Centroids & Moment of Inertia
The centroid, or centre of gravity, of any object is the point within that object from which the force of gravity appears to act. An object will remain at rest if it is balanced on any point along a vertical line passing through its centre of gravity. In terms of moments, the centre of gravity of any object is the point around which the moments of the gravitational forces completely cancel one another.
hanging rock
The centre of gravity of a rock (or any other three dimensional object) can be found by hanging it from a string. The line of action of the string will always pass through the center of gravity of the rock. The precise location of the center of gravity could be determined if one would tie the string around the rock a number of times and note each time the line of action of the string. Since a rock is a three dimensional object, the point of intersection would most likely lie somewhere within the rock and out of sight.
The centroid of a two dimensional surface (such as the cross-section of a structural shape) is a point that corresponds to the centre of gravity of a very thin homogeneous plate of the same area and shape. The planar surface (or figure) may represent an actual area (like a tributary floor area or the cross-section of a beam) or a figurative diagram (like a load or a bending moment diagram). It is often useful for the centroid of the area to be determined in either case.
Symmetry can be very useful to help determine the location of the centroid of an area. If the area (or section or body) has one line of symmetry, the centroid will lie somewhere along the line of symmetry. This means that if it were required to balance the area (or body or section) in a horizontal position by placing a pencil or edge underneath it, the pencil would be best laid directly under the line of symmetry.
bilateral symmetry
If a body (or area or section) has two (or more) lines of symmetry, the centroid must lie somewhere along each of the lines. Thus, the centroid is at the point where the lines intersect. This means that if it were required to balance the area (or body or section) in a horizontal position by placing a nail underneath it, the point of the nail would best be placed directly below the point where the lines of symmetry meet. This might seem obvious, but the concept of the centroid is very important to understand both graphically and numerically. The position of the centre of gravity for some simple shapes is easily determined by inspection. One knows that the centroid of a circle is at its centre and that of a square is at the intersection of two lines drawn connecting the midpoints of the parallel sides. The circle has an infinite number of lines of symmetry and the square has four.
centroids of hollow sections
The centroid of a section is not always within the area or material of the section. Hollow pipes, L shaped and some irregular shaped sections all have thier centroid located outside of the material of the section. This is not a problem since the centroid is really only used as a reference point from which one measures distances. The exact location of the centroid can be determined as described above, with graphic statics, or numerically.
The centroid of any area can be found by taking moments of identifiable areas (such as rectangles or triangles) about any axis. This is done in the same way that the centre of gravity can be found by taking moments of weights. The moment of an large area about any axis is equal to the algebraic sum of the moments of its component areas. This is expressed by the following equation:
Sum MAtotal = MA1 + MA2 + MA3+ …
The moment of any area is defined as the product of the area and the perpendicular distance from the centroid of the area to the moment axis. By means of this principle, we may locate the centroid of any simple or composite area.
Galway Maths Grinds 