Pressure, Pascal's Law of Pressure and Upthrust

Hydrostatic is the study of fluid at rest. A substance that can flow from one point to another is called fluid. The density of a substance is defined as mass per unit volume.

$$ \text {Density,} \rho = \frac mV $$

The relative density of a substance is the ratio of its density of water at 4^oC. It is also called the specific gravity.

$$ \text {Relative density,} \rho _r = \frac {\rho }{\rho _w} $$

Where ρ is the density of the substance and ρ_w is the density of water at 4⁰C.

Thrust

A force acting perpendicularly to a surface is called the thrust. For example, our weight on the ground, the weight of the liquid in a beaker at the bottom etc. are examples of thrust.

Pressure

Force per unit area acting normally on the surface is called the pressure. Mathematically,

$$ P =\frac FA $$

So, pressure is also defined as the thrust per unit area of a surface. The direction of the force resulting from the pressure is determined by the orientation of the surface, and therefore, fluid pressure acts as a scalar quantity.

The unit of pressure is N/m² in SI-units which is also called Pascal, Pa. the dimension of pressure is

[MLT^-2]/[L²] = [ML^-1T^-2]

Derivation of Pressure

Liquids exert pressure at the container due to their weight.

Suppose a liquid in a container such as water in a beaker as shown in the figure. To find the pressure at a point inside it, let us consider a horizontal surface X of area A at that point at a depth h from the free surface of the liquid.

The force acting normally on X is weight of the liquid column directly above it a height h and cross-sectional area A. Since the volume of this liquid column, V = A × h and its mass, m = ρ × V, where ρ is its density, so weight of the liquid is

\begin{align*} W &= mg = \rho V g\\ \text {As the weight acts normally on X, the pressure at the surface is} \\ P &= \frac FA = \frac WA &= \frac {\rho A hg}{A} \rho &= h\rho g \end{align*}

This is an expression for the pressure exerted by a liquid at a depth h. It can be shown that the pressure exerted by a liquid at the sides of the container is the same as the same depth downward.

Laws of Liquid Pressure

From the expression, P = hρg, the liquid pressure follows the following laws:

1. The pressure of a liquid is directly proportional to its density.
2. The pressure at a point inside a liquid is directly proportional to its depth from the free surface of the liquid.
3. Pressure at a point inside a liquid is same in all directions.
4. The pressure exerted horizontally by a liquid on sides of the container is called lateral pressure which has the same magnitude as the downward pressure.
5. The pressure of a liquid is independent of the shape of the vessel.

Pascal’s Law of Pressure

Pascal’s law of pressure is the law of transmission of liquid pressure. It states that when a pressure is applied to an enclosed liquid, the pressure is equally transmitted to every portion of it.

Suppose a vessel containing water with three opening X,Y and Z of different cross-sectional area A, 2A and A/2 respectively as shown in the figure. These opening areas are closed with three tight pistons to keep water in the vessel. When a force F is applied to X inward, the forces needed to keep the pistons at the same position in Y and Y are F/2 and 2F respectively.

That is, the pressure at each opening,

$$ P = \frac {2F}{2A} = \frac FA = \frac {F/2}{A/2} = \frac FA $$

So, pressure is equally transmitted in all parts of the vessel.

Applications of Pascal’s Law

1. Hydraulic press
   This is a simple device in which a small force is magnified many times as shown in the figure. Since pressure is transmitted equally throughout the liquid, a piston of the small cross-sectional area is used to exert a small force directly on a liquid and pressure P = F₁/A₁ is transmitted through connecting pipe to a large cylinder equipped with a larger piston area, A₂. So,
   \begin{align*} \text {Pressure, P} &= \frac {F_1}{A_1} = \frac {F_2}{A_2} \\ \text {or,} \: F_2 &= \frac {A_2}{A_1} \times F_1 \\ \end{align*}

Hence, a larger force is produced at the larger cylinder. The hydraulic press is a force multiplying device with multiplication factor equal to the ratio of the area of pistons.

2. Hydraulic jack
   Cars and heavy trucks are raised to convenient heights by the use of hydraulic jack in the motor workshop as shown in the figure. So that mechanics can do work under them. In such device, a slight pressure is transmitted through a liquid to act on a large surface producing sufficient force to lift up the vehicle.
3. Hydraulic brakes
   When a small force is applied by the foot on the brake plate, the applied pressure is transmitted through the brake oil to act on the larger area where pistons are made to move the brake shoes against the brake drum.
4. High-pressure water jet cutting
   Stones, slates, rubbers, forms, asbestos etc. are cut by the high pressure of water jet such as a pressure of 350 atmospheres. This is completely dust free technique.
5. Teeth sealing
   On teeth scaling, the tooth is hit fine jet of water at high pressure.

Upthrust

The upward force exerted by a fluid on an object which is completely or partially immersed in the fluid is called the upthrust or buoyancy. Because of the upthrust, we can easily lift up a heavy object in water and so, the object has lesser weight in water. When a body is completely immersed in water as shown in the figure, the pressure at its bottom B is greater than at its top T. So, a net upward force acts on a body due to the pressure difference and upthrust or buoyancy is produced in liquid.

Let W_a, be the weight of the object in air and W_w, the weight in water. Then, upthrust of the liquid,
\begin{align*} U &= \text {weight of the body in air} – \text {weight of the body in water} \\ &= W_a-W_w \end{align*}

So, upthrust in the loss in weight of the object in the fluid.