Equilibrium of a mechanical system (absolutely rigid body)

The equilibrium of a mechanical system is a state in which all points of a mechanical system are at rest with respect to the reference frame under consideration. If the frame of reference is inertial, the equilibrium is called absolute, if it is non-inertial - relative.

To find the equilibrium conditions for an absolutely rigid body, it is necessary to mentally divide it into big number small enough elements, each of which can be represented by a material point. All these elements interact with each other - these forces of interaction are called internal. In addition, external forces can act on a number of points of the body.

According to Newton's second law, for the acceleration of a point to be zero (and the acceleration of a point at rest to be zero), the geometric sum of the forces acting on that point must be zero. If the body is at rest, then all its points (elements) are also at rest. Therefore, for any point of the body, we can write:

$(F_i)↖(→)+(F"_i)↖(→)=0$,

where $(F_i)↖(→)+(F"_i)↖(→)$ is the geometric sum of all external and internal forces acting on the $i$-th element of the body.

The equation means that for the equilibrium of a body, it is necessary and sufficient that the geometric sum of all forces acting on any element of this body be equal to zero.

From the equation it is easy to obtain the first condition for the equilibrium of a body (system of bodies). To do this, it is enough to sum the equation over all elements of the body:

$∑(F_i)↖(→)+∑(F"_i)↖(→)=0$.

The second sum is equal to zero according to Newton's third law: the vector sum of all internal forces of the system is equal to zero, since any internal force corresponds to a force equal in absolute value and opposite in direction.

Hence,

$∑(F_i)↖(→)=0$

The first condition for the equilibrium of a rigid body (system of bodies) is the equality to zero of the geometric sum of all external forces applied to the body.

This condition is necessary but not sufficient. It is easy to verify this by remembering the rotating action of a pair of forces, the geometric sum of which is also equal to zero.

The second condition for the equilibrium of a rigid body is the equality to zero of the sum of the moments of all external forces acting on the body, relative to any axis.

Thus, the equilibrium conditions for a rigid body in the case of an arbitrary number of external forces look like this:

$∑(F_i)↖(→)=0;∑M_k=0$

Pascal's law

Hydrostatics (from the Greek hydor - water and statos - standing) is one of the subsections of mechanics that studies the equilibrium of a liquid, as well as the equilibrium of solids partially or completely immersed in a liquid.

Pascal's law is the basic law of hydrostatics, according to which the pressure on the surface of a liquid, produced by external forces, is transferred by the liquid equally in all directions.

This law was discovered by the French scientist B. Pascal in 1653 and published in 1663.

To verify the validity of Pascal's law, it is enough to do a simple experiment. Let's attach a hollow ball with many small holes to the tube with the piston. After filling the balloon with water, press the piston to increase the pressure in it. Water will begin to pour out, but not only through the hole that is in the line of action of the force applied by us, but through all the others too. Moreover, the pressure of water, due to external pressure, in all the streams that appear will be the same.

We will get a similar result if we use smoke instead of water. Thus, Pascal's law is valid not only for liquids, but also for gases.

Liquids and gases transmit the pressure exerted on them equally in all directions.

The transfer of pressure by liquids and gases in all directions at the same time is explained by the rather high mobility of the particles of which they are composed.

Pressure of a liquid at rest on the bottom and walls of a vessel (hydrostatic pressure)

Liquids (and gases) transmit in all directions not only external pressure, but also the pressure that exists inside them due to the weight of their own parts.

The pressure exerted by a fluid at rest is called hydrostatic.

We obtain a formula for calculating the hydrostatic pressure of a liquid at an arbitrary depth $h$ (in the vicinity of point A in the figure).

The pressure force acting from the overlying narrow column of liquid can be expressed in two ways:

1) as the product of the pressure $p$ at the base of this column and the area of ​​its section $S$:

2) as the weight of the same liquid column, i.e. the product of the mass $m$ of the liquid and the free fall acceleration:

The mass of a liquid can be expressed in terms of its density $p$ and volume $V$:

and the volume - through the height of the column and its cross-sectional area:

Substituting in the formula $F=mg$ the value of mass from $m=pV$ and volume from $V=Sh$, we get:

Equating the expressions $F=pS$ and $F=pVg=pShg$ for the pressure force, we get:

Dividing both sides of the last equality by the area $S$, we find the fluid pressure at depth $h$:

This is the formula hydrostatic pressure.

Hydrostatic pressure at any depth inside a liquid does not depend on the shape of the vessel in which the liquid is located, and is equal to the product of the density of the liquid, the gravitational acceleration and the depth at which the pressure is determined.

It is important to emphasize once again that the hydrostatic pressure formula can be used to calculate the pressure of a liquid poured into a vessel of any shape, including the pressure on the walls of the vessel, as well as the pressure at any point in the liquid directed from bottom to top, since the pressure at the same depth is the same in all directions.

With considering atmospheric pressure$p_0$, the formula for the pressure of a fluid at rest in the IRF at a depth of $h$ is written as follows:

hydrostatic paradox

The hydrostatic paradox is a phenomenon in which the weight of a liquid poured into a vessel may differ from the pressure force of the liquid on the bottom of the vessel.

In this case, the word "paradox" means an unexpected phenomenon that does not correspond to conventional ideas.

So, in vessels expanding upwards, the pressure force on the bottom is less than the weight of the liquid, and in narrowing vessels it is greater. In a cylindrical vessel, both forces are the same. If the same liquid is poured to the same height into vessels 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 the cylindrical vessel.

This follows from the fact that the pressure of a fluid at rest depends only on the depth under the free surface and on the density of the fluid: $p=pgh$ ( hydrostatic pressure formula). And since the area of ​​the bottom of all vessels is the same, then the force with which the liquid presses on the bottom of these vessels is the same. It is equal to the weight of the vertical column $ABCD$ of the liquid: $P=pghS$, here $S$ is the area of ​​the bottom (although the mass, and hence the weight, in these vessels are different).

The hydrostatic paradox is explained by Pascal's law - the ability of a fluid to transmit pressure equally in all directions.

From the hydrostatic pressure formula it follows that the same amount of water, being in different vessels, can exert different pressure on the bottom. Since this pressure depends on the height of the liquid column, it will be greater in narrow vessels than in wide ones. Thanks to this, even a small amount of water can create a very high pressure. In 1648, B. Pascal demonstrated this very convincingly. He inserted a narrow tube into a closed barrel filled with water and, going up to the balcony of the second floor, 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 stand it, and it cracked.

Law of Archimedes

Archimedes' law - the law of statics of liquids and gases, according to which any body immersed in a liquid (or gas) is affected by this liquid (or gas) buoyancy force equal to the weight of the liquid (gas) displaced by the body and directed vertically upwards.

This law was discovered by the ancient Greek scientist Archimedes in the III century. BC e. Archimedes described his research in the treatise On Floating Bodies, which is considered one of his last scientific works.

Below are the conclusions following from the law of Archimedes.

The action of liquid and gas on a body immersed in them

If you submerge an air-filled ball in water and release it, it will float. The same will happen with wood chips, cork and many other bodies. What force makes them float?

A body immersed in water is subjected to water pressure from all sides. At each point of the body, these forces are directed perpendicular to its surface. If all these forces were the same, the body would experience only all-round compression. But at different depths, the hydrostatic pressure is different: it increases with increasing depth. Therefore, the pressure forces applied to the lower parts of the body are greater than the pressure forces acting on the body from above.

If we replace all the pressure forces applied to a body immersed in water with one (resultant or resultant) force that has the same effect on the body as all these individual forces together, then the resulting force will be directed upwards. This is what makes the body float. This force is called buoyancy force, or Archimedean force(named after Archimedes, who first pointed to its existence and established what it depends on). In the figure, it is designated as $F_A$.

The Archimedean (buoyant) force acts on the body not only in water, but also in any other liquid, since in any liquid there is hydrostatic pressure, which is different at different depths. This force also acts in gases, due to which balloons and airships fly.

Due to the buoyancy force, the weight of any body in water (or in any other liquid) is less than in air, and less in air than in airless space. It is easy to verify this by weighing the weight with the help of a training spring dynamometer, first in the air, and then lowering it into a vessel with water.

Weight reduction also occurs when a body is transferred from vacuum to air (or some other gas).

If the weight of a body in a vacuum (for example, in a vessel from which air is pumped out) is equal to $P_0$, then its weight in air is equal to:

$P_(air)=P_0-F"_A,$

where $F"_A$ is the Archimedean force acting on given body in the air. For most bodies, this force is negligible and can be neglected, i.e., we can assume that $P_(air)=P_0=mg$.

The weight of the body in liquid decreases much more than in air. If the body weight in air is $P_(air)=P_0$, then the body weight in liquid is equal to $P_(liquid)= P_0 - F_A$. Here $F_A$ is the Archimedean force acting in the fluid. Hence it follows that

$F_A=P_0-P_(liquid)$

Therefore, in order to find the Archimedean force acting on a body in any liquid, this body must be weighed in air and in the liquid. The difference between the obtained values ​​will be the Archimedean (buoyant) force.

In other words, given the formula $F_A=P_0-P_(liquid)$, we can say:

The buoyant force acting on a body immersed in a liquid is equal to the weight of the liquid displaced by this body.

The Archimedean force can also be determined theoretically. To do this, suppose that a body immersed in a fluid consists of the same fluid in which it is immersed. We have the right to assume this, since the pressure forces acting on a body immersed in a liquid do not depend on the substance from which it is made. Then the Archimedean force $F_A$ applied to such a body will be balanced by the downward gravity $m_(l)g$ (where $m_(l)$ is the mass of fluid in the volume of this body):

But the force of gravity $m_(l)g$ is equal to the weight of the displaced fluid $R_l$, Thus,

Considering that the mass of a liquid is equal to the product of its density $р_Ж$ and its volume, the formula $F_(A)=m_(Ж)g$ can be written as:

$F_A=p_(x)V_(x)g$

where $V_l$ is the volume of the displaced fluid. This volume is equal to the volume of that part of the body that is immersed in the liquid. If the body is completely immersed in the liquid, then it coincides with the volume $V$ of the whole body; if the body is partially immersed in the liquid, then the volume $V_zh$ of the displaced liquid is less than the volume $V$ of the body.

The formula $F_(A)=m_(g)g$ is also valid for the Archimedean force acting in a gas. Only in this case, the density of the gas and the volume of the displaced gas, and not the liquid, should be substituted into it.

Based on the foregoing law of Archimedes can be formulated like this:

Any body immersed in a fluid (or gas) at rest is subject to a buoyant force acting on the side of this fluid (or gas). equal to the product the density of a liquid (or gas), the acceleration of free fall, and the volume of that part of the body that is immersed in the liquid (or gas).

Free oscillations of mathematical and spring pendulums

Free oscillations (or natural oscillations) are oscillations of an oscillatory system, performed only due to the initially reported energy (potential or kinetic) in the absence of external influences.

Potential or kinetic energy can be communicated, for example, in mechanical systems through an initial displacement or an initial velocity.

Freely oscillating bodies always interact with other bodies and together with them form a system of bodies called oscillatory system.

For example, a spring, a ball, and a vertical post, to which the upper end of the spring is attached, are included in an oscillatory system. Here the ball slides freely along the string (friction forces are negligible). If you take the ball to the right and leave it to itself, it will oscillate freely around the equilibrium position (point O) due to the action of the elastic force of the spring directed towards the equilibrium position.

Another classic example of a mechanical oscillatory system is mathematical pendulum. In this case, the ball performs free oscillations under the action of two forces: gravity and the elastic force of the thread (the Earth also enters the oscillatory system). Their resultant is directed to the equilibrium position. The forces acting between the bodies of an oscillatory system are called internal forces. Outside forces called the forces acting on the system from the bodies that are not included in it. From this point of view, free oscillations can be defined as oscillations in a system under the action of internal forces after the system is taken out of equilibrium.

The conditions for the occurrence of free oscillations are:

1. the emergence in them of a force that returns the system to a position of stable equilibrium after it has been taken out of this state;
2. no friction in the system.

Dynamics of free oscillations

Oscillations of a body under the action of elastic forces. The equation of oscillatory motion of a body under the action of the elastic force $F_(control)$ can be obtained taking into account Newton's second law ($F=ma$) and Hooke's law ($F_(control)=-kx$), where $m$ is the mass ball, $a$ - acceleration acquired by the ball under the action of the elastic force, $k$ - spring stiffness coefficient, $x$ - displacement of the body from the equilibrium position (both equations are written in projection onto the horizontal axis $Ox$). Equating the right sides of these equations and taking into account that the acceleration $a$ is the second derivative of the $x$ coordinate (displacement), we get:

This is differential equation the motion of a body oscillating under the action of an elastic force: the second derivative of the coordinate with respect to time (the acceleration of the body) is directly proportional to its coordinate, taken with the opposite sign.

