A liquid stands at the same level in the U-tube when at...

A liquid stands at the same level in the U-tube when at rest. If area of cross-section of both the limbs are equal, the difference in heights $h$ of the liquid in thetwo limbs of $U$-tube, when the system is given an acceleration $a$ in horizontal direction as shown, is (A) $\frac{g L^{2}}{a H}$ (B) $\frac{L a}{g}$ (C) $\frac{L^{2} a}{H g}$ (D) $\frac{H g}{a}$

A liquid stands at the same level in the U-tube when at rest. If area of cross-section of both the limbs are equal, the difference in heights $h$ of the liquid in thetwo limbs of $U$-tube, when the system is given an acceleration $a$ in horizontal direction as shown, is
(A) $\frac{g L^{2}}{a H}$
(B) $\frac{L a}{g}$
(C) $\frac{L^{2} a}{H g}$
(D) $\frac{H g}{a}$

Key Concepts

Non-Inertial Reference Frame

When a system accelerates, observers within the system are in a non?inertial frame where inertial forces appear. These fictitious forces arise due to the acceleration and modify the apparent forces acting on the fluid, making it essential to incorporate them in the analysis of hydrostatic equilibrium.

Effective Gravity

The combination of the true gravitational force (acting downward) and the inertial force (acting horizontally due to acceleration) results in an effective gravity. This effective acceleration vector determines the direction in which the pressure gradient in the fluid must balance, thereby tilting the liquid’s free surface.

Hydrostatic Pressure Distribution

In a fluid at rest (or in equilibrium in a non-inertial frame), the pressure variation is governed by the balance between the pressure gradient and the effective gravitational force. This principle is applied to determine how pressure changes with position in the system, which in turn leads to a difference in liquid heights in connected columns.

Free Surface Alignment

A key consequence of the effective gravitational field is that the free surface of the liquid must form a plane perpendicular to the effective gravity. This geometrical constraint explains why, upon horizontal acceleration, the liquid's surface in a U-tube shifts to different levels in the two limbs, with the height difference being a function of the acceleration and the dimensions of the system.

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