When they solve any problems in physics in which there are moving objects, they always talk about the forces of friction. They are either taken into account or neglected, but no one doubts the fact of their presence. In this article, we consider what the moment of friction forces is, and also present problems for the elimination of which we will use the acquired knowledge.
The force of friction and its nature
Everyone understands that if one body moves on the surface of another in absolutely any way (slides, rolls), then there always exists some force that prevents this movement. It is called dynamic friction force. The reason for its occurrence is due to the fact that any bodies have microscopic roughnesses on their surfaces. When two objects come into contact, their roughnesses begin to interact with each other. This interaction is both mechanical in nature (the peak falls into the cavity) and occurs at the level of atoms (dipole attraction, van der Waals, and others).
When the bodies in contact are at rest, in order to move them relative to each other, it is necessary to apply a force that is greater than that to maintain the sliding of these bodies at each other at a constant speed. Therefore, in addition to dynamic, static friction force is also considered.
Friction force properties and formulas for its calculation
The school physics course states that for the first time the laws of friction were set forth by the French physicist Guillaume Amonton in the 17th century. In fact, this phenomenon began to be studied at the end of the 15th century by Leonardo da Vinci, examining a moving object on a smooth surface.
The friction properties can be summarized as follows:
- friction force always acts against the direction of movement of the body;
- its value is directly proportional to the reaction of the support ;
- it does not depend on the contact area;
- it does not depend on the speed of movement (for low speeds).
These features of the phenomenon under consideration allow us to introduce the following mathematical formula for the friction force:
F = μ * N, where N is the support reaction, μ is the proportionality coefficient.
The value of the coefficient μ depends solely on the properties of surfaces that rub against each other. The table of values for some surfaces is shown below.
For rest friction, the formula is used the same as above, but the values of the coefficients μ for the same surfaces will be completely different (they are larger in magnitude than for sliding).
A special case is rolling friction when one body rolls (does not slip) over the surface of another. For force in this case, apply the formula:
F = f * N / R.
Here R is the radius of the wheel, f is the rolling coefficient, which according to the formula has a dimension of length, which distinguishes it from dimensionless μ.
Moment of power
Before answering the question of how to determine the moment of friction forces, it is necessary to consider the physical concept itself. By the moment of force M is understood a physical quantity, which is defined as the product of the shoulder by the value of the force F applied to it. Below is the picture.
Here we see that the application F to the shoulder d, which is equal to the length of the wrench, creates a torque leading to the loosening of the green nut.
Thus, for the moment of force, the formula is valid:
M = d * F.
Note that the nature of the force F does not matter: it can be electric, gravitational or caused by friction. That is, the determination of the moment of friction force will be the same as that given at the beginning of the paragraph, and the written formula for M remains valid.
When does the moment of forces appear due to friction?
This situation arises when three main conditions are satisfied:
- Firstly, there should be a rotating system around a certain axis. For example, it can be a wheel moving on asphalt, or a horizontally located gramophone record rotating on an axis.
- Secondly, there must be friction between the rotating system and some medium. In the examples above: rolling friction acts on the wheel when it interacts with the asphalt surface; if you put a record on a table and unwind it, then it will experience sliding friction on the table surface.
- Thirdly, the arising friction force should act not on the axis of rotation, but on the rotating elements of the system. If the force is central, that is, it acts on the axis, then the shoulder is zero, so it will not create a moment.
How to find the moment of friction force?
To solve this problem, you must first determine which rotating elements are affected by the friction force. Then it is necessary to find the distance from these elements to the axis of rotation and determine what the friction force acting on each element is equal to. After this, it is necessary to multiply the distances r i by the corresponding values of F i and add the results. As a result, the total moment of rotation friction forces is calculated by the formula:
M = ∑ n r i * F i .
Here n is the number of friction forces arising in the rotation system.
It is interesting to note that although M is a vector quantity, therefore, when adding the moments in scalar form, its direction should be taken into account. Friction always acts against the direction of rotation, therefore, every moment M i = r i * F i will have the same sign.
Next, we will solve two problems, where we use the above formulas.
Grinder disc rotation
It is known that when a disc of a grinder with a radius of 5 cm cuts metal, it rotates at a constant speed. It is necessary to determine what moment of force the electric motor of the device creates if the friction force on the disk metal is 0.5 kN.
Since the disk rotates at a constant speed, the sum of all the moments of forces that act on it is zero. In this case, we have only 2 points: from the electric motor and from the friction force. Since they act in different directions, we can write the formula:
M 1 - M 2 = 0 => M 1 = M 2 .
Since friction acts only at the point of contact of the disc of the grinder with metal, that is, at a distance r from the axis of rotation, its moment of force is equal to:
M 2 = r * F = 5 * 10 -2 * 500 = 25 N * m.
Since the electric motor creates the same modulo moment, we get the answer: 25 N * m.
Rolling wooden disc
There is a disk made of wood, its radius r is 0.5 meters. This disk begins to roll on a wooden surface. It is necessary to calculate what distance he is able to overcome if his initial rotation speed ω was 5 rad / s.
The kinetic energy of a rotating body is:
E = I * ω 2/2.
Here I is the moment of inertia. The rolling friction force will slow down the movement of the disk. The work done by her can be calculated using the following formula:
A = M * θ.
Here θ is the angle in radians that the disk will be able to turn during its movement. The body will roll until all its kinetic energy is spent on the work of friction, that is, it is possible to equate the written formulas:
I * ω 2/2 = M * θ.
The moment of inertia of the disk I is equal to m * r 2/2. To calculate the moment M of the friction force F, it should be noted that it acts along the edge of the disk at the point of contact with the wooden surface, that is, M = r * F. In turn, F = f * mg / r (the reaction force of the support N is equal to the weight of the disk mg). Substituting all these formulas into the last equality, we obtain:
m * r 2 * ω 2/4 = r * f * mg / r * θ => θ = r 2 * ω 2 / (4 * f * g).
Since the distance L traveled by the disk is related to the angle θ by the expression L = r * θ, we obtain the finite equality:
L = r 3 * ω 2 / (4 * f * g).
The value of f can be seen in the table for the coefficients of rolling friction. For a tree-tree pair, it is 1.5 * 10 -3 m. We substitute all values, we obtain:
L = 0.5 3 * 5 2 / (4 * 1.5 * 10 -3 * 9.81) ≈ 53.1 m.
To confirm the correctness of the resulting final formula, you can verify that the units of length are obtained.