When they solve the problem of moving objects, in some cases their spatial dimensions are neglected, introducing the concept of a material point. For another type of problem in which resting or rotating bodies are considered, it is important to know their parameters and the points of application of external forces. In this case, we are talking about the moment of forces relative to the axis of rotation. Consider this issue in the article.
The concept of the moment of force
Before giving the formula of the moment of force relative to the axis of rotation of the motionless, it is necessary to explain what phenomenon will be discussed. The figure below shows a wrench of length d, the force F is applied to its end. It is easy to imagine that the result of its action will be a counterclockwise rotation of the wrench and loosening of the nut.
According to the definition, the moment of force relative to the axis of rotation is the product of the shoulder (d in this case) and force (F), that is, you can write the following expression: M = d * F. It should immediately be noted that the above formula is written in scalar form, that is, it allows you to calculate the absolute value of the moment M. As can be seen from the formula, the unit of measurement of the magnitude under consideration is Newtons per meter (N * m).
Moment of force - vector quantity
As stated above, the moment M is actually a vector. To clarify this statement, we consider another figure.
Here we see a lever of length L, which is fixed on an axis (shown by an arrow). A force F is applied to its end at an angle Φ. It is easy to imagine that this force will cause the lever to rise. The formula for the moment in vector form in this case is written as: M¯ = L¯ * F¯, here the bar over the symbol means that the quantity in question is a vector. It should be clarified that L¯ is directed from the axis of rotation to the point of application of the force F¯.
The above expression is a vector product. Its resulting vector (M¯) will be directed perpendicular to the plane formed by L¯ and F¯. To determine the direction of the moment M¯ there are several rules (right hand, gimlet). In order not to memorize them and not get confused in the order of multiplying the vectors L¯ and F¯ (the direction M¯ depends on it), one simple thing to remember: the moment of force will be directed in such a way that if you look from the end of its vector, then the acting force F ¯ will rotate the lever counterclockwise. This direction of the moment is conditionally taken as positive. If the system rotates clockwise, then the resulting torque is negative.
Thus, in the case under consideration with the lever L, the quantity M¯ is directed upward (from the figure to the reader).
In scalar form, the formula for the moment is written in the form: M = L * F * sin (180-Φ) or M = L * F * sin (Φ) (sin (180-Φ) = sin (Φ)). According to the definition of sine, we can write the equation: M = d * F, where d = L * sin (Φ) (see the figure and the corresponding right-angled triangle). The last formula is similar to the one given in the previous paragraph.
The above calculations demonstrate how to work with vector and scalar values of the moments of forces in order to prevent errors.
The physical meaning of M¯
Since the two cases considered in the previous paragraphs are associated with rotational motion, we can guess what is the meaning of the moment of force. If the force acting on a material point is a measure of the increase in the speed of linear movement of the latter, then the moment of force is a measure of its rotational ability in relation to the system under consideration.
We give a clear example. Anyone opens the door, holding its handle. This can also be done by pushing the door in the handle area. Why does no one open it, pushing in the loop area? Very simple: the closer the force is applied to the hinges, the more difficult it is to open the door, and vice versa. The conclusion of the previous sentence follows from the formula for the moment (M = d * F), whence it can be seen that for M = const the values of d and F are inversely related.
Moment of Strength - Additive
In all the cases discussed above, there was only one acting force. In solving real problems, the situation is much more complicated. Usually several torsional forces act on systems that rotate or are in equilibrium, each of which creates its own moment. In this case, the solution of problems is reduced to finding the total moment of forces relative to the axis of rotation.
The total moment is found by the usual sum of the individual moments for each force, however, do not forget to use the correct sign for each of them.
Problem solving example
To consolidate the knowledge gained, it is proposed to solve the following problem: it is necessary to calculate the total moment of force for the system shown in the figure below.
We see that three forces act on the lever 7 m long (F1, F2, F3), and they have different points of application relative to the axis of rotation. Since the direction of the forces is perpendicular to the lever, there is no need to apply a vector expression for the torsion moment. You can calculate the total moment M using the scalar formula and without forgetting about setting the desired sign. Since the forces F1 and F3 tend to rotate the lever counterclockwise, and F2 clockwise, the torque will be positive for the former, and negative for the latter. We have: M = F1 * 7-F2 * 5 + F3 * 3 = 140-50 + 75 = 165 N * m. That is, the total moment is positive and is directed upwards (towards the reader).