Do Over: Integration Over a Region in a Plane

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.