The pressure at the left bottom due to left vertical pipe ${=\mathrm{P}}_{\mathrm{L}}={\mathrm{P}}_{\mathrm{a}\mathrm{t}\mathrm{m}}+\mathrm{h}\left(2\mathrm{\rho }\right)\mathrm{g}$

The pressure at the right bottom due to right vertical pipe $={\mathrm{P}}_{\mathrm{R}}={\mathrm{P}}_{\mathrm{a}\mathrm{t}\mathrm{m}}+\mathrm{h}\left(\mathrm{\rho }\right)\mathrm{g}$

and Pressure difference in the horizontal tube due to acceleration

${\mathrm{P}}_{\mathrm{L}\mathrm{R}}=\left(\frac{\mathrm{L}}{2}\right)\mathrm{\rho }\mathrm{a}+\left(\frac{\mathrm{L}}{2}\right)2\mathrm{\rho }\mathrm{a}$

${\mathrm{P}}_{\mathrm{L}\mathrm{R}}=\frac{3\mathrm{L\rho a}}{2}$

The pressure at the left bottom due to right vertical pipe and horizontal pipe

${\mathrm{P}}_{\mathrm{L}}={\mathrm{P}}_{\mathrm{R}}+{\mathrm{P}}_{\mathrm{L}\mathrm{R}}$

${\mathrm{P}}_{\mathrm{L}}={\mathrm{P}}_{\mathrm{a}\mathrm{t}\mathrm{m}}+\mathrm{h}\mathrm{\rho }\mathrm{g}+\frac{3\mathrm{L\rho a}}{2}$

The pressure at the left bottom due to left vertical pipe=Pressure at the left bottom due to right vertical pipe and horizontal pipe

${\mathrm{P}}_{\mathrm{a}\mathrm{t}\mathrm{m}}+\mathrm{h}\left(2\mathrm{\rho }\right)\mathrm{g}={\mathrm{P}}_{\mathrm{a}\mathrm{t}\mathrm{m}}+\mathrm{h}\mathrm{\rho }\mathrm{g}+\frac{3\mathrm{L\rho a}}{2}$

  $\Rightarrow \mathrm{h}=\frac{3\mathrm{aL}}{2\mathrm{g}}$