Hydrostatic Law Conditions

The hydrostatic law is a key principle in fluid mechanics, explaining how pressure changes within a stationary fluid. It typically states that the pressure difference ($ \Delta P $) between two points in a static fluid is proportional to the vertical distance ($h$) separating them, the fluid's density ($\rho$), and the acceleration due to gravity ($g$). The formula is often written as $ \Delta P = \rho g h $, indicating pressure increases linearly with depth.

Analyzing Hydrostatic Law Stability Under Motion

We need to determine where this law remains unchanged ('unaltered') despite external factors.

• Option 1: Holds unaltered for a liquid in a vessel subjected to constant rotation.

  In a fluid rotating constantly with angular velocity $\omega$, a centrifugal force acts outwards. In the reference frame of the rotating fluid, the pressure gradient is given by $ \nabla P = \rho (\vec{g} - \omega^2 \vec{r}_{perp}) $. If the rotation is about a vertical axis, the vertical component of the pressure gradient is $ \frac{\partial P}{\partial z} = \rho g $. This specific relationship concerning depth is maintained. Although rotation introduces a radial pressure variation ($ \frac{\partial P}{\partial r} = \rho \omega^2 r $) and changes the free surface shape to a paraboloid, the vertical hydrostatic relationship $ \frac{\partial P}{\partial z} = \rho g $ is sometimes considered the aspect of the 'hydrostatic law' that holds.

• Option 2: Holds unaltered for a liquid in a vessel subjected to constant acceleration.

  When a fluid system undergoes constant linear acceleration ($\vec{a}$), the effective gravity changes to $ \vec{g}_{eff} = \vec{g} - \vec{a} $. The pressure gradient follows $ \nabla P = \rho \vec{g}_{eff} $. Since the effective gravity is different from $ \vec{g} $, the standard hydrostatic law is altered.

• Option 3: Doesn’t hold for the atmosphere.

  The hydrostatic law is fundamental to understanding atmospheric pressure and density variations with altitude. It serves as a crucial approximation in atmospheric science.

• Option 4: Holds for a liquid in equilibrium only.

  This statement accurately reflects the primary definition of the hydrostatic law's applicability (fluids at rest). However, the question asks about conditions where it holds 'unaltered', prompting evaluation of non-equilibrium scenarios.

Rotation Effects on Hydrostatic Law

Considering the options, the scenario involving constant rotation (Option 1) is identified as the one where a key aspect of the hydrostatic law (the vertical pressure gradient) remains consistent, despite modifications to the overall pressure distribution due to centrifugal effects.