Does buoyancy exert the same or different upward force on two dissimilarly shaped, but same volume objects? Suppose I have two objects of exactly the same volume, one is a sphere and one is a cube.
If the objects are completely submerged in water, is the upward buoyancy force the same or different on these two objects?
Based on this link the objects would experience the same buoyant force. For me, this is counter-intuitive.
My thinking is that given that the cube is oriented in the water such that the bottom flat surface is horizontal, it seems that the cube would experience a greater upward force do the efficiency of the flat-bottom surface, as opposed to the round-bottom surface of the sphere.
I'm assuming I am incorrect or incomplete in my perspective and I would appreciate an explanation as to what I am missing?
This occurs to me - Is it that since both objects are the same volume, that all the upward and downward forces on each object equalize and so the average buoyant force is the same regardless of shape or object orientation in the water?
 A: Why do you consider the flat bottom to be 'more efficient'? Suppose you had a hemispherical object- by your logic, the buoyancy would depend on whether the object was in the water with its flat or curved surface uppermost, and perhaps if the flat surface was oriented vertically, the effect of the 'more efficient' surface would be to move the object sideways.
Mathematically, when you integrate the pressure over the surface enclosing a volume, the result is dependent only on the enclosed volume, and not on the shape of the surface.
A: Good question!
The Archimedes principle tells us that the buoyant force, or the force exerted by the fluid on an immersed object, is the same as the weight of the volume of liquid displaced by it (which is nothing but the volume of the object itself). You can calculate the buoyant force by replacing the object with an equivalent volume of water and then calculating its weight.

Is it that since both objects are the same volume, that all the upward
and downward forces on each object equalize, and so the average buoyant
force is the same regardless of shape or object orientation in the
water?

Yes. The buoyant force depends only on the object's volume, as the water only 'sees' the volume of the immersed object.
The force on the cube and the sphere may not be identical at any particular point, but they will always add up to the same value, equal to $\rho_wVg$.
What do I mean by that? I mean that if you take the exact distribution of the buoyant force on the surface of the sphere, it will look entirely different from that of the cube. But if you integrate the components of these forces over the entire surface, you will always arrive at the same value. So the buoyant force does not change if the immersed object is a cube or a sphere, or a human.
This is an interesting observation because it allows us to ignore the complexities of the object's shape while calculating its volume, buoyant force, and weight. Had we not done so, it would have turned out to be an arbitrarily challenging (and much more interesting) exercise.
This is the exact reason why I find the Archimedes principle interesting. It shows how we can easily tackle some seemingly complex problems by simply viewing them differently.
If you are interested in knowing more about how the Archimedes principle works in the first place, imagine a water unicorn (a version of the unicorn made out of water) immersed in a tub of water. Which is to say, we have a stationary tub of water. What all are the forces acting upon the water unicorn?

Notice that the water unicorn doesn't sink (because it's just water!).
So there must be some force equal and opposite to the weight of the water unicorn. We do not care about what all forces there are or where they act upon. The only thing that really matters is that the forces prevent the water unicorn from sinking, so they must add up to $\rho_wVg$. It is the resultant of these forces we refer to as buoyancy. In other words, the buoyant force always acts upwards on the center of mass of the water unicorn and has a magnitude of $\rho_wVg$. Notice that even if our unicorn curls up into a ball or a cube or turns around, its weight will not change, so neither will the buoyancy.



Now imagine that the water unicorn magically transformed into a normal unicorn.

The buoyant force of the water stays the same, but now the weight of the unicorn is significantly lesser than that of the water unicorn (because $\rho_u<\rho_w)$. This would mean that the two forces do not cancel, and there will be a net upward force that would propel the unicorn upwards.
What if our unicorn transforms into an iron unicorn? Its weight would be greater than the weight of the water unicorn (because $\rho_u>\rho_w$), and there will be a net downward force that would make it sink.

In short, the buoyant force does not depend on the material that constitutes the unicorn. It only depends on the weight of the water unicorn, which is independent of the shape or orientation of the unicorn.
You can also see that the only factor that decides whether an object sinks or floats is its average density. This is why we can use Archimedes to separate pure gold from impure gold and good seeds from bad seeds.
A: 
My thinking is that given that the cube is oriented in the water such
that the bottom flat surface is horizontal, it seems that the cube
would experience a greater upward force do the efficiency of the
flat-bottom surface, as opposed to the round-bottom surface of the
sphere.

The buoyant force has nothing to do with the shape or orientation of the object. It is only determined by the weight of the water displaced.

This occurs to me - Is it that since both objects are the same volume,
that all the upward and downward forces on each object equalize and so
the average buoyant force is the same regardless of shape or object
orientation in the water?

The buoyant force is guaranteed to be the same for both objects only as long as they are both kept submerged, unless the density (and thus weight) of both objects is the same. Otherwise, the upward buoyant force on each object will depend on the density of each object relative to the density of water.
An object having density less than water will rise until the weight of the water corresponding to the portion of the volume remaining submerged equals the weight of the object, at which point the downward force of the weight will equal the upward buoyant force. But that upward buoyant force will be less than that of the object completely submerged
An object having density equal to water will float completely submerged. The downward weight of the object exactly equals the upward buoyant force (weight of the volume of the water displaced).
Finally, an object having density greater than water will sink because its downward weight is greater than the maximum upward buoyant force.
Hope this helps.
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If you take a hollow cylinder and chuck it in the water, it will float, laying horizontally. It will experience the same buoyant force whichever way you orientate it (say you restrain it in a vertical mesh tube) because it will displace the same volume of water. But the Potential Energy will be at its lowest with the cylinder horizontal. You need to do work on it to make it float upright, despite the fraction of water that's displaced being the same. Orientation and shape can appear to have an effect on upthrust when they don't actually.
