📚 Centre of Gravity | 重心
The centre of gravity (CG) of an object is the point through which the entire weight of the object appears to act, no matter how the object is oriented. In mechanics, it is an extremely useful idea: instead of treating weight as millions of tiny gravitational pulls on every particle, we can replace the whole weight by a single downward force acting at the CG. This allows us to analyse equilibrium, moments, and stability much more simply.
物体的重心（centre of gravity, CG）是指：无论物体如何放置，物体的全部重力看起来都从这一点通过并作用。在力学中，这是一个非常有用的概念：我们不必把重力看成作用在成千上万个微粒上的微小引力，而是可以用一个作用在重心上的竖直向下的合力来代替整个重力。这样我们就能更简单地分析平衡、力矩和稳定性。
1. Definition of Centre of Gravity | 重心的定义
The centre of gravity is defined as the point at which the resultant gravitational force on a body acts. For a body in a uniform gravitational field, the weight can be represented by a single force W = mg acting vertically downward through this point. The position of the CG depends on the distribution of weight within the body.
重心被定义为物体所受合重力作用的那一点。在均匀重力场中，物体的重力可以用一个通过该点竖直向下的合力 W = mg 来表示。重心的位置取决于物体内部重量的分布。
For the purpose of calculating moments, the total weight of a body can be assumed to act at its centre of gravity. This means that if a body is supported at its CG, it will balance perfectly in any orientation because the weight produces no turning effect about that point.
在计算力矩时，可以假定物体的总重力作用在重心上。这意味着，如果一个物体在重心处被支撑，它在任何方向上都能完全平衡，因为重力对该点不产生转动效果。
W = mg
Moment about CG = W × 0 = 0
2. Centre of Gravity vs Centre of Mass | 重心与质心的区别
The centre of mass (CM) is the average position of all the mass in a body, while the centre of gravity is the average position of the gravitational force acting on that mass. In a uniform gravitational field, these two points coincide exactly. This is the situation assumed in almost all CIE A-Level Physics problems.
质心（centre of mass, CM）是物体中所有质量的平均位置，而重心是作用在该质量上的重力的平均位置。在均匀重力场中，这两个点完全重合。几乎所有 CIE A-Level 物理题都采用这一假设。
If the gravitational field is non-uniform, the pull per kilogram changes from one part of the body to another, so the CG may shift away from the CM. However, for objects near the Earth’s surface, the gravitational field can be treated as uniform, so no distinction is needed.
如果重力场不均匀，每千克质量受到的重力会因位置不同而变化，因此重心可能会偏离质心。但是，对于地球表面附近的物体，重力场可视为均匀场，因此不需要区分二者。
A practical consequence is that for regular, uniform objects, the CG lies at the geometric centre. For irregular or composite bodies, the CG must be found by experiment or by using moments.
一个实际结论是：对于规则且密度均匀的物体，重心位于几何中心。对于不规则或由多个部分组成的物体，则必须通过实验或力矩计算来寻找重心。
3. Centre of Gravity of Regular Shapes | 规则形状的重心
For a uniform body with a simple geometric shape, the centre of gravity is located at the geometric centre or centroid. The centroid of a shape is the point where its area or volume would balance in all directions.
对于密度均匀且形状规则的物体，重心位于几何中心或形心。形心是指该形状的面积或体积在所有方向上都能平衡的那个点。
Common examples are listed below. A uniform sphere has its CG at its centre. A uniform cylinder has its CG at the midpoint of its axis. A uniform cuboid has its CG at the intersection of its body diagonals. A uniform triangular lamina has its CG at the intersection of its medians, one-third of the way up from the base.
常见例子如下：均匀球体的重心在球心；均匀圆柱体的重心在轴线的中点；均匀长方体的重心在体对角线的交点；均匀三角形薄片的重心在中线的交点，位于从底边向上三分之一处。
| Shape | Position of CG |
| Uniform sphere | Centre of sphere |
| Uniform cylinder | Midpoint of the axis |
| Uniform cuboid | Intersection of body diagonals |
| Uniform triangular lamina | Intersection of medians, h/3 above base |
For a composite object made of two or more simple parts, the CG can be found by taking moments of the weights of the individual parts about a chosen origin. The general formula for the x-coordinate is:
对于由两个或更多简单部分组成的复合物体，可以通过对各部分的重力围绕选定原点取力矩来求重心。x 坐标的通用公式为：
x = (W₁x₁ + W₂x₂ + …) / (W₁ + W₂ + …)
Here W₁, W₂ are the weights of the parts, and x₁, x₂ are the x-coordinates of their individual centres of gravity. A similar equation applies to the y-coordinate.
