Thread tension. How to calculate tension in physics
In this problem it is necessary to find the ratio of the tension force to
Rice. 3. Solution of problem 1 ()
The stretched thread in this system acts on block 2, causing it to move forward, but it also acts on bar 1, trying to impede its movement. These two tension forces are equal in magnitude, and we just need to find this tension force. In such problems, it is necessary to simplify the solution as follows: we assume that the force is the only external force that makes the system of three identical bars move, and the acceleration remains unchanged, that is, the force makes all three bars move with the same acceleration. Then the tension always moves only one block and will be equal to ma according to Newton’s second law. will be equal to twice the product of mass and acceleration, since the third bar is located on the second and the tension thread should already move two bars. In this case, the ratio to will be equal to 2. The correct answer is the first one.
Two bodies of mass and , connected by a weightless inextensible thread, can slide without friction along a smooth horizontal surface under the action of a constant force (Fig. 4). What is the ratio of the thread tension forces in cases a and b?
Selected answer: 1. 2/3; 2. 1; 3. 3/2; 4. 9/4.
Rice. 4. Illustration for problem 2 ()
Rice. 5. Solution of problem 2 ()
The same force acts on the bars, only in different directions, so the acceleration in case “a” and case “b” will be the same, since the same force causes the acceleration of two masses. But in case “a” this tension force also makes block 2 move, in case “b” it is block 1. Then the ratio of these forces will be equal to the ratio of their masses and we get the answer - 1.5. This is the third answer.
A block weighing 1 kg lies on the table, to which a thread is tied, thrown over a stationary block. A load weighing 0.5 kg is suspended from the second end of the thread (Fig. 6). Determine the acceleration with which the block moves if the coefficient of friction of the block on the table is 0.35.
Rice. 6. Illustration for problem 3 ()
Let's write down a brief statement of the problem:
Rice. 7. Solution to problem 3 ()
It must be remembered that the tension forces and as vectors are different, but the magnitudes of these forces are the same and equal. Likewise, we will have the same accelerations of these bodies, since they are connected by an inextensible thread, although they are directed in different directions: - horizontally, - vertically. Accordingly, we select our own axes for each body. Let's write down the equations of Newton's second law for each of these bodies; when added, the internal tension forces are reduced, and we get the usual equation, substituting the data into it, we find that the acceleration is equal to .
To solve such problems, you can use the method that was used in the last century: the driving force in this case is the resultant external forces applied to the body. The force of gravity of the second body forces this system to move, but the force of friction of the block on the table prevents the movement, in this case:
Since both bodies are moving, the driving mass will be equal to the sum of the masses, then the acceleration will be equal to the ratio of the driving force to the driving mass 
This way you can immediately come to the answer.
At the top of two inclined planes, making angles with the horizon and , the block is fixed. On the surface of the planes with a friction coefficient of 0.2, bars kg and , connected by a thread thrown over a block, move (Fig. 8). Find the pressure force on the block axis.
Rice. 8. Illustration for problem 4 ()
Let's make a brief statement of the problem conditions and an explanatory drawing (Fig. 9):
Rice. 9. Solution to problem 4 ()
We remember that if one plane makes an angle of 60 0 with the horizon, and the second plane makes 30 0 with the horizon, then the angle at the vertex will be 90 0, this is an ordinary right triangle. A thread is thrown across the block, from which the bars are suspended; they pull down with the same force, and the action of the tension forces F H1 and F H2 leads to the fact that their resultant force acts on the block. But these tension forces will be equal to each other, they form a right angle with each other, so when adding these forces, you get a square instead of a regular parallelogram. The required force F d is the diagonal of the square. We see that for the result we need to find the tension force of the thread. Let's analyze: in which direction does the system of two connected bars move? The more massive block will naturally pull the lighter one, block 1 will slide down, and block 2 will move up the slope, then the equation of Newton’s second law for each of the bars will look like:
The solution of the system of equations for coupled bodies is performed by the addition method, then we transform and find the acceleration:
This acceleration value must be substituted into the formula for the tension force and find the pressure force on the block axis:
We found that the pressure force on the block axis is approximately 16 N.
We have reviewed various ways solving problems that many of you will need in the future to understand the principles of the design and operation of those machines and mechanisms that you will have to deal with in production, in the army, and in everyday life.
