📚 Conditions of Equilibrium in Rigid Body Statics | 刚体静力学的平衡条件
In A-Level Mathematics (Mechanics), rigid body statics investigates bodies that remain at rest under the action of forces. Unlike particle mechanics, where only force balance is required, a rigid body can also rotate; hence a complete equilibrium analysis demands that both the resultant force and the resultant moment acting on the body be zero.
在A-Level数学（力学）中，刚体静力学研究物体在力系作用下保持静止的问题。与质点力学不同，刚体还可能发生转动；因此，完整的平衡分析要求作用于刚体的合力与合力矩均为零。
1. The Rigid Body Model | 刚体模型
A rigid body is an idealised model in which the distance between any two particles remains constant, regardless of the forces applied. The body may translate or rotate, but it never deforms.
刚体是一种理想化模型，其中任意两个质点之间的距离在受力作用下始终保持不变。刚体可以发生平动或转动，但绝不会发生形变。
This distinction matters because the point of application of a force on a rigid body governs its turning effect. For a particle, all forces act through a single point; for a rigid body, moments must be taken into account as well.
这种区别之所以重要，是因为力的作用点决定了它对刚体的转动效应。对于质点，所有力都通过同一点作用；而对于刚体，必须同时考虑力矩。
2. First Condition – Translational Equilibrium | 第一条件——平动平衡
A rigid body is in translational equilibrium when the vector sum of all external forces acting on it is zero:
当作用在刚体上的所有外力的矢量总和为零时，刚体处于平动平衡状态：
ΣF = 0
Since force is a vector, this equation is equivalent to two independent scalar equations in two-dimensional problems:
由于力是矢量，在二维问题中，上述方程等效于两个独立的标量方程：
ΣFx = 0 and ΣFy = 0
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ΣFx = 0 requires that the algebraic sum of the horizontal components of all forces vanishes, preventing horizontal acceleration.
ΣFx = 0 要求所有力的水平分量之代数和为零，从而防止物体在水平方向产生加速度。
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ΣFy = 0 requires that the algebraic sum of the vertical components of all forces vanishes, preventing vertical acceleration.
ΣFy = 0 要求所有力的竖直分量之代数和为零，从而防止物体在竖直方向产生加速度。
If the body is initially at rest and ΣF = 0, its centre of mass will remain at rest. However, force balance alone does not guarantee that the body will not rotate.
若物体初始静止且 ΣF = 0，则其质心将保持静止。然而，仅满足力的平衡并不能保证物体不发生转动。
3. Second Condition – Rotational Equilibrium | 第二条件——转动平衡
A rigid body is in rotational equilibrium when the algebraic sum of the moments of all external forces about any point is zero:
当所有外力对任意一点的力矩之代数和为零时，刚体处于转动平衡状态：
ΣM = 0 (about any chosen point)
The phrase “about any point” is powerful: if moments balance about one point, they balance about every point in the plane. In practice we choose a convenient point — usually one through which an unknown force passes — to eliminate that unknown from the moment equation.
“关于任意一点”这一说法非常有力：若力矩对某一点平衡，则对平面内任意一点都平衡。实际操作中，我们选择方便的点——通常是某个未知力作用线通过的点——从而将该未知力从力矩方程中消去。
Both conditions must hold simultaneously. A body may have zero resultant force yet still spin due to a couple, or it may have zero resultant moment yet still translate. Complete equilibrium requires ΣF = 0 and ΣM = 0 together.
两个条件必须同时成立。物体可能合力为零但因力偶而旋转，也可能合力矩为零但仍发生平动。完整的平衡要求 ΣF = 0 与 ΣM = 0 同时满足。
4. Moment of a Force | 力矩
Before applying the second condition, we must define the moment of a force. The moment M of a force F about a point O is the product of the force and its perpendicular distance d from O to the line of action:
在应用第二条件之前，必须先定义力矩。力 F 对点 O 的力矩 M 等于力的大小与点 O 到力的作用线的垂直距离 d 之乘积：
M = F × d
The unit of moment is the newton-metre (N·m). A convention must be adopted for sign: for example, anticlockwise moments may be taken as positive and clockwise moments as negative, or vice versa — consistency is essential.
力矩的单位是牛顿·米（N·m）。必须约定正负号规则：例如，可规定逆时针力矩为正、顺时针力矩为负——关键在于前后一致。
It is critical to use the perpendicular distance, not the distance along a slanted object. For example, if a force is applied at an angle to a rod, the moment arm is the component of the distance perpendicular to the force’s line of action.
必须使用垂直距离，而不是沿倾斜物体的距离。例如，若力以某一角度作用于杆上，力臂应取距离在垂直于力的作用线方向上的分量。
5. Couples | 力偶
A couple consists of two equal and opposite parallel forces that do not share the same line of action. Although the resultant force of a couple is zero, it produces a definite turning effect.
力偶由两个大小相等、方向相反且作用线不重合的平行力构成。虽然力偶的合力为零，但它确实产生确定的转动效应。
The moment of a couple is equal to the magnitude of either force multiplied by the perpendicular distance between the two lines of action:
力偶矩等于其中一个力的大小乘以两力作用线之间的垂直距离：
M = F × d
A key property is that the moment of a couple is the same about every point in the plane. Since the net force is zero, a couple cannot be balanced by a single force; it can only be balanced by another couple of equal magnitude and opposite sense.
力偶矩的一个关键性质是：它对平面内任意一点的矩都相同。由于合力为零，力偶不能由单个力来平衡；只能由另一个大小相等、转向相反的力偶来平衡。
6. Types of Supports and Their Reactions | 支座类型与约束反力
Correctly identifying the reaction forces at supports is usually the hardest part of a statics problem. The following table summarises the common support types encountered in A-Level questions.
正确识别支座处的约束反力通常是静力学题目中最困难的部分。下表总结了A-Level试题中常见的支座类型。
| Support Type | 支座类型 | Reaction Description | 反力描述 |
| Smooth surface | 光滑接触面 | Single force normal to the surface | 垂直于接触面的单个力 |
| Rough surface | 粗糙接触面 | Normal reaction plus friction along the tangent | 法向反力加上沿切向的摩擦力 |
| Smooth pin or hinge | 光滑销钉或铰链 | Force of unknown magnitude and direction; resolve into two components |
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