Study of viscous friction forces. Viscous friction force Examples of manifestations of liquid viscosity

Mechanics of continuums

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Viscous friction force F, acting on the fluid, is proportional (in the simplest case of shear flow along a flat wall) to the speed of relative motion v bodies and areas S and inversely proportional to the distance between the planes h :

F → ∝ − v → ⋅ S h (\displaystyle (\vec (F))\propto -(\frac ((\vec (v))\cdot S)(h)))

The proportionality coefficient, depending on the nature of the liquid or gas, is called coefficient of dynamic viscosity. This law was proposed by Isaac Newton in 1687 and bears his name (Newton's law of viscosity). Experimental confirmation of the law was obtained at the beginning of the 19th century in Coulomb’s experiments with torsion balances and in the experiments of Hagen and Poiseuille with the flow of water in capillaries.

There is a qualitatively significant difference between the forces of viscous friction and dry friction, among other things, that a body in the presence of only viscous friction and an arbitrarily small external force will necessarily begin to move, that is, for viscous friction there is no static friction, and vice versa - under the influence of only viscous friction, a body that initially moved will never (in within the framework of a macroscopic approximation that neglects Brownian motion) will not stop completely, although the motion will slow down indefinitely.

Second viscosity

The second viscosity, or volumetric viscosity, is internal friction when momentum is transferred in the direction of movement. It affects only when taking into account compressibility and (or) taking into account the heterogeneity of the coefficient of the second viscosity in space.

If dynamic (and kinematic) viscosity characterizes pure shear deformation, then the second viscosity characterizes volumetric compression deformation.

Bulk viscosity plays a large role in the attenuation of sound and shock waves, and is determined experimentally by measuring this attenuation.

Gas viscosity

μ = μ 0 T 0 + C T + C (T T 0) 3 / 2. (\displaystyle (\mu )=(\mu )_(0)(\frac (T_(0)+C)(T+C))\left((\frac (T)(T_(0)))\ right)^(3/2).)

μ = dynamic viscosity in (Pa s) at a given temperature T,
μ 0 = reference viscosity in (Pa s) at some reference temperature T0,
T= set temperature in Kelvin,
T0= reference temperature in Kelvin,
C= Sutherland constant for the gas whose viscosity is to be determined.

This formula can be used for temperatures in the range 0< T < 555 K и при давлениях менее 3,45 МПа с ошибкой менее 10 %, обусловленной зависимостью вязкости от давления.

The Sutherland constant and reference viscosities of gases at various temperatures are given in the table below

