📚 Centres of Mass | 质心
The centre of mass is one of the ideas that turns messy mechanics problems into simple ones. A body of any shape, however complicated, behaves in translational motion as if all of its mass were gathered at a single point, and when you take moments about a pivot or hang a rigid body from a nail, the weight acts through that same point. This article develops the definition from first principles, collects the standard results for uniform laminas, shows how to handle composite bodies and cut-out holes, and finishes with the toppling and equilibrium questions that appear again and again in A-Level papers.
质心是把复杂力学问题化繁为简的关键概念之一。无论物体形状多么复杂，在分析平动时都可以把全部质量看作集中在某一点上；当你对支点取力矩，或把刚体悬挂在钉子上时，重力也都通过这同一点作用。本文将从头建立质心的定义，汇总均匀薄片的标准结论，说明如何处理组合体和被挖去的孔洞，最后讨论在 A-Level 试题中反复出现的倾覆与平衡问题。
1. Defining the Centre of Mass | 质心的定义
The centre of mass of a system of particles is the unique point that represents the weighted average position of all the mass in the system. For the purpose of analysing translational motion, and for calculating the moment of the total weight, the whole mass may be treated as if it were concentrated at this single point.
质点系的质心是表示系统中全部质量加权平均位置的唯一一点。在分析平动以及计算总重力的力矩时，可以把整个质量看成集中在这一个点上。
r̄ = Σ mᵢ rᵢ / Σ mᵢ
In Cartesian coordinates this vector equation becomes two scalar equations, one for each direction. The denominator is the total mass M of the system, so the position vector of the centre of mass is simply the sum of each mass multiplied by its own position vector, divided by the total mass.
在直角坐标系中，这个矢量方程变成两个标量方程，每个方向各一个。分母是系统的总质量 M，因此质心的位置矢量就是每个质量乘以其自身位置矢量之和，再除以总质量。
x̄ = Σ mᵢ xᵢ / M ȳ = Σ mᵢ yᵢ / M
Because the expressions are weighted averages, the centre of mass always lies closer to the more massive particles, and its position does not depend on where the coordinate origin is placed. If the origin is moved, every xᵢ changes but x̄ does not. A useful consequence is that the centre of mass need not lie inside the material of the body at all: the centre of mass of a ring is at its centre, in the empty space, and the centre of mass of a boomerang can lie outside the wood.
因为这些表达式是加权平均，所以质心总是更靠近质量较大的质点，而且它的位置与坐标原点选在哪里无关。原点移动时，每个 xᵢ 都会改变，但 x̄ 不变。一个有用的推论是：质心完全不必位于物体材料内部。圆环的质心在其圆心处的空白处，回旋镖的质心也可以落在木料之外。
If a body has a plane of symmetry, the centre of mass lies in that plane. If it has two or three mutually perpendicular planes of symmetry, the centre of mass lies on their line or point of intersection, which is why the centre of mass of a uniform sphere or cube is simply its geometric centre.
如果物体有一个对称面，质心就位于该平面内。如果它有两个或三个互相垂直的对称面，质心就位于它们的交线或交点上，这就是均匀球体或立方体的质心正好是其几何中心的原因。
2. Discrete Particles in One Dimension | 一维离散质点系
When the particles lie on a straight line, choose the x-axis along that line. Only one coordinate is then needed, and the calculation reduces to a single weighted average. The safest method is to draw a small table of mass and position before doing any arithmetic.
当各质点位于同一直线上时，取该直线为 x 轴，此时只需一个坐标，计算简化为一次加权平均。最稳妥的做法是先列出质量与位置的小表格，再做算术运算。
x̄ = Σ mᵢ xᵢ / Σ mᵢ
in is 2 × 0 + 3 × 4 + 5 × 10 = 62 kg m. Dividing gives the position of the centre of mass.
例题：质量为 2 kg 位于 x = 0，3 kg 位于 x = 4 m，5 kg 位于 x = 10 m。总质量为 2 + 3 + 5 = 10 kg，关于原点的一阶矩之和为 2 × 0 + 3 × 4 + 5 × 10 = 62 kg m。相除即得质心位置。
x̄ = 62 / 10 = 6.2 m
nswer by taking moments about the centre of mass itself: the 5 kg of mass to the left of x̄ has its own centre of mass at (2 × 0 + 3 × 4) / 5 = 2.4 m, which is 3.8 m from x̄, while the 5 kg particle on the right is also 3.8 m from x̄. The two moments cancel, as they must.
可以用对质心本身取矩的方法检验答案：x̄ 左侧的 5 kg 质量，其自身质心在 (2 × 0 + 3 × 4) / 5 = 2.4 m 处，距 x̄ 为 3.8 m；右侧 5 kg 质点距 x̄ 同样为 3.8 m。两边的力矩相互抵消，这正是必然的结果。
Common slip: if a particle is placed to the left of the origin, its coordinate is negative and the negative sign must be carried through the calculation. Mixing units, for example grams in one term and kilogrammes in another, destroys the answer even when the method is correct.
