Centres of Mass | 质心

📚 Centres of Mass | 质心

In mechanics, the centre of mass of a body is the point at which the entire mass may be considered to be concentrated for the purpose of analysing translational motion. When you take moments about a pivot, or when a rigid body is suspended, the weight acts through this single point.

在力学中，质心是可将物体全部质量视作集中于该点的位置，用于分析平动。当你对支点取力矩时，或刚体悬挂时，重力都通过这个点作用。

1. Defining 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 mass in the system. For translational motion and for calculating moments of the total weight, the whole mass can be treated as if it were located at this point.

质点系的质心是表示系统中所有质量加权平均位置的唯一一点。对于平动以及计算总重力的力矩，可以把整个质量看成集中在这个点上。

r̄ = Σ mᵢ rᵢ / Σ mᵢ

In Cartesian coordinates this becomes x̄ = Σ mᵢ xᵢ / Σ mᵢ and ȳ = Σ mᵢ yᵢ / Σ mᵢ. The denominator is the total mass M.

在直角坐标中，上式变为 x̄ = Σ mᵢ xᵢ / Σ mᵢ 和 ȳ = Σ mᵢ yᵢ / Σ mᵢ。分母为总质量 M。

Because the expressions are weighted averages, the centre of mass lies closer to the more massive particles and does not depend on the coordinate origin.

因为这些表达式是加权平均，所以质心更靠近质量较大的质点，并且与坐标原点无关。

2. Discrete Particles in One Dimension | 一维离散质点系

When particles lie on a straight line, choose the x-axis along that line. Then only the x-coordinate is needed:

当质点在同一直线上时，选择该直线为 x 轴。这时只需要计算 x 坐标：

x̄ = Σ mᵢ xᵢ / Σ mᵢ

Example: masses 2 kg at x = 0, 3 kg at x = 4 m, and 5 kg at x = 10 m have a total mass of 10 kg and

例：质量分别为 2 kg（x = 0）、3 kg（x = 4 m）和 5 kg（x = 10 m），总质量为 10 kg，并且

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