Which of the following laws states that pressure or intensity of pressure at a point in static fluid is equal in all directions ?
Try it first: pick an option, then read the explanation.
Asked in SSC JE 23.03.2021 (Morning Shift)
Detailed Explanation
In a fluid at rest, shear stresses are identically zero across all possible planes. Under this condition, normal compressive stresses (pressures) acting on any arbitrary inclined plane passing through a point are equal in magnitude.
Derivation using an infinitesimal triangular fluid wedge of dimensions \(dx, dz, ds\) at rest:
1. Balancing forces in the horizontal \(x\)-direction:
\[ p_x (dy \, dz) - p_s (dy \, ds) \sin\theta = 0 \]
Since \(ds \sin\theta = dz\), dividing by \(dy \, dz\) gives:
\[ p_x = p_s \]
2. Balancing forces in the vertical \(z\)-direction:
\[ p_z (dy \, dx) - p_s (dy \, ds) \cos\theta - \frac{1}{2} \rho g (dx \, dy \, dz) = 0 \]
Neglecting third-order differentials and noting \(ds \cos\theta = dx\):
\[ p_z = p_s \]
3. Combining the two results yields Pascal's law:
\[ p_x = p_y = p_z = p \]
Why “Pascal's law” Is Correct
Pascal's law specifically states that the pressure intensity at any point in a static fluid is invariant with orientation—it acts equally in all directions normal to the boundary.
Why the Other Options Are Wrong
- A. Law of conservation of energy: Conservation of energy relates kinetic, potential, and flow energies (Bernoulli's equation in fluids).
- B. Law of conservation of mass: Conservation of mass gives the continuity equation (∂ρ/∂t + ∇·(ρv) = 0).
- C. Newton's law: Newton's laws govern force and motion (F = ma) or viscous shear stress (τ = μ du/dy).
Key Point
Pascal's Law: Pressure at any point in a fluid at rest is isotropic (equal in all directions).
Key Concepts for Students
- Pascal's law holds strictly for static fluids or ideal (inviscid) fluids where shear stress is zero.
- In a moving viscous fluid, normal stresses in orthogonal directions are unequal (σ_x ≠ σ_y ≠ σ_z).
- Hydraulic presses, jacks, and brakes rely directly on Pascal's principle to multiply applied forces.
Practice more: this is question 10 in Hydrostatic Pressure MCQs – Part 1. Try the whole set and check every answer instantly.