Centre of Gravity | 重心

📚 Centre of Gravity | 重心

The centre of gravity (CG) of an object is the point through which its entire weight appears to act, no matter how the object is oriented. In a uniform gravitational field, the centre of gravity coincides with the centre of mass, and understanding its location is crucial for analysing equilibrium, stability and motion. This article covers the essential concepts and experimental techniques required for CIE A-Level Physics.

物体的重心是指无论物体如何放置，其全部重量似乎都作用于此的那个点。在均匀重力场中，重心与质心重合，理解重心的位置对于分析平衡、稳定性和运动至关重要。本文涵盖 CIE A-Level 物理所需的重点概念和实验方法。

1. Definition of Centre of Gravity | 重心的定义

The centre of gravity is defined as the point at which the entire weight of a body can be considered to act for the purpose of calculating the torque due to its weight. It is a vector point that accounts for the distribution of mass within the gravitational field. For a rigid body, if you support it at its centre of gravity, it will balance in any orientation.

重心被定义为：在计算重力产生的力矩时，可以认为物体全部重量都作用于此的那个点。这是一个矢量点，反映了质量在重力场中的分布。对于刚体而言，如果在重心处支撑物体，它在任何姿态下都能保持平衡。

2. Centre of Gravity vs Centre of Mass | 重心与质心的区别

In a uniform gravitational field, the centre of gravity and the centre of mass are exactly the same point. The centre of mass depends purely on the mass distribution, whereas the centre of gravity depends on both mass distribution and the variation of the gravitational field. For most problems on Earth, g is constant, so we use the terms interchangeably. However, if the gravitational field is non-uniform (e.g. a very tall structure), the centre of gravity may be slightly lower than the centre of mass.

在均匀重力场中，重心与质心是完全重合的点。质心仅取决于质量分布，而重心取决于质量分布和重力场的变化。在地球上大多数问题中，重力加速度 g 是常数，因此这两个术语可以混用。然而，如果重力场不均匀（例如一个极高的结构），重心可能略低于质心。

For CIE examinations, you are expected to know that centre of gravity = centre of mass for objects in a uniform field. The formula for the centre of mass of a set of particles is:

对于 CIE 考试，你应知道在均匀场中重心等于质心。一组质点的质心公式为：

r_cm = ( ∑ mₙ rₙ ) / ( ∑ mₙ )

where mₙ and rₙ are the mass and position vector of the n-th particle. This same expression gives the centre of gravity when g is constant.

其中 mₙ 和 rₙ 分别是第 n 个质点的质量和位置矢量。当 g 为常数时，同样的表达式给出重心的位置。

3. Centre of Gravity of Symmetrical Uniform Objects | 对称均匀物体的重心

For a uniform, homogeneous object with a regular geometric shape, the centre of gravity lies at the geometric centre. For example, a uniform sphere has its CG at its centre; a uniform rod has its CG at its midpoint; a uniform rectangular sheet has its CG at the intersection of its diagonals.

对于形状规则且均匀的物体，重心位于几何中心。例如，均匀球体的重心在其球心；均匀细杆的重心在其中点；均匀矩形薄片的重心在其对角线的交点。

If an object has an axis of symmetry, the centre of gravity will lie somewhere on that axis. Identifying symmetry helps quickly determine the CG in exam problems without complex calculations. Always state the assumption of ‘uniform density’ when using this method.

如果物体有对称轴，重心必定位于该轴上。利用对称性可以在考试中快速确定重心，而无需复杂的计算。使用这种方法时务必说明假设了“密度均匀”。

4. Finding Centre of Gravity: Suspension Method | 寻找重心：悬挂法

An experimental method to locate the centre of gravity of an irregularly shaped lamina is the suspension method. The lamina is freely suspended from a point near its edge. When it comes to rest, a plumb line is used to mark a vertical line downwards from the point of suspension. The centre of gravity must lie somewhere along this line. The process is repeated by suspending the lamina from a different point. The intersection of the two plumb lines gives the position of the centre of gravity.

测定不规则薄片重心的一种实验方法是悬挂法。将薄片从靠近边缘的某一点自由悬挂，待其静止时，用铅垂线从悬挂点向下标记一条垂直线。重心必定位于这条线上的某处。再换另一个悬挂点重复此过程，两条铅垂线的交点即为重心的位置。

This works because when the lamina is in equilibrium under gravity and the tension from the string, the resultant torque about the suspension point must be zero, forcing the CG to be vertically below the pivot. This technique is a classic CIE practical skill.

