Hydrostatic paradox deals with the pressure of a liquid at all points of the same horizontal level (depth).

It is defined as:

“The hydrostatic pressure at a certain horizontal level of a liquid is directly proportional to the distance of the horizontal level from the surface of the liquid”.

The hydrostatic paradox states that the height of water in a container is independent of the shape of the container.

The height of fluid relative to the base of the container determines the pressure, and the pressure equilibrium determines the shape of fluid. Let us consider two containers ‘a’ and ‘b’ as shown below:

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The pressure at a depth ‘h’ is the same at all horizontal points of the cylindrical container. Any element of fluid in the cylindrical container ‘a’ is always in equilibrium because the weight of the fluid element is balanced by the pressure difference of the fluid elements below it and above it.

For container ‘b’, the pressure of any fluid element at the edge is the same in the above side, like container ‘a’ but not in the below side.

The liquid pressure of the container is in equilibrium with outside atmospheric pressure. Hence, an equal pressure is exerted to the fluid within. This pressure is not enough to maintain equilibrium of fluid elements. Now, the question arises, how is the equilibrium of fluid at the edge explained?

This is explained as follows:

The walls of the container exert force on the fluid element based on the pressure at each point. In container ‘b’ the walls are slanted, and offer an upward force that stabilizes the fluid elements next to it.

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The above figure represents the free-body diagram of fluid elements at the edge. The wall of the container exerts a diagonal force on the fluid element. The horizontal component of this force is balanced by the pressure exerted by the fluid. The vertical component of his force is balanced by outside atmospheric pressure.

Hydrostatic law determines the amount of pressure exerted at any point of a given area of fluid, above a surface. It can also be defined as the total weight of this fluid on that surface.

Hydrostatic pressure is the increasing amount of pressure that is exerted on water as depth increases. The French scientist Blaise Pascal gave a principle which states that “If one part of an object in water is pressurized, that pressure is transmitted throughout the entire body of water without diminishing.

This principle forms the basis of hydraulic systems, and it is applied in hydraulic pump systems. In a hydraulic system, the pressure of a column of water is placed on one side, to exert that pressure on the other side of the column.

For example, if a downward force is applied into the left side of a u-shaped pipe (having a valve), the valve applies pressure on the left arm causing the plate to move on the right arm. This force is used to lift heavy loads like cars, trucks, boats, cranes and other vehicles.

Hydrostatic Paradox Expression:

The mathematical expression of Hydrostatic Paradox is given by:

P ∝ h

The formula for hydraulic fluid pressure is given by:

The fluid pressure at a depth h below the surface of any fluid is given by the formula-

P= Pa + 𝝆gh

Where,

P = pressure of the fluid at a depth h from the surface of the liquid/fluid.

Pa = atmospheric pressure.

𝝆 = mass density of the fluid/liquid.

g = acceleration due to gravity.

h = vertical height between the surface and the point.

Paradox definition- Paradox is an apparent contradiction to the physical descriptions of the universe. Some paradoxes are based on resolutions, while others are against the resolution and indicate flaws in the theory.

FAQ (Frequently Asked Questions)

1. What are the uses of hydraulics?

Ans- Hydraulics is used in many things. Here are some of the major uses of hydraulics.

a. The brakes of cars work on hydraulics. When you push the brake pedal, it further pushes the small piston. The piston applies some pressure on the brake fluid, which further presses the brake pads of the large pistons. When the brake pads come in contact with the brake drum, it slows down the car and eventually stops it.

Hydraulics is used in various applications like:

Heavy equipment

Airplanes and jet planes

Adjusting wings

Putting out/bringing in landing gear

Opening/closing doors

2. What are some examples of paradox?

Ans- Here is a list of paradox examples:

● Cool tropics paradox: A contradiction between ice-free periods of the Eocene and Cretaceous, modeled estimates of tropical temperatures during warm, and the lower temperature that proxies suggest were present.

● Irresistible force paradox: What happens when an unstoppable force hits an immovable object?

● Paradox of place: If everything that exists has a place, that place must have a place in turn, and so it should be infinitum.

● Paradox of the grain of millet: When a grain of millet falls it makes no sound, but when a thousand grains fall altogether they make sound, thus nothing become something.

3. What is the difference between hydrostatic fluid & hydrodynamic fluid?

Ans- The differences between hydrostatic fluid & hydrodynamic fluid are mentioned below:

a. Hydrostatic fluid

Hydrostatic systems transfer energy by pressure. The hydrostatic fluid is static with respect to the surface of the container and the piston. The oil and piston are also static with respect to each other. A given drop of oil pushes itself to transfer energy.

Examples: gear pumps, vane pumps, piston pumps, motors and hydraulic cylinders.

b. Hydrodynamic fluid

Hydrodynamic systems transfer energy by the relative motion and exchange of momentum between oil and the object. If one parameter is moving more than the other, its momentum or motion causes the other to accelerate or decelerate, thereby transferring energy.

Examples: centrifugal pumps, fans and fluid torque convertors.

4. Calculate the hydrostatic pressure of water at the bottom of a 7 meter pool. Given: density of water = 1000 kg/m^{3}.

Ans- To find hydrostatic pressure, the following formula is used:

P = ρ * g * d

Where,

ρ = density of the liquid,

g = gravity, and

d = depth of the liquid.

We have,

P = 1000 * 9.8 * 7 = 68,600 Pa of hydrostatic pressure at a depth of a 7 meter pool.