Consider two concentric thin wall steel tubes, the larger one closed at the bott

Question

Consider two concentric thin wall steel tubes, the larger one closed at the bottom and the inner one open. The tubes are filled with a fluid. A well fitting cylindrical plug that slides easily is dropped in the inner tube and a well fitting annular ring (doughnut) which slides easily is dropped between the two tubes. The weight of the doughnut plug (F2) is 3X the weight of the cylindrical plug (F1). After oscillating a little, the plugs settle down concentrically with one another. What must be the relationship between the radii of the tubes? (Ignore the wall thickness of the tubes).

Explanation / Answer

We know that pressure in a confined fluid is same

So if the height from bottom of both plugs are same,

then the pressure generated by each plug is same:

so

mg/(pi*r^2) = mg/(pi*(R^2-r^2))

ie same pressure ,R = outer radius,r = inner radius

=>r^2 = R^2-r^2

=>2*r^2 = R^2

=>R = r*sqrt(2)

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