📚 Forces: Vectors and Moments | 力：向量与力矩
In the CIE A-Level Physics syllabus, the topic "Forces: vectors and moments" brings together the vector nature of force, the conditions for translational and rotational equilibrium, and practical applications such as levers, beams and tipping. A strong grasp of these ideas is essential for solving statics problems, and the same ideas reappear later in dynamics, materials and circular motion.
在 CIE A-Level 物理大纲中，"力：向量与力矩"这一主题综合了力的向量性质、平动与转动平衡条件，以及杠杆、横梁和翻倒等实际应用。扎实掌握这些概念对解决静力学问题至关重要，而同一套思想在后续的动力学、材料学和圆周运动中也反复出现。
Examiners expect three skills to work together: representing a force correctly as a vector, resolving or combining forces accurately, and applying the principle of moments to a body that is in equilibrium. This article builds those skills step by step, then applies them to beams, levers, ladders, cranes and toppling blocks.
考官的期望是三项能力相互配合：把力正确地表示为向量、准确地对力进行合成与分解、以及对处于平衡状态的物体应用力矩原理。本文逐步建立这些能力，然后应用到横梁、杠杆、梯子、起重机和翻倒的木块等情境中。
1. Scalar and vector quantities | 标量与向量
Physical quantities in mechanics are divided into scalars and vectors. A scalar has magnitude only; examples include distance, speed, mass, energy, time and temperature. A vector has both magnitude and direction; examples include displacement, velocity, acceleration, force and momentum. Addition of scalars follows ordinary arithmetic, whereas addition of vectors must account for direction. Adding 3 N and 4 N gives 7 N only when the forces act along the same straight line; when they act at right angles the resultant is 5 N, because the resultant is the hypotenuse of a right-angled triangle.
力学中的物理量分为标量和向量。标量只有大小，例如距离、速率、质量、能量、时间和温度。向量既有大小又有方向，例如位移、速度、加速度、力和动量。标量相加遵循普通算术法则，而向量相加必须考虑方向。3 N 与 4 N 相加只有在两力沿同一直线时才等于 7 N；当二者互相垂直时合力为 5 N，因为合力是直角三角形的斜边。
| Quantity | 物理量 | Scalar / Vector | 标量或向量 | SI unit | 国际单位 |
|---|---|---|
| distance / displacement 距离 / 位移 | scalar / vector 标量 / 向量 | m |
| speed / velocity 速率 / 速度 | scalar / vector 标量 / 向量 | m s⁻¹ |
| mass / weight 质量 / 重量 | scalar / vector 标量 / 向量 | kg / N |
| energy, work 能量、功 | scalar 标量 | J |
| force, momentum 力、动量 | vector 向量 | N, kg m s⁻¹ |
| temperature 温度 | scalar 标量 | K |
In written work, a vector is often shown in bold type, such as F, while its magnitude is written in italic, F. On a diagram, an arrow carries three pieces of information: the point of application, the direction, and the length of the arrow represents the magnitude to a chosen scale. Never write a vector quantity without stating or drawing its direction.
在书面表达中，向量常用粗体表示，如 F，而其大小用斜体 F 表示。在示意图上，箭头携带三项信息：作用点、方向，以及箭头长度按选定比例代表的力的大小。书写向量时千万不要省略或漏画方向。
2. Vector addition and resolution | 向量的合成与分解
Two or more vectors are added using the triangle rule or the parallelogram rule. Place the tail of the second vector at the head of the first, and the resultant runs from the tail of the first to the head of the last. For vectors at right angles, the magnitude of the resultant follows Pythagoras and the direction follows trigonometry. For non-perpendicular vectors, either draw a scale diagram or resolve both into components.
两个或多个向量可以用三角形法则或平行四边形法则相加。把第二个向量的起点放在第一个向量的终点，合力便从第一个向量的起点指向最后一个向量的终点。对于互相垂直的向量，合力大小由勾股定理给出，方向由三角函数给出。对于不垂直的向量，可以画比例图，也可以把两者都分解成分量。
F = √Fₓ² + Fy² tan θ = Fy ÷ Fₓ
e same combined effect. Measuring θ from the x-axis, the components are Fₓ = F cos θ and Fy he slope and the other perpendicular to it usually makes the algebra much shorter than using horizontal and vertical axes.
