In this video I go through an example problem on how to find the center of gravity of a 3D composite body with varying density. Check out the steps below for more details.
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Steps to finding the center of mass of a composite body:
1) Establish a coordinate system for the object
2) Break the object up into simpler shapes (composite parts) that you can more easily find the centroid of
3) Find and list the coordinates of the centroid of each composite part
4) Calculate and list the area (or volume for 3D objects) of each composite part
5) Plug the coordinates of the centroid and the areas into the following equations
x ̅=(Σ x ̃*A)/(Σ A) y ̅=(Σ y ̃*A)/(Σ A) z ̅=(Σ z ̃*A)/(Σ A)
where x ̃,y ̃,and z ̃ are the x, y, and z coordinates respectively of the center of mass of a composite part and A is the area of a composite part. If you are dealing with a 3D object then replace A with the volume of each component part. Each coordinate is multiplied by the area or volume of that same composite shape. x ̅,y ̅,and z ̅ are the x, y, and z coordinates respectively of the entire area or object.
Notes:
- In order for the above process to work to find the centroid of an object, it must be of constant density. If the entire object is not of constant density then you break it up into components that are of constant density and in the equations you replace volume with mass or weight. Remember that mass is the volume multiplied by density.
- If the object is symmetrical about an axis, then centroid will lie on that axis of symmetry and you will not need to calculate the one of the coordinates of the centroid. For example, if a 2D object is symmetrical about the y axis then you will not need to calculate the x coordinate of the centroid x ̅.