Hydraulics inverted vessel with water experiments. New thought. See what the “Hydrostatic paradox” is in other dictionaries

21.11.2023

Consider three vessels of different shapes filled with liquid to the same level h c . All vessels are such that they have the same bottom area.

In accordance with the general formula for determining the force acting on a flat surface

you can calculate the force acting on the bottom of the vessel. For all three vessels, these forces will be the same and independent of the weight of the liquid in the vessel. But all the vessels will act on the support with different forces equal to the weight of the vessels with liquid. This fact is called hydrostatic paradox .

Basics of the theory of floating bodies

We will assume that in a liquid with density ρ immersed body with volume V . Let's choose a coordinate system, axis Z which we direct downwards, and the axis X And Y along the free surface. Let us consider the forces acting on the body from the liquid. All horizontal components, as established above, will be balanced. To determine the vertical components, we select an elementary cylindrical volume in a solid body with a cross-sectional area dS . Forces act on the end surfaces of this volume dF 1 from above and dF 2 below.

dF 1 will:

Vertical force component dF 2 will:

Integrating this expression over the area of the horizontal projection of the body, we obtain:

This expression is called Archimedes' law : a body immersed in a liquid loses as much weight as the liquid it displaces weighs. In other words, a body immersed in a liquid is subject to a buoyancy force equal to the weight of the liquid displaced by the body. This force is applied at a point called the displacement point.

Depending on the ratio of weight and buoyant force, three states of the body are possible:

if the weight is greater than the buoyant force, the body sinks,

if the weight is less than the buoyancy force, the body floats,

if the weight is equal to the buoyant force, the body floats.

It is directed vertically upward and applied at a point corresponding to the center of pressure called - center of displacement the amount of water displaced by a floating body - displacement.

Figure Body buoyancy A And 6 - the vessel is stable

The figure shows a diagram of the ship's hull with the following symbols: a-a-plane of navigation, limited by the waterline as a contour; oh-oh- swimming axis - an axis normal to the swimming plane and passing through the center of gravity of the body WITH.

There are three centers on the swimming axis: the center of gravity WITH, center of displacement D and metacenter M(the point of intersection of the swimming axis with the line of action of the Archimedean force).

The distance from the metacenter to the center of gravity of the body is called metacentric height h m . Taking the floating plane as the comparison plane, we characterize stability.

At h m > 0 the body position will be stable, with h m < 0 - unstable, and when h m =0 the body will be in a state of indifferent equilibrium.

Buoyancy and stability - key concepts of the theory of floating bodies. Buoyancy - This is the state of equilibrium of a solid body partially or completely immersed in a liquid. Stability - the ability of a floating body, unbalanced, to restore its original position after the cessation of the forces causing the roll. Bank - the position of a body in which the vertical plane of its symmetry is deviated from the vertical towards the earth's surface.

Between the weight ratios of a floating body G and its buoyant force Rv There are three possible states of a body immersed in a liquid.

If G > Rv, then the body sinks, since the resultant force G And Rv directed vertically downwards.

If G< Рв, the body floats in a semi-submerged state (surface swimming), and at the same time the resultant force G And Rv is directed vertically upward, so the body floats up until the new reduced buoyant force Pb is equal to the weight of the body G (G= Pv).

A body floats submerged if G= PV, it can be in stable or unstable equilibrium. For a body to be in balance, it is necessary that its center of gravity and the center of displacement lie on the same vertical.

If a floating body is exposed to external forces (wind, sharp turn), it will deviate from the equilibrium position (give a roll). When a body floats stable, the center of gravity is located below the center of displacement, and after the interaction of these forces ceases, the body returns to its previous position. During unstable swimming, the center of gravity of the body is located above the center of displacement. In this case, the body is taken out of equilibrium and cannot return to its original position. The state of indifferent equilibrium is characterized by the coincidence of the centers of gravity and displacement.

The buoyancy of a body is expressed by the formula

Where G water weight;  V - specific gravity of water; V - the volume of water displaced by a body.

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What is the hydrostatic paradox?

The hydrostatic paradox lies in the fact that the weight of a liquid poured into a vessel may differ from the pressure it exerts on the bottom of the vessel. Thus, in vessels that expand upward, the pressure force on the bottom is less than the weight of the liquid, and in vessels that narrow, it is greater. In a cylindrical vessel both forces are equal. If the same liquid is poured to the same height into vessels of different shapes, but with the same bottom area, then, despite the different weight of the poured liquid, the pressure force on the bottom is the same for all vessels and is equal to the weight of the liquid in a cylindrical vessel. This follows from the fact that the pressure of a fluid at rest depends only on the depth below the free surface and on the density of the fluid. The hydrostatic paradox is explained as follows. Since hydrostatic pressure is always normal to the walls of the vessel, the pressure force on the inclined walls has a vertical component, which compensates for the weight of the volume of liquid that is excessive against the cylinder in a vessel expanding upward and the weight of the volume of liquid that is missing against the cylinder in a vessel that narrows upward. The hydrostatic paradox was discovered by the French physicist Blaise Pascal (1623–1662).

