The Nature of Fluid Pressure
When we talk about fluids at rest, we are primarily concerned with how they exert force on surfaces. Unlike solids, which can support shear stress, fluids in equilibrium cannot resist tangential forces. Therefore, the force exerted by a fluid at rest on any surface is always directed normal to that surface.
Pressure is defined as the normal force exerted by a fluid per unit area. Mathematically, if a force dF acts on a small area dA, the pressure P is given by P = dF/dA. In the SI system, the unit is the Pascal (Pa), where 1 Pa = 1 N/m². Because fluids are continuous media, pressure at a point acts equally in all directions, provided the fluid is at rest.
Pressure Variation with Depth
In a stationary fluid column under the influence of gravity, pressure is not uniform; it increases with depth. Consider a small cylindrical element of fluid at a depth h. The weight of the fluid column above it must be supported by the pressure difference between the bottom and the top of the element.
The fundamental equation for pressure variation is P = P₀ + ρgh, where P₀ is the atmospheric pressure at the surface, ρ is the density of the fluid, g is the acceleration due to gravity, and h is the depth. This linear relationship implies that at any given horizontal level within a continuous body of static fluid, the pressure is constant.
“In a continuous fluid at rest, all points at the same horizontal level are at the same pressure, regardless of the shape of the container.”
Pascal’s Law and its Applications
Pascal’s Law states that a change in pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid and to the walls of the container. This principle is the cornerstone of many hydraulic systems used in engineering.
The most common application is the Hydraulic Lift. By applying a small force on a small piston, we can generate a large force on a larger piston because the pressure (Force/Area) remains constant across the system. If F₁/A₁ = F₂/A₂, then F₂ = F₁ × (A₂/A₁). This allows for massive mechanical advantage, enabling heavy objects to be lifted with minimal effort.
Buoyancy and Archimedes’ Principle
When an object is immersed in a fluid, it experiences an upward force known as the buoyant force. This phenomenon is explained by Archimedes’ Principle, which states that the buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object.
- Floating: The object is in equilibrium when the buoyant force equals the weight of the object (Weight = Buoyant Force).
- Sinking: The object sinks if its weight exceeds the maximum possible buoyant force (when fully submerged).
- Apparent Weight: An object submerged in a liquid appears lighter because the buoyant force acts in opposition to gravity. W_apparent = W_actual – F_buoyant.
Important Facts and Formulas
| Concept | Formula |
|---|---|
| Pressure | P = F / A |
| Hydrostatic Pressure | P = P₀ + ρgh |
| Buoyant Force | F_b = ρ_fluid × V_displaced × g |
| Pascal’s Law | F₁/A₁ = F₂/A₂ |
Key Points to Remember
- Pressure is a scalar quantity, but it exerts force in a direction normal to the surface.
- Atmospheric pressure at sea level is approximately 1.013 × 10⁵ Pa.
- Density (ρ) of water is 1000 kg/m³; this is a standard value used in most JEE problems.
- The buoyant force depends only on the volume of the displaced fluid, not the depth of the object, provided it is fully submerged.
- For an object floating, the fraction of volume submerged is given by the ratio of the densities: V_submerged / V_total = ρ_object / ρ_fluid.
- Barometers and manometers are common instruments used to measure atmospheric and gas pressures, respectively.
Quick Revision Summary
- Fluids at rest cannot support shear stress; force is always normal.
- Pressure increases linearly with depth in a static fluid column.
- Pascal’s Law allows for force multiplication in hydraulic systems.
- Buoyant force is equal to the weight of the fluid displaced (Archimedes’ Principle).
- Floating objects displace a weight of fluid equal to their own weight.
- Always check if the problem asks for “gauge pressure” (P – P₀) or “absolute pressure” (P).
- Vertical equilibrium equations (ΣF = 0) are the primary tools for solving statics problems.
