Rolling friction force without slipping. Equilibrium of a rigid body in the presence of rolling friction. Rolling resistance coefficient

Let the body of rotation located on the support be acted upon by: P - an external force trying to bring the body into a state of rolling or supporting rolling and directed along the support, N - pressing force and Rp - reaction force of the support.

If the vector sum of these forces is zero, then the axis of symmetry of the body moves uniformly and rectilinearly or remains motionless. Vector Ft=-P determines the rolling friction force opposing movement. This means that the downforce is balanced by the vertical component of the ground reaction, and the external force is balanced by the horizontal component of the ground reaction.

Ft·R=N·f

Hence the rolling friction force is equal to:

The origin of rolling friction can be visualized like this. When a ball or cylinder rolls along the surface of another body, it is slightly pressed into the surface of this body, and itself is slightly compressed. Thus, a rolling body always seems to be rolling up a hill. At the same time, sections of one surface are separated from another, and the adhesion forces acting between these surfaces prevent this. Both of these phenomena cause rolling friction forces. The harder the surfaces, the less indentation and the less rolling friction.

Designations:

Ft- rolling friction force

f- rolling friction coefficient, which has the dimension of length (m) (an important difference from the sliding friction coefficient should be noted μ , which is dimensionless)

R- body radius

N- pressing force

P- an external force trying to bring the body into a state of rolling or supporting rolling and directed along the support;

Rp- support reaction.

Friction is a physical phenomenon that a person struggles with in order to reduce it in any rotating and sliding parts of mechanisms, without which, however, the movement of none of these mechanisms is impossible. In this article we will consider from the point of view of physics what force is

What types of friction forces exist in nature?

First of all, let's consider what place rolling friction occupies among other friction forces. These forces arise as a result of the contact of two different bodies. These can be solid, liquid or gaseous. For example, the flight of an aircraft in the troposphere is accompanied by friction between its body and air molecules.

Considering exclusively rigid bodies, the forces of static, sliding and rolling friction are distinguished. Each of us has noticed: in order to move a box on the floor, it is necessary to apply some force along the surface of the floor. The value of the force that will bring the box out of rest will be equal in magnitude to the static friction force. The latter acts between the bottom of the box and the floor surface.

Once the box has begun its movement, a constant force must be applied to keep this movement uniform. This fact is due to the fact that between the contact of the floor and the box, the sliding friction force acts on the latter. As a rule, it is several tens of percent less than static friction.

If you put round cylinders of hard material under the box, it will become much easier to move it. A force will act on the cylinders rotating during movement under the box. It is usually much less than the previous two forces. That is why the invention of the wheel by mankind was a huge leap towards progress, because people were able to move much larger loads with the help of a small applied force.

The physical nature of rolling friction

Why does rolling friction occur? This question is not easy. To answer it, we should consider in detail what happens to the wheel and surface during the rolling process. First of all, they are not perfectly smooth - neither the surface of the wheel, nor the surface on which it rolls. However, this is not the main cause of friction. The main reason is the deformation of one or both bodies.

Any body, no matter what solid material it is made of, is deformed. The greater the weight of the body, the greater the pressure it exerts on the surface, which means it itself is deformed at the point of contact and deforms the surface. This deformation in some cases is so small that it does not exceed the elastic limit.

During the rolling of the wheel, the deformed areas restore their original shape after the contact with the surface ceases. However, these deformations are repeated cyclically with a new rotation of the wheel. Any cyclic deformation, even if it lies within the elastic limit, is accompanied by hysteresis. In other words, at the microscopic level, the shape of the body before and after deformation is different. The hysteresis of deformation cycles during the rolling of a wheel leads to a “spraying” of energy, which manifests itself in practice in the form of the appearance of a rolling friction force.

Rolling the perfect body

In this case, an ideal body means that it is indeformable. In the case of an ideal wheel, the area of its contact with the surface is zero (it touches the surface along a line).