分解是相反的过程：用一个向量的两个互相垂直的分量代替它，二者共同作用效果不变。从 x 轴量取角度 θ 时，分量为 Fₓ = F cos θ 与 Fy = F sin θ。在倾角为 θ 的斜面上，物体重力分解为沿斜面向下的 mg sin θ 和垂直于斜面的 mg cos θ。取一个轴沿斜面、另一个轴垂直斜面，通常比取水平和竖直方向使计算大为简化。
Fₓ = F cos θ Fy = F sin θ W∥ = mg sin θ W⊥ = mg cos θ
= 25.0 N. These two components together produce exactly the same effect as the original 50 N force, so they may be used in place of it in any subsequent equilibrium calculation.
例题：一个 50 N 的力与水平方向成 30° 斜向上作用。其水平分量为 50 cos 30° = 43.3 N，竖直分量为 50 sin 30° = 25.0 N。这两个分量共同产生的效果与原来的 50 N 力完全相同，因此在后续任何平衡计算中都可以用它们替代原来的力。
3. Free-body diagrams and common forces | 受力图与常见力
A free-body diagram shows one chosen object alone, with every force acting on it drawn as an arrow starting from the point where the force acts. Forces that the object exerts on other bodies must not appear. The diagram should show the weight acting at the centre of gravity, plus all contact forces from surfaces, strings and fluids. A clear free-body diagram is worth marks in its own right in practical and structured questions.
受力图只画出所选的一个物体，并把它受到的所有力都画成从作用点出发的箭头。物体对其他物体施的力不能出现在图中。图中应画出作用在重心上的重力，以及来自表面、绳索和流体的所有接触力。在实验题和结构化题目中，清晰的受力图本身就可以得分。
- Weight (W = mg) always acts vertically downwards through the centre of gravity. | 重力（W = mg）总是竖直向下，作用在重心上。
- Normal contact force (N) acts perpendicular to the surface of contact. | 法向接触力（N）垂直于接触面。
- Friction (f) acts along the surface, opposing relative motion or its tendency. | 摩擦力（f）沿接触面，阻碍相对运动或相对运动趋势。
- Tension (T) acts along the string or rod, pulling away from the object. | 张力（T）沿绳或杆方向，把物体向外拉。
- Drag and upthrust oppose motion through a fluid and act upwards in a fluid respectively. | 阻力与浮力分别阻碍流体中的运动、以及在流体中竖直向上作用。
Newton's third law pairs are frequently confused with balanced forces. A weight and a normal contact force on the same body are not a third-law pair, because they act on the same object and are usually unequal. A genuine third-law pair consists of two forces of the same type, equal in magnitude, opposite in direction, acting on two different bodies along the same line.
牛顿第三定律力对常被与平衡力混淆。同一物体上的重力与法向接触力不是第三定律力对，因为它们作用在同一物体上，而且大小通常不相等。真正的第三定律力对由两个同种类型的力组成，大小相等、方向相反、沿同一直线作用在两个不同物体上。
4. Moments and the principle of moments | 力矩与力矩原理
The moment of a force about a point is the product of the force and the perpendicular distance from the point to the line of action of the force. Its unit is the newton metre (N m), not the joule, because a moment is not energy. The perpendicular distance is the shortest distance from the pivot to the line of action; if the force is not perpendicular to the beam, use the perpendicular component.
力对某点的力矩等于力的大小与从该点到力的作用线的垂直距离的乘积。它的单位是牛顿米（N m），而不是焦耳，因为力矩不是能量。垂直距离是从支点到力的作用线的最短距离；如果力不垂直于横梁，就使用其垂直分量。
M = F × d
The principle of moments states that for a body in rotational equilibrium, the sum of the clockwise moments about any point equals the sum of the anticlockwise moments about the same point. Equivalently, the total moment about any point is zero. A convenient convention is to call anticlockwise moments positive and clockwise moments negative; the arithmetic then takes care of direction automatically.
力矩原理指出：处于转动平衡的物体，绕任意一点的顺时针力矩之和等于绕同一点的逆时针力矩之和。等价地说，绕任意一点的总力矩为零。一个方便的约定是把逆时针力矩取为正、顺时针力矩取为负，这样运算会自动处理方向。
ΣM = 0 (anticlockwise moments × 逆时针) = (clockwise moments × 顺时针)
Worked example: a uniform metre rule is pivoted at its centre. A 2.0 N weight hangs 0.30 m to the left of the pivot, and a second weight F the reaction at the pivot does not appear, because its distance from the pivot is zero. Choosing the pivot as the point about which to take moments is a standard way of eliminating an unknown force.
