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# 9.3: Centroid of a Body

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### 9.3: Centroid of a Body

The centroid is an important concept in engineering, physics, and mechanics. It is the geometric center of a body. It always lies within the body except in cases with holes or cavities. When the material that a body is composed of is uniform or homogeneous, the centroid coincides with its center of mass or the center of gravity.

For a homogeneous body with constant density, the centroid can usually be found using equations representing a balance of the moments of the body's volume. If the volume has two planes of symmetry, then the centroid must lie along the line of intersection between those two planes. Additionally, for areas lying in an x-y plane bounded by a curve, this calculation for obtaining the coordinates for centroidal location can be done using a rectangular strip for the differential area element, and then integrating it. Similarly, the centroid can also be obtained for a line segment on an x-y plane described by a thin curve y = f (x). The centroidal coordinates for a line is determined using the following expressions:

The knowledge of the location of the centroid of a body helps to calculate moments of inertia and other important properties in physics and mechanics. Knowing these properties helps engineers design stable structures that can withstand various loads without buckling and other problems that may arise during their lifetime.