Gas	C	T0	μ 0 Viscosity of liquids Dynamic viscosity τ = − η ∂ v ∂ n , (\displaystyle \tau =-\eta (\frac (\partial v)(\partial n)),) Viscosity coefficient η (\displaystyle \eta )(coefficient of dynamic viscosity, dynamic viscosity) can be obtained based on considerations of molecular movements. It's obvious that η (\displaystyle \eta ) will be less, the shorter the residence time t of the molecules. These considerations lead to an expression for the viscosity coefficient called the Frenkel-Andrade equation: η = C e w / k T (\displaystyle \eta =Ce^(w/kT)) Another formula representing the viscosity coefficient was proposed by Baczynski. As shown, the viscosity coefficient is determined by intermolecular forces depending on the average distance between the molecules; the latter is determined by the molar volume of the substance V M (\displaystyle V_(M)). Numerous experiments have shown that there is a relationship between molar volume and viscosity coefficient: η = c V M − b , (\displaystyle \eta =(\frac (c)(V_(M)-b)),) where c and b are constants. This empirical relationship is called Baczynski's formula. The dynamic viscosity of liquids decreases with increasing temperature and increases with increasing pressure. Kinematic viscosity In technology, in particular, when calculating hydraulic drives and tribotechnics, one often has to deal with the quantity: ν = η ρ , (\displaystyle \nu =(\frac (\eta )(\rho )),) and this quantity is called kinematic viscosity. Here ρ (\displaystyle \rho )- liquid density; η (\displaystyle \eta )- coefficient of dynamic viscosity (see above). Kinematic viscosity in older sources is often given in centistokes (cSt). In SI this value is translated as follows: 1 cSt = 1 mm 2 / (\displaystyle /) 1 c = 10 −6 m 2 / (\displaystyle /) c Conditional viscosity Conditional viscosity is a value that indirectly characterizes the hydraulic resistance to flow, measured by the flow time of a given volume of solution through a vertical tube (of a certain diameter). Measured in Engler degrees (named after the German chemist K. O. Engler), denoted by °ВУ. It is determined by the ratio of the time of flow of 200 cm 3 of the test liquid at a given temperature from a special viscometer to the time of flow of 200 cm 3 of distilled water from the same device at 20 ° C. Conditional viscosity up to 16 °ВУ is converted into kinematic according to the GOST table, and conditional viscosity exceeding 16 °ВУ, according to the formula: ν = 7 , 4 ⋅ 10 − 6 E t , (\displaystyle \nu =7,4\cdot 10^(-6)E_(t),) Where ν (\displaystyle \nu )- kinematic viscosity (in m 2 /s), and E t (\displaystyle E_(t))- conditional viscosity (in °VU) at temperature t. Newtonian and non-Newtonian fluids Newtonian fluids are those for which the viscosity does not depend on the rate of deformation. In the Navier-Stokes equation for a Newtonian fluid there is a viscosity law similar to the above (essentially a generalization of Newton’s law, or Navier-Stokes law): σ i j = η (∂ v i ∂ x j + ∂ v j ∂ x i) , (\displaystyle \sigma _(ij)=\eta \left((\frac (\partial v_(i))(\partial x_(j)) )+(\frac (\partial v_(j))(\partial x_(i)))\right),) Where σ i , j (\displaystyle \sigma _(i,j))- viscous stress tensor. η (T) = A ⋅ exp ⁡ (Q R T) , (\displaystyle \eta (T)=A\cdot \exp \left((\frac (Q)(RT))\right),) Where Q (\displaystyle Q)- activation energy of viscosity (J/mol), T (\displaystyle T)- temperature (), R (\displaystyle R)- universal gas constant (8.31 J/mol K) and A (\displaystyle A)- some constant. Viscous flow in amorphous materials is characterized by a deviation from the Arrhenius law: viscosity activation energy Q (\displaystyle Q) varies from a large value Q H (\displaystyle Q_(H)) at low temperatures (in a glassy state) by a small amount Q L (\displaystyle Q_(L)) at high temperatures (in a liquid state). Depending on this change, amorphous materials are classified as either strong when (Q H − Q L)< Q L {\displaystyle \left(Q_{H}-Q_{L}\right), or brittle when (Q H − Q L) ≥ Q L (\displaystyle \left(Q_(H)-Q_(L)\right)\geq Q_(L)). The fragility of amorphous materials is numerically characterized by the Doremus fragility parameter R D = Q H Q L (\displaystyle R_(D)=(\frac (Q_(H))(Q_(L)))): strong materials have R D< 2 {\displaystyle R_{D}<2} , while brittle materials have R D ≥ 2 (\displaystyle R_(D)\geq 2). The viscosity of amorphous materials is very accurately approximated by the biexponential equation: η (T) = A 1 ⋅ T ⋅ [ 1 + A 2 ⋅ exp ⁡ B R T ] ⋅ [ 1 + C exp ⁡ D R T ] (\displaystyle \eta (T)=A_(1)\cdot T\cdot \left\ cdot\left) with constant A 1 (\displaystyle A_(1)), A 2 (\displaystyle A_(2)), B (\displaystyle B), C (\displaystyle C) And D (\displaystyle D) associated with the thermodynamic parameters of connecting bonds of amorphous materials. In narrow temperature ranges close to the glass transition temperature T g (\displaystyle T_(g)) this equation is approximated by VTF-type formulas or compressed Kohlrausch exponentials. If the temperature is significantly below the glass transition temperature T< T g {\displaystyle T, the biexponential viscosity equation reduces to an Arrhenius-type equation η (T) = A L T ⋅ exp ⁡ (Q H R T) , (\displaystyle \eta (T)=A_(L)T\cdot \exp \left((\frac (Q_(H))(RT))\right) ,) with high activation energy Q H = H d + H m (\displaystyle Q_(H)=H_(d)+H_(m)), Where H d (\displaystyle H_(d)) -

