The current state of technology would have looked completely different if humanity had not learned in the distant past to use rolling friction force for its benefits. What it is, why it appears and how it can be calculated, these issues are considered in the article.
What is this rolling friction ?
It is understood as a physical force that appears in all cases when one object does not slip, but rolls over the surface of another. Examples of rolling friction force are the movement of a wooden cart wheel on a dirt road or the movement of a car wheel on asphalt, the rolling of metal ball and needle bearings along a steel axis, the movement of a paint roller on a wall, and so on.
In contrast to the static and sliding frictional forces caused by interactions at the atomic level of the rough surfaces of the body and the surface, the cause of the appearance of friction as a result of rolling is the deformation hysteresis.
Let us explain the named fact on the example of a wheel. When it is in contact with absolutely any solid surface, then in the contact zone it microdeforms in the elastic region. As soon as the wheel rotates a certain angle, this elastic deformation will disappear and the body will regain its shape. Nevertheless, as a result of the rolling of the wheel, the compression and restoration cycles are repeated, which are accompanied by energy loss and microscopic disturbances in the structure of the surface layers of the wheel. These losses are called hysteresis. When moving, they are manifested in the occurrence of rolling friction.
Rolling undeformable bodies
Consider the ideal case when the wheel, moving on an absolutely solid surface, does not undergo microdeformations. In this case, the zone of its contact with the surface will correspond to a straight segment whose area is zero.
When driving, four forces act on the wheel. This is the traction force F, the reaction force of the support N, the weight of the wheel P and the friction f r . The first three forces are central (act on the center of mass of the wheel), so they do not create torque. The force f r acts tangential to the wheel rim. The rolling friction moment is equal to:
M = f r * r.
Here, the radius of the wheel is indicated by the letter r.
The forces N and P act vertically, therefore, in the case of uniform motion, the friction force f r will be equal to the traction force F:
F = f r .
Any infinitely small force F will be able to overcome the value of f r , and the wheel will begin to move. This conclusion leads to the fact that in the case of a non-deformable wheel, the rolling friction force is zero.
Rolling deformable (real) bodies
In the case of real bodies, as a result of the deformation of the wheel, its area of support on the surface is not equal to zero. In a first approximation, it is a rectangle with sides l and 2 * d. Where l is the width of the wheel, which does not interest us much. The appearance of the rolling friction force is due precisely to the value 2 * d.
As in the case of a non-deformable wheel, the four forces mentioned above also act on the real object. All relations between them remain with the exception of one: the reaction force of the support as a result of deformation will not act through the axis on the wheel, but will be displaced relative to it by a distance d, that is, it will take part in creating torque. The formula for moment M in the case of a real wheel takes the form:
M = N * d - f r * r.
Zeroing the value of M is a condition for uniform rolling of the wheel. As a result, we arrive at the equality:
f r = d / r * N.
Since N is equal to the weight of the body, we obtain the final formula for the rolling friction force:
f r = d / r * P.
This expression contains a useful result: with an increase in the radius r of the wheel, the friction force f r decreases.
Coefficient of rolling resistance and coefficient of rolling
In contrast to the static and sliding friction forces, rolling is characterized by two coefficients dependent on each other. The first of these is the d value described above. It is called the coefficient of rolling resistance, because the greater its value, the greater the force f r . For wheels of trains, cars, metal bearings, the value of d lies within the tenths of a millimeter.
The second coefficient is the actual rolling coefficient. It is a dimensionless quantity and is equal to:
C r = d / r.
Many tables give precisely this value, since it is more convenient to use it for solving practical problems than the value of d. In most practical cases, the value of C r does not exceed several hundredths (0.01-0.06).
The rolling condition of real bodies
Above we got the formula for the force f r . We write it through the coefficient C r :
f r = C r * P.
It can be seen that its shape is similar to that for the rest friction force , in which instead of C r the value µ is used — the rest friction coefficient.
Traction force F will cause the wheel to roll only if it is greater than f r . However, the thrust F can also lead to sliding if it exceeds the corresponding rest force. Thus, the rolling condition of real bodies is that the force f r be less than the rest friction force.
In most cases, the coefficient µ is 1-2 orders of magnitude higher than the value of C r . However, in some situations (the presence of snow, ice, oily liquids, and dirt on the rolling surface), μ can become less than C r . In the latter case, wheel slippage will be observed.