例题：一根均匀米尺以中点为支点。在支点左侧 0.30 m 处挂 2.0 N 的砝码，在右侧 0.15 m 处放另一个重 F 的砝码。绕支点取力矩平衡：2.0 × 0.30 = F × 0.15，得 F = 4.0 N。注意支点处的反力没有出现，因为它到支点的距离为零。选择支点作为取矩点，是消去未知力的标准技巧。
5. Couples and torque | 力偶与转矩
A couple consists of two forces that are equal in magnitude, opposite in direction, and not along the same line. A couple has no resultant translational effect, because the two forces cancel, but it produces a pure turning effect. The moment of a couple, often called the torque, equals one force multiplied by the perpendicular distance between the lines of action.
力偶由两个大小相等、方向相反、且不在同一直线上的力组成。力偶没有平动效果，因为两个力相互抵消，但它产生纯粹的转动效果。力偶的力矩（常称为转矩）等于其中一个力乘以两个力的作用线之间的垂直距离。
torque = F × d (d = perpendicular separation of the two forces × 两力之间的垂直距离)
orces separated by 0.20 m give a torque of 15 × 0.20 = 3.0 N m. Turning a steering wheel is a practical couple: two 30 N forces applied tangentially at opposite sides of a wheel of diameter 0.36 m give a torque of 30 × 0.36 = 10.8 N m.
力偶的一个有用性质是：它在平面内对任意一点的力矩都相同，因此无需选择支点。例题：两个 15 N 的力相距 0.20 m，转矩为 15 × 0.20 = 3.0 N m。转动方向盘就是实际的力偶：在直径为 0.36 m 的方向盘两侧沿切线各施 30 N 的力，转矩为 30 × 0.36 = 10.8 N m。
6. Centre of gravity and weight | 重心与重力
the midpoint. Weight is the gravitational force on a mass, W = mg, and g is about 9.81 m s⁻² near the Earth' ical plumb line; repeating from a second point gives the centre of gravity where the lines cross.
物体的重心是物体的全部重力可以看作集中作用的点。对于均匀米尺或均匀细杆，重心在中点。重量是作用在质量上的重力，W = mg，在地球表面附近 g 约为 9.81 m s⁻²。对于不规则薄板，可以用实验方法找重心：把薄板从销钉上自由悬挂，画出竖直的重垂线；再从第二个悬挂点重复一次，两条线的交点就是重心。
W = mg x̄ = (m₁x₁ + m₂x₂ + ...) ÷ (m₁ + m₂ + ...)
as a point mass at its own centre of gravity and take moments about a convenient point. Worked example: a 2.0 kg mass is 0.20 m from the left end of a light rod and a 3.0 kg mass is 0.60 m from the same end. Taking moments about the left end, the centre of gravity is at x̄ 0.44 m from the left end. This is the same calculation as balancing moments, which is why the two ideas are usually examined together.
对于组合体，可把每一部分看作位于自身重心处的质点，然后绕一个方便的点取矩。例题：轻杆左端 0.20 m 处有 2.0 kg 的物体，左端 0.60 m 处有 3.0 kg 的物体。绕左端取矩，重心位于距左端 x̄ = (2.0 × 0.20 + 3.0 × 0.60) ÷ (2.0 + 3.0) = 0.44 m 处。这与力矩平衡是同一个计算，因此这两个概念常一起考查。
7. Conditions for equilibrium | 平衡条件
A body is in equilibrium when it has no resultant force and no resultant moment. Translational equilibrium requires the vector sum of all forces to be zero, which is most easily checked by resolving in two perpendicular directions. Rotational equilibrium requires the total moment about any point to be zero. A body at rest is in static equilibrium; a body moving with constant velocity is in dynamic equilibrium, and both satisfy the same two conditions.
当物体所受合力为零且合力矩也为零时，物体处于平衡状态。平动平衡要求所有力的向量和为零，最容易的做法是沿两个互相垂直的方向分解检查。转动平衡要求绕任意一点的总力矩为零。静止的物体处于静力平衡；以恒定速度运动的物体处于动力平衡，二者满足同样的两个条件。
ΣFₓ = 0 ΣFy = 0 ΣM = 0
If three non-parallel coplanar forces act on a body in equilibrium, their lines of action must meet at a single point,
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