Oscillations of a mathematical pendulum. To obtain the equation of oscillation of a mathematical pendulum, it is necessary to decompose the force of gravity $F_т=mg$ into normal $F_n$ (directed along the thread) and tangential $F_τ$ (tangential to the trajectory of the ball - circle) components. The normal component of the force of gravity $F_n$ and the elastic force of the thread $F_(control)$ in total give the pendulum a centripetal acceleration that does not affect the magnitude of the speed, but only changes its direction, and the tangential component $F_τ$ is the force that returns the ball to equilibrium position and causes it to oscillate. Using, as in the previous case, Newton's law for tangential acceleration - $ma_τ=F_τ$ and taking into account that $F_τ=-mgsinα$, we get:

The minus sign appeared because the force and the angle of deviation from the equilibrium position $α$ have opposite signs. For small deflection angles $sinα≈α$. In turn, $α=(s)/(l)$, where $s$ is the arc $OA$, $l$ is the length of the thread. Considering that $a_τ=s""$, we finally get:

The form of the equation $s""=(g)/(l)s$ is similar to the equation $x""=-(k)/(m)x$. Only here the parameters of the system are the length of the thread and the acceleration of free fall, and not the stiffness of the spring and the mass of the ball; the role of the coordinate is played by the length of the arc (i.e., the path traveled, as in the first case).

Thus, free vibrations are described by equations of the same type (subject to the same laws) regardless of physical nature forces that cause these vibrations.

The solution of the equations $x""=-(k)/(m)x$ and $s""=(g)/(l)s$ is a function of the form:

$x=x_(m)cosω_(0)t$(or $x=x_(m)sinω_(0)t$)

That is, the coordinate of a body that performs free oscillations changes over time according to the cosine or sine law, and, therefore, these oscillations are harmonic.

In the equation $x=x_(m)cosω_(0)t$ xt is the oscillation amplitude, $ω_(0)$ is the natural cyclic (circular) oscillation frequency.

The cyclic frequency and the period of free harmonic oscillations are determined by the properties of the system. So, for vibrations of a body attached to a spring, the following relations are true:

$ω_0=√((k)/(m)); T=2π√((m)/(k))$

The natural frequency is the greater, the greater the stiffness of the spring or the less mass of the load, which is fully confirmed by experience.

For a mathematical pendulum, the following equalities hold:

$ω_0=√((g)/(l)); T=2π√((l)/(g))$

This formula was first obtained and tested by the Dutch scientist Huygens (a contemporary of Newton).

The period of oscillation increases with the length of the pendulum and does not depend on its mass.

Particular attention should be paid to the fact that harmonic vibrations are strictly periodic (because they obey the law of sine or cosine) and even for a mathematical pendulum, which is an idealization of a real (physical) pendulum, are possible only at small oscillation angles. If the deflection angles are large, the load displacement will not be proportional to the deflection angle (the sine of the angle) and the acceleration will not be proportional to the displacement.

The speed and acceleration of a body that performs free oscillations will also perform harmonic oscillations. Taking the time derivative of the function $x=x_(m)cosω_(0)t$, we obtain an expression for the speed:

$x"=υ=-x_(m) sinω_(0)t=υ_(m)cos(ω_(0)t+(π)/(2))$

where $υ_(m)$ is the velocity amplitude.

Similarly, we obtain the expression for the acceleration a by differentiating $x"=υ=-x_(m) sinω_(0)t=υ_(m)cos(ω_(0)t+(π)/(2))$:

$a=x""=υ"-x_(m)ω_0^(2)cosω_(0)t=a_(m) cos(ω_(0)t+π)$

where $a_m$ is the acceleration amplitude. Thus, from the obtained equations it follows that the amplitude of the speed of harmonic oscillations is proportional to the frequency, and the acceleration amplitude is proportional to the square of the oscillation frequency:

$υ_(m)=ω_(0)x_m; a_m=ω_0^(2)x_m$

Oscillation phase

The oscillation phase is an argument of a periodically changing function that describes an oscillatory or wave process.

For harmonic vibrations

$X(t)=Acos(ωt+φ_0)$

where $φ=ωt+φ_0$ - oscillation phase, $А$ - amplitude, $ω$ - circular frequency, $t$ - time, $φ_0$ - initial (fixed) oscillation phase: at time $t=0$ $φ=φ_0$. The phase is expressed in radians.

The phase of a harmonic oscillation at a constant amplitude determines not only the coordinate of the oscillating body at any moment of time, but also the speed and acceleration, which also change according to the harmonic law (the speed and acceleration of harmonic oscillations are the first and second time derivatives of the function $X(t)= Acos(ωt+φ_0)$, which, as is known, again give sine and cosine). Therefore, it can be said that the phase determines the state of the oscillatory system at a given amplitude at any time.

Two oscillations with the same amplitudes and frequencies may differ from each other in phases. Since $ω=(2π)/(T)$, then

$φ-φ_0=ωt=(2πt)/(T)$

The ratio $(t)/(T)$ shows what part of the period has passed since the start of oscillations. Any value of time expressed in fractions of a period corresponds to a phase value expressed in radians. The solid curve is the dependence of the coordinate on time and simultaneously on the phase of oscillations (upper and lower values ​​on the x-axis, respectively) for a point that performs harmonic oscillations according to the law:

$x=x_(m)cosω_(0)t$

Here the initial phase is equal to zero $φ_0=0$. At the initial moment of time, the amplitude is maximum. This corresponds to the case of oscillations of a body attached to a spring (or a pendulum), which at the initial moment of time was taken away from the equilibrium position and released. It is more convenient to describe oscillations starting from an equilibrium position (for example, with a short push of a ball at rest) using the sine function:

As is known, $cosφ=sin(φ+(π)/(2))$, so the oscillations described by the equations $x=x_(m)cosω_(0)t$ and $x=sinω_(0)t$ differ from each other only in phases. The phase difference, or phase shift, is $(π)/(2)$. To determine the phase shift, you need to express the oscillating value through the same trigonometric function- cosine or sine. The dotted curve is shifted relative to the solid one by $(π)/(2)$.

Comparing the equations of free vibrations, coordinates, velocities and accelerations material point, we find that the velocity fluctuations are ahead in phase by $(π)/(2)$, and the acceleration fluctuations are ahead by $π$ of the displacement (coordinate) fluctuations.

damped vibrations

Attenuation of oscillations is a decrease in the amplitude of oscillations over time, due to the loss of energy by the oscillatory system.

Free vibrations are always damped vibrations.

Oscillation energy losses in mechanical systems are associated with its conversion into heat due to friction and environmental resistance.

Thus, the mechanical energy of pendulum oscillations is spent on overcoming the forces of friction and air resistance, while passing into internal energy.

The amplitude of the oscillations gradually decreases, and after a while the oscillations stop. Such fluctuations are called fading.

The greater the force of resistance to movement, the faster the oscillations stop. For example, vibrations in water stop faster than in air.

Elastic waves (mechanical waves)

Disturbances that propagate in space, moving away from their place of origin, are called waves.

Elastic waves are perturbations propagating in solid, liquid and gaseous media due to the action of elastic forces in them.

These environments are called elastic. A perturbation of an elastic medium is any deviation of the particles of this medium from its equilibrium position.

Take, for example, a long rope (or rubber tube) and attach one of its ends to the wall. Pulling the rope tight, with a sharp lateral movement of the hand, we will create a short-term disturbance at its loose end. We will see that this perturbation will run along the rope and, having reached the wall, will be reflected back.

The initial perturbation of the medium, leading to the appearance of a wave in it, is caused by the action in it of some foreign body, which is called wave source. This may be the hand of a person who hit the rope, a pebble that fell into the water, etc.

If the action of the source is of a short-term nature, then the so-called single wave. If the source of the wave makes a long oscillatory motion, then the waves in the medium begin to go one after another. A similar picture can be seen by placing a vibrating plate with a tip lowered into the water over a bath of water.

A necessary condition for the emergence of an elastic wave is the appearance at the moment of the occurrence of a perturbation of elastic forces that prevent this perturbation. These forces tend to bring neighboring particles of the medium closer together if they diverge, and move them away when they approach each other. Acting on particles of the medium that are more and more distant from the source, the elastic forces begin to take them out of their equilibrium position. Gradually, all particles of the medium, one after another, are involved in oscillatory motion. The propagation of these oscillations manifests itself in the form of a wave.

In any elastic medium, two types of motion simultaneously exist: oscillations of the particles of the medium and the propagation of a perturbation. A wave in which the particles of the medium oscillate along the direction of its propagation is called longitudinal, and the wave in which the particles of the medium oscillate across the direction of its propagation is called transverse.

Longitudinal wave

A wave in which oscillations occur along the direction of wave propagation is called longitudinal.

In an elastic longitudinal wave, perturbations are compressions and rarefactions of the medium. The compression deformation is accompanied by the appearance of elastic forces in any medium. Therefore, longitudinal waves can propagate in all media (in liquid, solid, and gaseous).

An example of the propagation of a longitudinal elastic wave is shown in the figure. The left end of a long spring suspended on threads is struck with a hand. From the impact, several turns approach each other, an elastic force arises, under the influence of which these turns begin to diverge. Continuing to move by inertia, they will continue to diverge, bypassing the equilibrium position and forming a rarefaction in this place. With a rhythmic impact, the coils at the end of the spring will either approach or move away from each other, that is, they will oscillate around their equilibrium position. These vibrations will gradually be transmitted from coil to coil along the entire spring. Condensations and rarefaction of coils will spread along the spring, or elastic wave.

transverse wave

Waves in which vibrations occur perpendicular to the direction of their propagation are called transverse.

In a transverse elastic wave, perturbations are displacements (shifts) of some layers of the medium relative to others. Shear deformation leads to the appearance of elastic forces only in solids: the shift of layers in gases and liquids is not accompanied by the appearance of elastic forces. Therefore, transverse waves can propagate only in solids.

plane wave

A plane wave is a wave whose direction of propagation is the same at all points in space.

In such a wave, the amplitude does not change with time (with distance from the source). Such a wave can be obtained if a large plate located in a continuous homogeneous elastic medium is made to oscillate perpendicular to the plane. Then all points of the medium adjacent to the plate will oscillate with the same amplitudes and the same phases. These oscillations will propagate in the form of waves in the direction of the normal to the plate, and all particles of the medium lying in planes parallel to the plate will oscillate with the same phases.

The locus of points at which the phase of the oscillations has the same value is called wave surface, or wave front.

From this point of view, a plane wave can be given the following definition.

A wave is called flat if its wave surfaces represent a set of planes parallel to each other.

The line normal to the wave surface is called beam. Wave energy is transferred along the rays. For plane waves, the rays are parallel lines.

The plane sine wave equation is:

$s=s_(m)sin[ω(t-(x)/(υ))+φ_0]$

where $s$ is the displacement of the oscillating point, $s_m$ is the amplitude of the oscillations, $ω$ is the cyclic frequency, $t$ is the time, $х$ is the current coordinate, $υ$ is the propagation velocity of the oscillations or wave velocity, $φ_0$ - initial phase of oscillations.

spherical wave

A wave is called spherical if its wave surfaces look like concentric spheres. The center of these spheres is called the center of the wave.

Rays in such a wave are directed along radii diverging from the center of the wave. In the figure, the source of the wave is a pulsating sphere.

The amplitude of oscillations of particles in a spherical wave necessarily decreases with distance from the source. The energy emitted by the source is evenly distributed over the surface of the sphere, the radius of which continuously increases as the wave propagates. The spherical wave equation has the form:

$s=(a_0)/(r)sin[ω(t-(r)/(υ))+φ_0]$

Unlike a plane wave, where $s_m=A$ is a constant value, in a spherical wave it decreases with distance from the wave center.

Wave length and speed

Any wave propagates with some speed. Under wave speed understand the propagation speed of the disturbance. For example, an impact on the end of a steel rod causes local compression in it, which then propagates along the rod at a speed of about $5$ km/s.

The speed of a wave is determined by the properties of the medium in which this wave propagates. When a wave passes from one medium to another, its speed changes.

Wavelength is the distance over which a wave propagates in a time equal to the period of oscillation in it.

Since the speed of the wave is a constant value (for a given medium), the distance traveled by the wave is equal to the product of the speed and the time of its propagation. Thus, to find the wavelength, it is necessary to multiply the speed of the wave by the period of oscillations in it:

where $υ$ is the wave speed, $T$ is the oscillation period in the wave, $λ$ ( Greek letter lambda) is the wavelength.

The formula $λ=υT$ expresses the relationship between the wavelength and its speed and period. Taking into account that the oscillation period in a wave is inversely proportional to the frequency $v$, i.e. $T=(1)/(v)$, we can obtain a formula expressing the relationship between the wavelength and its speed and frequency:

$λ=υT=υ(1)/(v)$

The resulting formula shows that the speed of a wave is equal to the product of the wavelength and the frequency of oscillations in it.