Viscous friction force occurs between layers of the same solid body (liquid or gas). The force of viscous friction depends on the relative speed of displacement of individual layers of gas or liquid relative to each other. For example, viscous friction occurs when a liquid or gas flows through pipes at a speed (Fig. 2.3).

The speed of the liquid layers decreases as they approach the pipe walls. Speed difference ratio
in two close layers located at a distance
, is called the average velocity gradient.

In accordance with Newton's equation, the modulus of the average force of viscous friction

(2.54)

Where – viscosity coefficient, S – area of interacting layers of the medium located at a distance ∆x from each other.

The viscosity coefficient depends on the state of aggregation and temperature of the substance.

Viscosity coefficient

Resistance force
occurs when solid bodies move in a liquid or gas. The modulus of the resistance force is proportional to the density of the medium , cross-sectional area of a moving body S and the square of its speed

, (2.55)

G de
[kg/m] – coefficient of resistance of the medium.

A body moving in a medium experiences the action of viscous friction force (F tr) and resistance force (F resist). At low speeds, the resistance force is less than the force of viscous friction, and at high speeds it significantly exceeds it (Fig. 2.4).

At a certain speed the forces F tr and F resist become equal in magnitude.

The resistance force of the medium depends on the shape of the moving body. A body shape in which the drag force is small is called streamlined. Rockets, airplanes, cars and other machines moving at high speeds in the air or in water are given a streamlined, drop-shaped shape.

2.6.5.Elastic force. Hooke's law.

P When external forces act on a body, elastic and inelastic deformation occurs.

With elastic deformation, the body completely restores its shape and size after the action of external forces ceases. During inelastic deformation, the shape and dimensions of the body are not restored.

Elastic deformation of the spring.

When the spring is stretched (Fig. 2.14) by an amount relative to its equilibrium state (x 0 = 0), an elastic force arises , which returns the spring to its previous position after the cessation of the external force. Modulus of elastic force arising at linear tension or compression of a spring is determined by Hooke's law.

, (2.56)

Where – projection of the elastic force on the x-axis, the minus sign takes into account the opposite directions of the force and spring movement
.

Rod deformation

Rod long l 0 and cross section S under the action of forces And perpendicular to its ends in opposite directions it is deformed (stretched or compressed) (Figure 2.15). The deformation of the rod is determined by the relative value

(2.57)

where ∆ l =l - l 0 , l- length of the rod after deformation.

Experience shows that

, (2.58)

where α is the elasticity coefficient of the rod,

=σ – normal voltage, measured in
(pascal).

Along with the elasticity coefficient a, to characterize the elastic properties of bodies under normal stresses, they use Young's modulus E = 1/a, which, like voltage, is measured in pascals.

Relative elongation (compression) and Young’s modulus in accordance with equalities (2.13 and 2.14) are determined from the relations:

,
. (2.59)

Young's modulus is equal to the normal stress at which the deformation of the rod Dl is equal to its original lengthl 0. In reality, at such stresses, the destruction of the rod occurs.

Solving equation (2.58) for F , and substituting instead of e=Dl/l 0 ,a= 1/E, we obtain a formula for determining the force deforming a rod with cross-section S by the amount

, (2.60)

Where is a constant coefficient for the rod, which, in accordance with Hooke’s law, corresponds to the elasticity coefficient of the rod during compression and tension.

When a tangential (tangential) stress is applied to the rod

forces F 1 and F 2 applied parallel to opposite faces with area S of a rectangular rod cause shear strain(Figure 2.16).

If the action of forces is uniformly distributed over the entire surface of the corresponding face, then a tangential stress arises in any section parallel to these faces
. Under the influence of stresses, the body is deformed so that one face moves relative to the other by a certain distance A. If the body is mentally divided into elementary layers parallel to the faces, then each layer will be shifted relative to the layers adjacent to it.

