A U-tube containing a liquid is accelerated horizontally...

A U-tube containing a liquid is accelerated horizontally with a constant acceleration $a .$ If the separation between the two vertical limbs is $l$, then the difference in the heights of the liquid in the two arms is
(a) zero
(b) $l$
(c) $\frac{l a}{g}$
(d) $\frac{l a}{g}$

Key Concepts

Hydrostatic Equilibrium

Hydrostatic equilibrium refers to the balance between the pressure gradient in a fluid and the body forces (including both actual gravitational force and the pseudo force due to acceleration). In such problems, calculating the pressure distribution using the effective gravitational field is crucial to determine variations in the liquid surface levels.

Liquid Surface Orientation

In an accelerated container, the free surface of the liquid adjusts to be perpendicular to the effective gravitational field. This principle explains why the liquid levels in the U-tube become different: the inclined free surface creates a height difference that is proportional to the acceleration and the separation between the tube limbs.

Non-Inertial Reference Frame

When a system is accelerated, it becomes a non-inertial reference frame where pseudo forces must be taken into account. This concept is essential for analyzing problems in accelerated frames because the standard form of Newton’s laws must be modified to include fictitious forces that arise due to the acceleration of the frame.

Effective Gravity

In an accelerated frame, the actual gravitational acceleration and the acceleration of the frame combine to form an effective gravity vector. This effective gravity alters the apparent direction and magnitude of the gravitational pull experienced by the fluid, which is key to understanding how pressure gradients develop within the liquid.

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