这里 W₁、W₂ 是各部分的重量，x₁、x₂ 是各部分各自重心的 x 坐标。y 坐标也有类似公式。
4. Locating Centre of Gravity Experimentally | 实验确定重心
For a thin, irregular lamina, the centre of gravity can be located using a simple suspension method. The lamina is hung freely from a small hole or pivot. When it comes to rest, the CG must lie vertically below the pivot, because the weight and the tension in the suspension string must produce zero total moment.
对于薄的、形状不规则的薄片，可以用简单的悬挂法寻找重心。将薄片从小孔或支点自由悬挂。当它静止时，重心一定位于支点的正下方，因为重力与悬挂绳的张力必须产生零合力矩。
A plumb line is drawn from the pivot downward on the lamina. The lamina is then suspended from a different pivot, and a second plumb line is drawn. The point where the two lines intersect is the centre of gravity. Repeating with a third pivot gives a check.
从支点向下在薄片上画出铅垂线。然后将薄片从另一个不同的支点悬挂，再画第二条铅垂线。两条线的交点就是重心。用第三个支点重复实验可以检验结果。
This method relies on the fact that, when a body is suspended at equilibrium, the line of action of the weight passes through the pivot. Therefore any plumb line drawn from the pivot contains the CG.
该方法依据的事实是：当物体悬挂并处于平衡状态时，重力的作用线通过支点。因此从支点画出的任何铅垂线都包含重心。
5. Equilibrium and Centre of Gravity | 平衡与重心
An object is in equilibrium when the resultant force on it is zero and the resultant moment about any point is zero. This means the object is not accelerating linearly and not rotating. The position of the centre of gravity is central to checking the moment condition.
当物体所受合力为零，且关于任意点的合力矩也为零时，物体处于平衡状态。这意味着物体没有线性加速度，也没有转动。重心的位置对检验力矩条件至关重要。
For an object resting on a horizontal surface, the weight acts downward from the CG. The surface exerts an upward normal reaction force. If the normal reaction is distributed over the base, we can replace it by a single resultant force acting at a point directly below the CG when the object is symmetrical.
对于静止在水平表面上的物体，重力从重心竖直向下作用。表面对物体施加向上的法向反作用力。如果法向反作用力分布在底面上，当物体对称时，我们可以用一个作用在重心正下方某点的合力来代替它。
If the vertical line through the CG falls within the base, the object can be in stable equilibrium. If this line falls outside the base, the weight produces a net turning effect about the edge of the base, so the object will topple.
如果通过重心的竖直线落在底面范围内，物体可以处于稳定平衡。如果这条线落在底面之外，重力就会对底面边缘产生净转动效果，因此物体会倾倒。
ΣF = 0 and ΣM = 0
For stability: line of action of W must pass through the base
6. Stability: Stable, Unstable, Neutral | 稳定性：稳定、不稳定、随遇
Stability describes how an object behaves when it is given a small displacement from equilibrium. There are three main types: stable equilibrium, unstable equilibrium, and neutral equilibrium. The change in height of the centre of gravity is the key indicator.
稳定性描述物体在受到微小扰动离开平衡位置后的表现。主要有三种类型：稳定平衡、不稳定平衡和随遇平衡。重心高度的变化是关键指标。
In stable equilibrium, a small displacement raises the centre of gravity. The weight then produces a restoring moment that returns the object to its original position. A cone standing on its base is a good example.
在稳定平衡中，微小扰动会使重心升高。重力于是产生恢复力矩，使物体回到原来位置。一个底面朝下站立的圆锥体就是一个好例子。
In unstable equilibrium, a small displacement lowers the centre of gravity. The weight then produces a turning moment that moves the object further away from equilibrium. A cone balanced on its tip is unstable.