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Discovery of Pascal's law

According to Pascal's instructions, a strong oak barrel was filled to the brim with water and tightly closed with a lid.
The end of a vertical glass tube of such length was inserted into a small hole in the lid that its end was at the level of the second floor.

Going out onto the balcony, Pascal began filling the tube with water.
He had not even managed to pour out a dozen glasses when suddenly, to the amazement of the onlookers who surrounded the barrel, the barrel burst with a crash.
She was torn apart by an incomprehensible force.
Pascal is convinced: yes, the force that breaks the barrel does not at all depend on the amount of water in the tube.
It's all about the height to which the tube was filled.

Thus he comes to the discovery of the law that bears his name.

Task

If we assume the height of the water in the tube is 4 meters (second floor balcony),
the diameter of the barrel is 0.8 m, and the height of the barrel is 0.8 m.
What force breaks the barrel?

Solution:

At the surface of the water in the barrel under the lid, this pressure is
P = pgh,
where p is the density of water,
g - free fall acceleration,
h is the height of the water column in the tube.
Multiplying the resulting pressure by the diametrical cross-sectional area of the barrel
(S = D*H, H - barrel height),
we get the force that broke the strong oak walls of the barrel.

P = pg (h + H/2)DH = 27.6 kN.

Hydrostatic paradox is a property of liquids, which consists in the fact that the force of gravity of a liquid poured into a vessel may differ from the force with which this liquid acts on the bottom of the vessel. The reason for the hydrostatic paradox is that the liquid presses not only on the bottom, but also on the walls of the vessel. The weight of the liquid in the vessel will be equal to the sum of the altitude components of the pressure over the entire internal area of the vessel. If, for example, a vessel has areas of the inner surface on which the pressure is directed upward, these areas will contribute to the weight with a minus sign. The static pressure of the liquid on the bottom will be greater than the weight of the liquid divided by the bottom area.

In 1648, the paradox was demonstrated by B. Pascal. He inserted a narrow tube into a closed barrel filled with water and, going up to the second floor balcony, poured a mug of water into this tube. Due to the small thickness of the tube, the water in it rose to a great height, and the pressure in the barrel increased so much that the fastenings of the barrel could not withstand it, and it cracked.

Types of fluid movement (flow). Basic concepts: trajectory, streamline, stream tube, elementary stream.

A streamline is a curved line for any point of which, at a selected moment in time, the local velocity vector is directed tangentially (we are not talking about the normal velocity component, since it is equal to zero).

Current line is an elementary stream of flow, the cross-sectional area of which is infinitely small.

The totality of all the streamlines that pass through each point of the flow contour forms a surface called a stream tube. Inside this tube, the liquid contained in it moves, which is called a trickle.

A trickle is considered elementary if the contour under consideration is infinitely small, and finite if the contour has a finite area.

The cross section of the stream, which is normal at each point to the streamlines, is called the living cross section of the stream. Depending on the finiteness or infinite smallness, the area of the stream is usually denoted by ω and dω, respectively.

A certain volume of liquid that passes through the live section per unit time is called the flow rate of the stream Q.

Trajectory is the path traversed by a given particle of liquid in space over a certain period of time.

Fluid flow in general can be unsteady (unsteady) or steady (stationary).

Unsteady motion– one in which at any point in the flow the speed and pressure change over time, i.e. u And P depend not only on the coordinates of the point in the flow, but also on the moment of time at which the characteristics of the movement are determined, i.e.:

And .

An example of unsteady motion may be the flow of liquid from an emptying vessel, in which the level of liquid in the vessel gradually changes (decreases) as the liquid flows out.

Steady motion– one in which at any point in the flow the speed and pressure do not change over time, i.e. u And P depend only on the coordinates of a point in the flow, but do not depend on the moment of time at which the motion characteristics are determined:

And ,

and therefore , , , .

An example of steady-state motion is the flow of liquid from a vessel with a constant level that does not change (remains constant) as the liquid flows out.

16. Types of fluid flows, flow characteristics: open section, wetted perimeter, hydraulic radius, flow rate, average speed.

The set of elementary streams of liquid represents flow liquids. The following types of flows (or types of fluid movements) are distinguished:

Pressure flows (pressure movements)- these are those when the flow is limited by solid walls on all sides, while at any point in the flow the pressure differs from atmospheric pressure, usually to a greater extent, but can also be less than atmospheric. The movement in this case occurs due to the pressure created, for example, by a pump or water tower. The pressure along the pressure flow is usually variable. Such movement occurs in all hydraulic drives of technological equipment, water pipelines, heating systems, etc.

