Accelerating U-tube

Classical Mechanics Level pending

A U-tube with a very small diameter, vertical sections of height h and horizontal section of length 2h is filled with mercury of density ρ such that the mercury only occupies the horizontal section. The U-tube is capped tightly at both ends and then accelerated to the right such that the mercury column occupies a length of \( \frac { 5h }{ 3 } \) inside the horizontal section. Atmospheric pressure is \({ P }_{ 0 }\). If the acceleration of the tube can be expressed as \( \frac { { AP }_{ 0 }+B\rho gh }{ C\rho h } \), where A, B and C are positive integers, find the values of C-(A+B).

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