During shear deformation, any straight line initially perpendicular to the layers will deviate by a certain angle φ. the tangent of which is called the relative shift

, (2.61)

where b is the height of the face. During elastic deformations, the angle φ is very small, so we can assume that
And
.

Experience shows that the relative shear is proportional to the tangential stress

, (2.62)

where G is the shear modulus.

Shear modulus depends only on the properties of the material and is equal to the tangential stress at an angle φ = 45˚. The shear modulus, like Young's modulus, is measured in pascals (Pa). Shifting a rod by an angle causes force

=GSφ, (2.63)

where G·S – coefficient of elasticity of the rod during shear deformation.

Viscosity(internal friction) ( English. viscosity) is one of the transfer phenomena, the property of fluid bodies (liquids and gases) to resist the movement of one part of them relative to another. The mechanism of internal friction in liquids and gases is that chaotically moving molecules transfer momentum from one layer to another, which leads to equalization of velocities - this is described by the introduction of a friction force. The viscosity of solids has a number of specific features and is usually considered separately. The basic law of viscous flow was established by I. Newton (1687): When applied to liquids, viscosity is distinguished:

Dynamic (absolute) viscosity µ – a force acting on a unit area of a flat surface that moves at a unit speed relative to another flat surface located at a unit distance from the first. In the SI system, dynamic viscosity is expressed as Pa×s(pascal second), non-system unit P (poise).
Kinematic viscosity ν – dynamic viscosity ratio µ to liquid density ρ .

ν= µ / ρ ,

ν , m 2 /s – kinematic viscosity;
μ , Pa×s – dynamic viscosity;
ρ , kg/m 3 – liquid density.

Viscous friction force

This is the phenomenon of the occurrence of tangential forces that prevent the movement of parts of a liquid or gas relative to each other. Lubrication between two solids replaces dry sliding friction with sliding friction of layers of liquid or gas against each other. The speed of particles in the medium changes smoothly from the speed of one body to the speed of another body.

The force of viscous friction is proportional to the speed of relative motion V bodies, proportional to area S and inversely proportional to the distance between the planes h.

F=-V S / h,

The proportionality coefficient, depending on the type of liquid or gas, is called coefficient of dynamic viscosity. The most important thing about the nature of viscous friction forces is that in the presence of any force, no matter how small, the bodies will begin to move, that is, there is no static friction. Qualitatively significant difference in forces viscous friction from dry friction

If a moving body is completely immersed in a viscous medium and the distances from the body to the boundaries of the medium are much greater than the dimensions of the body itself, then in this case we speak of friction or medium resistance. In this case, sections of the medium (liquid or gas) directly adjacent to the moving body move at the same speed as the body itself, and as they move away from the body, the speed of the corresponding sections of the medium decreases, becoming zero at infinity.

The resistance force of the medium depends on:

its viscosity
on body shape
on the speed of movement of the body relative to the medium.

For example, when a ball moves slowly in a viscous fluid, the friction force can be found using the Stokes formula:

F=-6 R V,

Gas viscosity

The viscosity of gases (the phenomenon of internal friction) is the appearance of friction forces between layers of gas moving relative to each other in parallel and at different speeds. The viscosity of gases increases with increasing temperature

The interaction of two layers of gas is considered as a process during which momentum is transferred from one layer to another. The frictional force per unit area between two layers of gas, equal to the impulse transmitted per second from layer to layer through a unit area, is determined by Newton's law:

τ=-η dν / dz

Where:
dν/dz- velocity gradient in the direction perpendicular to the direction of movement of the gas layers.
The minus sign indicates that the momentum is transferred in the direction of decreasing velocity.
η - dynamic viscosity.

η= 1 / 3 ρ(ν) λ, where:

ρ - gas density,
(ν) - arithmetic average speed of molecules
λ - the average free path of molecules.

