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

Do Over: Integration Over a Region in a Plane

Throughout the semester we have covered a variety of topics and how their mathematical orientation applies to real world scenarios. One topic we discussed, and I would like to revisit, is integration over a region in a plane which involves calculating a double integral. Integrating functions of two variables allows us to calculate the volume under the function in a 3D space. You can see a more in depth description and my previous example in my blog post, https://ukyma391.blogspot.com/2021/09/integration-for-over-regions-in-plane_27.html . I want to revisit this topic because in my previous attempt my volume calculations were incorrect, and my print lacked structural stability. I believed this print and calculation was the topic I could most improve on and wanted to give it another chance. What needed Improvement? The function used previously was f(x) = cos(xy) bounded on [-3,3] x [-1,3]. After solving for the estimated and actual volume, it was difficult to represent in a print...
Post a Comment
Read more

Do Over: Ruled Surfaces

Why to choose this project to repeat For the do over project, I would like to choose the ruled surfaces. I don't think my last project was creative, and the 3D printed effect was not very satisfactory. In the previous attempts, all the lines are connected between a straight line and a circle. This connection structure is relatively uncomplicated. The printed model has too many lines, resulting in too dense line arrangement. The gaps between lines are too small, and the final effect is that all the lines are connected into a curved surface, which is far from the effect I expected. What to be improved In this do over project, I would like to improve in two aspects. Firstly, a different ruled surface is chosen. In the previous model, one curve is a unit circle on the \(x-y\) plane, and the ruled surface is a right circular conoid. In this do over project, it is replaced by two border lines. Each borderline is in the shape of an isosceles right triangl...
Post a Comment
Read more

Ruled Surfaces : Trefoil

Ruled Surfaces : Trefoil A ruled surface is a surface that consists straight lines, called rulings, which lie upon the surface. These surfaces are formed of a set of points that are "swept" by a straight line. This is relatively intuitive once you see a good visual, but can be a bit abstract without that concrete example. A very basic example of a ruled surface is a cylinder; if we have a straight line and move it in a circle we create a cylinder made entirely of straight line. Note that the surface will only be a cylinder if all the lines are parallel. If the lines are not parallel we can create hyperboloids and cones depending on how much we have rotated. The rotation we are describing here is not a simple turning action, but more of a twisting motion—less like rotating a can by turning it and more like wringing out a washcloth by twisting it. Specifically, a cylinder is essentially two circles connected by rulings, if we keep one of the circles...
Post a Comment
Read more