How to calculate tension in physics. Field of an infinitely charged thread Tension force of the thread
popular definition
Strength is action, which can change the state of rest or movement body; therefore, it can accelerate or change the speed, direction, or direction of motion of a given body. Against, tension- this is the state of a body subject to the action of opposing forces that attract it.
She is known as tensile force, which, when exposed to an elastic body, creates tension; This last concept has various definitions, which depend on the branch of knowledge from which it is analyzed.
Ropes, for example, allow forces to be transferred from one body to another. When two equal and opposite forces are applied at the ends of a rope, the rope becomes taut. In short, tensile forces are each of these forces that supports the rope without breaking .
Physics And engineering talk about mechanical stress, to denote the force per unit area surrounding a material point on the surface of a body. Mechanical stress can be expressed in units of force divided by units of area.
Voltage is also a physical quantity that drives electrons through a conductor into a closed electrical circuit that causes electric current to flow. In this case, the voltage can be called voltage or potential difference .
On the other side, surface tension of a liquid is the amount of energy required to reduce its surface area per unit area. Consequently, the liquid exerts resistance, increasing its surface area.
How to find the tension force
Knowing that force tension is force, with which a line or string is tensioned, the tension can be found in a static type situation if the angles of the lines are known. For example, if the load is on a slope and a line parallel to the slope prevents the load from moving downward, the tension is resolved, knowing that the sum of the horizontal and vertical components of the forces involved must add up to zero.
First step to do this calculation- draw a slope and place a block of mass M on it. The slope increases on the right, and at one point it meets a wall, from which a line runs parallel to the first. and tie the block, holding it in place and creating a tension T. Next you should identify the angle of inclination with the Greek letter, which may be "alpha", and the force it exerts on the block with the letter N, since we are talking about normal strength .
From the block vector should be drawn perpendicular to the slope and up to represent the normal force, and one down (parallel to the axis y) to display gravity. Then you start with formulas.
To find strength F = M is used. g , Where g is his constant acceleration(in the case of gravity this value is 9.8 m/s^2). The unit used for the result is newton, which is denoted by N. In the case of a normal force, it must be expanded into vertical and horizontal vectors using the angle it makes with the axis x: to calculate the up vector g is equal to the cosine of the angle, and for the vector in the direction to the left, towards the bosom of this.
Finally, the left-hand side of the normal force must be equal to the right-hand side of the stress T, finally resolving the stress.
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life
The etymological origin of the word life is found in Latin. Specifically, it comes from the word vita, which in turn comes from the Greek term bios. They all mean life. The concept of life can be defined from different approaches. The most common concept is related
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The Latin word ocŭlus comes from the eye, a concept that refers to the organ that provides vision in animals and humans. The term, in any case, has other meanings. As an organ, the eye can detect luminosity and convert its changes into a nerve impulse that is interpreted by the brain. Although it's de
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The first necessary step in uncovering the meaning of the term "soundtrack" is to determine the etymological origin of the two words that form it: A group that seems to come from Germanic or Frankish, depending on what it means. Sonora, which comes from Latin. Specifically, it is the result of combining the verb "sonare", which can be translated as "making noise", and the suffix "-oro", which is equivalent to "fullness". Group concept
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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 block 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 to 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.
A block is fixed at the top of two inclined planes making angles and with the horizon. 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 looked at various ways to solve problems that many of you will find useful in the future in order 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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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, otherwise it is called inhomogeneous. 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 the technical system, respectively, and Obviously,
where is the modulus of elasticity of the thread material or
Let the diameters of the thread 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 for the equilibrium of a 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.
Problem 10048
A disk-shaped block with a mass of m = 0.4 kg rotates under the action of the tension force of a thread, to the ends of which weights of masses m 1 = 0.3 kg and m 2 = 0.7 kg are suspended. Determine the tension forces T 1 and T 2 of the thread on both sides of the block.
Problem 13144
A light thread is wound on a homogeneous solid cylindrical shaft of radius R = 5 cm and mass M = 10 kg, to the end of which a load of mass m = 1 kg is attached. Determine: 1) the dependence s(t), according to which the load moves; 2) the tension force of the thread T; 3) dependence φ(t), according to which the shaft rotates; 4) angular velocity ω of the shaft t = 1 s after the start of movement; 5) tangential (a τ) and normal (a n) accelerations of points located on the surface of the shaft.
Problem 13146
A weightless thread is thrown through a stationary block in the form of a homogeneous solid cylinder with a mass m = 0.2 kg, to the ends of which bodies with masses m 1 = 0.35 kg and m 2 = 0.55 kg are attached. Neglecting friction in the axis of the block, determine: 1) acceleration of the load; 2) the ratio T 2 /T 1 of the thread tension forces.
Problem 40602
A thread (thin and weightless) is wound around a hollow thin-walled cylinder of mass m. Its free end is attached to the ceiling of an elevator moving downward with acceleration a l. The cylinder is left to its own devices. Find the acceleration of the cylinder relative to the elevator and the tension force of the thread. During movement, consider the thread vertical.
Problem 40850
A mass weighing 200 g is rotated on a thread 40 cm long in a horizontal plane. What is the tension force of the thread if the load makes 36 revolutions in one minute?
Problem 13122
A charged ball of mass m = 0.4 g is suspended in the air on a silk thread. A charge q of different and equal magnitude is brought from below to it at a distance of r = 2 cm. As a result, the tension force of the thread T increases by n = 2.0 times. Find the amount of charge q.
Problem 15612
Find the ratio of the modulus of the tension force of the thread of the mathematical pendulum in the extreme position with the modulus of the tension force of the thread of the conical pendulum; the lengths of the threads, the masses of the weights and the angles of deflection of the pendulums are the same.
Problem 16577
Two small identical balls, each weighing 1 µg, are suspended on threads of equal length and touching. When the balls were charged, they separated by a distance of 1 cm, and the tension force on the thread became equal to 20 nN. Find the charges of the balls.
Problem 19285
Establish a law according to which the tension force F of the thread of a mathematical pendulum changes over time. The pendulum oscillates according to the law α = α max cosωt, its mass m, length l.
Problem 19885
The figure shows a charged infinite plane with a surface plane of charge σ = 40 μC/m 2 and a similarly charged ball with mass m = l g and charge q = 2.56 nC. The tension force of the thread on which the ball hangs is...
