A spherical marble of radius $1\,$cm is stuck in a circular hole slightly smaller than its own radius (for calculation purposes , both are equal) at the bottom of a filled bucket of height $10\,$cm. Find the force on the marble due to the water.
I have always been troubled by problems like this. Does the marble not displace a certain volume of fluid? Should a buoyant force not act on it? However, in this problem, the answer happens to equal the product of the pressure, and the projection area....
And, when I came across this similar problem :-
A steel ball is floating in a trough of mercury. If we fill the empty part of the trough with water, what happens to the steel ball?
The answer to this one is that the steel ball rises.
Here, instead of multiplying the pressure and area of projection, and arguing that a net downward force acts, we argue that the steel ball displaces water, and causes an upward buoyant force to act.
My question is, when does one know which force to apply?