Skip to main content
Center of Mass : Pear Slice

Finding the center of mass of a solid object or of a solid plane can seem daunting; however, in the most simplistic of terms it is just balancing an said object or plane. It is easy for us to physically balance many things: a spoon on your finger, a book on your hand, or a teeter-totter when you were a child. Balancing larger or more complex objects cannot usually be accomplished by feel alone; that is where finding the center of mass of a solid object or plane comes into play. Additionally, when constructing a building or other structure knowing the center of mass would be vital for structural stability.

Method

Finding the center of mass involves finding the mass of the solid as a whole, finding the moments of the solid with respect to the x and y axis. We will denote the mass as \(M)\), the moment with respect to the x-axis as \( M_x \), and the moment with respect to the y-axis \(M_y\). Given the function of our solid \(f(x)\) we will find the mass and moments in the following manner: \[{ M = \rho \int^{b}_a f(x) dx \\ M_x = \rho \int^{b}_a \frac{1}{2} f(x)^2 \\ M_Y = \rho \int^{b}_a xf(x) dx , }\] where \(a\) and \(b\) are the bounds on the region within which the solid lies.

The actual point within the coordinate plane that is the center of mass is denoted by \( (x̄, ȳ )\), where \[ x̄ = \frac{M_y}{M} \] and \[ ȳ = \frac{M_x}{M}.\]

As fall approaches pears are returning to the grocery stores. Since pears are a fairly irregular fruit, in that they hard difficult to balance upright, I thought it would be the perfect example. It would be a bit difficult to find the center of mass for a whole pear, but a horizontal slice would be fairly easy. So we begin witha region on a coordinate plane bounded by \[{ y = \sqrt{\frac{x^3(1-x)}{1.667^2} \\ y = - \sqrt{ \frac{x^3(1-x)}{1.667^2} }\]. which gives a region that looks like the following:

When integrated we get \( M= 0.235572 \), \( M_x = 0.147233 \), and \( M_y = 0.0359856 \). This gives us a center of mass at \( x̄ , ȳ) = (0.62500,0.15278) \). This point is slightly off center which was un expected. The purple point in the diagram above represents the actual center of mass, while the green point is a center of mass that is actually centered. While surprising a solid plane in the shape of the diagram above does balance at the center of mass marked in purple.

Labels:

Comments

Post a Comment

Popular posts from this blog

The Approximation of a Solid of Revolution

Most math teachers I've had have been able to break down Calculus into two very broad categories: derivatives and integrals. What is truly amazing, is how much you can do with these two tools. By using integration, it is possible to approximate the shape of a 2-D function that is rotated around an axis. This solid created from the rotation is known as a solid of revolution. To explain this concept, we will take a look at the region bounded by the two functions: \[ f(x) = 2^{.25x} - 1 \] and \[ g(x) = e^{.25x} - 1 \] bounded at the line y = 1. This region is meant to represent a cross section of a small bowl. While it may not perfectly represent this practical object, the approximation will be quite textured, and will provide insight into how the process works. The region bounded by the two functions can be rotated around the y-axis to create a fully solid object. This is easy enough to talk about, but what exactly does this new solid look like? Is...
8 comments
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