質心

質量中心簡稱質心，指物質系統上被認為質量集中於此的一個假想點，质心的位置矢量是质点组中各个质点的位置矢量$r_i$根据其对应质量加权平均之后的平均矢量。質心不一定要在有重力場的系統中才會有意義，而重心則否。值得注意的是，除非重力場是均勻的，否則同一物質系統的質心與重心通常不在同一假想點上。对于密度均匀、形状对称分布的物体，其质心位于其几何中心处。

在一個N維空間中的質量中心，座標系計算公式為：

$r_m = {\sum {m_i} {r_i} \over \sum m_i}$

其中：

• $r$表示某一坐標軸
• $m_i$表示物質系統中，某i質點的質量
• $r_i$表示物質系統中，某i質點的座標。

動畫展演

雙星互繞時它們的質心位置：

兩顆星體質量差不多，例如休神星。	兩顆星體質量不同，例如冥王星與冥衛一。	兩顆星體質量有很大的不同，例如地球與月球。	兩顆星體質量有極大的不同，例如太陽與地球。
兩顆星體以橢圓軌道互繞，此狀況通常稱為聯星。

重心

重力作用的平均位置，定義各質點位置相對於質心乘上各質點的重力所產生的合力矩為零。

均勻重力場

在地球表面附近，重力場可被認定為均勻且平行向下，所以重心會等同於質心。 在物理學，使用「質心」來表示質量分布的好處，從以合力來考慮連續體的重力可以看出。Consider a body of volume V with density ρ(r) at each point r in the volume. In a parallel gravity field the force f at each point r is given by,

$\mathbf{f}(\mathbf{r}) = -dm\, g\vec{k}= -\rho(\mathbf{r})dV\,g\vec{k},$

其中dm是在點r的質量，g 是重力加速度，以及k 是定義垂直方向的單位向量。 Choose a reference point R in the volume and compute the resultant force and torque at this point,

$\mathbf{F} = \int_V \mathbf{f}(\mathbf{r}) = \int_V\rho(\mathbf{r})dV( -g\vec{k}) = -Mg\vec{k},$

and

$\mathbf{T} = \int_V (\mathbf{r}-\mathbf{R})\times \mathbf{f}(\mathbf{r}) = \int_V (\mathbf{r}-\mathbf{R})\times (-g\rho(\mathbf{r})dV\vec{k} )= \left(\int_V \rho(\mathbf{r}) (\mathbf{r}-\mathbf{R})dV \right)\times (-g\vec{k}) .$

If the reference point R is chosen so that it is the center of mass, then

$\int_V \rho(\mathbf{r}) (\mathbf{r}-\mathbf{R})dV =0,$

which means the resultant torque T=0. Because the resultant torque is zero the body will move as though it is a particle with its mass concentrated at the center of mass.

非均勻重力場

常用於天體力學

平行場

Some of the inhomogeneity in a gravitational field may be modeled by a variable but parallel field: g(r) = g(r)n, where n is some constant unit vector. Although a non-uniform gravitational field cannot be exactly parallel, this approximation can be valid if the body is sufficiently small. The center of gravity may then be defined as a certain weighted average of the locations of the particles composing the body. Whereas the center of mass averages over the mass of each particle, the center of gravity averages over the weight of each particle:

$\mathbf{r}_\mathrm{cg} = \frac{1}{W} \sum_i w_i \mathbf{r}_i,$

where $\mathbf{w}_\mathrm{i}$ is the (scalar) weight of the Template:Mvarth particle and Template:Mvar is the (scalar) total weight of all the particles. This equation always has a unique solution, and in the parallel-field approximation, it is compatible with the torque requirement.

A common illustration concerns the Moon in the field of the Earth. Using the weighted-average definition, the Moon has a center of gravity that is lower (closer to the Earth) than its center of mass, because its lower portion is more strongly influenced by the Earth's gravity.

球形場

If the external gravitational field is spherically symmetric, then it is equivalent to the field of a point mass Template:Mvar at the center of symmetry r. In this case, the center of gravity can be defined as the point at which the total force on the body is given by Newton's Law:

$\frac {GmM (\mathbf{r}_\mathrm{cg} - \mathbf{r})} {|\mathbf{r}_\mathrm{cg} - \mathbf{r}|^3} = \mathbf{F},$

where Template:Mvar is the gravitational constant and Template:Mvar is the mass of the body. As long as the total force is nonzero, this equation has a unique solution, and it satisfies the torque requirement. A convenient feature of this definition is that if the body is itself spherically symmetric, then r_cg lies at its center of mass. In general, as the distance between r and the body increases, the center of gravity approaches the center of mass.

Another way to view this definition is to consider the gravitational field of the body; then r_cg is the apparent source of gravitational attraction for an observer located at r. For this reason, r_cg is sometimes referred to as the center of gravity of Template:Mvar relative to the point r.

參見

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