Gravity flows (gravity-free movements) differ in that the flow has a free surface under atmospheric pressure. Non-pressure movement occurs under the influence of gravity of the fluid flow itself. The pressure in such flows is approximately the same and differs from atmospheric pressure only due to the depth of the flow. An example of such movement could be the flow of water in a river, canal, or stream.

Free jet has no solid walls. Movement occurs under the influence of inertial forces and the weight of the fluid. The pressure in such a flow is almost equal to atmospheric pressure. An example of a free stream is the flow of liquid from a hose, tap, etc.

In hydraulics, the following flow characteristics are distinguished: live section, wetted perimeter, hydraulic radius, flow rate, average speed.

Live section flow is a surface (cross section) normal to all stream lines intersecting it and lying inside the fluid flow. The open cross-sectional area is designated by the letter Y. For an elementary stream of liquid, the concept is used live section of an elementary stream(cross section of the stream perpendicular to the streamlines), the area of which is denoted by dY.

Wetted perimeter flow - the line along which the liquid comes into contact with the surfaces of the channel in a given living section. The length of this line is indicated by the letter c.

In pressure flows, the wetted perimeter coincides with the geometric perimeter, since the fluid flow comes into contact with all solid walls.

Hydraulic radius R flow is a quantity often used in hydraulics, representing the ratio of the open cross-sectional area S to the wetted perimeter c:

Fluid flow rate (liquid flow rate)– the amount of liquid flowing per unit time through the live cross-section of the flow.

Average fluid flow velocity Vav in a given section, this is a flow velocity that does not actually exist, the same for all points of a given living section, with which the liquid would have to move in order for its flow rate to be equal to the actual one.

Hydrostatic paradox

lies in the fact that the weight of a liquid poured into a vessel may differ from the pressure it exerts on the bottom of the vessel. Thus, in vessels expanding upward ( rice. ) the force of pressure on the bottom is less than the weight of the liquid, and in converging areas it is greater. In a cylindrical vessel both forces are equal.

If the same liquid is poured to the same height into vessels of different shapes, but with the same bottom area, then, despite the different weight of the poured liquid, the pressure force on the bottom is the same for all vessels and is equal to the weight of the liquid in a cylindrical vessel. This follows from the fact that the pressure of a fluid at rest depends only on the depth below the free surface and on the density of the fluid. G. p. is explained by the fact that since hydrostatic pressure R always normal to the walls of the vessel, the pressure force on the inclined walls has a vertical component p 1, which compensates for the excess weight against the cylinder 1 volume of liquid in the vessel 3 and the weight of the missing one against the cylinder 1 volume of liquid in the vessel 2 . G. p. was discovered by the French physicist B. Pascal (See Pascal).

Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .

See what the “Hydrostatic paradox” is in other dictionaries:

The weight of a liquid poured into a vessel may differ from the force of pressure it exerts on the bottom of the vessel. Thus, in vessels that expand upward, the pressure force on the bottom is less than the weight of the liquid, and in vessels that narrow, it is greater. In a cylindrical vessel both forces are equal.... ... Big Encyclopedic Dictionary

The point is that the weight of a liquid poured into a vessel may differ from the force of pressure it exerts on the bottom of the vessel. Thus, in vessels that expand upward (Fig.), the pressure force on the bottom is less than the weight of the liquid, and in vessels that narrow, it is greater. In cylindrical... ... Physical encyclopedia

Hydrostatic paradox is a phenomenon in which the weight of liquid poured into a vessel may differ from the force of pressure on the bottom. Causes Scheme of Pascal's experiment The reason for the hydrostatic paradox is that the liquid gives ... Wikipedia

Phys. the law by virtue of which the pressure on the bottom in vessels of different shapes, but with the same size, is filled. with the same liquid to the same height, in the same way, despite the difference in the amount of liquid. Dictionary of foreign words,... ... Dictionary of foreign words of the Russian language

The weight of the liquid poured into the vessel may differ from the force of pressure of the liquid on the bottom of the vessel. Thus, in vessels that expand upward (Fig.), the pressure force on the bottom is less than the weight of the liquid, and in vessels that narrow, it is greater. In cylindrical in the vessel both forces are equal.... ... Natural science. encyclopedic Dictionary- (Pascal’s law) is formulated as follows: The pressure exerted on a liquid (or gas) in any one place on its boundary, for example, by a piston, is transmitted without change to all points of the liquid (or gas). The law is named after the French scientist Blaise... ... Wikipedia