Let us characterize the forces that act on a non-deformable wheel. Firstly, there are two vertical forces: body weight P and N. Both forces pass through the center of mass (wheel axis), therefore they do not take part in the creation of torque. For them you can write:

Secondly, there are two horizontal forces: the external force F, which pushes the wheel forward (it passes through the center of mass), and the rolling friction force f r. The latter creates a torque M. For them we can write the following equalities:

Here r is the radius of the wheel. These equalities contain a very important conclusion. If the frictional force f r is infinitesimal, it will still create a torque that will cause the wheel to move. Since the external force F is equal to f r, any infinitesimal value of F will cause the wheel to roll. This means that if the rolling body is ideal and does not experience deformation during movement, then there is no need to talk about any rolling friction force.

All existing bodies are real, that is, they experience deformation.

Rolling of a real body

Now consider the situation described above only for the case of real (deformable) bodies. The area of contact between the wheel and the surface will no longer be zero, it will have some finite value.

Let's do a force analysis. Let's start with the action of vertical forces, that is, weight and reaction of the support. They are still equal to each other, that is:

However, the force N now acts vertically upward not through the wheel axis, but is slightly displaced from it by a distance d. If we imagine the area of contact of the wheel with the surface as the area of a rectangle, then the length of this rectangle will be the thickness of the wheel, and the width will be equal to 2*d.

Now let's move on to considering horizontal forces. The external force F still does not create a torque and is equal to the friction force f r in absolute value, that is:

The moment of force leading to rotation will create friction f r and reaction of the support N. Moreover, these moments will be directed in different directions. The corresponding expression looks like:

In the case of uniform motion, the moment M will be equal to zero, so we get:

The last equality, taking into account the formulas written above, can be rewritten as follows:

In fact, we have obtained the main formula for understanding the rolling friction force. Later in the article we will analyze it.

Rolling resistance coefficient

This coefficient has already been introduced above. A geometric explanation was also given. We are talking about the value of d. Obviously, the greater this value, the greater the moment created by the reaction force of the support, which prevents the movement of the wheel.

The rolling resistance coefficient d, in contrast to the coefficients of static and sliding friction, is a dimensional value. It is measured in units of length. In tables it is usually given in millimeters. For example, for train wheels rolling on steel rails, d = 0.5 mm. The value of d depends on the hardness of the two materials, the load on the wheel, the temperature and some other factors.

Rolling friction coefficient

It should not be confused with the previous coefficient d. The rolling friction coefficient is designated by the symbol C r and is calculated using the following formula:

This equality means that the value of Cr is dimensionless. It is this that is given in a number of tables containing information about the type of friction under consideration. This coefficient is convenient to use for practical calculations, since it does not require knowledge of the wheel radius.

The value of Cr in the vast majority of cases is less than the coefficients of friction and rest. For example, for car tires moving on asphalt, the value of C r is within a few hundredths (0.01 - 0.06). However, it increases significantly when flat tires move on grass and sand (≈0.4).

Analysis of the resulting formula for force fr

Let us write again the formula for rolling friction force obtained above:

From the equality it follows that the larger the diameter of the wheel, the less force F should be applied in order for it to start moving. Now we write this equality through the coefficient C r, we have:

As you can see, the friction force is directly proportional to the weight of the body. In addition, with a significant increase in weight P, the coefficient Cr itself changes (it increases due to an increase in d). In most practical cases, C r lies within a few hundredths. In turn, the value of the sliding friction coefficient lies within a few tenths. Since the formulas for rolling and sliding friction forces are the same, rolling turns out to be advantageous from an energy point of view (the force f r is an order of magnitude smaller than the sliding force in most practical situations).

Rolling condition

Many of us have encountered the problem of car wheels slipping when driving on ice or mud. Why is this happening? The key to answering this question lies in the relationship between the absolute values of the rolling and static friction forces. Let's write out the formula for rolling again:

When the force F is greater than or equal to the rolling friction, then the wheel will begin to roll. However, if this force sooner exceeds the value of static friction, then the wheel will slip earlier than it will roll.

Thus, the sliding effect is determined by the ratio of the coefficients of static friction and rolling friction.

Ways to prevent car wheel slippage

The rolling friction of a car wheel located on a slippery surface (for example, on ice) is characterized by the coefficient C r = 0.01-0.06. However, values of the same order are characteristic of the static friction coefficient.