此方法之所以有效，是因为当薄片在重力和细绳拉力作用下处于平衡时，对悬挂点的合力矩必须为零，从而迫使重心位于悬挂点正下方。这是一项经典的 CIE 实验技能。

5. Centre of Gravity and Stability | 重心与稳定性

The stability of an object resting on a surface depends on the position of its centre of gravity relative to its base of support. An object is more stable when its centre of gravity is as low as possible and the area of its base is as large as possible. If a vertical line drawn from the centre of gravity falls within the base, the object will return to its original position after a small tilt – it is in stable equilibrium.

静止于平面上的物体的稳定性取决于其重心相对于支撑面的位置。重心越低、底座面积越大，物体就越稳定。如果从重心引出的竖直线落在支撑底面以内，则物体在小角度倾斜后能回到原位——它处于稳定平衡。

As soon as the vertical line from the CG passes outside the base, the weight produces a torque that causes the object to topple. This concept is frequently tested with blocks on inclined planes or loaded vehicles.

一旦从重心引出的竖直线超出底面，重量就会产生一个使物体翻倒的力矩。这个概念常常在斜面方块或装载车辆的问题中进行考查。

6. Types of Equilibrium | 平衡的种类

There are three types of equilibrium: stable, unstable and neutral. A body is in stable equilibrium if, when slightly displaced, it experiences a restoring torque that brings it back. Its centre of gravity rises when displaced. In unstable equilibrium, a slight displacement causes a torque that moves the object further away; the centre of gravity falls. In neutral equilibrium, displacement does not change the height of the centre of gravity, so the object stays in its new position.

平衡有三种类型：稳定平衡、不稳定平衡和随遇平衡。如果物体轻微偏离后有回复力矩使其回到原位，则处于稳定平衡，此时重心在偏离后会升高。不稳定平衡下，轻微偏离会产生使物体继续远离的力矩，重心会降低。随遇平衡下，偏离不会改变重心的高度，因此物体会停留在新位置。

Common examples: a cone resting on its base (stable), a cone balanced on its tip (unstable), a sphere on a horizontal table (neutral). Exam questions may ask you to sketch the position of the CG and the line of action of weight.

常见例子：锥体底面着地（稳定平衡），锥体尖端平衡（不稳定平衡），球体放在水平桌上（随遇平衡）。考试中可能要求你绘出重心位置和重力作用线的示意图。

7. Centre of Gravity and Torque | 重心与力矩

When calculating the torque due to the weight of an extended body, the weight can be considered to act at a single point – the centre of gravity. This greatly simplifies rotational equilibrium problems. For a body in equilibrium, not only must the net force be zero, but the net torque about any point must also be zero. The weight, acting through the CG, contributes a torque equal to the product of the weight and the horizontal distance from the pivot to the line of action of the weight.

在计算刚体重力产生的力矩时，可以认为重力集中作用于一个点——重心。这大大简化了转动平衡问题。对于一个处于平衡状态的物体，不仅合外力为零，对任意点的合外力矩也必须为零。重力通过重心作用，产生的力矩等于重力大小乘以支点到重力作用线的水平距离。

A typical CIE problem involves a uniform rod hinged at one end, held horizontally by a string. The weight acts at the midpoint, and taking moments about the hinge allows you to find the tension. Always draw an arrow for weight from the centre of gravity.

典型的 CIE 题目涉及一端铰接的均匀杆，用绳子水平拉住。其重力作用于中点，对铰点求矩即可求出绳子拉力。始终记住从重心处画出重力的箭头。

8. Centre of Gravity and Gravitational Potential Energy | 重心与重力势能

The gravitational potential energy (GPE) of an extended object in a uniform field is given by GPE = mgh, where h is the height of the centre of gravity above a chosen reference level. This is because the total weight mg can be thought of as being concentrated at the CG. When analysing energy changes in toppling or tilting problems, you only need to track the vertical displacement of the centre of gravity.

均匀场中，刚体的重力势能由 GPE = mgh 给出，其中 h 是重心在所取参考水平面以上的高度。这是因为总重量 mg 可以视为集中作用于重心。在分析倾倒或倾斜问题中的能量变化时，只需考查重心在竖直方向上的位移。

For example, when a uniform rod of length L pivoted at one end falls from vertical to horizontal, the CG drops by L/2, so the loss in GPE is mg(L/2). This principle is frequently used in conjunction with conservation of energy.

