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A hollow thin hemispherical shell of mass M and radius R rests with its flat face down on a horizontal rubber surface (which provides friction). A small tube is inserted through a hole at the top of the hemisphere and water is poured in. The density of water is rho. Find the maximum height h (measured from the flat base) to which water can be filled in the tube before the hemisphere tips or slides.
- h = M / (pi * R² * rho) + 2*R/3
- h = M / (pi * R² * rho) + R/3
- h = M / (2*pi * R² * rho) + 2*R/3
- h = M / (pi * R² * rho) + R
Correct answer: h = M / (pi * R² * rho) + 2*R/3
Solution
The water column in the tube creates pressure. The hemisphere remains on the surface as long as the rubber friction and normal force balance the system. The critical condition is when the net upward force of water pressure on the inside of the curved shell equals M*g (the shell's weight), since the water weight itself is supported by the base of the projected cylinder. The upward force on hemisphere from water = rho*g*h*(pi*R²) - weight of water inside hemisphere. Weight of water in hemisphere = rho*g*(2/3)*pi*R³. Force balance: rho*g*h*(pi*R²) - rho*g*(2/3)*pi*R³ = M*g. Solving: h = M/(pi*R²*rho) + 2R/3.
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