Wavelength is the spatial period of the wave. On a wave graph, wavelength is defined as the distance between the two closest points of the harmonic traveling wave, which are in the same phase of oscillations. The drawing is, as it were, instantaneous photographs of waves in an oscillating elastic medium at times $t$ and $t+∆t$. The $x$ axis coincides with the direction of wave propagation, and the displacements $s$ of oscillating particles of the medium are plotted on the y-axis.

The frequency of oscillations in the wave coincides with the frequency of oscillations of the source, since the oscillations of particles in the medium are forced and do not depend on the properties of the medium in which the wave propagates. When a wave passes from one medium to another, its frequency does not change, only the speed and wavelength change.

Interference and diffraction of waves

Wave interference (from lat. inter - mutually, between themselves and ferio - I hit, hit) - mutual strengthening or weakening of two (or more) waves when they are superimposed on each other while simultaneously propagating in space.

Usually, the interference effect is understood as the fact that the resulting intensity at some points in space is greater, at others - less than the total intensity of the waves.

Wave interference- one of the main properties of waves of any nature: elastic, electromagnetic, including light, etc.

Interference of mechanical waves

The addition of mechanical waves - their mutual superposition - is easiest to observe on the surface of the water. If you excite two waves by throwing two stones into the water, then each of these waves behaves as if the other wave does not exist. Sound waves from different independent sources behave similarly. At each point in the medium, the oscillations caused by the waves simply add up. The resulting displacement of any particle of the medium is an algebraic sum of displacements that would occur during the propagation of one of the waves in the absence of the other.

If two coherent harmonic waves are simultaneously excited in water at two points $O_1$ and $O_2$, then ridges and troughs will be observed on the surface of the water, which do not change with time, i.e., there will be interference.

The condition for the occurrence of the maximum intensity at some point $M$, located at distances $d_1$ and $d_2$ from the sources of waves $O_1$ and $O_2$, the distance between which is $l<< d_1$ и $l << d_2$, будет:

where $k = 0,1,2,...$, and $λ$ is the wavelength.

The amplitude of oscillations of the medium at a given point is maximum if the difference between the paths of two waves that excite oscillations at this point is equal to an integer number of wavelengths and provided that the phases of the oscillations of the two sources coincide.

The path difference $∆d$ is understood here as the geometric difference in the paths that waves travel from two sources to the point under consideration: $∆d=d_2-d_1$. With a path difference $∆d=kλ$, the phase difference of two waves is equal to an even number $π$, and the oscillation amplitudes will add up.

Minimum condition is an:

$∆d=(2k+1)(λ)/(2)$

The amplitude of the oscillations of the medium at a given point is minimal if the difference between the paths of the two waves that excite oscillations at this point is equal to an odd number of half-waves and provided that the phases of the oscillations of the two sources coincide.

The phase difference of the waves in this case is equal to an odd number $π$, i.e., the oscillations occur in antiphase, therefore, they are damped; the amplitude of the resulting oscillation is zero.

Interference Energy Distribution

As a result of interference, the energy is redistributed in space. It concentrates in the highs due to the fact that it does not enter the lows at all.

Wave diffraction

Diffraction of waves (from Latin diffractus - broken) - in the original narrow sense - the rounding of obstacles by waves, in the modern - wider - any deviations in the propagation of waves from the laws of geometric optics.

Wave diffraction manifests itself especially clearly in cases where the dimensions of the obstacles are smaller than or comparable to the wavelength.

The ability of waves to bend around obstacles can be observed on sea waves that easily bend around a stone, the dimensions of which are small compared to the wavelength. Sound waves are also able to bend around obstacles, thanks to which we hear, for example, the signal of a car located around the corner of a house.

The phenomenon of wave diffraction on the surface of water can be observed if a screen with a narrow slit is placed in the path of the waves, the dimensions of which are smaller than the wavelength. Behind the screen, a circular wave propagates, as if an oscillating body, the source of the waves, was located in the opening of the screen. According to the Huygens-Fresnel principle, this is how it should be. Secondary sources in a narrow gap are located so close to each other that they can be considered as one point source.

If the dimensions of the slit are large compared to the wavelength, then the wave passes through the slit, almost without changing its shape, only barely noticeable curvature of the wave surface is visible at the edges, due to which the wave also penetrates into the space behind the screen.

Sound (sound waves)

Sound (or sound waves) are oscillatory motions of particles of an elastic medium propagating in the form of waves: gaseous, liquid or solid.

The word "sound" is also understood as sensations caused by the action of sound waves on a special sense organ (hearing organ or, more simply, ear) of a person and animals: a person hears a sound with a frequency of $ 16 Hz to $ 20 $ kHz. The frequencies in this range are called sound.

So, the physical concept of sound implies elastic waves not only of those frequencies that a person hears, but also lower and higher frequencies. The first are called infrasound, second- ultrasound. The highest frequency elastic waves in the range of $10^(9) - 10^(13)$ Hz belong to hypersound.

You can “hear” sound waves by making a long steel ruler clamped in a vise tremble. However, if a large part of the ruler protrudes above the vise, then, having caused its oscillations, we will not hear the waves generated by it. But if you shorten the protruding part of the ruler and thereby increase the frequency of its oscillations, then the ruler will start to sound.

Sound sources

Any body vibrating at a sound frequency is a source of sound, since waves propagating from it arise in the environment.

There are both natural and artificial sources of sound. One of the artificial sources of sound, the tuning fork, was invented in 1711 by the English musician J. Shore for tuning musical instruments.

A tuning fork is a bent (in the form of two branches) metal rod with a holder in the middle. By hitting one of the branches of the tuning fork with a rubber mallet, we will hear a certain sound. The branches of the tuning fork begin to vibrate, creating alternating compression and rarefaction of the air around them. Propagating through the air, these perturbations form a sound wave.

The standard vibration frequency of the tuning fork is $440$ Hz. This means that for $1$ from its branches $440$ of vibrations are made. They are invisible to the eye. If, however, you touch the sounding tuning fork with your hand, you can feel its vibration. To determine the nature of the vibrations of the tuning fork, a needle should be attached to one of its branches. Having made the tuning fork sound, we draw a needle connected to it along the surface of a smoked glass plate. A trace in the form of a sinusoid will appear on the plate.

To amplify the sound emitted by the tuning fork, its holder is mounted on a wooden box, open on one side. This box is called resonator. When the tuning fork vibrates, the vibration of the box is transmitted to the air in it. Due to the resonance that occurs when the box is properly sized, the amplitude of the forced oscillations of the air increases, and the sound is amplified. Its amplification is also facilitated by an increase in the area of ​​the radiating surface, which occurs when the tuning fork is connected to the box.

Something similar happens in such musical instruments as guitar, violin. By themselves, the strings of these instruments create a faint sound. It becomes loud due to the presence of a body of a certain shape with a hole through which sound waves can escape.

Sound sources can be not only oscillating solid bodies, but also some phenomena that cause pressure fluctuations in the environment (explosions, flight of bullets, howling wind, etc.). The most striking example of such phenomena is lightning. During a thunderstorm, the temperature in the lightning channel rises to $30,000°$C. The pressure rises sharply, and a shock wave appears in the air, gradually turning into sound vibrations (with a typical frequency of $60$ Hz), propagating in the form of thunder.

An interesting sound source is the disk siren invented by the German physicist T. Seebeck (1770-1831). It is a disk connected to an electric motor with holes located in front of a strong air stream. As the disk rotates, the air flow through the holes is periodically interrupted, resulting in a sharp characteristic sound. The frequency of this sound is determined by the formula $v=nk$, where $n$ is the disk rotation frequency, $k$ is the number of holes in it.

By using a siren with multiple rows of holes and an adjustable disc speed, sounds of different frequencies can be obtained. The frequency range of sirens used in practice is usually from $200$ Hz to $100$ kHz and higher.

These sound sources got their name by the name of half-birds, half-women, who, according to ancient Greek myths, lured sailors on ships with their singing, and they crashed on the coastal rocks.

Sound receivers

Sound receivers are used to perceive sound energy and convert it into other types of energy. Sound receivers include, in particular, the hearing apparatus of humans and animals. In technology, sound is received mainly by microphones (in air), hydrophones (in water), and geophones (in the earth's crust).

In gases and liquids, sound waves propagate in the form of longitudinal compression and rarefaction waves. Compression and rarefaction of the medium arising from vibrations of the sound source (bell, string, tuning fork, telephone membrane, vocal cords, etc.) reach the human ear after some time, causing the eardrum to make forced oscillations with a frequency corresponding to the frequency of the sound source . Trembling of the tympanic membrane is transmitted through the system of bones to the endings of the auditory nerve, irritates them and thereby causes certain auditory sensations in a person. Animals also respond to elastic vibrations, although they perceive waves of other frequencies as sound.

The human ear is a very sensitive instrument. We begin to perceive sound already when the amplitude of oscillations of air particles in a wave turns out to be equal to only the radius of an atom! With age, due to the loss of elasticity of the eardrum, the upper limit of frequencies perceived by a person gradually decreases. Only young people are able to hear sounds with a frequency of $20$ kHz. On average, and even more so at an older age, both men and women cease to perceive sound waves whose frequency exceeds $ 12-14 $ kHz.

People's hearing deteriorates as a result of prolonged exposure to loud sounds. Working near high-powered aircraft, in very noisy factory floors, frequenting discos, and excessive use of audio players adversely affect the acuity of sound perception (especially high-frequency ones) and, in some cases, can lead to hearing loss.

Sound volume

Loudness is a subjective quality of the auditory sensation that allows sounds to be placed on a scale from quiet to loud.

The auditory sensations that various sounds cause in us largely depend on the amplitude of the sound wave and its frequency, which are the physical characteristics of the sound wave. These physical characteristics correspond to certain physiological characteristics associated with our perception of sound.

The loudness of a sound is determined by its amplitude: the greater the amplitude of oscillations in a sound wave, the greater the volume.

So, when the vibrations of a sounding tuning fork decay, along with the amplitude, the volume of the sound also decreases. Conversely, by hitting the tuning fork harder and thereby increasing the amplitude of its oscillations, we will also cause a louder sound.

The loudness of a sound also depends on how sensitive our ear is to that sound. The human ear is most sensitive to sound waves with a frequency of $1-5$ kHz. Therefore, for example, a high female voice with a frequency of $1000$ Hz will be perceived by our ear as louder than a low male voice with a frequency of $200$ Hz, even if the amplitudes of vibrations of the vocal cords are the same.

The loudness of the sound also depends on its duration, intensity and on the individual characteristics of the listener.

sound intensity is the energy carried by a sound wave in $1$s through a surface with an area of ​​$1m^2$. It turned out that the intensity of the loudest sounds (which cause a sensation of pain) exceeds the intensity of the weakest sounds accessible to human perception by $10 trillion times! In this sense, the human ear turns out to be a much more advanced device than any of the usual measuring instruments. None of them can measure such a wide range of values ​​(for instruments, the measurement range rarely exceeds $100$).

The unit of loudness is called sleep. A muffled conversation has a volume of $1$. The ticking of a clock is characterized by a volume of about $0.1$ son, a normal conversation is $2$ son, the sound of a typewriter is $4$ son, and a loud street noise is $8$ son. In the blacksmith's shop, the volume reaches $64$ sone, and at a distance of $4$ m from a running jet engine - $264$ sone. Sounds even louder begin to cause pain.

Pitch

In addition to loudness, sound is characterized by height. The pitch of a sound is determined by its frequency: the higher the frequency of vibrations in a sound wave, the higher the sound. Low frequency vibrations correspond to low sounds, high frequency vibrations correspond to high sounds.

So, for example, a bumblebee flaps its wings with a lower frequency than a mosquito: for a bumblebee it is $220 flaps per second, and for a mosquito it is $500-600$. Therefore, the flight of a bumblebee is accompanied by a low sound (buzz), and the flight of a mosquito is accompanied by a high sound (squeak).

A sound wave of a certain frequency is otherwise called a musical tone, so pitch is often referred to as pitch.

The main tone mixed with several vibrations of other frequencies forms a musical sound. For example, violin and piano sounds can include up to $15-20$ of different vibrations. Its timbre depends on the composition of each complex sound.

The frequency of free vibrations of a string depends on its size and tension. Therefore, by stretching the strings of the guitar with the help of pegs and pressing them to the neck of the guitar in different places, we change their natural frequency, and, consequently, the pitch of the sounds they make.

The nature of sound perception largely depends on the layout of the room in which speech or music is heard. This is explained by the fact that in closed rooms, the listener perceives, in addition to direct sound, also a continuous series of repetitions quickly following each other, caused by multiple reflections of sound from objects in the room, walls, ceiling and floor.

sound reflection

At the boundary between two different media, part of the sound wave is reflected, and part travels further.

When sound passes from air to water, $99.9%$ of the sound energy is reflected back, but the pressure in the sound wave transmitted into the water is almost $2$ times greater than in air. The auditory apparatus of fish reacts precisely to this. Therefore, for example, screams and noises above the surface of the water are a sure way to scare away marine life. These screams will not deafen a person who is under water: when immersed in water, air plugs will remain in his ears, which will save him from sound overload.

