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Hydrostatic paradox is a part of Fluid Dynamics and deals with the liquid pressure at every depth and horizontal level.
- A paradox is a statement that is logically self-contradictory or contradicts one's expectations.
- It is a statement that leads to an apparently self-contradictory or logically undesirable conclusion despite apparently valid reasoning from actual premises.
- The hydrostatic paradox explains the pressure at a horizontal level in the fluid which occurs to be proportional to the vertical distance to the fluid’s surface.
| Table of Contents |
Key Terms: Hydrostatics, Hydrodynamics, Fluid Pressure, Pascal’s Law, Pressure formula, Fluids, Weight, Atmospheric pressure, Acceleration due to gravity, Density
Hydrostatic Paradox
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The hydrostatic paradox states that
The barometric formula depends on the fluid height and not only on its length and width.
- Moreover, at a certain large height, one may achieve any pressure.
- This precise feature of hydrostatics is mentioned as the hydrostatic paradox.
- Simon Stevin, a Dutch physicist, was the first to mathematically explain the paradox.
- In 1916, Richard Glazebrook mentioned the hydrostatic paradox while describing an arrangement that he attributed to Pascal.
- A heavyweight W is supported by a board of area A, which is supported by a fluid bladder connected to a vertical tube with a cross-sectional area α.
- Furthermore, pouring water of weight w down the tube would result in the heavy weight being raised.
Example of Hydrostatic ParadoxThree vessels X, Y, and Z of diverse shapes consist of different volumes of liquid, but all utilize a similar pressure (P) at all points at the same horizontal level.
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Also check:
| Chapter Related Concepts | ||
|---|---|---|
| Bernoulli’s principle | Mechanical Properties of Fluids | Stoke’s Law |
| Hydraulic Machines | Pascal’s Law | Mechanical Properties of Fluids Formula |
Hydrostatic Law
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Hydrostatic law defines the amount of pressure exerted at any point of a certain area of fluid, above a surface.
- It will be defined as the total weight of this fluid on that surface.
- Hydrostatic pressure is the amount of pressure that is applied to water as depth rises.
- The French scientist Blaise Pascal gave a principle.
- It states that If one part of an object in water is pressurized, that pressure is communicated through the entire body of water without diminishing.
- This principle forms the center of hydraulic methods, and it is applied in hydraulic pump systems.
- In a hydraulic system, the pressure of a column of water is located on one side, to exert that pressure on another side of the column.
- For example, if a downward force is applied to the left side of a U-shaped pipe the valve puts pressure on the left arm initiating the plate to transfer to the right arm.
- This force is used to lift heavy loads like cars, trucks, boats, cranes, and vehicles.
Fluid Pressure Formula
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The pressure at the depth h under the surface of a fluid is known by
P = Pa + ρgh
Where,
- P is the pressure of depth h from the surface of liquid or fluid.
- Pa is the atmospheric pressure.
- ρ being the mass density of the fluid or liquid.
- g is acceleration by gravity.
- h is the vertical height of the surface point.
Pascal (Pa) is the SI unit of pressure
An individual shall calculate the trend of variation of liquid pressure conferring to depth with the assistance of the formula.
Example of Water Pressure at DepthExample: What is the pressure of a swimmer at 10 m underneath the surface:
Explanation: Given
Formula used: P = Pa + ρgh
P = Pa + ρgh ⇒ P = 1.01×105 Pa + 1000 kg/m3 × 9.8 m/s2 × 10 m = 199000 Pa ⇒ P ≅ 2 atm
P = Pa + ρgh ⇒ P = 1.01×105 Pa + 1000 kg/m3 × 9.8 m/s2 × 10 m = 199000 Pa ⇒ P ≅ 2 atm
P = Pa + ρgh ⇒ P = 1.01×105 Pa + 1000 kg/m3 × 9.8 m/s2 × 10 m = 199000 Pa ⇒ P ≅ 2 atm It can be observed that, though the lakes are of dissimilar shapes, the pressure a swimmer experiences at 10 meters under the surface stays the same. This offers a strong insight that the height of the fluid column matters in evaluating liquid pressure and not the shape of the container or volume of the container the cross-sectional part, or the base area. In the above-mentioned example, the pressure at 10 m depth of the water column is 100% further than equated to the surface.