Bibliography
- Tikhomirova S.A., Yavorsky B.M. Physics (basic level) - M.: Mnemosyne, 2012.
- Gendenshtein L.E., Dick Yu.I. Physics 10th grade. - M.: Mnemosyne, 2014.
- Kikoin I.K., Kikoin A.K. Physics-9. - M.: Education, 1990.
Homework
- What law do we use when composing equations?
- What quantities are the same for bodies connected by an inextensible thread?
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Consider an endless thread carrying a charge uniformly distributed along its length. The charge concentrated on an infinite thread is, of course, also infinite, and therefore it cannot serve as a quantitative characteristic of the degree of charge of the thread. Such a characteristic is taken to be “ linear charge density" This value is equal to the charge distributed on a piece of thread of unit length:
Let's find out what the field strength is created by a charged thread at a distance A from it (Fig. 1.12).
Rice. 1.12.
To calculate the intensity, we again use the principle of superposition of electric fields and Coulomb’s law. Let's select an elementary section on the thread dl.The charge is concentrated in this area dq= t dl, which can be considered point-like. At point A such a charge creates a field (see 1.3)
Based on the symmetry of the problem, we can conclude that the desired field strength vector will be directed along a line perpendicular to the thread, that is, along the axis X. Therefore, the addition of tension vectors can be replaced by the addition of their projection onto this direction.
(1.7)
Rice. (1.12 b) allows us to draw the following conclusions:
Thus
. (1.9)
Using (1.8) and (1.9) in equation (1.7), we obtain
Now, to solve the problem, it remains to integrate (1.10) over the entire length of the thread. This means that angle a will vary from to .
In this problem, the field has cylindrical symmetry. The field strength is directly proportional to the linear charge density on the thread t and inversely proportional to the distance A from the thread to the point where the tension is measured.
Lecture 2 “Gauss’s theorem for electric field»
Lecture outline
Electric field strength vector flux.
Gauss's theorem for the electric field.
Application of Gauss's theorem to calculate electric fields.
Field of an infinite charged thread.
Field of an infinite charged plane. Field of a parallel-plate capacitor.
Field of a spherical capacitor.
We finished the first lecture by calculating the field strength of an electric dipole and an infinitely charged thread. In both cases, the principle of superposition of electric fields was used. Now let's turn to another method for calculating the intensity, based on Gauss's theorem for the electric field. This theorem deals with the flow of a tension vector through an arbitrary closed surface. Therefore, before proceeding to the formulation and proof of the theorem, we will discuss the concept of “vector flow”.
Electric field strength vector flux
Let us select a flat surface in a uniform electric field (Fig. 2.1.). This surface is a vector numerically equal to the surface area D S and directed perpendicular to the surface
Rice. 2.1.
But a unit normal vector can be directed either in one or the other direction from the surface (Fig. 2.2.). Arbitrarily Let's choose the positive direction of the normal as shown in Fig. 2.1. A-priory The flow of the electric field strength vector through a selected surface is the scalar product of these two vectors:
Rice. 2.2.
If the field is generally inhomogeneous, and the surface S, through which the flow should be calculated, is not flat, then this surface is divided into elementary sections, within which the tension can be considered unchanged, and the sections themselves are flat (Fig. 2.3.) The flow of the tension vector through such an elementary section is calculated by the definition of flow
Here E n = E∙ cosa - projection of the tension vector onto the normal direction. Full flow across the entire surface S we find by integrating (2.3) over the entire surface
(2.4)
Rice. 2.3.
Now let's imagine closed surface in an electric field. To find the flow of the tension vector through such a surface, we perform the following operations (Fig. 2.4.):
Divide the surface into sections. It is important to note that in case closed Only the “outer” normal of a surface is considered positive.
Let's calculate the flow at each elementary section:
Note that a vector “flowing” from a closed surface creates a positive flow, while a vector “flowing” creates a negative flow.
To calculate the total flux of the tension vector through the entire closed surface, all these fluxes need to be algebraically added, that is, equation (2.3) must be integrated over closed surfaces S
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In mechanics, a thread is understood as a material system of one dimension, which, under the influence of applied forces, can take the shape of any geometric line. A thread that does not offer resistance to bending and torsion is called an ideal or absolutely flexible thread. An ideal thread can be extensible or inextensible (an extreme abstraction). In the following, in the absence of special instructions, the term “flexible thread” or simply “thread” will be understood as an ideal inextensible or extensible thread.