When sound passes from water to air, $99.9%$ of energy is again reflected. But if the sound pressure increased during the transition from water to air, now, on the contrary, it sharply decreases. It is for this reason that a person above the water does not hear the sound that occurs under water when one stone strikes another.

This behavior of sound on the border between water and air gave reason to our ancestors to consider the underwater world as a “world of silence”. Hence the expression "mute like a fish." However, even Leonardo da Vinci suggested listening to underwater sounds by putting your ear to an oar lowered into the water. Using this method, you can see that the fish are actually quite talkative.

Echo

The reflection of the sound also explains the echo. Echoes are sound waves reflected from some obstacle (buildings, hills, trees) and returning to their source. We hear an echo only when the reflected sound is perceived separately from the spoken one. This happens when sound waves reach us, successively reflected from several obstacles and separated by a time interval $t > 50-60$ ms. Then there is a multiple echo. Some of these phenomena have become world famous. So, for example, the rocks located in the form of a circle near Adersbach in the Czech Republic repeat $7$ syllables in a certain place, and in Woodstock Castle in England the echo clearly repeats $17$ syllables!

The word "echo" is associated with the name of the mountain nymph Echo, who, according to ancient Greek mythology, was unrequitedly in love with Narcissus. From longing for her beloved, Echo dried up and turned to stone so that only a voice remained of her, capable of repeating the endings of words spoken in her presence.

Why is there no echo in a small apartment? After all, in it the sound should be reflected from the walls, ceiling, floor. The fact is that the time $t$ during which the sound travels a distance, say, $s=6m$, propagating at a speed of $υ=340$ m/s, is equal to:

$t=(s)/(υ)=(6)/(340)=0.02c$

And this is much less than the time ($0.06$ s) needed to hear the echo.

The increase in the duration of a sound caused by its reflections from various obstacles is called reverberation. Reverb is great in empty rooms where it leads to boominess. Conversely, rooms with upholstered walls, draperies, curtains, upholstered furniture, carpets, as well as those filled with people absorb sound well, and therefore reverberation in them is negligible.

Sound speed

Sound propagation requires an elastic medium. Sound waves cannot propagate in a vacuum because there is nothing to vibrate there. This can be verified by a simple experiment. If an electric bell is placed under a glass bell, then as the air is pumped out from under the bell, the sound from the bell will become weaker and weaker until it stops altogether.

It is known that during a thunderstorm we see a flash of lightning and only after a while hear thunder. This delay occurs due to the fact that the speed of sound in air is much less than the speed of light coming from lightning.

speed of sound in air was first measured in 1636 by the French scientist M. Mersenne. At a temperature of $20°$C, it is equal to $343$ m/s, i.e. $1235$ km/h. Note that it is to this value that the speed of a bullet fired from a Kalashnikov assault rifle decreases at a distance of $800$ m. The muzzle velocity of the bullet is $825$ m/s, which is much higher than the speed of sound in air. Therefore, a person who hears the sound of a shot or the whistle of a bullet need not worry: this bullet has already passed him. The bullet outruns the sound of the shot and reaches its victim before the sound arrives.

The speed of sound in gases depends on the temperature of the medium: with an increase in air temperature, it increases, and with a decrease, it decreases. At $0°$С, the speed of sound in air is $332$ m/s.

Sound travels at different speeds in different gases. The larger the mass of gas molecules, the lower the speed of sound in it. Thus, at a temperature of $0°$C, the speed of sound in hydrogen is $1284$ m/s, in helium - $965$ m/s, and in oxygen - $316$ m/s.

The speed of sound in liquids, as a rule, is greater than the speed of sound in gases. The speed of sound in water was first measured in 1826 by J. Colladon and J. Sturm. They conducted their experiments on Lake Geneva in Switzerland. On one boat they set fire to gunpowder and at the same time struck a bell lowered into the water. The sound of this bell, lowered into the water, was caught on another boat, which was at a distance of $14$ km from the first one. The speed of sound in water was determined from the time interval between the flash of the light signal and the arrival of the sound signal. At a temperature of $8°$C, it turned out to be $1440$ m/s.

Speed ​​of sound in solids more than liquids and gases. If you put your ear to the rail, then after hitting the other end of the rail, two sounds are heard. One of them reaches the ear along the rail, the other - through the air.

Earth has good sound conductivity. Therefore, in the old days, during a siege, “hearers” were placed in the fortress walls, who, by the sound transmitted by the earth, could determine whether the enemy was digging to the walls or not. Putting their ear to the ground, they also watched the approach of the enemy cavalry.

Solid bodies conduct sound well. Because of this, people who have lost their hearing are sometimes able to dance to music that reaches the auditory nerves not through the air and outer ear, but through the floor and bones.

The speed of sound can be determined by knowing the wavelength and frequency (or period) of oscillation:

$υ=λv, υ=(λ)/(T)$

infrasound

Sound waves with a frequency less than $16$ Hz are called infrasound.

The human ear does not perceive infrasonic waves. Despite this, they are able to have a certain physiological effect on a person. This action is explained by resonance. The internal organs of our body have rather low natural frequencies: the abdominal cavity and chest - $5-8$ Hz, the head - $20-30$ Hz. The average value of the resonant frequency for the whole body is $6$ Hz. Having frequencies of the same order, infrasonic waves make our organs vibrate and, at very high intensity, can lead to internal hemorrhages.

Special experiments have shown that irradiating people with sufficiently intense infrasound can cause loss of balance, nausea, involuntary rotation of the eyeballs, etc. For example, at a frequency of $4-8$ Hz, a person feels the movement of internal organs, and at a frequency of $12$ Hz - illness.

They say that once the American physicist R. Wood (who was known among his colleagues as a great original and a merry fellow) brought to the theater a special apparatus emitting infrasonic waves, and turning it on, directed it to the stage. No sound was heard, but the actress had a tantrum.

The resonant effect of low-frequency sounds on the human body also explains the exciting effect of modern rock music, saturated with repeatedly amplified low frequencies of drums and bass guitars.

Infrasound is not perceived by the human ear, but some animals can hear it. For example, jellyfish confidently perceive infrasonic waves with a frequency of $8-13$ Hz, which occur during a storm as a result of the interaction of air currents with the crests of sea waves. Reaching the jellyfish, these waves in advance (for $15$ hours!) "Warn" about the approaching storm.

Sources of infrasound lightning discharges, shots, volcanic eruptions, working jet aircraft engines, wind flowing around the crests of sea waves, etc. can serve. Infrasound is characterized by low absorption in various media, as a result of which it can propagate over very long distances. This allows you to determine the location of strong explosions, the position of the firing gun, control underground nuclear explosions, predict tsunamis, etc.

Ultrasound

Elastic waves with a frequency above $20$ kHz are called ultrasound.

Ultrasound in the animal world. Ultrasound, like infrasound, is not perceived by the human ear, but some animals are able to emit and perceive it. So, for example, dolphins, thanks to this, confidently navigate in muddy water. By sending and receiving returned ultrasonic pulses, they are able to detect even a small pellet carefully lowered into the water at a distance of $20-30$ m. Ultrasound also helps bats that can't see well or see nothing at all. By emitting ultrasonic waves with the help of their hearing aid (up to $250 per second), they are able to navigate in flight and successfully catch prey even in the dark. It is curious that some insects developed a special defensive reaction in response to this: certain species of night butterflies and beetles also turned out to be able to perceive ultrasounds emitted by bats, and, having heard them, they immediately fold their wings, fall down and freeze on the ground.

Ultrasonic signals are also used by some whales. These signals allow them to hunt squid in the complete absence of light.

It has also been established that ultrasonic waves with a frequency of more than $25$ kHz cause pain in birds. This is used, for example, to scare away seagulls from drinking water reservoirs.

The use of ultrasound in technology. Ultrasound is widely used in science and technology, where it is obtained using various mechanical (for example, a siren) and electromechanical devices.

Sources of ultrasound are installed on ships and submarines. By sending short pulses of ultrasonic waves, you can catch their reflections from the bottom or any other objects. The delay time of the reflected wave can be used to judge the distance to the obstacle. The echo sounders and sonar used in this case make it possible to measure the depth of the sea, solve various navigational tasks (swimming near rocks, reefs, etc.), carry out fishing reconnaissance (detect schools of fish), and also solve military tasks (search for enemy submarines, non-periscope torpedo attacks, etc.).

In the industry, the reflection of ultrasound from cracks in metal castings is used to judge defects in products.

Ultrasounds crush liquid and solid substances, forming various emulsions and suspensions.

Using ultrasound, it is possible to solder aluminum products, which cannot be done using other methods (since there is always a dense layer of oxide film on the surface of aluminum). The tip of an ultrasonic soldering iron not only heats up, but also oscillates with a frequency of about $20$ kHz, due to which the oxide film is destroyed.

The conversion of ultrasound into electrical vibrations, and then into light, allows sound vision to be realized. With the help of sound vision, you can see objects in water that is opaque to light.

In medicine, ultrasound is used to weld broken bones, detect tumors, carry out diagnostic studies in obstetrics, etc. The biological effect of ultrasound (leading to the death of microbes) makes it possible to use it for pasteurizing milk and sterilizing medical instruments.

Topics of the USE codifier: fluid pressure, Pascal's law, Archimedes' law, conditions for navigation of bodies.

In hydro- and aerostatics, two questions are considered: 1) the equilibrium of liquids and gases under the action of forces applied to them; 2) the equilibrium of solids in liquids and gases.

When a medium is compressed, elastic forces arise in it, called pressure forces. Pressure forces act between adjacent layers of the medium, on solid bodies immersed in the medium, as well as on the bottom and walls of the vessel.

The pressure force of the medium has two characteristic properties.

1. The pressure force acts perpendicular to the surface of the selected element of the medium or solid body. This is explained by the fluidity of the medium: elastic forces do not arise in it with a relative shift of the layers, therefore there are no elastic forces tangential to the surface.

2. The force of pressure is evenly distributed over the surface on which it acts.

A natural quantity that arises in the process of studying the pressure forces of a medium is pressure.

Let a force act on the surface of the area, which is perpendicular to the surface and uniformly distributed over it. Pressure is the quantity

The unit of pressure is the pascal (Pa). 1 Pa is the pressure exerted by a force of 1 N on a surface of 1 m2.

It is useful to remember the approximate value of normal atmospheric pressure: Pa.

hydrostatic pressure.

Hydrostatic is the pressure of a stationary fluid caused by gravity. Find the formula for the hydrostatic pressure of a liquid column.

Suppose that a vessel with a bottom area is filled with liquid up to a height (Fig. 1). The density of the liquid is

The volume of the liquid is , so the mass of the liquid is . The pressure force of the liquid on the bottom of the vessel is the weight of the liquid. Since the fluid is stationary, its weight is equal to the force of gravity:

Dividing the force by the area, we get the fluid pressure:

This is the formula for hydrostatic pressure.

So, at a depth of 10 m, water exerts pressure Pa, approximately equal to atmospheric pressure. We can say that atmospheric pressure is approximately equal to 10 m of water column.

For practice, such a large height of the liquid column is inconvenient, and real liquid manometers are mercury. Let's see how high a mercury column (kg/m) must be to create a similar pressure:

This is why the millimeter of mercury (mmHg) is widely used to measure atmospheric pressure.

Pascal's law.

If you put a nail vertically and hit it with a hammer, the nail will transmit the action of the hammer vertically, but not sideways. Solid bodies, due to the presence of a crystal lattice, transmit the pressure produced on them only in the direction of the force.

Liquids and gases (recall that we call them media) behave differently. In environments, Pascal's law is valid.

Pascal's law. The pressure exerted on a liquid or gas is transmitted to any point of this medium without change in all directions.

(In particular, the same pressure force acts on a platform placed inside a liquid at a fixed depth, no matter how you turn this platform.)

For example, a diver at depth experiences pressure. Why? According to Pascal's law, water transfers atmospheric pressure unchanged to depth, where it is added to the hydrostatic pressure of the water column.

An excellent illustration of Pascal's law is the experience with Pascal's ball. This is a ball with many holes connected to a cylindrical vessel ( fig. 2)

If you pour water into a vessel and move the piston, water will spray out of all the holes. This just means that water transmits external pressure in all directions.

The same is observed for gas: if the vessel is filled with smoke, then when the piston moves, wisps of smoke will again come out of all the holes at once. Therefore, the gas also transmits pressure in all directions.

You use Pascal's law every day when you squeeze toothpaste out of a tube. Namely, you squeeze the tube in the transverse direction, and the paste moves perpendicular to your effort - in the longitudinal direction. Why? Your pressure is transmitted inside the tube in all directions, in particular - towards the opening of the tube. That's where the paste comes out.

Hydraulic Press.

Hydraulic Press - This is a device that gives a gain in strength. That is, by applying a relatively small force in one place of the device, it is possible to obtain a much larger force in another place.

The hydraulic press is shown in fig. 3 . It consists of two communicating vessels with different cross-sectional area and closed by pistons. There is liquid in the vessels between the pistons.

