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A beaker of radius R holds water filled to height h. Water has density rho, surface tension T, and the atmospheric pressure is P0. Consider a vertical diametral plane ABCD slicing through the water column. Find the magnitude of the force that the water on one side of this plane exerts on the water on the other side.

1. |2*P0*R*h + pi*R²*rho*g*h - 2*R*T|
2. |2*P0*R*h + R*rho*g*h² - 2*R*T|
3. |P0*pi*R² + R*rho*g*h² - 2*R*T|
4. |P0*pi*R² + R*rho*g*h² + 2*R*T|

Correct answer: |2*P0*R*h + R*rho*g*h² - 2*R*T|

Solution

Integrating pressure over the vertical rectangular section (width 2R, height h): atmospheric gives P0*(2R*h), water's gauge pressure gives (rho*g*h/2)*(2R*h) = R*rho*g*h². Surface tension pulls inward along the top line of length 2R, contributing 2R*T. The net magnitude combines these.

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