Viscosity of some gases (at 0°C)

Liquid viscosity

Liquid viscosity- this is a property that manifests itself only when a fluid moves, and does not affect fluids at rest. Viscous friction in liquids obeys the law of friction, which is fundamentally different from the law of friction of solids, because depends on the friction area and the speed of fluid movement.
Viscosity– the property of a liquid to resist the relative shear of its layers. Viscosity manifests itself in the fact that with the relative movement of layers of liquid, shear resistance forces arise on the surfaces of their contact, called internal friction forces, or viscous forces. If we consider how the velocities of different layers of liquid are distributed across the cross section of the flow, we can easily notice that the further away from the walls of the flow, the greater the speed of particle movement. At the walls of the flow, the fluid velocity is zero. This is illustrated by a drawing of the so-called jet flow model.

A slowly moving layer of liquid “brakes” an adjacent layer of liquid moving faster, and vice versa, a layer moving at a higher speed drags (pulls) along a layer moving at a lower speed. Internal friction forces appear due to the presence of intermolecular bonds between moving layers. If we select a certain area between adjacent layers of liquid S, then according to Newton's hypothesis:

F=μ S (du / dy),

μ - coefficient of viscous friction;
S– friction area;
du/dy- velocity gradient

Magnitude μ in this expression is dynamic viscosity coefficient, equal to:

μ= F / S 1 / du / dy , μ= τ 1/du/dy,

τ – tangential stress in the liquid (depends on the type of liquid).

Physical meaning of the viscous friction coefficient- a number equal to the friction force developing on a unit surface with a unit velocity gradient.

In practice it is more often used kinematic viscosity coefficient, so called because its dimension lacks the designation of force. This coefficient is the ratio of the dynamic coefficient of viscosity of a liquid to its density:

ν= μ / ρ ,

Units of viscous friction coefficient:

N·s/m 2 ;
kgf s/m 2
Pz (Poiseuille) 1(Pz)=0.1(N s/m 2).

Fluid Viscosity Property Analysis

For dropping liquids, viscosity depends on temperature t and pressure R, however, the latter dependence appears only with large changes in pressure, on the order of several tens of MPa.

The dependence of the coefficient of dynamic viscosity on temperature is expressed by a formula of the form:

μ t =μ 0 e -k t (T-T 0),

μ t - coefficient of dynamic viscosity at a given temperature;
μ 0 - coefficient of dynamic viscosity at a known temperature;
T - set temperature;
T 0 - temperature at which the value is measured μ 0 ;
e

The dependence of the relative coefficient of dynamic viscosity on pressure is described by the formula:

μ р =μ 0 e -k р (Р-Р 0),

μ R - coefficient of dynamic viscosity at a given pressure,
μ 0 - coefficient of dynamic viscosity at a known pressure (most often under normal conditions),
R - set pressure;
P 0 - pressure at which the value is measured μ 0 ;
e – the base of the natural logarithm is equal to 2.718282.

The effect of pressure on the viscosity of a liquid appears only at high pressures.

Newtonian and non-Newtonian fluids

Newtonian fluids are those for which the viscosity does not depend on the rate of deformation. In the Navier-Stokes equation for a Newtonian fluid, there is a viscosity law similar to the above (in fact, a generalization of Newton’s law, or Navier’s law).

Goal of the work: study of the phenomenon of viscous friction and one of the methods for determining the viscosity of liquids.

Devices and accessories: balls of various diameters, micrometer, calipers, ruler.

Elements of theory and experimental method

All real liquids and gases have internal friction, also called viscosity. Viscosity manifests itself, in particular, in the fact that the movement that has arisen in a liquid or gas gradually ceases after the cessation of the causes that caused it. From everyday experience, for example, it is known that in order to create and maintain a constant flow of liquid in a pipe, there must be a pressure difference between the ends of the pipe. Since in steady flow the fluid moves without acceleration, the need for pressure forces to act indicates that these forces are balanced by some forces that inhibit the movement. These forces are the forces of internal friction.

There are two main modes of liquid or gas flow:

1) laminar;

2) turbulent.