The principle of operation of a hydraulic press is very simple and based on Pascal's law.

Let be the area of ​​the small piston and be the area of ​​the large piston. Let's put pressure on the small
piston with force. Then under the small piston in the liquid there will be pressure:

According to Pascal's law, this pressure will be transmitted unchanged in all directions to any point in the liquid, in particular - under a large piston. Therefore, a force will act on the large piston from the liquid side:

The resulting ratio can be rewritten as follows:

We see that more is as many times as more. For example, if the area of ​​the large piston is 100 times the area of ​​the small piston, then the force on the large piston will be 100 times the force on the small piston. This is how a hydraulic press gives a gain in strength.

Law of Archimedes.

We know that wood does not sink in water. Therefore, the force of gravity is balanced by some other force acting on the piece of wood from the side of the water vertically upwards. This force is called
pushing or Archimedean by force. It acts on any body immersed in a liquid or gas.

Find out the cause of the Archimedean force. Consider a cylinder with cross-sectional area and height , immersed in a liquid of density . The bases of the cylinder are horizontal. The upper base is at a depth of , the lower one is at a depth (Fig. 4).

Rice. 4.

Pressure forces act on the side surface of the cylinder, which only lead to compression of the cylinder. These forces can be ignored.

At the level of the upper base of the cylinder, the fluid pressure is equal to . The pressure force acting vertically downwards acts on the upper base.

At the level of the lower base of the cylinder, the fluid pressure is equal to . The pressure force directed vertically upwards acts on the lower base (Pascal's law!).

Since , then , and therefore there is a resultant of pressure forces directed upwards. This is the Archimedean force. We have:

But the product is equal to the volume of the cylinder. We finally get:

. (1)

This is the formula for the Archimedean force. The Archimedean force arises due to the fact that the pressure of the liquid on the lower base of the cylinder is greater than on the upper one.

Formula (1) can be interpreted as follows. The product is the mass
liquid whose volume is equal to . But then , where is the weight of the liquid taken in volume . Therefore, along with (1) we have:

. (2)

In other words, the Archimedean force acting on the cylinder is equal to the weight of the fluid, the volume of which coincides with the volume of the cylinder.

Formulas (1) and (2) are also valid in the general case, when a volume body immersed in a liquid or gas has any shape, and not just the shape of a cylinder (of course, in the case of a gas, this is the density of the gas). Let's explain why this happens.

Let's allocate mentally in the environment some volume of the arbitrary form. This volume is in balance: it does not sink and does not float. Consequently, the force of gravity acting on the medium located inside the volume we selected is balanced by the forces of pressure on the surface of our volume from the rest of the medium - after all, the lower elements of the surface have more pressure than the upper ones.

In other words, the resultant of the forces of hydrostatic pressure on the surface of the selected volume - the Archimedean force - is directed vertically upwards and is equal to the weight of the medium in this volume.

The force of gravity acting on our volume is applied to its center of gravity. This means that the Archimedean force must also be applied to the center of gravity of the selected volume. Otherwise, gravity and the Archimedean force form a pair of forces that will cause the rotation of our volume (and it is in equilibrium).

And now let's replace the selected volume of the medium with a solid body of the same volume and the same shape. It is clear that the forces of pressure of the medium on the surface of the body will not change, since the configuration environment surrounding the body. Therefore, the Archimedean force will continue to be directed vertically upward and equal to the weight of the medium, taken in volume. The point of application of the Archimedean force will be the center of gravity of the body.

Law of Archimedes. A body immersed in a liquid or gas is acted upon by a buoyant force directed vertically upwards and equal to the weight of the medium, the volume of which is equal to the volume of the body.

Thus, the Archimedean force is always found by the formula (1) . Note that this formula does not include either the density of the body or any of its geometric characteristics - with a fixed volume, the value of the Archimedean force does not depend on the substance and shape of the body.

So far, we have considered the case of complete immersion of the body. What is the Archimedean force for partial immersion? No buoyant force acts on that part of the body that is above the surface of the liquid. If this part is mentally cut off, then the magnitude of the Archimedean force will not change. But then we will get a completely immersed body, the volume of which is equal to the volume of the immersed part of the original body.

This means that a body partially immersed in a liquid is affected by a buoyant force equal to the weight of the liquid, the volume of which is equal to the volume of the immersed part of the body. Formula (1) is also valid in this case, only the volume of the entire body must be replaced by the volume of the submerged part of the submersion:

Archimedes discovered that a body completely immersed in water displaces a volume of water equal to its own volume. The same fact holds for other liquids and gases. Therefore, we can say that any body immersed in a liquid or gas is affected by a buoyant force equal to the weight of the medium displaced by the body.

Swimming tel.

Consider a body of density and a liquid of density . Let us assume that the body is completely immersed in a liquid and released.

From this point on, only gravity and Archimedean force act on the body. If the volume of the body is , then

There are three possibilities for further movement of the body.

1. The force of gravity is greater than the Archimedean force: , or . In this case, the body sinks.

2. The force of gravity is equal to the Archimedean force: , or . In this case, the body remains motionless in the state indifferent balance.

3. The force of gravity is less than the Archimedean force: , or . In this case, the body floats, reaching the surface of the liquid. With further ascent, the volume of the submerged part of the body will begin to decrease, and with it the Archimedean force. At some point, the Archimedean force will equal the force of gravity (equilibrium position). By inertia, the body will float further, stop, and begin to sink again. . . Damped oscillations will occur, after which the body will remain floating in the equilibrium position (), partially immersed in the liquid.

Thus, the condition of body floating can be written as an inequality: .

In the fourth task of the exam in physics, we test our knowledge of communicating vessels, the forces of Archimedes, Pascal's law, moments of forces.

Theory for assignment No. 4 USE in physics

Moment of power

Moment of force is a quantity that characterizes the rotational action of a force on a rigid body. The moment of force is equal to the product of the force F at a distance h from the axis (or center) to the point of application of this force and is one of the main concepts of dynamics: M 0 = Fh.

Distanceh commonly referred to as the shoulder of strength.

In many problems of this section of mechanics, the rule of moments of forces that are applied to a body, conventionally considered a lever, is applied. The equilibrium condition of the lever F 1 / F 2 \u003d l 2 / l 1 can be used even if more than two forces are applied to the lever. In this case, the sum of all moments of forces is determined.

Law of communicating vessels

According to the law of communicating vessels in open communicating vessels of any type, the fluid pressure at each level is the same.

At the same time, the pressures of the columns above the liquid level in each vessel are compared. The pressure is determined by the formula: p=ρgh. If we equate the pressures of the columns of liquids, we get the equality: ρ 1 gh 1 = ρ 2 gh 2. From this follows the relation: ρ 1 h 1 = ρ 2 h 2, or ρ 1 / ρ 2 \u003d h 2 / h 1. This means that the heights of the liquid columns are inversely proportional to the density of the substances.

Strength of Archimedes

Archimedean or buoyant force occurs when some solid body is immersed in a liquid or gas. A liquid or gas tends to occupy the place “taken away” from them, therefore they push it out. The Archimedes force only works when the force of gravity acts on the body mg

The Archimedes force is traditionally referred to as F A.

Analysis of typical options for tasks No. 4 USE in physics

Demo version 2018

Solution algorithm:

1. Remember the rule of moments.
2. Find the moment of force created by load 1.
3. We find the shoulder of the force that will create load 2 when it is suspended. We find its moment of force.
4. We equate the moments of forces and determine the desired value of the mass.
5. We write down the answer.

Decision:

The first version of the task (Demidova, No. 1)

The moment of force acting on the lever on the left is 75 N∙m. What force must be applied to the lever on the right to keep it in balance if its arm is 0.5 m?

Solution algorithm:

1. We introduce the notation for the quantities that are given in the condition.
2. We write out the rule of moments of force.
3. We express force through the moment and shoulder. Calculate.
4. We write down the answer.

Decision:

1. To bring the lever into balance, moments of forces M 1 and M 2 applied to the left and right are applied to it. The moment of force on the left is conditionally equal to M 1 = 75 N∙m. The arm of the force on the right is equal to l= 0.5 m
2. Since it is required that the lever be in equilibrium, then by the rule of moments M 1 = M 2. Insofar as M 1 =F· l, then we have: M 2 =F∙ l.
3. From the resulting equality, we express the force: F\u003d M 2 /l= 75/0.5=150 N.

The second version of the task (Demidova, No. 4)

Archimedean or buoyant force occurs when some solid body is immersed in a liquid or gas. A liquid or gas tends to occupy the place “taken away” from them, therefore they push it out. The Archimedes force only works when gravity acts on the body mg. In weightlessness, this force does not arise.

Thread tension force T occurs when the thread is trying to stretch. It does not depend on whether gravity is present.

If several forces act on a body, then when studying its motion or equilibrium state, the resultant of these forces is considered.

Solution algorithm:

1. We translate the data from the condition into SI. We enter the tabular value of water density necessary for solving.
2. We analyze the condition of the problem, determine the pressure of liquids in each vessel.
3. We write down the equation of the law of communicating vessels.
4. We write down the answer.

Decision:

The third version of the task (Demidova, No. 20)

Solution algorithm:

1. We analyze the condition of the problem, determine the pressure of liquids in each vessel.
2. We write down the equality of the law of communicating vessels.
3. We substitute the numerical values ​​of the quantities and calculate the desired density.
4. We write down the answer.

Natural science is so human, so true,
that I wish good luck to everyone who gives himself to him ...
Johann Wolfgang von Goethe

We owe Archimedes the foundation of the doctrine of the equilibrium of liquids.
Joseph Louis Lagrange

BOX OF QUALITATIVE TASKS IN PHYSICS
ARCHIMEDE'S FORCE

Didactic materials on physics for students and their parents ;-) and, of course, for creative teachers.
For those who love to learn!

I bring to your attention 55 quality tasks in physics on the topic: "Archimedean force". Let's give credit to the integration: in the first lines... biophysical material; according to the tradition of green pages, we will not disregard fiction and illustrative material;-) and also accompany the tasks with informative notes and comments - for the curious, we will give detailed answers to some problems.
And more ;-) legendary tale of Archimedes' challenge with the golden crown.

Task #1
Most algae (for example, spirogyra, kelp, etc.) have thin, flexible stems. Why don't algae need strong, hard stems? What happens to algae if you release water from the reservoir in which they are located?

For the curious: Many aquatic plants remain upright, despite the extreme flexibility of their stems, because large air bubbles are enclosed at the ends of their branches, playing the role of floats.
water chestnut chilim. Curious aquatic plant chilim (water chestnut) grows in the backwaters of the Volga, in lakes and estuaries. Its fruits (water nuts) reach a diameter of 3 cm and have a shape similar to a sea anchor with or without a few sharp horns. This "anchor" serves to keep the young germinating plant in a suitable place. When the chilim fades, heavy fruits begin to form under water. They could drown the plant, but just at that time on the petioles of the leaves swellings are formed - a kind of "rescue belts". Thus, the volume of the underwater part of the plants increases, and, consequently, the buoyancy force increases. This achieves a balance between the weight of the fruit and the buoyancy force generated by the swelling.

Otto Wilhelm Thome(Otto Wilhelm Thome; 1840–1925) was a German botanist and illustrator. Author of a collection of botanical illustrations "Flora of Germany, Austria and Switzerland (Flora von Deutschland, Österreich und der Schweiz)", 1885

§ For flower growers, I suggest that you admire the flower portraits on the green page "Reinagle George Philip (botanical illustrations)".

Task #2
In mammals living on land, strong limbs are adapted for movement, but in marine mammals (whales, dolphins), fins and a tail are sufficient for movement. Explain why.

Answer: Archimedean force is an important natural factor that determines the structure of the skeleton of marine mammals. Since a buoyant (Archimedean force) acts on a creature living in water, its weight in liquid is less than in air by the value of this force. Thus, a “light” whale in the water, a dolphin, does not need strong limbs for movement, for this purpose they have enough fins and a tail.

Task #3
What role does the swim bladder play in fish?

For the curious: The density of living organisms inhabiting the aquatic environment differs very little from the density of water, so their weight is almost completely balanced by the Archimedean force. Thanks to this, aquatic animals do not need such massive skeletons as terrestrial ones. The role of the swim bladder in fish is interesting. This is the only body part of the fish that has noticeable compressibility; By squeezing the bubble with the efforts of the pectoral and abdominal muscles, the fish changes the volume of its body and, thereby, the average density, due to which it can regulate the depth of its diving within certain limits.

Task #4
How does a whale regulate its diving depth?

Answer: Whales regulate their diving depth by increasing and decreasing their lung capacity.

Archibald Thorburn(Archibald Thorburn; 05/31/1860 - 10/09/1935) - Scottish illustrator.

§ For lovers of animalistics, I recommend to look at the green page “Mystery Paintings by Artist Stephen Gardner” and count the tails of whales ;-)

Task #5
Although the whale lives in water, it breathes with lungs. Despite the presence of lungs, the whale will not live even an hour if it accidentally finds itself aground or on land. Why?