When calculating the strength of a thread, calculating the surface forces acting on the thread, as well as in a number of other cases, it is necessary to take into account the transverse dimensions of the thread. Therefore, when speaking about the one-dimensionality of a thread, we, of course, mean that the transverse dimensions are small compared to the length and that they do not violate the properties of an ideal thread listed above.
The ideal thread model represents some abstraction, but in many cases yarn and threads (in the process of their manufacture), cables, chains and ropes quite satisfactorily correspond to this model. Plane problems in the mechanics of certain belts and shells are sometimes reduced to this same model. Therefore, the theory of an ideal thread is of great practical importance.
Let the thread, under the influence of forces applied to it, take on a certain equilibrium configuration.
The position of each point of a stretched or inextensible thread will be determined by arc coordinate 5, measured from a fixed point of the thread, for example, point A (Fig. 1.1). Let us select a segment of the thread with length and mass. The density of a stretched thread at a point (sometimes called linear density) is the limit of the ratio, provided that the point tends along the thread to point M:
In general, the linear density of the thread depends on the selected point, i.e.
If, before stretching, the density of the thread was the same at all points, then the thread is called homogeneous, in otherwise- heterogeneous. With this definition of the linear density of the thread, its heterogeneity can be caused by heterogeneity of the material or different cross-sectional area of the thread.
Let the thread be in equilibrium under the action of distributed forces. Let's make a mental cut at the point of the thread and consider the force with which the part of the thread located in the direction of the positive arc coordinate (in Fig. 1.2, the right part of the thread) acts on the other (left) part of the thread. It is obvious that this force, called the tension of the thread, is directed along a common tangent to the thread at a point (this statement will be proven in § 1.2). Naturally, the left side of the thread acts on the right side with
the same in magnitude, but with a force directed in the opposite direction, i.e., force
Each point of the thread has its own tension. Therefore, in equilibrium, the tension of the thread will be a function of the arc coordinate
If we introduce a unit tangent vector then we have
where is the thread tension modulus.
The normal thread tension o is determined, as usual, by the equality
Here is the cross-sectional area of the thread.
Let the length of the thread element be before stretching and after stretching it becomes equal. Since the stretch of the thread depends on the normal stress, the ratio represents a certain function a
By specifying the function, we will obtain the corresponding law of stretching, for example, elastic, plastic stretching, etc. Let us dwell in more detail on the elastic stretching of a homogeneous thread according to Hooke’s law, when the equality is satisfied
where is the elastic modulus of the thread. Using equality (1.3), we obtain
where a is the specific relative elongation of the thread. If the thread is inextensible, then
Note that the modulus of elasticity of the thread has the dimension of ordinary force: in the International System of Physical Units in technical system accordingly, it is obvious that
where is the modulus of elasticity of the thread material or
Let the diameters of the thread be before and after stretching. Then the relative change in the diameter of the thread is determined by the equality
Assuming that the thread is isotropic and that the distension is subject to Hooke’s law, we will have
where is Poisson's ratio. Using equalities (1.4) and (1.6), we find the value of the thread diameter after stretching
As a rule, the value is negligible compared to unity. Therefore, the change in the diameter of the thread when it is stretched is usually neglected (at least for steel cables) and it is believed that for a stretched cable
Let us consider a thread that is subject to forces distributed along its length, for example, gravity, force
wind pressure, etc. We denote the main vector of forces acting on the thread element by and assume that it is applied to the point located in the shallows (Fig. 1.3). The force per unit length of the thread, or the intensity of the distributed forces, is called the expression
From here, up to terms of higher order, we relatively obtain
The dimension of force per unit length of thread differs from the dimension of ordinary force: in the system it is equal in the technical system -
Distributed forces acting on a thread can be divided into mass and surface. The first include forces that depend on the mass of the thread, such as gravity and inertia. Surface forces, for example, pressure forces of the oncoming flow, do not depend on the mass of the thread (they can depend on the area of the longitudinal diametrical section of the thread, i.e., on its diameter, the speed of the oncoming flow and other factors).
Let us dwell in more detail on mass forces. If we denote the force per unit length, then the force per unit mass of the thread will be determined by the equality
In particular, for gravity we will have
where is the acceleration of gravity, the force of gravity per unit length of the thread. For a homogeneous unstretched thread, the force is numerically equal to the weight of a unit length of the thread.