To avoid the risk of wheel slipping, use special “winter” tires into which metal spikes are screwed. The latter, crashing into the ice surface, increase the coefficient of static friction.

Another way to increase static friction is to modify the surface on which the wheel moves. For example, by sprinkling it with sand or salt.

The rolling of bodies on a flat surface is a very common type of mechanical motion. However, solving specific problems associated with rolling bodies, as a rule, causes difficulties that could be largely avoided if, at the very beginning of studying this topic, the concept of rolling friction force was more clearly defined. The fact is that when rolling bodies one has to deal with three different types of friction forces: the force of static friction (some authors use “adhesion”), sliding friction and rolling friction (in the narrow sense). Only the last two forces are associated with the dissipation of mechanical energy (i.e., the conversion of mechanical energy into heat). The static friction force, although it plays a role in the dynamics of motion, does not perform mechanical work. The habit, or the established stereotype of solving problems associated with replacing a force distributed over a surface with its resultant at a certain point of application, leads in the case of rolling friction to a number of “paradoxes” that can be avoided by abandoning the unambiguous interpretation of this force. A number of authors of classic textbooks on physics for universities, as a rule, avoid considering this issue. Believing that the rolling friction forces under normal conditions are small, the authors of textbooks and problem books, when considering problems of rolling bodies with and without slipping, as a rule, limit themselves to the remark that the rolling friction forces can be neglected, without assessing the significance of such a simplification. Indeed, this approach allows us to solve a number of problems quite simply and effectively. In this case, in a number of cases the law of conservation of mechanical energy is used. However, a simple analysis reveals that when bodies are forced to roll on a horizontal surface the static friction force can be directed in any direction and can even vanish, which is impossible for rolling friction forces in the narrow sense. In this situation, the question even arises: in comparison with what force can the rolling friction force be neglected? The problem of forced rolling is quite instructive and we will discuss its solution here. A cylinder of mass and radius is placed on a horizontal rough surface. There is a pulley of radius on the cylinder. A thread is wound on the pulley, which is pulled by the end with a constant force. Let us study the dependence of the static friction force on the radius of the pulley and find out the conditions under which rolling and sliding will occur. Rolling friction forces in the narrow sense will, as is customary, be considered negligible.


Rice.. Forced rolling of the cylinder.		Rice.. Graph of the dependence of the rolling friction force on the contact area on the point of application of the external force.

The forces acting on the cylinder are shown in Fig. . Having written the equation of translational and rotational motion in the absence of slipping:

We obtain an expression for the static friction force:

The graph of the obtained dependence is presented in Fig. . There will be no slipping until ( -- friction coefficient), i.e. at

If the force is applied at a distance from the center, there will be no sliding at any arbitrarily small coefficient of friction. When a force is applied near the center of a rolling body, the resulting static friction force is practically equal in magnitude and opposite in direction to the applied external force. If an external force is applied at a distance from the center of the rolling cylinder, the static friction force will be directed in the same direction as the external force. This interesting circumstance illustrates our idea expressed in the introduction. Part of the mechanisms of rolling friction force is due to physical processes occurring at the contact area. In particular, one of the important characteristics of these processes is the true stress distribution on it. The analyzed problem clearly demonstrates that the distribution of stresses at the contact area fundamentally depends on the method of application of force, i.e. on rolling conditions. It is natural to expect that the rolling friction force will significantly depend on these conditions. Problems of this type require clarification to explain a number of observed effects that occur during rolling. As an example, consider the features of the movement of billiard balls. Let's consider the following question: how to hit a billiard ball with a cue so that the friction force of the ball on the cloth makes it move: a) accelerated; b) slowly; c) evenly. To simplify the analysis, we assume that the blow is struck horizontally with the cue in a vertical plane passing through the center of the ball and the point of contact with the surface of the billiard table (Fig.).


Rice.. Hitting a billiard ball.		Rice.. Different directions of sliding friction force depending on the nature of the movement of the billiard ball.