For the curious: The largest representatives of the order of cetaceans - blue whales. The mass of the blue whale reaches 130 tons; largest land animal elephant has a mass of 3 to 6 tons(like the language of some whales ;-) At the same time, the whale is able to develop a very decent speed in the water up to 20 knots. The force of gravity acting on the whale is estimated at millions of newtons, but in the water it is supported by the Archimedean force and the whale in the water is weightless. On land, the enormous force of gravity will press the whale to the ground. The whale's skeleton is not adapted to withstand this weight, the whale will not even be able to breathe, since in order to inhale it must expand the lungs, that is, raise the muscles surrounding the chest. Under the influence of such a huge force, breathing significantly worsens, blood vessels are pinched, and the whale dies.

Knot - unit of speed equal to one nautical mile per hour. It is used in nautical and aviation practice. By international definition, one knot is equal to 1,852 km/h.

Task #6
How to adjust diving depth cephalopod nautilus pompilius(lat. Nautilus pompilius)?

Answer: Cephalopods from the nautilus genus live in shells separated by partitions into separate chambers, the animal itself occupies the last chamber, while the rest are filled with gas. When the nautilus wants to sink to the bottom, it fills the shell with water, it becomes heavy and sinks easily. To float to the surface, the nautilus pumps gas into its hydrostatic "balloons", it displaces the water, and the shell floats. Liquid and gas are under pressure in the shell, so the mother-of-pearl house does not burst even at a depth of seven hundred meters, where nautiluses sometimes swim. The steel tube would flatten here, and the glass would turn into a snow-white powder. Nautilus manages to avoid death only thanks to the internal pressure that is maintained in its tissues, and to keep its house intact by filling it with an incompressible liquid. Everything happens, as in a modern deep-sea boat - a bathyscaphe, for which nature received a patent five hundred million years ago ;-)

Nautilus pompilius(lat. Nautilus pompilius) is a species of cephalopod molluscs of the genus Nautilus. It usually lives at a depth of up to 400 meters. It lives off the coast of Indonesia, the Philippines, New Guinea and Melanesia, in the South China Sea, the northern coast of Australia, western Micronesia and western Polynesia. Nautiluses lead a benthic lifestyle, collecting dead animals and large organic remains - that is, nautiluses are marine scavengers.

Kondakov Nikolai Nikolaevich(1908-1999) - Soviet biologist, candidate of biological sciences, animal painter. His main contribution to biological science was his drawings of various representatives of the fauna. These illustrations have been included in many publications, such as TSB (Great Soviet Encyclopedia), Red Book of the USSR, in animal atlases and teaching aids.

For the curious: At cuttlefish- an animal from the class cephalopods(the closest relative of squids and octopuses), vestigial internal calcareous shell contains numerous cavities. To regulate buoyancy, the cuttlefish pumps water out of its skeleton and allows gas to fill the emptied cavities, that is, it acts on the principle of water tanks in a submarine. The main way of movement of cuttlefish, octopuses, squids is jet, but this is a topic for another box of quality problems in physics ;-)
Microscopic radiolarians they have droplets of oil in their protoplasm, with the help of which they regulate their weight and thanks to which they rise and fall into the sea.
Siphonophores zoologists call a special group of intestinal animals. Like jellyfish, they are free-swimming marine animals. However, unlike the former, they form complex colonies with a very pronounced polymorphism. At the very top of the colony there is usually a bubble containing gas, with the help of which the entire colony is kept in the water column and moved. The gas is produced by special glands. This bubble sometimes reaches a length of 30 cm.

Rudimentary organs, rudiments(from Latin rudimentum - germ, fundamental principle) - organs that have lost their main significance in the process of evolutionary development of the organism.
Polymorphism - multiplicity, the presence in the same species of organisms of several different forms.

Illustrations from Ernst Haeckel's book
"Art Forms of Nature (Kunstformen der Natur)", 1904

cephalopods Gamochonia		Siphonophores Siphonophorae		deep sea radiolarians Phaeodaria

Ernst Heinrich Philipp August Haeckel(Ernst Heinrich Philipp August Haeckel; 1834–1919) was a German naturalist and philosopher.
"Art Forms of Nature (Kunstformen der Natur)"- lithographic book Ernst Haeckel originally published between 1899 and 1904 in sets of 10 prints, a full version of 100 prints appeared in 1904.

Task #7
Why do ducks and other waterfowl submerge little when swimming?

Answer: An important factor in the life of waterfowl is the presence of a thick, impermeable layer of feathers and down, which contains a significant amount of air; due to this peculiar air bubble surrounding the entire body of the bird, its average density is very low. This explains the fact that ducks and other waterfowl do not submerge much when swimming.

Task #8
"Meshchorskaya side", 1939

“... Water rats live in deep holes on the banks of these rivers. There are rats completely gray with old age. If you quietly follow the hole, you can see how the rat is catching fish. She crawls out of the hole, dives very deep and swims out with a terrible noise ... To make it easier to swim, water rats gnaw off a long stalk of the kugi and swim holding it in their teeth. The stalk of the coogee is full of air cells. He perfectly holds on the water not even such a weight as a rat ... "
Explain the measure taken by water rats to facilitate swimming.

Answer: Body buoyancy- its ability to float at a given load, having a predetermined immersion. Buoyancy margin - additional load, which corresponds to the weight of the liquid in the volume of the surface part of the floating body. The buoyancy of the body is determined by the law of Archimedes.
Law of Archimedes is formulated as follows: a buoyant force acts on a body immersed in a liquid or gas, equal to the weight of the amount of liquid or gas that is displaced by the immersed part of the body. Based on the law of Archimedes, it can be concluded that for a body to float, it is necessary that the weight of the liquid displaced by this body be equal to or exceed the weight of the body itself.
The enterprising water rat, unfamiliar with the law of Archimedes, successfully used it for its unselfish, but beneficial purposes ...

Kuga- the popular name of some aquatic plants of the sedge family, mainly lake reeds. The stems of the lake reed, like many other aquatic plants, are very loose, porous - they are densely penetrated by a network of air channels and therefore have excellent buoyancy.

Task #9
"Steppe. History of one trip", 1888. Anton Pavlovich Chekhov
“... Egorushka also undressed, but did not go down the bank, but ran up and flew from a height of one and a half sazhens. Describing an arc in the air, he fell into the water, sank deep, but did not reach the bottom; some force, cold and pleasant to the touch, picked him up and carried him back upstairs.
What kind of force "cold and pleasant to the touch" are we talking about?

For the curious: Sazhen - old Russian measure of length, first mentioned in Russian sources at the beginning of the 11th century. In the XI-XVII centuries, there was a sazhen of 152 and 176 cm. This was the so-called fly fathom, determined by the span of a person’s hands from the end of the fingers of one hand to the end of the fingers of the other.
So-called oblique sazhen- measuring 216 and 248 cm - was determined by the distance from the fingers of the outstretched hand to the foot of the opposite leg. Under Peter I, Russian measures of length were equalized with English ones. The size of a sazhen was determined to be 7 English feet, or 84 inches. This corresponded to 3 arshins, or 48 inches, which equaled 213.35 cm.

1 fathom= 1/500 versts = 3 arshins = 12 spans = 48 versts = 2.1336 meters

It is interesting that the the word "sazhen" comes from the Old Slavonic verb "squeeze" (walk wide). In Ancient Russia, not one, but many different fathoms were used. We have already met with the flywheel and oblique fathom, the turn has come for some other fathoms:

1 fathom ≈ 1.83 meters
1 Greek fathom ≈ 2.304 meters
1 masonry sazhen ≈ 1.597 meters
1 pipe fathom ≈ 1.87 meters (this fathom was used to measure the length of pipes in salt mines)
1 church fathom ≈ 1.864 meters
1 royal sazhen ≈ 1.974 meters

However, there are also square and cubic fathoms. The amount of something measured by such a measure: fathom of earth(sazhen square); fathom of firewood(sazhen cubic).

Task #10
"Grandfather Mazai and hares", 1870. Nikolay Alekseevich Nekrasov
“A knotty log floated past,
Sitting, and standing, and lying in a layer,
A dozen hares were saved on it
"I would take you - but sink the boat!"
It’s a pity for them, however, but it’s a pity for the find -
I got hooked on a knot
And he dragged a log behind him ... "
Explain why the hares could sink the boat. What is meant by displacement and carrying capacity of a ship? What is a waterline?

For the curious: Waterline- this is the line along which the calm surface of the water comes into contact with the hull of a ship or other floating vessel. The waterline can be of different types (constructive, calculated, operating, cargo).
Load waterline is of great practical importance. Before this mark became mandatory, many ships were lost in the fleets of the whole world. The main reason for the loss of ships is overload, due to the desire to obtain additional profit from transportation, which was exacerbated by the difference in water density (depending on its temperature and salinity, the ship's sediment can vary significantly). The first precedent in modern history is the British Load Line (Load Line) Act of 1890, under which the minimum allowable freeboard was set not by the shipowner, but by a government agency.

Illustrations by Alexei Nikanorovich Komarov
to Nikolai Alekseevich Nekrasov's poem "Grandfather Mazai and Hares"

... I see one small island - Hares on it gathered in a crowd ...		Instantly my team fled, Only two couples left on the boat ...

Komarov Alexey Nikanorovich(1879–1977) is considered the founder of the Russian animalistic school. Aleksey Nikanorovich Komarov illustrated scientific and children's books, created drawings for stamps, postcards, and visual aids. Several generations of children grew up learning from textbooks with his wonderful drawings.

Task #11
Where is the carrying capacity of the same barge greater - in river or sea water?

Answer: The density of river water is less than sea water, since the density of ordinary water is 1000 kg / m 3, and salt water is 1030 kg / m 3. So the strength of Archimedes in sea water will be greater. That is, in sea water, a barge can lift a load with greater gravity and not sink. This means that the carrying capacity of the same barge in sea water is greater.

Task #12
Submarines sailing in the northern seas are often covered with a thick layer of ice while on the surface of the water. Is it easier or more difficult to submerge the boat in the presence of such an additional ice load?

Task #13
For submarines, a depth is set below which they must not sink. What explains the existence of such a limit?

Answer: The deeper the submarine sinks, the more pressure will be experienced by its walls. Since there is a limit to the strength of the boat structure, there is also a limit to the depth of its immersion.

For the curious:
What design features do submarines have?
Submarines play an important role in all navies - warships capable of diving to a considerable depth (over 100 meters) and moving there hidden from the enemy.
Submarines must be able to float and submerge, as well as sail below the surface of the water. Since the volume of the boat remains unchanged in all cases, in order to perform these maneuvers, the boat must have a device for changing its weight. This device consists of a number of ballast compartments in the hull of the boat, which, using special devices, can be filled with outboard water (in this case, the weight of the boat increases and it sinks) or freed from water (in this case, the weight of the boat decreases and it floats).
Note that a small excess or lack of water in the ballast compartments is enough for the boat to sink to the very bottom of the sea or float to the surface of the water. It often happens that in a certain layer under water, the density of water changes rapidly with depth, increasing from top to bottom. Near the level of such a layer, the equilibrium of the boat is stable. Indeed, if the boat, being at this level, for any reason, sinks a little deeper, then it falls into an area of ​​\u200b\u200bhigher water density. The supporting force increases and the boat will begin to float, returning to its original depth. If the boat rises for any reason, then it will fall into an area of ​​\u200b\u200blesser density of water, the supporting force will decrease, and the boat will return to its original level. Therefore, submariners call such layers " liquid soil": the boat can “lie” on it, maintaining balance indefinitely, while in a homogeneous environment this is not possible and in order to maintain a given depth, the boat must constantly change the amount of ballast, taking or displacing water from the ballast compartments, or must all the time move by maneuvering the depth rudders.

Raising the State Flag of the USSR at the North Pole the crew of the submarine "Leninsky Komsomol", 1962 Pen Sergey Varlenovich, 1985 Central Naval Museum, St. Petersburg

For the curious: "Lenin Komsomol", originally K-3 - the first Soviet nuclear submarine, project 627. The name "Leninsky Komsomol" was inherited by the submarine from the diesel submarine of the same name "M-106" of the Northern Fleet, which died in one of the military campaigns in 1943.
In July 1962, for the first time in the history of the Soviet Navy, she made a long trip under the ice of the Arctic Ocean, during which she twice passed the point of the North Pole. Under command Lev Mikhailovich Zhiltsov July 17, 1962 for the first time in the history of the Soviet submarine fleet surfaced near the North Pole. The crew of the ship hoisted the State Flag of the USSR near the Pole in the ice of the Central Arctic.
In 1991, she was withdrawn from the Northern Fleet. After a series of dark days and a still incomplete reconstruction, it was decided to convert the Leninsky Komsomol submarine into a museum. They say that on the Neva they are already looking for a place for her eternal parking. Perhaps it will be next to the legendary Aurora ...

