When at rest, a liquid stands at the same level in the tubes...

When at rest, a liquid stands at the same level in the tubes shown in figure. But as indicated a height difference $h$ occurs when the system is given an acceleration a towards the right. Here, $h$ is equal to (a) $\frac{a L}{2 g}$ (b) $\frac{g L}{2 a}$ (c) $\frac{g L}{a}$ (d) $\frac{a L}{g}$

When at rest, a liquid stands at the same level in the tubes shown in figure. But as indicated a height difference $h$ occurs when the system is given an acceleration a towards the right. Here, $h$ is equal to
(a) $\frac{a L}{2 g}$
(b) $\frac{g L}{2 a}$
(c) $\frac{g L}{a}$
(d) $\frac{a L}{g}$

Key Concepts

Hydrostatic Equilibrium in Accelerating Fluids

Hydrostatic equilibrium in a fluid requires that the pressure remains constant along a surface. In an accelerating system, even though the fluid experiences additional apparent forces, the free surface adjusts itself so that it is perpendicular to the net force. This condition determines how much the liquid’s surface will tilt and hence the vertical displacement over a given horizontal distance.

Non-Inertial Reference Frames

When an observer is in an accelerating frame, such as a container moving with acceleration, fictitious or inertial forces must be introduced to correctly apply Newton’s laws. This means that even though the container is not at rest in an inertial frame, the fluid surfaces can be analyzed as if a modified gravitational field is present, incorporating both the real gravitational force and the fictitious force due to acceleration.

Effective Gravity

Effective gravity is the vector sum of the true gravitational acceleration and the acceleration of the system (often introduced as a fictitious force). In an accelerating frame, the direction of the effective gravity determines the orientation of equilibrium surfaces. The liquid surface aligns perpendicular to this effective gravity, resulting in a tilt that reflects the ratio of the acceleration to gravitational acceleration.

Tilt of a Liquid Surface in Accelerated Systems

When a container holding a liquid accelerates, the free surface of the liquid tilts so that it becomes perpendicular to the effective gravitational field. This tilt can be quantified by the angle whose tangent is equal to the ratio of the horizontal acceleration to the gravitational acceleration. This principle is key in understanding fluid behavior under acceleration.

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