Application of the integral: Momentum and center of mass of a flat plate

A flat sheet can be considered any flat geometric figure, that is to say that it lacks thickness, this means that the flat sheet will have area and not volume.

This makes us think that if density is equal to mass over volume: D = M/V, for a flat sheet density will be equal to mass over area: D = M/A.

Since the objective is to calculate the mass of a flat density, we clear mass from the equation D = M/A and we are left with:

M = D x A

For the case of this post we will denote the density not by the letter D, but we will replace it by the Greek letter rho (𝜌).

Another important consideration we can make is that the flat sheet has an irregular shape whose density 𝜌 is uniform and whose area is going to be bounded by the graphs y = f(x) ; y = g(x) as shown in the following figure:

Definition of Moments and center of mass of a flat plate

Let f and g be continuous functions such that 𝑓(𝑥) ≥ 𝑔(𝑥)𝑓𝑜𝑟 [𝑎;𝑏] and consider the uniform density flat sheet 𝜌 bounded by the graphs 𝒚=𝒇(𝒙);𝒚=𝒈(𝒙) 𝒂𝒏𝒅 𝒂≤𝒙≤𝒃

[1] Moment with respect to the x-axis:

[2] Moment with respect to the y-axis:

[3] Center of mass :

[4] Flat sheet mass with uniform density:

Exercise: Calculate the center of mass of a flat sheet of uniform density.

Find the center of mass of the sheet of uniform density 𝜌 bounded by the graphs of f(x) = 4-x² and the x-axis.

Solution: To find the region of the plane that represents a uniform density sheet, I will plot the function f(x) = 4-x² using GeoGebra software:

As can be seen in the previous image, the graph is symmetrical with respect to the y-axis, so we can conclude that the coordinate of the center of mass with respect to the x-axis is equal to zero, so it only remains to calculate the coordinate of the center of mass with respect to the y-axis:

We calculate the mass of the flat sheet as follows:

Since the graph is symmetric about the y-axis, we can take the interval of [0;2] and multiply by two:

Now we proceed to calculate the moment at x (Mx):

Therefore, the coordinate ofof the center of mass of the flat sheet represented by f(x) = 4-x² is:

Therefore the point in the plane that represents the center of mass or equilibrium point of the plane sheet of uniform density is:

Bibliographic Reference

Calculus with Analytic Geometry by Ron Larson, Robert, P. Hostetler and Bruce H. Edwards. Volume I. Eighth Edition. McGraw Hill. Año 2006