At first glance, it may seem strange that after an impact the ball can move accelerated on the table, since it is generally accepted that friction forces are always directed in the direction opposite to the movement. In fact, depending on the impact conditions, the friction force can be directed both along the speed of movement and against it (). Indeed, as a result of the impact, the ball acquires both translational and rotational motion. There are three different situations possible here. 1. If the speed of translational motion is less than the linear speed of rotational motion of points on the surface of the ball, then the ball moves with slipping and a sliding friction force appears, directed towards the movement, increasing the speed of translational motion and decreasing the speed of rotational motion until these speeds are equal. After this, the loss of mechanical energy of the ball during its rolling will be determined by the rolling friction force in the narrow sense. 2. If the speed of translational motion is greater than the speed of rotational motion, the ball will move slowly. 3. When the ball rolls, followed by a gradual loss of energy due to the action of rolling friction forces. The necessary impact conditions (see Fig.) are found from the equations of the dynamics of translational and rotational motions (without taking into account rolling friction forces):

Where is the moment of inertia of the ball. From here:

Due to the fact that the initial values of the translational and rotational velocities are equal to zero, we have:

Let us now consider the problem of the collision of billiard balls under various conditions. More precisely, we will determine the conditions under which, when a moving ball collides with another (stationary) ball: 1) both balls begin to move forward (rolling impact); 2) the attacking ball stopped, and the resting one began to move forward; 3) the incoming ball rolled back after the impact (pull hit). As before, we will neglect the rolling friction force of the balls both during the movement of the balls and during their interaction. The first case occurs with high impacts when the ball moves with rotation in the direction of movement. During an elastic collision, the balls exchange translational impulses and the second ball begins to slide at the speed of the first. In this case, the sliding friction force will reduce the speed of translational movements and increase the speed of rotational movements until the moment when they become equal and the ball rolls. The moving ball will stop, but since it was spinning, the sliding friction force will continue to act forward and the ball will begin to move again. In order to produce a collision of balls of the “impact with a guy” type, it is necessary that the sliding ball rotates in the opposite direction to the case considered above. Finally, in order to realize a collision with stopping the incoming ball, it is necessary that its translational and rotational speeds simultaneously become zero after the impact. In practice this is possible, but a theoretical explanation in this case will require taking into account the rolling friction force. It should be noted that in previous situations, taking into account the rolling friction force during collisions can lead to a significant modification of the solution. Track.:

> Rolling without slipping

Consider movement without slipping. Read about the role of angular and linear velocity, how translational and rotational motions work, formulas.

Rolling without slipping can be divided into rotational and translational movements.

Learning Objective

Learn to distinguish between two different movements where rolling occurs without sliding.

Main points

Rolling without sliding is much easier to understand if you break it down into translational and rotational motions.
When an object rolls on a plane without sliding, the point of contact between them does not move.
The speed v of a sliding object is directly related to the angular speed ω. Mathematically expressed as v = ωR, (R is the radius of the object and v is the linear velocity).

Terms

Angular velocity is a vector quantity that characterizes the movement of a body in a circular motion. It is equal to the angular velocity and is directed perpendicular to the plane.
Linear velocity is a vector quantity that displays the rate of change of position over time of the center of mass.

If from the very beginning the object turns over without being towed, then we can talk about rolling without slipping. To understand this, let's look at an example with a wheel on a flat horizontal surface.

Movement without slipping is much easier to understand if we distinguish in it the movement of the center of mass with a linear speed v and the rotational movement around the center with an angular speed w.

Rolling motion displays a combination of rotational and translational motions

When an object rolls on a plane without sliding, the point of contact does not move. If we imagine that the wheel is moving with speed v, then it is noticeable that it must also move around its axis with angular speed ω.

The angular velocity of the body (ω) is directly proportional to the speed of movement. You may have noticed: the faster the car accelerates, the more revolutions the wheels make. To calculate the exact relationship between linear and angular speeds, we can take the case where the wheel is displaced by a distance x when turning at an angle θ.

