Moment set

This article deals with the moment theorem from statics for the addition of moments. In the context of the moment with the spin (angular momentum / angular momentum) see conservation of angular momentum .

The moment theorem or Varignon's theorem is a law from mechanics that is used in particular in statics . In French- and English-speaking countries it is partly named after Pierre de Varignon (1654–1722). The moment theorem is:

"The sum of the moments of the forces of a spatial force system is equal to the moment of the resultant of this force system for the same reference point."

Or expressed mathematically:

${\displaystyle F_{R}\cdot a_{R}=F_{1}\cdot a_{1}+F_{2}\cdot a_{2}+\cdots +F_{n}\cdot a_{n}}$

With

${\displaystyle F_{R}}$	:	resulting power
${\displaystyle a_{R}}$	:	Distance between the line of action of the resulting force and the reference point
${\displaystyle F_{1},F_{2},\ldots ,F_{n}}$	:	individual forces
${\displaystyle a_{1},a_{2},\ldots ,a_{n}}$	:	Distances of the lines of action of the individual forces from the reference point

It must be noted that the resulting force is not the vector sum of the individual forces, but results from the individual forces through geometric considerations (for example with the force parallelogram ). This is related to the fact that the vector sum of several forces usually cannot be assigned a point of application and therefore no distance is known. A couple of forces can not be simplified to a resulting force, but only replaced by its moment.

The magnitude of the moment of the resultant depends on the reference point.

Applications

Moment equilibrium

An important application of the theorem of moments is found when establishing the equilibrium of moments . The set of moments then allows forces that are inclined to the coordinate axes to be split into two forces that are perpendicular to one another and each parallel to a coordinate axis, and then to add the moments of the force components. In technical mechanics it is common to add the moments of the individual force components instead of the moments of the forces. This is mathematically easier because the individual force components either do not appear in the equation of the moment equilibrium because their lever arm is zero, or they are fully included in the equation because they are perpendicular to the lever arm. On the other hand, there is the effort to determine the individual force components, but this usually has to be done anyway to check the balance of forces , so that the effort does not increase.

Resulting power

The moment theorem can also be used to reduce a force system that consists of numerous forces to a single resulting force (without a resulting moment). For this purpose, first of all, the moment of all individual forces is formed with regard to a single, but arbitrary point and these moments are then added. The forces are then shifted to the reference point and can be combined there to form a resulting force. If the lines of action of the individual forces do not intersect at a single point, this is not possible without adding offset moments to the system. The system of the resulting force (at the reference point) and the total moment is called the Dyname . The amount and direction of the resulting force are therefore fixed, but not its point of application. The resultant has no moment effect with regard to the chosen reference point, but the individual forces do. The set of moments can then be used to determine the distance between the force and the reference point.

Focus

The calculation of centers of mass is actually a special case of the calculation of resulting forces. The individual forces are the weight forces of the individual masses.

Individual evidence

1. ↑ Hibbeler: Technische Mechanik - Statik , Pearson, 12th edition, 2012, p. 149.
2. ↑ Hans Albert Richard, Manuela Sander: Technische Mechanik - Statik , Springer, 5th edition, 2016, 157.Similar
   also in Böge: Technische Mechanik , Springer, 31st edition, 2015, p. 38.
3. ↑ Hibbeler: Technische Mechanik - Statik , Pearson, 12th edition, 2012, p. 149.
4. ^ Hans Albert Richard, Manuela Sander: Technische Mechanik - Statik , Springer, 5th edition, 2016, 172.
5. ↑ Böge: Technische Mechanik , Springer, 31st edition, 2015, pp. 39, 77 f.
6. ^ Hans Albert Richard, Manuela Sander: Technische Mechanik - Statik , Springer, 5th edition, 2016, 172.
7. ↑ Böge: Technische Mechanik , Springer, 31st edition, 2015, pp. 39, 77 f.