Center of gravity

Japanese: 重心 - じゅうしん

Mathematical terminology

Center of gravity of a triangle <br/>The three medians of a triangle intersect at one point. That point is called the center of gravity of the triangle. The three medians are each divided internally by the center of gravity in a ratio of 2 to 1. The center of gravity, orthocenter, and circumcenter of a triangle are on a straight line. The center of gravity of a triangle is also the center of gravity in the mechanical sense when the three vertices are considered to be point masses of equal mass. The center of gravity of a triangle is also the mechanical center of gravity when the triangle is considered to be a plate of uniform density. However, the center of gravity of a triangle does not generally coincide with the center of gravity in the physical sense when the perimeter of the triangle is considered to be a collection of point masses of equal mass.

Center of gravity of a tetrahedron <br/>The four lines connecting each vertex of a tetrahedron to the center of gravity of the triangle that is the opposing face intersect at one point. This point is called the center of gravity of the tetrahedron. At this point, the line segments connecting the vertices to the centers of gravity of the opposing faces are divided internally in a ratio of 3 to 1. The center of gravity of a tetrahedron is the mechanical center of gravity when the four vertices are considered to be mass points of equal mass, and is also the mechanical center of gravity when the tetrahedron is considered to be a rigid body with uniform density.

Center of gravity of n points The center of gravity of n points on a plane or in space can be determined inductively as follows: When the center of gravity of k points has been determined, the point that divides the line segment connecting the k +1th point and the center of gravity of the k points internally in a ratio of k to 1 is the center of gravity of the k +1 points. The center of gravity of 2 points is the midpoint, and then the centers of gravity of 3 points, 4 points, ... are determined successively. This is also the mechanical center of gravity when the n points are considered as mass points of equal mass.

[Toshio Shibata]

Physics terminology

The position of the center of gravity is determined by the average position of the mass points that make up an object, in the sense described below. For this average, we take the weighted average, with the mass of each mass point as a burden. The sum of the moments of gravity acting on each mass point of an object about the center of gravity is zero. Therefore, we can think of the gravitational forces acting on each mass point as concentrating at the center of gravity. It is because of this property that it is called the center of gravity. When an object moves, the product of the object's total mass and the acceleration of the center of gravity is equal to the resultant force of external forces. In general, the movement of an object can be considered to consist of the movement of the center of gravity and the relative movement of each mass point of the object with respect to the center of gravity.

When discussing the collision of two particles, a coordinate system fixed in the laboratory (laboratory system) is sometimes used, but a coordinate system in which the center of gravity of the two particles is the coordinate origin (center of gravity system) is also used. When using the center of gravity system, the momentum (the product of mass and velocity) of the two particles is always equal in magnitude and direction, but points in opposite directions.

[Yoshiro Kainuma]

Center of gravity in mathematics
©Shogakukan ">

Center of gravity in mathematics

Center of gravity in physics
No matter where an object is supported, at O, O, or O, the center of gravity G is directly below it. Gravity and resistance are balanced, and the object comes to rest .

Center of gravity in physics

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

数学用語

三角形の重心
三角形の三つの中線は1点で交わる。その点を三角形の重心という。三つの中線は重心でそれぞれ2対1の比に内分される。三角形の重心、垂心、外心は1直線上にある。三角形の重心は三つの頂点を等質量の質点と考えた場合の力学的な意味での重心でもある。また、三角形の重心は、三角形を密度が均一な板と考えたときの力学的重心でもある。しかし、三角形の重心は、三角形の周を等質量の質点の集まりと考えたときの物理的な意味での重心とは一般には一致しない。

四面体の重心
四面体の各頂点と相対する面である三角形の重心とを結ぶ四つの直線は1点に交わる。この点を四面体の重心という。この点で頂点と対面の重心とを結ぶ線分は3対1の比に内分される。四面体の重心は四つの頂点を等質量の質点と考えたときの力学的重心であり、また、四面体を密度が均一の剛体と考えたときの力学的重心でもある。

n個の点の重心
平面上あるいは空間にあるn個の点の重心とは次のように帰納的に定められる。k個の点の重心が定まったとき、k＋1番目の点とk個の点の重心とを結ぶ線分をk対1に内分する点がk＋1個の点の重心である。2点の重心はその中点であることから始めて逐次3点、4点、……の重心が定められる。これはn個の点を等質量の質点と考えたときの力学的重心でもある。

［柴田敏男］

物理学用語

物体を構成する質点の、以下に述べるような意味での平均的な位置が重心の位置を与える。この平均としては、各質点の質量を重荷とする重荷平均をとる。物体の各質点に働く重力の重心の周りのモーメントの和はゼロである。したがって、各質点に働く重力が重心に集まって働いていると考えてもよい。重心とよぶのは、この性質のためである。物体が運動するとき、物体の全質量と重心の加速度との積は、外力の合力に等しい。一般に物体の運動は、重心の運動と、物体の各質点が重心に対して行う相対運動とからなる、とみなすことができる。

　2個の粒子の衝突を論ずるにあたって、実験室に固定した座標系（実験室系）を用いることもあるが、二つの粒子の重心を座標原点とするような座標系（重心系）を用いることもある。重心系を用いると、二つの粒子の運動量（質量と速度の積）はつねにその大きさと方向が等しく、向きが逆向きになる。

［飼沼芳郎］

数学における重心

物理学における重心

出典　小学館　日本大百科全書(ニッポニカ)日本大百科全書(ニッポニカ)について　情報 | 凡例

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