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If there is an object of a random shape and is submerged completely under water, would center of gravity of that object and the center of buoyancy coincide? Note that the object has uniform density. Only the shape is not uniform and random

I was reading about this topic when this question popped up in my head. I was reading some answers on the internet (of how center of gravity and center of buoyancy were different) on quora and there people had written that they would not coincide in case the object has non uniform density. However, will they still coincide if the density is uniform but the shape is different?

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  • $\begingroup$ When you say the object is submerged completely under water, is it being held under water by an external force, is it under the water but floating, or is it sinking? $\endgroup$
    – Bob D
    Commented Jul 11, 2021 at 19:04

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(Quick answer: Yes)

The definition of the center of buoyancy $B$ is that it is the center of mass of the displaced fluid. Hence, to find it, one looks at the undisturbed fluid (i.e. before the object is immersed) and focuses on the volume that will be occupied by the object. This will be an object-shaped volume of fluid, and its center of mass is $B$.

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In your example, the fluid is water, which is of uniform density, and the object is submerged. For any two identically-shaped bodies having different but uniform densities, the centers of mass are located at the same place (relative to the shape's boundary). This means the object-shaped volume of fluid has the same center of mass ($B$) as the body, and so the two coincide.

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Yes. For a completely submerged object, every particle of the object is being bouyed by a force equal to the weight of the fluid displaced. This "field of force" is acting like an acceleration field and being summed at the center of mass. As the system moves through the fluid, it may be reoriented by skin friction to an improved streamlined path. Portions with higher skin friction are oriented behind the object relative to its path of travel through the fluid.

A floating boat is floating because the center of bouyancy is located above the center of mass. The greater the distance between the two, the greater the "righting" moment if there is an upsetting of the floating objects orientation (wave action). Loading of the mass of the cargo (fish and water for example) in a boat higher relative to the center of bouyancy and it becomes unstable to being upset by waves (potentially taking on more water thereby increasing the upsetting mass). Continued loading of the mass above the center of bouyancy and the boat capsizes.

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