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Also check:
| Other Concepts | ||
|---|---|---|
| Surface tension | Venturi-meter | Viscosity |
| Surface Energy | Maxwell Equations | Wavelength Formula |
Things to Remember
- The hydrostatic paradox states that for any container of the same height having a different base, the pressure at the base is the same.
- Pascal's law states that pressure applied everywhere in a confined incompressible fluid is spread equivalently in each direction through the fluid.
- Fluid statics deals with fluids where there is no relevant motion amongst the fluid particles.
- Normally, this comprises two conditions: when the fluid is at rest and when it changes like a rigid solid.
- The pressure in a liquid at a particular depth is termed hydrostatic pressure.
- The hydrostatic paradox is usually calculated as the hydrostatic equation which is P = ρgh.
Sample Questions
Ques. What do you understand about the hydrostatic paradox? (2 Marks)
Ans. Hydrostatic paradox acts as the liquid pressure on each point being at a similar depth or horizontal level. Besides, this paradox states that the pressure at a particular fluid horizontal level should be turned out to be proportional to the vertical distance to the fluid’s surface. The mathematical expression is P ∝ h.
Ques. What is the reason for the hydrostatic paradox? (3 Marks)
Ans. To apprehend the hydrostatic paradox, ruminate three vessels X, Y, and Z of diverse shapes, having a changed volume of liquid. On filling it with liquid, the horizontal liquid level in all vessels stays the same even when there is a variance in the shape of the vessel. The reason behind this particular mechanism is that the liquid pressure needs to be similar at the bottom of the fluid pressure in order to have a similar depth at all points.
Ques. Name a few uses of hydraulics. (5 Marks)
Ans. Hydraulics has its usage in many things and a few of the major uses of hydraulics. The brakes of cars work on hydraulics and when you push the brake pedal, it pushes the small piston. The piston applies pressure on the brake fluid, which presses the brake pads of great pistons. When the brake pads come in interaction with the brake drum, it slows down the car and ultimately stops it.
Hydraulics is used in several applications:
- Heavy equipment
- Airplanes and jet planes
- Modifying wings
- Setting out or getting in the landing gear
- Opening or closing doors
Ques. What is Hydrostatistics? (2 Marks)
Ans. A branch of physics dealing with the characteristics of at-rest fluids, especially pressure in a fluid or pressure applied by a fluid on an immersed body.
Ques. What is the practical use of hydrostatic pressure? (1 Mark)
Ans. It helps in the measurement of blood pressure and the determination of the properties of other body fluids.
Ques. State difference in hydrostatic fluid and hydrodynamic fluid? (5 Marks)
Ans. The differences between hydrostatic fluid and hydrodynamic fluid are mentioned below:
- Hydrostatic fluid- Hydrostatic systems transfer energy by pressure. The hydrostatic fluid is static following the surface of the container and the piston. The oil and piston are static concerning each other. A particular drop of oil drives itself to transmit energy. A few examples are gear pumps, vane pumps, piston pumps, motors, and hydraulic cylinders.
- Hydrodynamic fluid- Hydrodynamic systems transmit energy by the related motion and interchange of momentum between oil and the object. If one parameter is moving faster than the other, its momentum or motion affects the other to accelerate or decelerate by transmitting energy. A few examples are centrifugal pumps, fans, and fluid torque converters.
Ques. Calculate the hydrostatic pressure of water at the bottom of a 7-meter pool. The density of water is 1000 kg. (3 Marks)
Ans. To find hydrostatic pressure, the formula used is as follows:
P = ρgd
Where,
- ρ = being the density
- g = being the gravity
- d = and depth of the liquid.
We have,
P is equal to 1000 x 9.8 x 7 = 68,600 Pa of the hydrostatic pressure of depth of a 7-meter pool.
Ques. Explain the application of the hydraulic lift. (2 Marks)
Ans. The Hydraulic lift uses the principle of Pascal's Law. The force is functional on the reduced one and the purpose is lifted to keep on the greater piston. When we need to lift a box or load, an equal pressure sufficient to lift the load to a suitable height is applied on the reduced piston which is transferred to every direction in the liquid. Therefore, the load is lifted and it works for a hydraulic lift.
Ques. What is the purpose of a barometer? (1 Mark)
Ans. The Barometer is a device used to measure pressure.
Ques. What are the reasons affecting the upthrust? (2 Marks)
Ans. The Upthrust force acting on a body is known by ρ×v×g, which depends on reasons
- The density of the liquid in which the body is submerged
- The submerged volume of the solid
- Acceleration through gravity.
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