例如，一根长度为 L 的均匀杆，一端铰接并从竖直位置落至水平，重心下降 L/2，因此重力势能减少 mg(L/2)。这一原理常与能量守恒结合使用。

9. The Role of Base Area | 底面积的作用

The base of support (or stance) of an object is the area enclosed by the extreme points of contact with the ground. A larger base area provides a wider region within which the vertical line through the CG can move without causing toppling. This is why a stool with widely spaced legs is more stable than one with legs close together, even if the CG is at the same height.

物体的支撑底面（或立足面）是指与地面接触的最外沿点所围成的区域。较大的底面积提供了一个更宽的区域，使通过重心的竖直线能够在该区域内移动而不致翻倒。这就是为什么腿间距较宽的椅子比腿间距较窄的椅子更稳定，即使重心高度相同。

Combining a low CG and a large base gives maximum stability. CIE questions might ask you to explain why a double-decker bus is tested on a tilting platform to ensure the CG stays within the wheelbase.

低重心与大底面积相结合能获得最大稳定性。CIE 考题可能会要求你解释为什么双层巴士需在倾斜平台上进行测试，以确保重心保持在轮距之内。

10. Worked Example: Locating Centre of Gravity of a Lamina | 实例：确定薄片的重心

Example: A uniform rectangular card of width 10 cm and height 15 cm has a circular hole of radius 2 cm cut out, with its centre 6 cm from the left edge and 8 cm from the bottom edge. Find the centre of gravity of the card.

例题：一张宽度 10 cm、高度 15 cm 的均匀矩形卡片上挖去一个半径为 2 cm 的圆孔，孔心距左边缘 6 cm、距下边缘 8 cm。求卡片的重心位置。

We treat the card as a combination of the whole rectangle plus a ‘negative mass’ for the removed circle. The original CG is at the geometric centre (5 cm, 7.5 cm). The area of the rectangle is 150 cm²; the area of the hole is approximately 12.57 cm². Using the principle of moments for the x-coordinate: (150 × 5.0 – 12.57 × 6.0) / (150 – 12.57) = 4.92 cm from the left. Similarly, the y-coordinate is (150 × 7.5 – 12.57 × 8.0) / (137.43) = 7.48 cm from the bottom. The CG has shifted slightly towards the side with more material.

我们将卡片看作完整矩形加上被挖去的圆所代表的“负质量”。原重心在几何中心 (5 cm, 7.5 cm)。矩形面积 150 cm²；孔面积约 12.57 cm²。对 x 坐标用力矩原理：(150 × 5.0 – 12.57 × 6.0) / (150 – 12.57) = 4.92 cm（距左边缘）。同理 y 坐标：(150 × 7.5 – 12.57 × 8.0) / (137.43) = 7.48 cm（距下边缘）。重心略微偏向质量较多的一侧。

This method of removing a shape and treating it as negative mass is standard at A-Level. Always draw a clear diagram and define a coordinate system.

这种挖去形状并作为负质量处理的方法是 A-Level 的标准做法。务必画出清晰的示意图并定义坐标系。

11. Exam Tips and Common Misconceptions | 考试提示与常见误解

Misconception: ‘The centre of gravity is always inside the material of the object.’ This is false; for a ring or a boomerang, the CG lies in empty space. Misconception: ‘A body topples as soon as it is tilted.’ In reality, it returns if the CG’s vertical line stays within the base. Exam tip: When a diagram is rotated, don’t assume the CG is at the geometric centre unless the object is uniform. Always mention ‘assuming the object is uniform’ if you rely on symmetry. Practise moments calculations where weight acts at the CG. Use correct units and show all steps.

误解: “重心总是在物体材料的内部。”这是错误的；对于圆环或回形镖，重心位于空处。误解: “物体一倾斜就会翻倒。”事实上，只要重心的竖直线仍落在底面内，物体就会回复。考试提示: 当示意图被旋转时，不要想当然认为重心在几何中心，除非物体是均匀的。如果依靠对称性，务必说明“假设物体是均匀的”。多多练习重量作用于重心的力矩计算。使用正确单位并展示所有步骤。

In CIE structured questions, you may be asked to define the centre of gravity, describe an experiment to find it, or analyse stability in a practical context. Precise diagrams and clear reasoning earn high marks.

在 CIE 结构性问题中，你可能会被要求定义重心、描述确定重心的实验或分析实际情境中的稳定性。精确的示意图和清晰的推理能赢得高分。

Published by TutorHao | Physics Revision Series | aleveler.com

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