-31 from the Knot Atlas. This knot is very unique and contains some very interesting properties that we are going to look into. The Crossing Number For my knot today, I chose knot 9-31 from the Knot Atlas. This knot contains 9 crossings! The Unknotting Number The unknotting number is exactly what is sounds like. This is the minimum number of times the knot must be passed through itself to untie it. Luckily, the Knot Atlas is super useful and provides the unknotting number for us, but I still...
Post a Comment
Read more

Do Over: Integration for Over Regions in the Plane

Introduction Earlier in the semester, we visited the topic of integration for over region regions in the plane. This was our first experience with taking integration in the third dimension. To do this, we had to use double integrals to calculate the volume of a region between two surfaces. This topic seems relatively straightforward and anyone who has taken a higher calculus class knows of double integrals. Today, we are going to revisit this topic for a couple of reasons. Why Double Integrals Again? As mentioned before, I am revisiting this topic for a couple of reasons. One of the main reasons is due to how the 3D print turned out. Due to the function, I chose, the final rectangular prism was stand alone. When printing, this resulted in a sloppy prism that lacked structural integrity. The image below is a reference to my original design that includes the prism that had the issue. The main cause of ...
Post a Comment
Read more

Middle of Mass Measured from Moment

The center of mass of an object is the point that lies in the average position of mass in that object. For a normal polygon, that point is in the middle of the shape. However, if the shape is irregular, has concave angles, or has holes, the center of mass can be far from the middle of the object. Two things come into factor when determining the center of mass of an object. Weight and distance. Obviously, if one side of an object has more mass than the other, the center of mass will be on the heavier side. But, distance plays a role, too. If two sides of an object have the same mass, but one is farther from the middle, the center of mass will be on the side that has farther mass. Multiplying both of these factors together gives the moment of an object. Dividing the moment by the overall mass gives the location of the center of mass.   The reason I chose my shape is to mimic how those toy balancing birds work. The toy is a model bird with its wings outstretched. The ...
Post a Comment
Read more