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24.2 CENTROIDS
À pThis is also known as the geometric centre, or first moment of area.
The fundamental definition for Centroids is given below, although a more efficient method for problems with simple geometries is discussed later.
When finding centroids, look for symmetry, the centroids will be in the center of symmetrical sections.
A simple example of application for these relationships is given below,
Lets try a more practical problem using the centroid, ([Hibbeler, 1992], prob. 9-16, pg. )
24.2.1 Finding Centroids Using Composite Shapes
(|This method basically uses tables, or obvious centroids for basic shapes in a complex shape. By using a summation of centroids, weighted by areas/volumes divided by total area.
A simple example for a two dimensional problem is given below, ([Hibbeler, 1992], prob. 9-48, pg. )
Next, lets consider a problem using the composite method in 3D volumes, ([Hibbeler, 1992], prob 9-68, pg. )
An example of a rouhly distributed load can be seen below,
24.2.2 Practice Problems
(|1. The 500 Kg plate below is to be used in an industrial machine. The basic shape is rectangular, but there is an angled arch cut in the bottom. Considering the centroid, and the applied force, determine the magnitude of the force (P) that is required to pull out the wedge if the both sides of the wedge have a coefficient of friction of 0.2.
2. The plate below has a triangular hole. We are asked to select a base width `b' for the triangle so that the reaction at roller B is 200N. The plate material weighs 1Kg per square meter.
3. We want to design two similar supports for a new sculpture. This means that the supports should be positioned to carry the same load. Find the distance of the second support from the first, and indicate the loads to be supported by each if the plate weights 1000kg per square meter.
4. An object is perched on the far end of a lever and we want to determine if it will tip, or slip off, or remain in place if the coefficient of friction is 0.2. (Note: the 2m distance to the right is to the centroid of the block)
5. The 500 Kg plate below is to be used in an industrial machine. The basic shape is rectangular, but there is an arch cut in the bottom. Considering the centroid, and the applied force, determine the magnitude of the force (P) that is required to pull out the wedge if both sides of the wedge have a coefficient of friction of 0.2.
6. A thin flat plate is suspended by means of a screw eye and 2 cords, AC and BC, as depicted in the diagram below. Given that the plate has a mass of 30kg per square meter of surface area, determine the reaction force at point O and the tension forces in the cords.
7. Find the x-y centroid of the shape below which is essentially rectangular, except for the shape cut out of the bottom (the function is given).
8. What is the x-y centroid of the shape below if the plate is homogenous?
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