Rotation around the axis or points of various objects is one of the important types of motion in technology and in nature, which is studied in the course of physics. The dynamics of rotation, in contrast to the dynamics of linear motion, operates with the concept of the moment of a particular physical quantity. This article is devoted to the question of what is the moment of forces.
The concept of the moment of force
Every bicycle lover at least once in his life spun the wheel of his “iron horse” with his hand. If the described action is carried out with the hand on the tire, it is much easier to spin the wheel than holding the spokes closer to the axis of rotation. This simple action is described in physics using a moment of force or torque.
What is a moment of power? You can answer this question if you imagine a system that can rotate around the O axis. If at some point P a force vector F¯ is applied to the system, then the moment of the acting force F¯ will be equal to:
M¯ = [OP¯ * F¯].
That is, the moment M¯ is a vector quantity equal to the product of the vector force F¯ by the radius vector OP¯.
The written formula allows us to note an important fact: if the external force F¯ is applied at any angle to any point of the axis of rotation, then it does not create a moment.
The absolute value of the moment of force
In the previous paragraph, we considered the definition of what the moment of force is relative to the axis. Now turn to the figure below.
A rod having a length L is shown here. On one side, it is fixed through a swivel joint to a vertical wall. The other end of the shaft is free. The force F¯ has an effect on this end. The angle between the rod and the force vector is also known. It is equal to φ.
Torque is determined through the vector product. The modulus of such a product is equal to the product of the absolute values of the vectors by the sine of the angle between them. Applying trigonometric formulas, we arrive at the following equality:
M = L * F * sin (φ).
Turning again to the figure above, we can rewrite this equality in the following form:
M = d * F, where d = L * sin (φ).
The value of d, which is equal to the distance from the force vector to the axis of rotation, is called the lever of force. The larger the value of d, the greater the moment created by the force F.
The direction of the moment of force and its sign
The study of the question of what is a moment of power cannot be complete if you do not consider its vector nature. Recalling the properties of a vector product, it is safe to say that the moment of force will be perpendicular to a plane built on factor vectors.
The specific direction M¯ is uniquely determined if the so-called gimlet rule is applied. It sounds simple: rotating the gimlet in the direction of the circular motion of the system, the direction of the moment of force is determined by the progressive movement of the gimlet.
If you look at the rotating system along its axis, then the vector of the moment of force applied to the point can be directed both at the reader and from him. In this regard, in quantitative calculations, the concept of a positive or a negative moment is used. In physics, it is accepted to consider positive that moment of force that leads to the rotation of the system counterclockwise.
What is the meaning of M¯?
This refers to the physical meaning. Indeed, it is known in the mechanics of linear motion that force is a measure of the ability to impart linear acceleration to a body. By analogy, the moment of force of a point is a measure of the possibility of reporting angular acceleration of the system. The moment of force is the cause of the appearance of angular acceleration and is directly proportional to it.
It is easy to understand the different possibilities of making a rotation or turning, if you remember that the door opens easier if you push it away from the door hinges, that is, in the area of the handle. Another example: any more or less heavy object is easier to hold if you press your hand to the body than to hold it on an outstretched arm. Finally, loosening the nut is easier if you use the long wrench. In these examples, the moment of force changes due to a decrease or increase in leverage.
It is appropriate to give an analogy of a philosophical nature, taking as an example the book of Eckhart Tolle, "The Power of the Moment Now." The book belongs to the psychological genre and teaches you to live without stress at the moment of your life. Only the current moment makes sense, only during it all actions are performed. Considering the named idea of the book “The Power of the Moment Now”, we can say that the torque in physics accelerates or decelerates the rotation at the current moment in time. Therefore, the main equation of moments has the following form:
dL = M * dt.
Where dL is the change in angular momentum over an infinitely small period of time dt.
The importance of the concept of moment of force for statics
Many are familiar with leverage tasks of various kinds. In practically all these problems of statics, it is required to find the equilibrium conditions of the system. To find these conditions, it is easiest to use the concept of moment of force.
If the system does not move and is in equilibrium, then the sum of all the moments of force relative to the axis, point or chosen support should be equal to zero, that is:
∑ i = 1 n M i ¯ = 0.
Where n is the number of forces acting.
Recall that the absolute values of the moments M i should be substituted into the equality above taking into account their sign. The reaction force of the support, which is considered as the axis of rotation, does not create torque. Below is a video that explains the topic of this paragraph of the article.
A moment of power and his work
Many readers have noticed that the moment of force is calculated in Newtons per meter. This means that it has the same dimension as work or energy in physics. Nevertheless, the concept of the moment of force is a vector quantity, and not a scalar one; therefore, the moment M¯ cannot be considered work. However, he can perform the work, which is calculated by the following formula:
A = M * θ.
Where θ is the central angle in radians that the system has rotated in a known time t.