In a laminar flow mode, the flow of liquid (gas) can be divided into thin layers, each of which moves in the general flow at its own speed and does not mix with other layers. Laminar flow is stationary.

In a turbulent regime, the flow becomes unsteady - the speed of particles at each point in space changes randomly all the time. In this case, intensive mixing of the liquid (gas) occurs in the flow.

Let us consider the laminar flow regime. Let us select two layers in the flow with an area S, located at a distance ∆ Z from each other and moving at different speeds V 1 and V 2 (Fig. 1). Then a viscous friction force arises between them, proportional to the velocity gradient D V/D Z in a direction perpendicular to the direction of flow:

Where the coefficient μ is by definition called viscosity or coefficient of internal friction, D V=V 2-V 1.

From (1) it is clear that viscosity is measured in pascal seconds (Pa s).

It should be noted that viscosity depends on the nature and state of the liquid (gas). In particular, the viscosity value can significantly depend on temperature, as is observed, for example, in water (see Appendix 2). Failure to take this dependence into account in practice in a number of cases can lead to significant discrepancies between theoretical calculations and experimental data.

In gases, viscosity is caused by the collision of molecules (see Appendix 1); in liquids, by intermolecular interaction, which limits the mobility of molecules.

The viscosity values of some liquid and gaseous substances are given in Appendix 2.

As already noted, the flow of a liquid or gas can occur in one of two modes - laminar or turbulent. The English physicist Osborne Reynolds established that the nature of the flow is determined by the value of the dimensionless quantity

Where is a quantity called kinematic viscosity, V– speed of the fluid (or body in the fluid), D– some characteristic size. If liquid flows in a pipe under D understand the characteristic cross-sectional size of this pipe (for example, diameter or radius). When a body moves in a liquid under D understand the characteristic size of this body, for example the diameter of a ball. With values Re< 1000 the flow is considered laminar, when Re> 1000 the flow becomes turbulent.

One of the methods for measuring the viscosity of substances (viscometry) is the falling ball method, or Stokes method. Stokes showed that for a ball moving at a speed V in a viscous medium, the viscous friction force acts equal to , Where D - diameter of the ball.

Consider the movement of a ball as it falls. According to Newton's second law (Fig. 2)

Where F— viscous friction force, — Archimedes force, — gravity force, ρ AND And ρ are the densities of the liquid and the material of the balls, respectively. The solution to this differential equation will be the following dependence of the ball speed on time:

Where V 0 is the initial speed of the ball, and

There is a speed of steady motion (at T>>τ). The value is the relaxation time. This value shows how quickly a stationary motion mode is established. It is usually believed that when T≈3τ movement is practically no different from stationary. Thus, by measuring the speed VU, you can calculate the viscosity of the liquid. Note that the Stokes formula is applicable at Reynolds numbers less than 1000, that is, in a laminar regime of fluid flow around a ball.

A laboratory installation for measuring the viscosity of liquids using the Stokes method is a glass vessel filled with the liquid being tested. Balls are thrown from above, along the axis of the cylinder. There are horizontal marks at the top and bottom of the vessel. By using a stopwatch to measure the time of movement of the ball between the marks and knowing the distance between them, the speed of the steady motion of the ball is found. If the cylinder is narrow, then corrections must be made to the calculation formula to account for the influence of the walls.

Taking these corrections into account, the formula for calculating viscosity will take the form:

Where L — distance between marks, D - diameter of the inner part of the vessel.

Work order

1. Using a caliper, measure the inner diameter of the vessel, using a ruler - the distance between the horizontal marks on the vessel, and using a micrometer - the diameters of all balls used in the experiment. Consider the acceleration due to gravity to be 9.8 m/s2. The density of the liquid and the density of the substance of the balls are indicated in a laboratory setup.

2. By lowering the balls one by one into the liquid, measure the time it takes each of them to travel between the marks. Enter the results into the table. The table indicates the experiment number, the diameter of the ball and its travel time, as well as the result of calculating the viscosity for each experiment.

The difference between viscous friction and dry friction is that it can go to zero simultaneously with speed. Even with a small external force, a relative velocity can be imparted to the layers of a viscous medium.

