File Name: center of mass and centroid .zip
In this section, we consider centers of mass also called centroids, under certain conditions and moments.
- Center of Gravity, Center of Mass, Centroids
- 6.6: Moments and Centers of Mass
- Centers of Mass and Moments of Inertia
We have already discussed a few applications of multiple integrals, such as finding areas, volumes, and the average value of a function over a bounded region. In this section we develop computational techniques for finding the center of mass and moments of inertia of several types of physical objects, using double integrals for a lamina flat plate and triple integrals for a three-dimensional object with variable density. The density is usually considered to be a constant number when the lamina or the object is homogeneous; that is, the object has uniform density.
Center of Gravity, Center of Mass, Centroids
The center of mass for an object can be thought as the point about which the entire mass of the object is equally distributed. In general, the center of mass and moments of a lamina can be determined using double integrals. However, in certain special cases when the density only depends on one coordinate, the calculations can be performed using single integrals. In this case the centroid of the lamina is determined by formulas. In case of constant density, the centroid has the following coordinates. The centroid of a triangle is the point of intersection of the medians of the triangle.
6.6: Moments and Centers of Mass
This page references the formulas for finding the centroid of several common 2D shapes. In the figures, the centroid is marked as point C. Its position can be determined through the two coordinates x c and y c , in respect to the displayed, in every case, Cartesian system of axes x,y. General formulas for the centroid of any area are provided in the section that follows the table. The centroid of an object represents the average location of all particles of the object. It can be defined for objects of any dimension, such as lines, areas, volumes or even higher dimension objects.
The centroid of a volume can be thought of as the geometric center of that shape. If this volume represents a part with a uniform density like most single material parts then the centroid will also be the center of mass, a point usually labeled as 'G'. Just as with the centroids of an area, centroids of volumes and the center of mass are useful for a number of situations in the mechanics course sequence, including the analysis of distributed forces, simplifying the analysis of gravity which is itself a distributed force , and as an intermediate step in determining mass moments of inertia. Just as with areas, the location of the centroid or center of mass for a variety of common shapes can simply be looked up in tables, such as the table provided in the right column of this website. However, we will often need to determine the centroid or center mass for other shapes and to do this we will generally use one of two methods. On this page we will only discuss the first method, as the method of composite parts is discussed in a later section. The tables used in the method of composite parts however are derived via the first moment integral, so both methods ultimately rely on first moment integrals.
Engineering Mechanics — Statics 9. HCM City Univ. Fundamental Problems Fundamental Problems - F9. Composite Body - Example 9. Fundamental Problems - F9.
Centers of Mass and Moments of Inertia
Using a single integral we were able to compute the center of mass for a one-dimensional object with variable density, and a two dimensional object with constant density. With a double integral we can handle two dimensions and variable density. You may want to review the concepts in section 9. The key to the computation, just as before, is the approximation of mass.
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