在不稳定平衡中，微小扰动会使重心降低。重力于是产生使物体进一步远离平衡位置的转动力矩。一个尖端支撑平衡的圆锥体是不稳定的。
In neutral equilibrium, a small displacement does not change the height of the centre of gravity, so no restoring or toppling moment is produced. A sphere rolling on a horizontal surface is in neutral equilibrium.
在随遇平衡中，微小扰动不会改变重心的高度，因此不会产生恢复力矩或倾倒力矩。在水平面上滚动的球体处于随遇平衡。
| Equilibrium type | CG height after displacement | Example |
| Stable | Increases | Cone on its base |
| Unstable | Decreases | Cone on its tip |
| Neutral | Unchanged | Sphere on a flat surface |
7. Toppling and Conditions for Tipping | 倾倒与翻倒条件
An object resting on a horizontal surface will topple if the vertical line through its centre of gravity falls outside its base. At the point of toppling, this line passes exactly through the edge of the base, and the normal reaction from the surface acts entirely at that edge.
静止在水平表面上的物体，如果通过其重心的竖直线落在底面之外，就会倾倒。在即将倾倒的瞬间，这条线恰好通过底面的边缘，表面对物体的法向反作用力也完全作用在该边缘上。
Consider a uniform cuboid of width b and height 2h, so its CG is at a height h above the base. If the cuboid is tilted slowly about one edge, it will topple when the line from the edge to the CG becomes vertical. The critical tilt angle θ is given by:
考虑一个宽度为 b、高度为 2h 的均匀长方体，其重心位于底面上方 h 处。如果长方体绕一条边缓慢倾斜，当从边到重心的线变为竖直时，它将倾倒。临界倾斜角 θ 由下式给出：
tan θ = (b / 2) / h
This formula comes from comparing the horizontal distance from the edge to the CG, which is b/2, with the vertical height h. If the object is tilted by an angle greater than θ, the line of action of the weight falls outside the base and the object tips over.
该公式源于将边缘到重心的水平距离 b/2 与竖直高度 h 进行比较。如果物体的倾斜角大于 θ，重力的作用线就会落在底面之外，物体就会翻倒。
A lower centre of gravity and a wider base both increase θ, making the object harder to topple. This is why racing cars have a low CG and a wide wheel track, while tall narrow objects tip easily.
较低的重心和较宽的底面都会增大 θ，使物体更难倾倒。这就是为什么赛车重心低、轮距宽，而高而窄的物体容易翻倒。
8. Centre of Gravity in Vehicles | 车辆中的重心
Vehicle stability depends strongly on the height of the centre of gravity above the road and the width between the wheels. A high centre of gravity increases the risk of rollover during cornering because the centripetal force acting at the CG can create a moment about the outer wheels.
车辆的稳定性在很大程度上取决于重心相对于路面的高度以及车轮之间的宽度。重心越高，转弯时发生侧翻的风险越大，因为作用在重心上的向心力会绕外侧车轮产生一个力矩。
When a vehicle turns, a horizontal centripetal force acts toward the centre of the turn. If this force and the weight produce a resultant force whose line of action falls outside the wheel base, the vehicle will roll over. The maximum safe speed for a curve depends on the CG height and the track width.
当车辆转弯时，水平向心力指向弯道中心。如果该力与重力产生的合力作用线落在轮距之外，车辆就会侧翻。弯道的最大安全速度取决于重心高度和轮距宽度。
Double-decker buses and lorries are tested on tilting platforms to ensure they do not topple at a specified angle. A low CG is also important when a vehicle is fully loaded: passengers or cargo placed high up raise the CG and reduce stability.
双层巴士和卡车都要在倾斜平台上进行测试，以确保它们在规定角度下不会翻倒。车辆满载时低重心也很重要：乘客或货物放在高处会抬高重心，降低稳定性。
In liquid tankers, the centre of gravity can shift as the liquid sloshes from side to side. This dynamic shift can suddenly move the line of action of the weight outside the base, so tankers often have internal baffles to limit liquid movement.