Task #14
"Amphibian Man", 1927. Alexander Romanovich Belyaev
“Dolphins are much heavier on land than in water. In general, everything is more difficult for you here. Even your own body. It's easier to live in water... ...And you'll sink to the bottom... It's like you're swimming in thick, blue air. Quiet. You don't feel your body. It becomes free, light, obedient to your every movement ... "
Is the author of the novel right? Explain the answer.

Alexander Romanovich Belyaev(03/16/1884 - 01/06/1942) - Soviet science fiction writer, one of the founders of Soviet science fiction literature. Among his most famous novels: "Professor Dowell's Head", "Amphibian Man", "Ariel" ...
If you haven't read it yet, I highly recommend it ;-)

§ I recommend to readers of the green pages a very entertaining and informative biophysical material that lifts the veil of secrecy over some features of the organization of dolphins: the anti-turbulent properties of the skin and an unsurpassed sonar ... on the green page of "Secrets of the Dolphin".

Task #15
In what water and why is it easier to swim: sea or river?

Answer: It is easier to swim in sea water, since a large buoyant force will act on a body immersed in sea water due to the fact that the density of sea water is greater than the density of river water.

Task #16
Why can we easily pick up our friend or a rather heavy stone in the water?

Task #17
A piece of marble weighs as much as a copper weight. Which of these bodies is easier to keep in water?

Answer: The density of marble is less than the density of copper, therefore, with the same mass, marble has a larger volume, which means that a large buoyant force will act on it and it is easier to keep it in water than a copper weight.

Task #18
Walking along the shore strewn with sea pebbles hurts with bare feet. And in the water, plunging deeper than the belt, walking on small stones does not hurt. Why?

Task #19
Swimming in a river with a muddy bottom, you can see that the legs get stuck more in the mud in a shallow place than in a deep one. Explain why.

Answer: As we dive deeper, we displace more water. According to the law of Archimedes, a large buoyant force will act on us in this case.

Task #20
Why are diving shoes equipped with heavy lead soles?

Answer: To increase the weight of the diver and give him more stability while working in the water. Heavy lead soles help the diver overcome the buoyancy of the water.

Task #21
Why does an empty glass bottle float on the surface of water, while a filled one sinks?

Answer: An empty glass bottle is immersed in water to a depth at which the volume of displaced water in terms of gravity is equal to the gravity of the bottle, which corresponds to the condition of bodies floating on the water surface. If the bottle is filled with water, the volume displaced will decrease and the bottle will sink.

Task #22
A brick sinks in water, while a dry pine log floats up. Does this mean that a large buoyant force acts on the log?

Task #23
"Dead Head", 1928. Alexander Romanovich Belyaev
“Morel rose, but the water soon reached the ankles of the legs and was constantly rising. His raft definitely didn't float. Maybe he got caught on something? At least one of its edges must rise! ... the raft was still resting on the bottom ...
"But what the hell is the matter?" Morel yelled angrily. He took a piece of iron wood lying on the shore, from which the raft was made, threw it into the water and immediately exclaimed:
“Is there another donkey like me in the world?” The stump sank like a stone. The iron tree was too heavy and could not float on the water.
Tough lesson! Lowering his head, Morel looked at the boiling river, in the waters of which so much effort and labor had been buried.
Can there be stones that float in water like wood and trees whose wood sinks in water like stone? Where can you find floating rocks, and where is sinking wood? What are both used for?

For the curious: When milk boils, foam rises. During volcanic eruptions, foam is also formed in boiling lava, but only stone. Freezing, this stone foam forms pumice. It is so light that it does not sink in water. As an abrasive pumice is applied for grinding metal and wood, polishing stone products, and is also used for hygienic removal of rough skin of the feet. Pumice deposits have been known since ancient times in the Aeolian Islands in the Tyrrhenian Sea north of Sicily. Significant pumice deposits are located in Kamchatka and in Transcaucasia (in Armenia near Yerevan). Wood birch Schmidt, temir-agacha, saxaul so thick and heavy drowning in water. Saxaul grows in semi-deserts and deserts of Asia; it is not suitable for construction, but it is an excellent fuel: in terms of its calorie content, saxaul approaches coal.
The hero of Alexander Belyaev's story, Professor Joseph Morel, received a scientific mission to Brazil, and ... it may very well be that he used trunks to build a raft caesalpinia ironwood (Brazilian ironwood), or maybe ... trunks guaiac (backout) tree- the wood of which sinking in water.

"Meshchorskaya side", 1939
Konstantin Georgievich Paustovsky
“There are a lot of lakes in the meadows. Their names are strange and varied: Quiet, Bull, Hotets, Ramoina, Kanava, Staritsa, Muzga, Bobrovka, Selyanskoye Lake and, finally, Langobardskoe.
At the bottom of Hotz lie black bog oaks.
What is bog oak and what is its density?

For the curious: In ancient times, majestic oak forests grew on the shores of Lake Hottsa. Water from year to year eroded and washed away the shores of the lake, and mighty oaks full of strength sank into the water (the density of wood of a live (or freshly cut) oak is 1020-1070 kg / m 3, and the density of water is 1000 kg / m 3). Oaks went under water, time passed, sand and silt washed the trunks of mighty oaks with a multi-meter layer. If the majority of trees in such conditions are doomed to fleeting and complete destruction, then the oak is just starting its second life. After a few hundred years, it reaches a delightful maturity and is awarded the honorary title - stained!
Such durability, as well as the inimitable color of bog oak, are caused by the reactions of tannin (tannic acid) with water containing metal salts (for example, iron). Depending on the amount of metal salts contained in lake or river water and the amount of tannins contained in wood, for a long time (from 200 to 2000 years or more ...) a specific coloration of bog oak wood took place - in colors from outrageous - ash- silvery with a pinkish-gray tint ... to a mystical blue-black with purple streaks. Real bog or peat oak is usually found during excavations of drained lakes and swamps. This is a very rare and expensive wood, which is sometimes not inferior to iron in terms of strength.
In historical descriptions, you can find the name of bog oak as "ebony" and "iron tree". It is characteristic that in Russia there was no concept of "cabinet maker" - craftsmen working with elite wood were called "blackwoods".
The wood of dried, prepared for processing, bog oak has a fairly high density (750-850 kg / m 3) compared to ordinary oak (650-760 kg / m 3).

Oaks in Old Peterhof Shishkin Ivan Ivanovich, 1891

Shishkin Ivan Ivanovich(01/25/1832–03/20/1898) - Russian landscape painter, academician, professor, head of the landscape workshop of the Imperial Academy of Arts, one of the founding members of the Association of Traveling Art Exhibitions.

Task #24
Why do air bubbles rise quickly in water?

Answer: The buoyant force acting on an air bubble in water is many times greater than the weight of the bubble itself (the gas compressed in the bubble). Rising up, the bubble enters the layers of water with less pressure, the bubble expands, the supporting force increases, and the speed of its ascent increases.

Task #25
In what gases could a soap bubble filled with helium rise?

Task #26
If a soap bubble with air inside it is placed in an open vessel filled with carbon dioxide, the bubble does not sink to the bottom of the vessel. Explain the phenomenon.

Answer: A soap bubble filled with air will float for some time on an invisible surface of carbon dioxide in a vessel.

Problem #27
The flask filled with hydrogen was turned upside down. Will hydrogen come out of the flask?

Task #28
Explain why the volume of hydrogen contained in the shell of a balloon increases as it rises.

Carnicero Antonio(Antonio Carnicero; 1748–1814) was a Spanish neoclassical painter.
Hot air balloon(fr. Montgolfiere) - a balloon with a shell filled with hot air. Named after surname the inventors of the Mongolf brothers e - Joseph-Michel and Jacques-Etienne. The first flight was made in France in the city of Annonay on June 5, 1783.
November 21, 1783 - a significant date in the history of aeronautics(in 2013 it is also round - 230 years ;-) On this day, two brave Frenchmen: Pilatre de Rozier and the Marquis d'Arlande, for the first time in history, flew in a balloon of the Montgolfier brothers.

Problem #29
In which case does a homemade paper balloon filled with hot air have more lift: when the guys launched it in the school building or in the school yard, where it was pretty cool?

Answer: The lift force of a balloon is equal to the difference between the weight of the air in the balloon and the weight of the gas filling the balloon. The greater the difference in the densities of air and gas filling the balloon, the greater the lifting force. Therefore, the lifting force of the ball is greater on the street, where the air is less heated.

Task #30
What explains the maximum height ("ceiling") for the balloon, which he is not able to overcome?

Answer: The decrease in air density with the height of the balloon.

Jacob Alt(Jacob Alt; 09/27/1798–09/30/1872) was an Austrian landscape painter, graphic artist and lithographer.

Task #31
A saucepan upside down floats in a vessel of water. Will the water level in the vessel change with the temperature of the air surrounding the pan? (Ignore the thermal expansion of the water, pot, and vessel.)

Answer: The water level in the vessel will not change. Since the weight of the contents in the vessel will not change with a change in the temperature of the air surrounding the pan, the force of water pressure on the bottom of the vessel will not change either.

Task #32
Why is it impossible to extinguish burning kerosene by pouring water on it? How should you stew?

Answer: The water will sink down and will not close the access of air (oxygen necessary for combustion) to the kerosene.

Task #33
One bottle contains vegetable oil and vinegar. How can any of these liquids be poured from a bottle?

Answer: The oil floats on top of the vinegar. To pour the oil, you just need to tilt the bottle. To pour vinegar, you need to close the bottle with a cork, turn it upside down, then open the cork just enough to pour out the right amount of vinegar.

Problem #34
Lactometer - a device for determining the fat content of milk - is a sealed glass tube floating in a liquid in a vertical position due to the load placed in its lower part. The markings on the tube show the fat content of the milk. In which milk - whole or skimmed (less fat) - should the lactometer sink deeper? Why?

Answer: The lactometer sinks deeper in whole milk. The density of higher fat milk is lower.

Problem #35
Half a liter of vegetable oil floats on the surface of the water in a bucket. How to collect most of the oil in a bottle without any tools and without touching the bucket?

Answer: The bottle is filled with water, closed with a finger, turned upside down and lowered with its neck into a layer of oil. If you remove your finger, the water will flow out of the bottle, and oil will enter the bottle in its place. You can also lower the empty bottle into the water in a vertical position so that the edge of the neck is at the level of the oil.

Problem #36
To clean rye seeds from poisonous horns, ergot seeds are immersed in a twenty percent aqueous solution of table salt. The ergot horns float up, but the rye remains at the bottom. What does this indicate?

Answer: The density of poisonous ergot horns is less, and the grain density is greater than the density of the solution.

Problem #37
A strong solution of table salt was poured into the vessel, and clean water was carefully poured on top. If a raw chicken egg is placed in a vessel, it will stay on the border between the solution and pure water. Explain the phenomenon.

Answer: The density of pure water is less than the average density of the egg, so it sinks in it. The density of the salt solution is greater than the density of the egg, so it floats in it.

Problem #38
Take a saucer and lower it edgewise into the water, it will sink. If the saucer is carefully lowered upside down into the water, it floats on the surface. Why?

Answer: Porcelain or faience has a higher density than water, so when the saucer is lowered with an edge, it sinks. When the bottom of the saucer is lowered into the water, it is immersed in water to such a depth at which the volume of displaced water in terms of gravity is equal to the gravity of the saucer, which corresponds to the condition of bodies floating on the surface of the water.

Problem #39
On the cups of equal-armed scales are two identical glasses, filled to the brim with water. A wooden block floats in one glass. What position are the scales in?

Answer: In balance.

Task #40
Two identical weights are suspended from the ends of an equal-arm lever. What happens if one weight is placed in water and the other in kerosene?

Answer: The balance will be broken.

Task #41
Brass and glass balls are balanced on the beam of equal-arm balances. Will the equilibrium be disturbed if the device is placed in an airless space (in carbon dioxide, in water)?

Answer: A glass ball will descend in the void, a brass ball in carbon dioxide and water.

Task #42
What material should the weights be made of so that when weighing accurately, it would be possible not to correct for weight loss in air?

Answer: The weights must be made from the same material as the body to be weighed.

Task #43
Will the water in communicating vessels be at the same level if a wooden spoon floats on its surface in one of the vessels?

Answer: Since a wooden spoon is in equilibrium on the surface of the water, its weight is equal to the weight of the water displaced by it. Therefore, if the spoon were replaced with water, then it would occupy a volume equal to the volume of the immersed part of the spoon, and the water level would not change. Therefore, the water in the communicating vessels will be at the same level.

Task #44
A massive ball of ice is frozen to the bottom of a vessel with water. How will the level of water in the vessel change when the ice melts? Will the force of water pressure on the bottom of the vessel change?

Answer: will go down; decrease. The density of ice is less than the density of water, so the volume of an ice ball is greater than the volume of water formed from this ball. It follows that the level of water in the vessel will decrease.

