Typical tasks of the "Statics" section of physics are the problems of equilibrium of bodies. In this case, the fact that the sum of the external forces is zero is not enough information to determine the state of the body in question, therefore, use another quantity, which is called the moment of force. What is it equal to and how to use it to solve equilibrium problems is considered in this article.
The meaning of the concept
The definition of the moment of force or torsion, as it is often called, can be given as follows: this is a physical quantity equal to the product of the acting force applied to a certain point of the body P by the arm of the force equal to the distance from the axis of attachment of this body O to point P. Presentation of this definition is an illustration.
Here we see that the action of the force F on a key of length d (shoulder) will lead to a tightening of the nut, which plays the role of the O axis for this system.
As a result, we can write that the modulus of the moment of force is equal to the following expression: M = d * F. From where it can be seen, the larger the arm d, the lesser force F will be required to create a certain torsion moment M. The SI unit of measure for this quantity is newtons meter (N * m).
M¯ is a vector
We have not completely examined the question of what the moment of force is equal to about an axis. The fact is that the acting force can be applied to the lever at a certain angle. And for this case, the formula for calculating the value of M. was not given. In fact, the latter is a vector, not a scalar. It is determined through the vector product. One vector is known - it is the external force F¯. Another directional segment is OP¯ (in accordance with the points introduced in the previous paragraph). What is the moment power in this case? It can be written as the following expression: M¯ = OP¯ * F¯ (the line above each factor is a vector icon).
The product module of two vectors is equal to the absolute length of each of them by the sine of the angle between them. Since the angle between the force and the shoulder in the paragraph above is 90 o , the vector formula is translated into a scalar form by simply lowering the vector icon. If the marked angle is not right, then the product of the moduli of the vectors should be multiplied by the sine of the corresponding angle. This situation is shown in the figure below.
It can be seen here that the force F is directed at an angle Φ, which means that the absolute value of the moment of force can be calculated as follows:
M = L * F * sin (Φ).
Here L is no longer a lever of force, however, L * sin (Φ) = d (this equality follows from the definition of the sine of the angle Φ), and the scalar expression for the moment can again be written in the form given in the first paragraph of the article.
The direction of the vector M¯
So, we have learned to calculate what the moment of force is equal to the axis, but since it is also a vector, we should get acquainted with the question of how to determine its direction. Of course, for this you can use the right-hand rule, which is valid for any vector product, but the resulting vector depends on the direction of the multiplied segments and the order in which they follow in the product, that is, the result OP¯ * F¯ and F¯ * OP¯ will be vectors with in the opposite direction.
Therefore, it is recommended to remember the following rule: if we imagine that the resulting force F led to the rotation of the body around its axis counterclockwise, it means that the observer looks at the system from the top of the vector M¯, which in this case is considered positive. On the contrary, if the force F should cause the body to rotate clockwise, then the observer looks at the moment of force along the direction of its action, here the module M¯ is a negative value.
The figure above shows the vector M¯ taking into account the direction of rotation in the system. Its module, as stated above, is positive, i.e. + M.
Multiple Force Systems
Until now, examples have been cited in which rotation is carried out by only one force. In the general case, there can be several such forces. How to calculate the resulting torsion moment? The value of M is additive, that is, to calculate it, you can summarize the moment of each force, but you must observe the sign (+ M or -M).
Solving the two-body equilibrium problem
Now we solve a simple problem. The figure below shows two people in equilibrium. It is necessary to determine how many times a yellow person sits farther from the axis of rotation than a green one, knowing that his weight is 1.5 times less than the weight of his partner.
First of all, it is necessary to determine what forces act in the system. There are three of them: two gravity forces of each person and the reaction force of the axis support . Since the shoulder for the latter is zero, it does not create a torsion moment. It remains to consider two forces of gravity.
Since the system is in equilibrium, the total moment of gravity is zero. Let P 1 , P 2 and x 1 , x 2 be the weights and distances from the axis for the yellow and green people, respectively. The weight of a yellow person creates a counterclockwise rotation, which means that its torsion moment is positive. For a green person, this value will be negative. Then we have: P 1 * x 1 - P 2 * x 2 = 0. From where we get: P 2 / P 1 = x 1 / x 2 = 1.5.
Thus, a yellow person sits 1.5 times further from the axis than green.