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

A U-tube containing a liquid is accelerated horizontally with a constant acceleration $a_{0} .$ If the separation between the vertical limbs is $l$, find the difference in the heights of the liquid in the two arms.

Key Concepts

Equipotential Surfaces

Equipotential surfaces are surfaces within a fluid where the pressure remains constant. In a situation involving an accelerating reference frame, these surfaces are oriented perpendicular to the effective gravitational field. Understanding equipotential surfaces is essential as they dictate the shape and orientation of the fluid's free surface in equilibrium.

Non-Inertial Reference Frame

A non-inertial reference frame is one that is accelerating relative to an inertial (non-accelerating) frame. In such frames, fictitious forces, like the inertial force, must be introduced to account for observed phenomena. This concept is crucial when analyzing systems where the observer is in a moving frame, as it allows the proper application of Newton’s laws by accounting for additional apparent forces.

Effective Gravitational Field

In an accelerating frame, the actual gravitational field combines vectorially with the fictitious inertial acceleration, resulting in an effective gravitational field. This effective field better describes the direction and magnitude of the perceived force acting on objects within the accelerating system, and it defines the orientation that forces and pressures in a fluid follow.

Hydrostatic Equilibrium

Hydrostatic equilibrium refers to a state in a fluid where the pressure gradient balances all body forces, such as gravity and any fictitious forces arising in an accelerated frame. Under hydrostatic equilibrium, the fluid’s free surface aligns perpendicular to the effective gravitational field, ensuring that the pressure on the free surface remains constant.

Free Surface Inclination

The free surface of a liquid in an accelerating frame will tilt so that it becomes perpendicular to the effective gravitational field. This inclination leads to a difference in height across the liquid’s surface over a horizontal span, with the magnitude of the tilt being related to the ratio of the horizontal acceleration to the gravitational acceleration. This concept explains why a liquid in a U-tube exhibits a height difference between its arms when subject to horizontal acceleration.

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