Focal point

This article deals with the principle of centroid from mechanics. To determine the center of gravity of homogeneous bodies, surfaces or lines see Geometric Center of Gravity

The center of gravity theorem (also the center of mass theorem) is a proposition from mechanics . It says that the center of mass (center of gravity) of a system , which consists of any number of individual point masses , moves as if the masses of all individual mass points were united in it and all forces that act on the mass points from outside at their respective positions work, taken together would only work on him. The principle of centroid also applies to spatially extended bodies, since these can be thought of as being composed of mass points.

Internal forces , d. H. Forces between the individual mass points of the system, however, have no effect on the movement of the center of gravity. Is the sum of all individual masses, and the vector sum of the externally acting on the mass points forces then applies to the acceleration of gravity , the second Newtonian law : ${\displaystyle m_{s}}$ ${\displaystyle {\vec {F}}_{\mathrm {ges} }^{\mathrm {ext} }}$ ${\displaystyle {\vec {a}}_{s}}$

${\displaystyle m_{s}\;{\vec {a}}_{s}={\vec {F}}_{\mathrm {ges} }^{\mathrm {ext} }}$.

This can be imagined as if the total mass of a system was united in the center of gravity and all external forces acted on it together, regardless of their actual points of application. The movement of the center of gravity is thus neither influenced by internal forces nor by external force pairs (the movement of the individual points does ).

If there are no external forces at all, the system is said to be (mechanically) closed. For a closed system, regardless of which forces the individual bodies belonging to the system mutually exert on one another, the center of gravity of the system moves uniformly in a straight line. The center of gravity theorem thus corresponds to the inertia law extended to systems of several point masses ( first Newton's law ). It is then equivalent to the law of conservation of momentum . The center of gravity moves uniformly in a straight line even when external forces act, but these cancel each other out in the resulting total force . ${\displaystyle {\vec {F}}_{\mathrm {ges} }^{\mathrm {ext} }=0}$

Examples

• Impinges on a flat surface, a body elastically to a different, equally heavy body rested before, both thereafter move so that its center of gravity without changing its rectilinear movement continues (conservation of momentum).
• If forces act on an extended body at rest at different points whose vector sum is zero, the body's center of gravity remains at rest. However, the forces can exert a torque and thus cause a rotary movement.
• A spacecraft can only accelerate in space using the recoil principle . If a rocket was at rest in a certain frame of reference before the engines were fired, the common center of gravity of the recoil mass and rocket mass remains at rest afterwards. See recoil drive .

Derivation

If the individual mass points of the system are numbered consecutively, then the equation of motion applies to each of them ${\displaystyle i=1,2,\dots ,n}$

${\displaystyle m_{i}{\vec {a}}_{i}={\vec {F}}_{i}^{\mathrm {ext} }+\sum _{j=1 \atop j\neq i}^{n}{\vec {F}}_{ji}^{\mathrm {int} }}$.

Therein is the force that acts on the mass point from outside and the inner force that the mass point exerts on it. According to Newton's third law, the following applies . ${\displaystyle {\vec {F}}_{i}^{\mathrm {ext} }}$ ${\displaystyle i}$ ${\displaystyle {\vec {F}}_{ji}^{\mathrm {int} }}$ ${\displaystyle j}$ ${\displaystyle {\vec {F}}_{ji}^{\mathrm {int} }=-{\vec {F}}_{ij}^{\mathrm {int} }}$

The sum of all the individual equations of motion is:

${\displaystyle \sum _{i=1}^{n}m_{i}{\vec {a}}_{i}=\sum _{i=1}^{n}{\vec {F}}_{i}^{\mathrm {ext} }+\sum _{i,j=1 \atop j\neq i}^{n}{\vec {F}}_{ji}^{\mathrm {int} }}$.

This is already the equation of motion of the center of gravity given above, because

${\displaystyle \sum _{i}m_{i}\,{\vec {a}}_{i}=m_{s}\,{\vec {a}}_{s}}$

and

${\displaystyle \sum _{i,j \atop i\neq j}{\vec {F}}_{ji}^{int}=0}$.

The first of these equations results directly from the equation, which defines the center of gravity through the positions of the individual mass points, by deriving it twice according to time : ${\displaystyle {\vec {r}}_{s}}$ ${\displaystyle {\vec {r}}_{i}}$

${\displaystyle \sum _{i}m_{i}\,{\vec {r}}_{i}=m_{s}\,{\vec {r}}_{s}\quad \Longrightarrow \quad \sum _{i}m_{i}\,{\vec {v}}_{i}=m_{s}\,{\vec {v}}_{s}\quad \Longrightarrow \quad \sum _{i}m_{i}\,{\vec {a}}_{i}=m_{s}\,{\vec {a}}_{s}}$.

The second equation applies because in the double sum of the internal forces for each summand the summand with the interchanged indices also appears, which together results in zero. ${\displaystyle {\vec {F}}_{ji}^{\mathrm {int} }}$

literature

• Nolting: Basic course: Theoretical Physics, 2, Analytical Mechanics , Verlag Zimmermann-Neufang, 3rd edition 1993, ISBN 3-922410-21-9 .

Individual evidence

1. ^ Nolting: Classical Mechanics , Springer, 10th edition, 2013, p. 268.
   Fließbach: Theoretische Physik I - Mechanik , Springer, 7th edition, 2015, p. 26.
   General and specific for example of the wheel : Jürgen Dankert, Helga Dankert: Technische Mechanik , Springer, 7th edition, 2013, p. 570.
2. ↑ Gerthsen: Physik , Springer, 24th edition, 2015, p. 25.
3. ↑ Henz, Langehanke: Paths through Theoretical Mechanics 1 , Springer, 2016, p. 141.
4. ↑ Straumann: Theoretische Mechanik , Springer, 2nd edition, 2015, p. 22.