常见失误：如果质点位于原点左侧，其坐标为负，负号必须贯穿整个计算。单位混用（例如一项用克、另一项用千克）会毁掉答案，即便方法本身正确。
3. Discrete Particles in Two Dimensions | 二维离散质点系
In two dimensions the same idea is applied twice, once for each coordinate direction. It is important to understand that this is not an approximation: the horizontal position of the centre of mass depends only on the horizontal coordinates and the masses, and the vertical position only on the vertical coordinates and the masses, so the two calculations can be done independently.
在二维情形下，同样的思想应用两次，每个坐标方向各一次。重要的是要明白这并非近似：质心的水平位置只取决于各质点的水平坐标与质量，竖直位置只取决于竖直坐标与质量，因此两个计算可以独立进行。
x̄ = Σ mᵢ xᵢ / M ȳ = Σ mᵢ yᵢ / M
6 + 3 × 0 = 12 kg m, and the y-moments give 1 × 0 + 2 × 0 + 3 × 6 = 18 kg m. Hence the centre of mass is at the point shown.
例题：质量 1 kg 位于 (0, 0)，2 kg 位于 (6, 0)，3 kg 位于 (0, 6)，长度单位为米。总质量为 6 kg。x 方向的一阶矩为 1 × 0 + 2 × 6 + 3 × 0 = 12 kg m，y 方向的一阶矩为 1 × 0 + 2 × 0 + 3 × 6 = 18 kg m，因此质心位于下式所示位置。
x̄ = 12 / 6 = 2.0 m ȳ = 18 / 6 = 3.0 m
Notice that the centre of mass (2.0, 3.0) coincides with none of the three particles. This is normal for a system of a few discrete masses; the centre of mass is a mathematical point associated with the distribution of mass, not with any particular lump of matter.
注意质心 (2.0, 3.0) 与三个质点中的任何一个都不重合。对于由少数几个离散质量组成的系统，这是正常现象；质心是与质量分布相联系的数学点，而非与任何一块具体物质相联系。
4. Centre of Mass and Centre of Gravity | 质心与重心
The centre of gravity of a body is the point through which the resultant weight of the body acts, whatever the orientation of the body. It is defined by taking moments of the weight of each element, so the position of the centre of gravity depends on the gravitational field as well as on the distribution of mass.
物体的重心是物体的合重力所通过的那一点，且与物体的取向无关。它是通过对各微元的重力取矩来定义的，因此重心的位置既取决于质量分布，也取决于重力场。
r̄gravity = Σ wᵢ rᵢ / Σ wᵢ = Σ mᵢ gᵢ rᵢ / Σ mᵢ gᵢ
If the gravitational field strength g is the same at every point of the body, then g cancels top and bottom and the centre of gravity coincides with the centre of mass. In all A-Level problems this condition holds, so the two terms may be used interchangeably. They differ only in extreme situations, such as a very tall structure for which g varies measurably with height, or a body so large that the direction of g changes across it.
如果物体各点处的重力场强度 g 都相同，则 g 在分子分母中约去，重心与质心重合。在所有 A-Level 题目中这一条件都成立，因此两个术语可以互换使用。只有在极端情形下二者才不同，例如极高的建筑中 g 随高度有明显变化，或者物体大到各处 g 的方向都不同。
Physically the distinction matters in another way too. The centre of mass is also the point that responds to a resultant external force in Newton's second law for a rigid body, and it is the point that moves at constant velocity when no external force acts. When a spanner spins through the air after being thrown, it is the centre of mass that follows a smooth parabola while the rest of the tool rotates about it.
物理上二者还有另一层区别。质心也是刚体在牛顿第二定律中响应合外力的那个点，还是无外力作用时保持匀速直线运动的那个点。扳手被抛出手后在空气中旋转时，正是质心沿平滑的抛物线运动，而扳手的其余部分绕它转动。
5. Standard Results for Uniform Laminas | 均匀薄片的标准结果
A lamina is a flat sheet of negligible thickness, and a uniform lamina has constant mass per unit area. For such bodies the standard results below may be quoted directly in an examination; they save a great deal of integration. In every case the position is measured from the stated reference point.
薄片是厚度可忽略的平板，均匀薄片指单位面积质量处处相等。对于这类物体，下列标准结论在考试中可以直接引用，可省去大量积分工作。每种情形下的位置都从所给参考点量起。
| Body | 物体 | Centre of mass | 质心位置 |
|---|---|
| Uniform rod | 均匀细棒 | Midpoint | 中点 |
| Rectangular or parallelogram lamina | 矩形或平行四边形薄片 | Intersection of diagonals | 对角线交点 |
| Triangular lamina | 三角形薄片 | Intersection
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