In an incompressible static fluid column, the rate of increase of pressure in the vertical direction is equal to weight density of fluid at that point. Which of the following is the law for this statements ?
Try it first: pick an option, then read the explanation.
Asked in SSC JE 23.03.2021 (Morning Shift)
Detailed Explanation
The variation of pressure within a static fluid column is governed by the equilibrium of gravity and pressure forces on a fluid element. This foundational principle is termed the Hydrostatic Law.
Consider a small cylindrical fluid element of cross-sectional area \(dA\) and height \(dz\) at depth \(z\) in a static fluid:
1. Balancing vertical forces on the element:
\[ p \cdot dA - (p + dp) \cdot dA - \rho g \, dA \, dz = 0 \]
\[ -dp \cdot dA = \rho g \, dA \, dz \;\implies\; \frac{dp}{dz} = -\rho g = -w \]
2. Expressing with respect to depth \(h\) measured vertically downward (\(h = -z\)):
\[ \frac{dp}{dh} = w = \rho g \]
Where \(w\) (or \(\gamma\)) is the specific weight (weight density) of the fluid.
Why “Hydrostatic law” Is Correct
The statement is the exact verbatim definition of the Hydrostatic Law: the rate of increase of pressure intensity in the vertically downward direction equals the weight density (specific weight \(w = \rho g\)) of the fluid.
Why the Other Options Are Wrong
- A. Bernoulli's law: Bernoulli's law applies to moving fluids in steady, frictionless, incompressible streamline flow.
- B. Viscosity law: Newton's law of viscosity governs viscous shear stress in flowing fluids: τ = μ(du/dy).
- D. Pascal's law: Pascal's law states that pressure at a point in a fluid at rest is isotropic (equal in all directions).
Key Point
Hydrostatic Law: dp/dh = w = ρg (pressure increases with depth at the rate of specific weight).
Key Concepts for Students
- Pressure in a static fluid is independent of horizontal coordinates (∂p/∂x = 0, ∂p/∂y = 0).
- Surfaces of constant pressure (isobars) in a static fluid under gravity are always horizontal planes.
- Integrating dp = ρg dh for constant density yields the familiar relation p = ρgh = wh.
Practice more: this is question 8 in Hydrostatic Pressure MCQs – Part 1. Try the whole set and check every answer instantly.