Fluid Mechanics Principles: Pressure, Statics, and Buoyancy Calculations

Fundamental Concepts of Fluid Statics

Definition of Pressure

The Pressure (P) is defined as the ratio of the exerted Force (F) to the surface Area (S) over which it acts:

$$P = \frac{F}{S}$$

Hydrostatic Pressure

Hydrostatic Pressure is the pressure exerted by a liquid at all points within it.

Fundamental Equation of Fluid Statics

The pressure at a depth $h$ in a fluid of density $\rho$ under gravity $g$ is given by:

$$P = \rho g h$$

Communicating Vessels Principle

When several containers of different shapes containing the same liquid are connected at the bottom, the height of the liquid surface is identical in all vessels.

Application of Communicating Vessels

This principle is fundamental to water distribution systems for communities.

Determining Relative Density

By balancing pressures in two columns of different liquids ($\rho_1, \rho_2$) at heights $h_1$ and $h_2$:

$$\rho_1 g h_1 = \rho_2 g h_2$$

Simplifying by removing the gravitational acceleration ($g$):

$$\frac{\rho_1}{\rho_2} = \frac{h_2}{h_1}$$

Pascal's Principle

Pressure applied to an enclosed incompressible fluid is transmitted undiminished to every portion of the fluid and the walls of the containing vessel.

Applications of Pascal's Principle

The primary application is the hydraulic press, used when large forces are needed (e.g., for deforming or cutting materials).

Hydraulic Press Relationship

The relationship between forces ($F$) and piston areas ($S$) is based on equal pressure ($P_1 = P_2$):

$$\frac{F_1}{S_1} = \frac{F_2}{S_2}$$

Note: The area of a circular piston is calculated as $S = \pi R^2$.

Atmospheric Pressure Measurement

Atmospheric pressure ($P_{atm}$) is typically measured using a barometer (e.g., Torricelli's experiment), where the pressure balances the weight of a fluid column:

$$P_{atm} = \rho g h$$

Archimedes' Principle and Buoyancy

Buoyancy (Archimedes' Principle): Any body wholly or partially submerged in a fluid experiences an upward buoyant force ($E$) equal to the weight of the fluid displaced.

Origin of the Buoyant Force

The buoyant force arises from the difference in hydrostatic pressure acting on the top ($P_1$) and bottom ($P_2$) surfaces of the submerged object:

• Force on top: $F_1 = P_1 S$
• Force on bottom: $F_2 = P_2 S$

The resultant upward force ($R$) is the difference between these parallel and opposite forces:

$$R = F_2 - F_1 = (P_2 - P_1) S$$

Derivation of Buoyant Force Magnitude

Using the fundamental equation of fluid statics, where $P_2 - P_1 = \rho_{fluid} g h$ (and $h$ is the height of the object):

$$R = (\rho_{fluid} g h) S = \rho_{fluid} g (S h)$$

Since $S h$ equals the volume of the object ($V_{object}$), which is the volume of the displaced fluid ($V_{displaced}$):

$$R = E = \rho_{fluid} g V_{object}$$

This buoyant force ($E$) is equal to the weight of the displaced fluid ($W_{displaced}$).

Apparent Weight

The buoyant force ($E$) acts opposite to the weight ($W$) of the submerged object. The resulting force is the apparent weight ($W_{apparent}$):

$$W_{apparent} = W - E$$

Conditions for Flotation

Flotation depends on the comparison between the object's weight ($W$) and the buoyant force ($E$):

1. If $W < E$: The object rises and floats until the buoyant force equals the weight (partial submergence).
2. If $W = E$: The object remains in equilibrium, suspended at the point where it is placed within the fluid.
3. If $W > E$: The object sinks.