A body rolling down a distance x on a plane without slipping

In mathematics, the length of an arc is equal to the angle of the segment multiplied by the radius of the object (R). It follows that the arc length of a wheel rotated by θ reaches Rθ. Since the wheel is in constant contact with the surface, the arc length is also x. It turns out:

Don't forget that x and θ depend on time, so let's take their derivatives:

Here v is similar in linear speed, and - angular speed ω. Now you can simplify everything:


Number of rotational kinematics
Angular acceleration
Rotational kinematics
Dynamics
Rotational kinetic energy
Conservation of Angular Momentum
Vector nature of rotational kinematics
Problem Solving
Linear and rotational quantities
Energy conservation

The name defines the essence.

Japanese proverb

The rolling friction force, as centuries of human experience shows, is approximately an order of magnitude less than the sliding friction force. Despite this, the idea of a rolling bearing was formulated by Virlo only in 1772.

Let's consider the basic concepts of rolling friction. When a wheel rolls on a stationary base and, when turning through an angle, its axis (point 0) shifts by an amount, then such a movement is called clean rolling without slipping. If the wheel (Fig. 51) is loaded with a force N, then in order to make it move it is necessary to apply a torque. This can be accomplished by applying a force F to its center. In this case, the moment of force F relative to point O 1 will be equal to the moment of rolling resistance.

Fig.51. Pure rolling circuit

If the wheel (Fig. 51) is loaded with a force N, then in order to make it move it is necessary to apply a torque. This can be accomplished by applying a force F to its center. In this case, the moment of force F relative to point O 1 will be equal to the moment of rolling resistance.

Rolling friction coefficient is the ratio of the driving moment to the normal load. This quantity has the dimension of length.

Dimensionless characteristic - rolling resistance coefficient is equal to the ratio of the work of the driving force F on a unit path to the normal load:

where: A is the work of the driving force;

Length of a single path;

M - moment of driving force;

The angle of rotation of the wheel corresponding to the path.

Thus, the expression for the coefficient of friction during rolling and sliding is different.

It should be noted that the adhesion of a rolling body to the track should not exceed the friction force, otherwise rolling will turn into sliding.

Let's consider the movement of a ball along the track of a rolling bearing (Fig. 52a). Both the largest diametrical circle and smaller circles of parallel sections are in contact with the track. The path traveled by a point on circles of different radii is different, that is, slippage occurs.

When a ball or roller rolls along a plane (or internal cylinder), contact occurs at a point or along a line only theoretically. In real friction units, under the influence of work loads, deformation of the contact zone occurs. In this case, the ball is in contact in a certain circle, and the roller is in contact in a rectangle. In both cases, rolling is accompanied by the formation and destruction of frictional bonds, as with sliding friction.

The roller, due to the deformation of the raceway, travels a path shorter than the length of its circumference. This is clearly noticeable when a rigid steel cylinder rolls on a flat elastic rubber surface (Fig. 52b). If the load causes only elastic deformations e, then the rolling trace is restored. During plastic deformations, the raceway remains.

Fig.52. Rolling: a - a ball on a track, b - a cylinder on an elastic base

Due to the inequality of paths (along the circumference of the roller and along the supporting surface), slippage occurs.

It has now been established that the reduction of sliding friction (from slipping) by improving the quality of processing of contact surfaces or the use of lubricants almost does not occur. It follows that the rolling friction force is caused to a greater extent not by slipping, but by energy dissipation during deformation. Since the deformation is mainly elastic, rolling friction losses are the result of elastic hysteresis.

Elastic hysteresis consists in the dependence of deformation under the same loads on the sequence (multiplicity) of influences, that is, on the loading history. Part of the energy is stored in the deformable body and when a certain energy threshold is exceeded, wear particles are separated - destruction. The greatest losses occur when rolling on a viscoelastic base (polymers, rubber), the smallest - on a high-modulus metal (steel rails).

The empirical formula for determining the rolling friction force is:

where: D is the diameter of the rolling body.

Analysis of the formula shows that the friction force increases:

With increasing normal load;

With a decrease in the size of the rolling body.

As the rolling speed increases, the friction force changes little, but wear increases. Increasing the driving speed due to the wheel diameter reduces the rolling friction force.