Barycenter

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In physics, the barycenter of a material body coincides with its center of mass when the body is homogeneous (uniform density) or when the distribution of matter in the body has certain properties, such as like symmetry. It is distinguished from the centroid in geometry, where the centroid of a surface contained in a plane geometric figure is a point such that any line that passes through it divides said segment into two parts with equal momentum with respect to said segment. straight.

If one of the two orbiting bodies is much more massive than the other and the bodies are relatively close to each other, the barycenter will usually lie inside the more massive object. In this case, instead of the two bodies appearing to orbit a point between them, the less massive body will appear to orbit the more massive body, while the more massive body can be observed to wobble slightly. This is the case for the Earth-Moon system, whose barycenter is on average 4,671 kilometers (2,902.4 mi) from the center of the Earth, which is 75% of the Earth's radius of 6,378 kilometers (3,963.1 mi).. When the two bodies are of similar masses, the barycenter will usually lie at a point between them and both bodies will orbit around it. This is the case for Pluto and Charon, one of Pluto's natural satellites, as well as many binary asteroids and binary stars. When the less massive object is far away, the center of gravity can be located outside of the more massive object. This is the case of Jupiter and the Sun; despite the fact that the Sun is a thousand times more massive than Jupiter, its barycenter is slightly away from the Sun due to the relatively large distance between them.

Two-body problem

Main article: Problem of the two bodies

The barycenter is one of the foci of the elliptical orbit of each body in question. This is an important concept in the fields of astronomy and astrophysics. If a is the semimajor axis of the system, r₁ is the semimajor axis of the primary's orbit around the barycenter, and r₂ = a − r₁ is the semimajor axis of the orbit of the secondary. When the barycenter is located within the more massive body, that body will appear to "wobble"; instead of following a perceptible orbit. In a simple two-body case, the distance from the center of the primary to the centroid, r₁, is given by:

{displaystyle r_{1}=acdot {frac {m_{2}}{m_{1}+m_{2}}}={frac {a}{1+{frac {m_{1}}{m_{2}}}}}}

where:

r₁ is the distance from the center of the body 1 to the baricentro

a is the distance between the centers of the two bodies

m₁ and m₂ are the masses of the two bodies.

Calculation of the center of gravity

Let A₁, …, A_n n points, and m₁, …, m_n n numbers (m as mass). Then the centroid of the (A_i, m_i) is the point G defined as follows:

{displaystyle {overrightarrow {OG,}}={frac {sum {m_{i}{overrightarrow {O!A_{i}}}}}{sum {m_{i}}}}={frac {m_{1}{overrightarrow {O!A_{1}}}+...+m_{n}{overrightarrow {O!A_{n}}}}{m_{1}+...+m_{n}}},quad {mbox{con}}quad sum {m_{i}}neq 0,.}

This definition does not depend on point O, which can be any point. If the origin of the plane or space is taken, the coordinates of the barycenter are obtained as a weighted average by the m_i of the coordinates of the points A_i:

{displaystyle x_{G}={frac {sum {m_{i}x_{i}}}{sum {m_{i}}}}={frac {m_{1}x_{1}+...+m_{n}x_{n}}{m_{1}+...+m_{n}}},.}

The previous definition is equivalent to the following formula, more practical for vector calculus, since it does without fractions (it is obtained by taking O = G):

{displaystyle sum _{i=1}^{n}{m_{i}{overrightarrow {G!A_{i}}}}={vec {0}}quad {mbox{o bien}}quad m_{1}{overrightarrow {G!A_{1}}}+...+m_{n}{overrightarrow {G!A_{n}}}={vec {0}},.}

Related concepts

An isobaricenter (iso: same) is a barycenter with all masses equal to each other; it is usual in such a case to take them equal to 1. If the masses are not specified, the barycenter defaults to the isobaricenter.

The barycenter coincides with the physical concept of the center of mass of a material body as long as the body is homogeneous. The coincidence of the center of gravity and the center of mass allows to locate the former in a simple way. If we take a surface cut out of cardboard and hold it vertically from any of its points, it will rotate until the center of gravity (barycenter) is located exactly vertical to the holding point; Marking said vertical on the cardboard and repeating the process holding from a second point, we will find the center of gravity at the point of intersection.

Physical concept

In Physics, the centroid, the center of gravity and the center of mass can, under certain circumstances, coincide with each other, although they designate different concepts. The centroid is a purely geometric concept that depends on the shape of the system; the center of mass depends on the distribution of matter, while the center of gravity depends on the gravitational field.

Let us consider a material body:

For the center of the body to match the mass center, the body must have uniform density or a distribution of matter that presents certain properties, such as symmetry.
For a body mass center to match the center of gravity, the body must be under the influence of a uniform gravitational field.

A concave figure can have its centroid at a point outside the figure itself.

Algebraic Properties

The algebraic properties of the centroid are:

Homogeneity: the baricentro does not change if all the masses are multiplied by the same factor k.

Formalmente: bar {(A)₁, m₁..., (A)_n, m_n} = bar {A₁, km₁..., (A)_n, km_n}.

Association: the baricentro can be calculated by regrouping points, i.e. by introducing partial bariacentres.

For example, if D = bar {(A, a), (B, b)} (with a + b) then bar {(A, a), (B, b), (C, c)} = bar {(D,a + b(C, c)} (a + b + c ì 0)

Examples

Example 1

Given the center of mass of a triangle ABC. Let I = bar { (B, 1), (C, 1)}, then G = bar {(A, 1), (B, 1), (C, 1)} = bar {(A, 1), (I, 2)}, which means that G is in the segment [A,I], one-third of the way from I.

Example 2

Baricentro of a median (see example 2)

The barycenter can be defined in mathematics with negative coefficients. Since there are no negative masses, what physical meaning can be attached to these calculations? Here is a very simple example: a "half moon" as shown in the figure, made up of a yellow circle, with center B, in which we have eliminated another circle with a radius twice as small, with center A. He wonders about the center of mass of that croissant.

The calculation is very simplified if we consider the "half moon" like a juxtaposition of two disks, a large one with positive mass, and a small one with negative mass. The masses are proportional to the areas (uniform density), which would give a mass of 4 for the first disk, and -1 for the second. So G = bar {(A, -1), (B, 4)}.

Geometric calculation of the centroid

The geometric calculation (with ruler and compass) of the centroid of a polygon (regular or irregular), with n vertices, can be done as follows:

The polygon is decomposed into disjoint triangles and quadrilaterals (that do not have vertices in common). The centroids of these triangles and quadrilaterals are calculated, and the corresponding polygon is formed.

It can be proved that this algorithm has logarithmic order.

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