在液罐车中，随着液体左右晃动，重心会发生偏移。这种动态偏移会突然使重力作用线移到底面之外，因此液罐车通常装有内部挡板以限制液体运动。
9. Worked Examples | 例题解析
Example 1: A uniform metre rule of weight 2.0 N is pivoted at the 30 cm mark. A 4.0 N weight is hung at the 10 cm mark. Find the position at which a 1.0 N weight must be hung to balance the rule horizontally.
例题1：一根重 2.0 N 的均匀米尺在 30 cm 刻度处支起。在 10 cm 刻度处悬挂一个 4.0 N 的砝码。求必须将 1.0 N 的砝码悬挂在什么位置，才能使米尺水平平衡。
The weight of the rule acts at its centre, the 50 cm mark, which is 20 cm from the pivot. The 4.0 N weight is 20 cm from the pivot on the other side. Taking moments about the pivot:
米尺的重力作用在其中心，即 50 cm 刻度处，距支点 20 cm。4.0 N 砝码在支点另一侧 20 cm 处。对支点取力矩：
Clockwise moment = 1.0 N × d
Anticlockwise moment = 4.0 N × 0.20 m + 2.0 N × 0.20 m = 1.2 N m
For balance, clockwise moment = anticlockwise moment, so 1.0 × d = 1.2, giving d = 1.2 m. Since d is measured from the pivot and the weight must be on the right-hand side, the position is 30 cm + 120 cm = 150 cm, which is off the rule. Therefore no position on the rule can balance this system; an extension would be needed.
平衡时顺时针力矩等于逆时针力矩，因此 1.0 × d = 1.2，解得 d = 1.2 m。由于 d 是从支点向右测得的距离，该位置为 30 cm + 120 cm = 150 cm，已超出米尺范围。因此米尺上没有任何位置能使该系统平衡，需要加长杆。
Example 2: A uniform rectangular block is 0.40 m wide and 0.80 m high. Calculate the maximum angle to which it can be tilted before it topples, assuming it does not slide.
例题2：一个均匀矩形块宽 0.40 m，高 0.80 m。假设它不滑动，计算它能倾斜而不倾倒的最大角度。
The CG is at the centre, so its height above the base is h = 0.40 m. The horizontal distance from the edge to the CG is b/2 = 0.20 m. Using tan θ = (b/2) / h:
重心位于几何中心，因此它距底面的高度为 h = 0.40 m。从边缘到重心的水平距离为 b/2 = 0.20 m。利用 tan θ = (b/2) / h：
tan θ = 0.20 / 0.40 = 0.50
θ = tan⁻¹(0.50) ≈ 26.6°
So the block can be tilted by up to about 26.6° from the vertical before the line of action of its weight passes outside the base.
因此，该矩形块可以在竖直方向的基础上倾斜约 26.6°，之后其重力作用线将落在底面之外。
10. Exam Tips and Common Mistakes | 考试技巧与常见错误
When defining centre of gravity, do not simply say it is ‘the point where the weight acts’. A complete definition should mention that it is the point through which the entire weight of the body appears to act, regardless of the body’s orientation.
在定义重心时，不要只说它是“重力作用的点”。完整的定义应提到：它是物体的全部重力似乎通过的那一点，无论物体如何放置。
Always draw the line of action of the weight from the CG vertically downward when checking stability or toppling. Many students forget to mark this line and then make mistakes with moments.
在检查稳定性或倾倒问题时，一定要从重心竖直向下画出重力的作用线。许多学生忘记画出这条线，导致力矩计算出错。
Remember that the normal reaction on a tilted object does not always act through the centre of the base. At the point of toppling, it acts at the edge, so taking moments about that edge can simplify the problem.
请记住，倾斜物体上的法向反作用力并不总是通过底面中心。在即将倾倒时，它作用在边缘上，因此绕该边缘取力矩可以简化问题。
Finally, for composite bodies, use the principle of moments to find the CG. Take moments about a convenient origin, and remember that the total weight is the sum of the weights of the parts. Always give coordinates with units where appropriate.
最后，对于复合物体，要利用力矩原理求重心。选择一个方便的原点取力矩，并记住总重力等于各部分重力之和。在适当的地方一定要写出坐标和单位。
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