Resistance force when moving in a viscous medium

Note 1

In addition to friction forces, when moving in liquid and gaseous media, resistance forces of the medium arise, which manifest themselves much more significantly than friction forces.

The behavior of liquid and gas in relation to the manifestations of friction forces is no different. Therefore, the characteristics given below apply to both conditions.

Definition 1

The action of the resistance force that arises when a body moves in a viscous medium is due to its properties:

absence of static friction, that is, the movement of a floating multi-ton ship using a rope;
the dependence of the drag force on the shape of the moving body, in other words, on its streamlining to reduce the drag forces;
dependence of the absolute value of the resistance force on speed.

Definition 2

There are certain patterns to which both the frictional forces and the resistance of the environment are subject, with the total force being symbolically designated as the friction force. Its value depends on:

body shape and size;
the state of its surface;
speed relative to the medium and its property called viscosity.

To depict the dependence of the friction force on the speed of the body relative to the medium, use the graph in Figure 1.

Picture 1 . Graph of friction force versus speed relative to the medium

If the speed value is small, then the resistance force is directly proportional to υ, and the friction force increases linearly with speed:

F t r = - k 1 υ (1) .

The presence of a minus sign means the direction of the friction force in the opposite direction relative to the direction of speed.

At a high speed, the linear law transitions to a quadratic one, that is, the friction force increases proportionally to the square of the speed:

F t r = - k 2 υ 2 (2) .

If in the air the dependence of the drag force on the square of the speed decreases, we speak of speeds with values of several meters per second.

The magnitude of the friction coefficients k 1 and k 2 depends on the shape, size and condition of the surface of the body and the viscous properties of the medium.

Example 1

If we consider a skydiver’s long jump, then his speed cannot constantly increase; at a certain moment it will begin to decline, at which the resistance force will become equal to the force of gravity.

The speed value at which law (1) makes the transition to (2) depends on the same reasons.

Example 2

Two metal balls of different masses fall from the same height with no initial velocity. Which ball will fall faster?

Given: m 1, m 2, m 1 > m 2

Solution

During the fall, both bodies gain speed. At a certain moment, the downward movement is carried out at a steady speed, at which the value of the resistance force (2) is equal to the force of gravity:

F t r = k 2 υ 2 = m g.

We obtain the steady speed using the formula:

υ 2 = m g k 2 .

Consequently, a heavy ball has a higher steady-state speed of fall than a light one. Therefore, reaching the earth's surface will happen faster.

Answer: a heavy ball will reach the ground faster.

Example 3

A skydiver flies at a speed of 35 m/s before the parachute opens, and afterward at a speed of 8 m/s. Determine the tension force of the lines when the parachute opens. The parachutist's mass is 65 kg, the free fall acceleration is 10 m/s2. Denote the proportionality of F t r relative to υ.

Given: m 1 = 65 kg, υ 1 = 35 m/s, υ 2 = 8 m/s.

Find: T - ?

Solution

Drawing 2

Before deployment, the parachutist had a speed of υ 1 = 35 m/s, that is, his acceleration was equal to zero.

According to Newton's second law we get:

0 = m g - k υ 1 .

It's obvious that

After the parachute has opened, its υ changes and becomes equal to υ 2 = 8 m/s. From here Newton's second law takes the form:

0 - m g - k υ 2 - T .

To find the tension force of the slings, you need to transform the formula and substitute the values:

T = m g 1 - υ 2 υ 1 ≈ 500 N.

Answer: T = 500 N.

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Encyclopedic YouTube

Second viscosity

Gas viscosity

Viscosity of liquids

Dynamic viscosity

Kinematic viscosity

Conditional viscosity

Newtonian and non-Newtonian fluids

2.6.5.Elastic force. Hooke's law.

Viscous friction force

Gas viscosity

Viscosity of some gases (at 0°C)

Liquid viscosity

Fluid Viscosity Property Analysis

Newtonian and non-Newtonian fluids

Resistance force when moving in a viscous medium