How Does Hydrostatic Paradox Apply in Daily Life?
When we break the word hydrostatic we get ‘hydro’ which means water and ‘static’ means something fixed or unchanged. Again the literal meaning of paradox is ‘contradiction’. So to sum it all up, the hydrostatic paradox is a state when containers with different shapes but have the same base, when filled with liquid of same height, then the force applied by the liquid on the container will be the same. However, the amount of water in each container would be different.
Break the words into smaller parts and you will understand the concepts by the means of the words themselves.
Keeping this in mind, the faculties at Vedantu have compiled all the details related to the topic in this particular article to give a better learning experience and to help the students perform well in their exams.
What will you learn from the article?
Hydrostatic Paradox - An introduction
Definition
Hydrostatic law
Hydrostatic Pressure
Frequently asked questions
Definition and meaning of Hydrostatic paradox
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:
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 on the above side, like container ‘a’ but not on the below side.
The liquid pressure of the container is in equilibrium with outside atmospheric pressure. Hence, equal pressure is exerted on the fluid within. This pressure is not enough to maintain the 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.
(Image will be uploaded soon)
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
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 the 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 to 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
Hydraulic Pressure
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.
Solved Example
Ques-Calculate the hydrostatic pressure of water at the bottom of a 7-meter pool. Given: density of water = 1000 kg/m3.
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.
Do you know?
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.
The shape of the container and the mass has no impact on the level of pressure of water
Hydraulic Pressure formula P= Pa + 𝝆gh
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FAQs on What Is the Hydrostatic Paradox?
1. What is the hydrostatic paradox in simple terms?
The hydrostatic paradox is the observation that the pressure exerted by a liquid at the bottom of a container depends only on the height of the liquid column and not on the container's shape or the total volume of the liquid. This seems counter-intuitive because one might expect a wider container with more liquid to exert more pressure, but this is not the case.
2. What is the fundamental principle that explains the hydrostatic paradox?
The paradox is explained by the formula for hydrostatic pressure: P = hρg. In this formula, 'P' is the pressure, 'h' is the vertical height (or depth) of the fluid, 'ρ' (rho) is the fluid's density, and 'g' is the acceleration due to gravity. Since pressure only depends on height 'h' and density 'ρ' (for a given fluid), the pressure at the base will be identical for any container shape, as long as the liquid height is the same.
3. Can you provide a real-world example of the hydrostatic paradox?
A classic example involves three vessels with the same base area but different shapes: one is a standard cylinder, another narrows towards the top, and the third widens. If all three are filled with water to the exact same level, the pressure measured at the bottom of each vessel will be precisely the same. Consequently, the total force on the base (Pressure × Area) is also identical, even though the total weight of water in each vessel is vastly different.
4. Why is this phenomenon called a 'paradox' if it has a clear scientific explanation?
It is termed a 'paradox' because it presents an apparent contradiction to our everyday intuition. Logically, we might assume that a greater weight of liquid should always result in greater pressure at the base. The paradox lies in the conflict between this intuitive (but incorrect) belief and the actual physical principle that pressure at a depth is uniform and independent of the liquid's total weight or container shape.
5. How can the force on the base be the same if the weight of the liquid is different in each container?
The container's walls exert forces that resolve this apparent contradiction. In a vessel with sloping sides, the walls exert both horizontal and vertical forces on the fluid.
- In a container that narrows upwards, the slanted walls provide an upward force, supporting some of the liquid's weight.
- In a container that widens upwards, the slanted walls exert a downward force on the liquid. This downward force is transmitted to the base, adding to the force from the liquid's weight.
6. For which class and chapter is the hydrostatic paradox relevant in the CBSE curriculum?
The hydrostatic paradox is a key concept taught in Class 11 Physics, specifically within Chapter 9: Mechanical Properties of Fluids, as per the CBSE syllabus for the 2025-26 session. It serves as a crucial application of the principles of fluid pressure.
7. What is the key difference between hydrostatics and hydrodynamics?
The primary distinction is the state of the fluid being studied:
- Hydrostatics deals with fluids that are at rest or in static equilibrium. It focuses on concepts like pressure, buoyancy, and principles such as Pascal's Law.
- Hydrodynamics is the study of fluids that are in motion. It covers topics like fluid flow, viscosity, turbulence, and principles like Bernoulli's theorem.
8. What is the importance of understanding hydrostatics in engineering and daily life?
Understanding hydrostatics is fundamental to many practical applications. Its principles are used in:
- The design of dams and submarines to withstand extreme pressure at great depths.
- The operation of hydraulic systems, such as car brakes and industrial lifts, which use Pascal's law to multiply force.
- Explaining why the water level in communicating vessels is always the same, a principle used in water level indicators.
- Understanding buoyancy, which governs why ships and icebergs float.
