Rotation around an axis or a point of various objects is one of the important types of movement 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 one or another physical quantity. This article is devoted to the question of what is the moment of forces.
The concept of moment of force
Every cyclist at least once in his life spun the wheel of his "iron horse" by hand. If the described action is performed by holding the tire with your hand, then it is much easier to spin the wheel than by holding the spokes closer to the axis of rotation. This simple action is described in physics as a moment of force or torque.
What is a moment of force? You can answer this question if you imagine a system that can rotate around the axis O. 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¯ and the radius vector OP¯.
The written formula allows us to note an important fact: if an external force F¯ is applied at any angle to any point of the rotation axis, then it does not create a moment.
Absolute value of moment of force
In the previous paragraph, we considered the definition of what is the moment of force about the axis. Now let's look at the picture below.
Here is a rod with a length L. On the one hand, it is fixed through a hinged joint on a vertical wall. The other end of the rod is free. A force F¯ acts on this end. The angle between the rod and the force vector is also known. It is equal to φ.
The 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 and the sine of the angle between them. Applying trigonometric formulas, we arrive at the following equality:
M=LFsin(φ).
Referring again to the figure above, we can rewrite this equality in the following form:
M=dF, where d=Lsin(φ).
The value d, which is equal to the distance from the force vector to the axis of rotation, is called the lever of force. The greater the value of d, the greater the moment will be created by the force F.
Direction of the moment of force and its sign
Studying the question of what ismoment of force cannot be complete unless its vector nature is considered. Remembering the properties of the cross product, we can say with confidence that the moment of force will be perpendicular to the plane built on multiplier vectors.
The specific direction of M¯ is uniquely determined by applying the so-called gimlet rule. It sounds simple: by rotating the gimlet in the direction of the circular motion of the system, the direction of the moment of force is determined by the translational movement of the gimlet.
If you look at a rotating system along its axis, then the vector of the moment of force applied to a point can be directed both towards the reader and away from him. In this regard, in quantitative calculations, the concept of a positive or negative moment is used. In physics, it is customary to consider positive the moment of force that leads to the rotation of the system counterclockwise.
What is the meaning of M¯?
Meaning the physical meaning. Indeed, in the mechanics of linear motion, it is known 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 ability to communicate the angular acceleration of the system. The moment of force is the cause of the angular acceleration and is directly proportional to it.
The different possibilities of making a rotation or turn are easy to understand if you remember that the door opens more easily if it is pushed 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 at arm's length. Finally, unscrewing the nut is easier if you use a long wrench. In the above examples, the moment of force is changed by decreasing or increasing the lever of force.
Here it is appropriate to give an analogy of a philosophical nature, taking as an example the book by Eckhart Tolle "The Power of the 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 has meaning, only during it all actions are performed. Considering the named idea of the book "The Force of the Moment Now" it can be said that the torque in physics accelerates or slows down the rotation at the current moment of time. Therefore, the main moment equation has the following form:
dL=Mdt.
Where dL is the change in angular momentum over an infinitesimal time interval dt.
Importance of the concept of moment of force for statics
Many people are familiar with tasks involving leverage of various kinds. In almost all of these problems of statics, it is required to find the conditions for the equilibrium of the system. The easiest way to find these conditions is to use the concept of moment of force.
If the system is not moving and is in equilibrium, then the sum of all moments of forces about the axis, point or selected support must be equal to zero, that is:
∑i=1Mi¯=0.
Where n is the number of acting forces.
Recall that the absolute values of the moments Mi must be substituted into the equation above withconsidering their sign. The reaction force of the support, which is considered as the axis of rotation, does not create a torque. Below is a video that explains the topic of this paragraph of the article.
Moment of force and its 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. However, the concept of the moment of force is a vector quantity, not a scalar one, so the moment M¯ cannot be considered work. However, he can do the work, which is calculated by the following formula:
A=Mθ.
Where θ is the central angle in radians that the system has rotated in a known time t.