Since the mass of the thread does not change when stretched, we will have
From here, using equality (1.3), we obtain
Thus, the mass forces per unit length of the tensile thread can be represented by the equality
Surface forces per unit length are usually proportional to the diameter of the thread
where the proportionality coefficient X depends on various factors (for example, flow speed, medium density, etc.). As already noted, in the overwhelming majority of cases, the change in the diameter of the tensile thread can be neglected, and then the number in the last formula should be considered constant. For tensile threads, the modulus of elasticity of which is very small, it is possible that a change in the diameter of the thread must be taken into account. Then you should use formula (1.8).
In the general case, the force per unit length of the thread depends on the arc coordinate of the point of the latter’s position in space, the direction of the tangent or normal to the thread and tension. Indeed, the density and, therefore, the force of gravity of a non-uniform thread depend on the position of the point on the thread, i.e. from its arc coordinate The force of hydrostatic pressure is directed normal to the thread and its module is proportional to the height of the level, i.e. this force depends on the coordinates of the point. From formula (1.15) it follows that the analytical expression for the force per unit length of a stretched thread clearly includes the modulus
tension Therefore, if we consider drinking in a rectangular coordinate system, then in the general case we will have Fig. 1.4.
If the ends of the thread are fixed, then these equalities can serve to determine the reactions of the fastening points. Most often, there are threads with two fixed ends, less often - threads with one fixed and one free end, and the value of the force applied to the free end is specified or can be determined from additional information (its position is usually unknown). More complex boundary conditions are also encountered. Many of them will be considered when studying specific problems. In addition to the direct conditions on the boundaries, geometric (one or more) parameters must be specified, for example, the length of the thread, the sag, etc. We will conditionally refer to these elements as boundary conditions.
Now we can formulate the main problem of the equilibrium of an ideal thread: the forces acting on the thread (distributed and concentrated), the law of tension of the thread are given, and the required number of boundary conditions are found. It is required to determine the form of equilibrium of the thread, its tension at any point and the change in length (for tensile threads).
In conclusion, we note that when solving specific problems, the main difficulties arise, as a rule, when integrating differential equations balance of the thread. However, it should be borne in mind that in many cases the equilibrium equations of a thread can be integrated relatively easily, and the greatest difficulties arise when constructing a solution that satisfies the boundary conditions.
It is necessary to know the point of application and direction of each force. It is important to be able to determine which forces act on the body and in what direction. Force is denoted as , measured in Newtons. In order to distinguish between forces, they are designated as follows
Below are the main forces operating in nature. It is impossible to invent forces that do not exist when solving problems!
There are many forces in nature. Here we consider the forces that are considered in the school physics course when studying dynamics. Other forces are also mentioned, which will be discussed in other sections.
Gravity
Every body on the planet is affected by Earth's gravity. The force with which the Earth attracts each body is determined by the formula
The point of application is at the center of gravity of the body. Gravity always directed vertically downwards.
Friction force
Let's get acquainted with the force of friction. This force occurs when bodies move and two surfaces come into contact. The force occurs because surfaces, when viewed under a microscope, are not as smooth as they appear. The friction force is determined by the formula:
The force is applied at the point of contact of two surfaces. Directed in the direction opposite to movement.
Ground reaction force
Let's imagine a very heavy object lying on a table. The table bends under the weight of the object. But according to Newton's third law, the table acts on the object with exactly the same force as the object on the table. The force is directed opposite to the force with which the object presses on the table. That is, up. This force is called the ground reaction. The name of the force "speaks" support reacts. This force occurs whenever there is an impact on the support. The nature of its occurrence at the molecular level. The object seemed to deform the usual position and connections of the molecules (inside the table), they, in turn, strive to return to their original state, “resist.”
Absolutely any body, even a very light one (for example, a pencil lying on a table), deforms the support at the micro level. Therefore, a ground reaction occurs.
There is no special formula for finding this force. It is denoted by the letter , but this force is simply a separate type of elasticity force, so it can also be denoted as
The force is applied at the point of contact of the object with the support. Directed perpendicular to the support.
Since the body is represented as a material point, force can be represented from the center
Elastic force
This force arises as a result of deformation (change in the initial state of the substance). For example, when we stretch a spring, we increase the distance between the molecules of the spring material. When we compress a spring, we decrease it. When we twist or shift. In all these examples, a force arises that prevents deformation - the elastic force.