Problem #45
A piece of ice floats in a glass filled to the brim with water. Will the water overflow when the ice melts? What happens if the glass contains not water, but: 1) a denser liquid (for example, very salty water), 2) a less dense liquid (for example, kerosene)?

Answer: According to the law of Archimedes, the weight of floating ice is equal to the weight of the water displaced by it. Therefore, the volume of water formed when the ice melts will be exactly equal to the volume of water displaced by it, and the water level in the glass will not change. If there is a liquid in the glass that is denser than water, then the volume of water formed after the ice melts will be greater than the volume of liquid displaced by the ice, and the water will overflow. Conversely, in the case of a less dense liquid, after the ice melts, the level will drop.

Task #46
A piece of ice floats in a vessel filled with water with a steel ball frozen into it. Will the water level in the vessel change when the ice melts? Make a detailed explanation.

Answer: Will go down. A piece of ice with a steel ball weighs more than a piece of ice of the same volume, therefore, it is immersed in water deeper than a pure piece of ice, and displaces a larger volume of water than that which will be taken up by the water formed when the ice melts. When the ice melts, the water level will drop. The ball will then fall to the bottom, but its volume will remain the same, and it does not directly change the water level.

Problem #47
A piece of ice floats in a vessel filled with water, containing an air bubble. Will the water level in the vessel change when the ice melts?

Answer: In the presence of an air bubble, ice weighs less than a solid piece of ice of the same volume and, therefore, is immersed in water to a lesser depth. However, since the weight of the air can be neglected, the water level in the vessel will not change.

Problem #48
A block of ice floats in a vessel filled with water. How will the depth of immersion of the bar in water change if kerosene is poured over the water?

Answer: will decrease. With the addition of kerosene on top of the water, the pressure on the lower edge of the bar increases.

Problem #49
A block of ice floats in a vessel filled with water, on which lies a wooden ball. The density of the substance of the ball is less than the density of water. Will the water level in the vessel change when the ice melts?

Answer: Will not change. A block of ice and a ball float in the ode. This means that they displace as much water as they weigh. Since after the ice melts, the weight of the contents in the vessel will not change, since the force of water pressure on the bottom of the vessel will not change either. This means that the water level in the vessel will remain the same.

Problem #50
The density of a body is determined by weighing it in air and water. When a small body is immersed in water, air bubbles are retained on its surface, due to which an error is obtained in determining the density. Is the density value more or less obtained in this case?

Answer: Adhering air bubbles slightly increase body weight, but significantly increase its volume. Therefore, the density value is smaller.

Problem #51
Explain the essence of the work of water sedimentation tanks. Why does the settling of water lead to the purification of water from substances insoluble in it? But what about soluble impurities?

Answer: Every particle in water is affected by gravity and the Archimedean force. If the first of them is greater than the second, then under the action of their resultant particle sinks to the bottom, then the water after settling becomes drinkable.

Problem #52
Ancient Greek scientist Aristotle to prove the weightlessness of air, he weighed an empty leather bag and the same bag filled with air. In both cases, the readings of the scales were the same. Why is Aristotle's conclusion that air has no weight wrong?

Answer: Because the weight of the air bag increased by as much as the buoyant force of the air acting on the inflated bag increased. To prove the gravity of air, it would be enough to pump air out of some vessel or pump it into a strong vessel.

Aristotle(384 BC–322 BC) – Ancient Greek philosopher. Student Plato. From 343 BC e. - mentor Alexander the Great. The most influential of the dialecticians of antiquity; founder of formal logic. Aristotle developed many physical theories and hypotheses based on the knowledge of the time. Actually myself the term "physics" was introduced by Aristotle.
Rembrandt Harmenszoon van Rijn(Rembrandt Harmenszoon van Rijn; 1606-1669) - Dutch artist, draftsman and engraver, great master of chiaroscuro, the largest representative of the golden age of Dutch painting.

Problem #53
Under terrestrial conditions, various methods are used to train and test astronauts in a state of weightlessness. One of them is as follows: a person in a special spacesuit is immersed in a pool of water in which he does not sink and does not float. Under what condition is this possible?

Answer: This is possible provided that the force of gravity acting on a person in a space suit will be balanced by the Archimedean force.

Problem #54
What conclusion can be drawn about the magnitude of the Archimedean force by conducting appropriate experiments on the Moon, where the force of gravity is six times less than on Earth?

Answer: The same as on Earth: a body immersed in a liquid (or gas) is affected by a buoyant force (Archimedean force) equal to the weight of the liquid (or gas) displaced by this body.

Problem #55
Will a steel key sink in water under weightless conditions, for example, on board an orbital station, inside which normal atmospheric air pressure is maintained?

Answer: The key can be located at any point in the liquid, since neither gravity nor the Archimedean force acts on the key under weightlessness.

The legendary tale of Archimedes' challenge with the golden crown

Archimedes(287 BC–212 BC) was an ancient Greek mathematician, physicist and engineer from Syracuse. He made many discoveries in geometry. He laid the foundations of mechanics, hydrostatics, the author of a number of important inventions.

Thinking Archimedes Domenico Fetti 1620

Domenico Fetti(c. 1589-1623) - Italian painter of the Baroque era.

The legendary tale of Archimedes' challenge with the golden crown transmitted in various forms. The Roman architect Vitruvius, reporting on the discoveries of various scientists that struck him, gives the following story:

“As for Archimedes, of all his numerous and varied discoveries, the discovery that I will tell about seems to me made with boundless wit.
During his reign in Syracuse, Hiero, after the successful completion of all his activities, vowed to donate a golden crown to the immortal gods in some temple. He agreed with the master on a high price for the work and gave him the amount of gold he needed by weight. On the appointed day, the master brought his work to the king, who found it perfectly executed; after weighing, the crown was found to correspond to the given weight of gold.
After that, a denunciation was made that part of the gold was taken from the crown and the same amount of silver was mixed in instead. Hiero was angry that he had been tricked, and, not finding a way to convict this theft, asked Archimedes to think carefully about it. He, immersed in thoughts on this issue, somehow accidentally came to the bathhouse and there, sinking into the bathtub, noticed that such an amount of water was flowing out of it, what was the volume of his body immersed in the bathtub. Finding out the value of this fact, he, without hesitation, jumped out of the bath with joy, ran home naked and in a loud voice informed everyone that he had found what he was looking for. He ran and shouted the same thing in Greek: "Eureka, Eureka" (Found, found!).
Then, based on his discovery, but, they say, he made two ingots, each of the same weight as the crown, one of gold, the other of silver. Having done this, he filled the vessel to the very brim and lowered a silver ingot into it, and ... an appropriate amount of water flowed out. Taking out the ingot, he poured the same amount of water into the vessel ..., measuring the poured water sextarium so that, as before, the vessel was filled with water to the very brim. So he found what weight of silver corresponds to what specific volume of water.
Having made such a study, he lowered the gold ingot in the same way ..., and, adding the same measure of the spilled amount of water, found on the basis of a smaller amount sextants water, how much less volume the ingot occupies.

Then the same method was used to determine the volume of the crown. She displaced more water than a gold bar, and the theft was proven.

Sextarius (sextarius)- Roman measure of volume, equal to 0.547 l
Sextant (sextans)- Roman measure of mass, equal to 54.6 g(1 sextant = 2 ounces; weight of 1 sextant = 0.53508 N)

And now, attention, question: Is it possible to calculate the amount of gold replaced by silver in the crown using the method of Archimedes?

Answer: According to the data that Archimedes had, he was only entitled to assert that the crown was not pure gold. But to establish exactly how much gold was concealed by the master and replaced by silver, Archimedes could not. This would be possible if the volume of an alloy of gold and silver were strictly equal to the sum of the volumes of its constituent parts. In fact, only a few alloys have this property. As for the volume of an alloy of gold and silver, it is less than the sum of the volumes of the metals included in it. In other words, the density of such an alloy is greater than the density obtained as a result of the calculation according to the rules of simple mixing. Another thing is if gold were replaced not by silver, but by copper: the volume of an alloy of gold and copper is exactly equal to the sum of the volumes of its constituent parts. In this case, the method of Archimedes, described in the above story, gives an unmistakable result.

Quite often this story is associated with the discovery of the law of Archimedes, although it concerns the method determining the volume of bodies of irregular shape and methods determining the specific gravity of bodies by measuring their volume by immersion in a liquid.

I wish you success in your decision
quality problems in physics!

Literature:
§ Katz Ts.B. Biophysics at physics lessons
Moscow: Enlightenment publishing house, 1988
§ Zhytomyr S.V. Archimedes
Moscow: Enlightenment publishing house, 1981
§ Gorev L.A. Entertaining experiments in physics
Moscow: Enlightenment publishing house, 1977
§ Lukashik V.I. Physics Olympiad
Moscow: Enlightenment publishing house, 1987
§ Perelman Ya.I. Do you know physics?
Domodedovo: VAP publishing house, 1994
§ Tulchinsky M.E. Qualitative problems in physics
Moscow: Enlightenment publishing house, 1972
§ Erdavletov S.R., Rutkovsky O.O. Interesting geography of Kazakhstan
Alma-Ata: Mektep publishing house, 1989.

Liquids and gases, according to which, on any body immersed in a liquid (or gas), a buoyant force acts from this liquid (or gas), equal to the weight of the liquid (gas) displaced by the body and directed vertically upwards.

This law was discovered by the ancient Greek scientist Archimedes in the III century. BC e. Archimedes described his research in the treatise On Floating Bodies, which is considered one of his last scientific works.

The following are the findings from Archimedes' law.

The action of liquid and gas on a body immersed in them.

If you submerge an air-filled ball in water and release it, it will float. The same will happen with wood chips, cork and many other bodies. What force makes them float?

A body immersed in water is subjected to water pressure from all sides (Fig. a). At each point of the body, these forces are directed perpendicular to its surface. If all these forces were the same, the body would experience only all-round compression. But at different depths, the hydrostatic pressure is different: it increases with increasing depth. Therefore, the pressure forces applied to the lower parts of the body turn out to be greater than the pressure forces acting on the body from above.

If we replace all pressure forces applied to a body immersed in water with one (resulting or resultant) force that has the same effect on the body as all these individual forces together, then the resulting force will be directed upwards. This is what makes the body float. This force is called the buoyant force, or Archimedean force (after Archimedes, who first pointed out its existence and established what it depends on). On the image b it is labeled as F A.

The Archimedean (buoyant) force acts on the body not only in water, but also in any other liquid, since in any liquid there is hydrostatic pressure, which is different at different depths. This force also acts in gases, due to which balloons and airships fly.

Due to the buoyancy force, the weight of any body in water (or in any other liquid) is less than in air, and less in air than in airless space. It is easy to verify this by weighing the weight with the help of a training spring dynamometer, first in the air, and then lowering it into a vessel with water.

Weight reduction also occurs when a body is transferred from vacuum to air (or some other gas).

If the weight of a body in a vacuum (for example, in a vessel from which air is pumped out) is equal to P0, then its weight in air is:

,

where F´ A is the Archimedean force acting on a given body in air. For most bodies, this force is negligible and can be neglected, i.e., we can assume that P air =P 0 =mg.

The weight of the body in liquid decreases much more than in air. If the weight of the body in the air P air =P 0, then the weight of the body in the fluid is P liquid \u003d P 0 - F A. Here F A is the Archimedean force acting in the fluid. Hence it follows that

Therefore, in order to find the Archimedean force acting on a body in any liquid, this body must be weighed in air and in the liquid. The difference between the obtained values ​​will be the Archimedean (buoyant) force.

In other words, taking into account formula (1.32), we can say:

The buoyant force acting on a body immersed in a liquid is equal to the weight of the liquid displaced by this body.

The Archimedean force can also be determined theoretically. To do this, suppose that a body immersed in a fluid consists of the same fluid in which it is immersed. We have the right to assume this, since the pressure forces acting on a body immersed in a liquid do not depend on the substance from which it is made. Then the Archimedean force applied to such a body F A will be balanced by the downward force of gravity mwellg(where m f is the mass of liquid in the volume of a given body):

But the force of gravity is equal to the weight of the displaced fluid R f. Thus.

Given that the mass of a liquid is equal to the product of its density ρ w on volume, formula (1.33) can be written as:

where Vwell is the volume of the displaced fluid. This volume is equal to the volume of that part of the body that is immersed in the liquid. If the body is completely immersed in the liquid, then it coincides with the volume V of the whole body; if the body is partially immersed in the liquid, then the volume Vwell volume of displaced fluid V bodies (Fig. 1.39).

Formula (1.33) is also valid for the Archimedean force acting in a gas. Only in this case, it is necessary to substitute the density of the gas and the volume of the displaced gas, and not the liquid, into it.

In view of the foregoing, Archimedes' law can be formulated as follows:

On any body immersed in a liquid (or gas) at rest, a buoyant force acts from this liquid (or gas), equal to the product of the density of the liquid (or gas), the free fall acceleration and the volume of that part of the body that is immersed in the liquid ( or gas).