Hooke's law
The elastic force is directed opposite to the deformation.
Since the body is represented as a material point, force can be represented from the center
When connecting springs in series, for example, the stiffness is calculated using the formula
When connected in parallel, the stiffness
Sample stiffness. Young's modulus.
Young's modulus characterizes the elastic properties of a substance. This is a constant value that depends only on the material, its physical condition. Characterizes the ability of a material to resist tensile or compressive deformation. The value of Young's modulus is tabular.
More about properties solids.
Body weight
Body weight is the force with which an object acts on a support. You say, this is the force of gravity! The confusion occurs in the following: indeed, often the weight of a body is equal to the force of gravity, but these forces are completely different. Gravity is a force that arises as a result of interaction with the Earth. Weight is the result of interaction with support. The force of gravity is applied at the center of gravity of the object, while weight is the force that is applied to the support (not to the object)!
There is no formula for determining weight. This force is designated by the letter.
The support reaction force or elastic force arises in response to the impact of an object on the suspension or support, therefore the weight of the body is always numerically the same as the elastic force, but has the opposite direction.
The support reaction force and weight are forces of the same nature; according to Newton’s 3rd law, they are equal and oppositely directed. Weight is a force that acts on the support, not on the body. The force of gravity acts on the body.
Body weight may not be equal to gravity. It may be more or less, or it may be that the weight is zero. This condition is called weightlessness. Weightlessness is a state when an object does not interact with a support, for example, the state of flight: there is gravity, but the weight is zero!
It is possible to determine the direction of acceleration if you determine where the resultant force is directed
Please note that weight is force, measured in Newtons. How to correctly answer the question: “How much do you weigh”? We answer 50 kg, not naming our weight, but our mass! In this example, our weight is equal to gravity, that is, approximately 500N!
Overload- ratio of weight to gravity
Archimedes' force
Force arises as a result of the interaction of a body with a liquid (gas), when it is immersed in a liquid (or gas). This force pushes the body out of the water (gas). Therefore, it is directed vertically upward (pushes). Determined by the formula:
In the air we neglect the power of Archimedes.
If the Archimedes force is equal to the force of gravity, the body floats. If the Archimedes force is greater, then it rises to the surface of the liquid, if less, it sinks.
Electrical forces
There are forces of electrical origin. Occur when there is electric charge. These forces, such as the Coulomb force, Ampere force, Lorentz force, are discussed in detail in the section Electricity.
Schematic designation of forces acting on a body
Often a body is modeled as a material point. Therefore, in diagrams, various points of application are transferred to one point - to the center, and the body is depicted schematically as a circle or rectangle.
In order to correctly designate forces, it is necessary to list all the bodies with which the body under study interacts. Determine what happens as a result of interaction with each: friction, deformation, attraction, or maybe repulsion. Determine the type of force and correctly indicate the direction. Attention! The amount of forces will coincide with the number of bodies with which the interaction occurs.
The main thing to remember
1) Forces and their nature;
2) Direction of forces;
3) Be able to identify the acting forces
There are external (dry) and internal (viscous) friction. External friction occurs between contacting solid surfaces, internal friction occurs between layers of liquid or gas when they relative motion. There are three types of external friction: static friction, sliding friction and rolling friction.
Rolling friction is determined by the formula
The resistance force occurs when a body moves in a liquid or gas. The magnitude of the resistance force depends on the size and shape of the body, the speed of its movement and the properties of the liquid or gas. At low speeds of movement, the drag force is proportional to the speed of the body
At high speeds it is proportional to the square of the speed
Let's consider the mutual attraction of an object and the Earth. Between them, according to the law of gravity, a force arises 
Now let's compare the law of gravity and the force of gravity 
The magnitude of the acceleration due to gravity depends on the mass of the Earth and its radius! Thus, it is possible to calculate with what acceleration objects on the Moon or on any other planet will fall, using the mass and radius of that planet.
The distance from the center of the Earth to the poles is less than to the equator. Therefore, the acceleration of gravity at the equator is slightly less than at the poles. At the same time, it should be noted that the main reason for the dependence of the acceleration of gravity on the latitude of the area is the fact of the Earth’s rotation around its axis.
As we move away from the Earth's surface, the force of gravity and the acceleration of gravity change in inverse proportion to the square of the distance to the center of the Earth.
