📚 The torque of a couple | 力偶的力矩
In A-Level Physics, you often encounter situations where two forces act together to produce rotation without any tendency to cause linear motion. This special pair of forces is known as a couple, and the measure of its turning effect is called the torque of a couple. Understanding this concept is essential for analysing rigid body equilibrium and rotational dynamics in the CIE syllabus.
在A-Level物理中，经常遇到一对力共同作用产生转动而不引起平动的情况。这一对特殊的力称为力偶，其转动效应的量度就是力偶的力矩。理解这一概念对于分析刚体平衡和转动力学在CIE考试大纲中至关重要。
1. What is a Couple? | 什么是力偶？
A couple consists of two parallel forces that are equal in magnitude, opposite in direction, and do not share the same line of action. Because the forces are equal and opposite, their resultant force is zero, meaning no linear acceleration occurs. However, the separation of their lines of action creates a net turning effect.
力偶由两个大小相等、方向相反、作用线平行但不共线的力组成。由于这两个力大小相等、方向相反，合力为零，因此不会产生线性加速度。然而，它们作用线的分离产生了净转动效应。
For example, when you turn a steering wheel with both hands, each hand applies a force in opposite directions along different lines. These two forces form a couple that rotates the wheel without pushing the car sideways.
例如，当你用双手转动方向盘时，每只手沿不同作用线施加方向相反的力。这两个力构成一个力偶，使方向盘转动而不将汽车推向一侧。
2. Characteristics of a Couple | 力偶的特征
The defining features of a couple are: 'balanced forces' (net force zero), 'parallel but non-collinear lines of action', and 'pure rotation'. Unlike a single force that tends to produce linear motion along with rotation about a fixed point, a couple exclusively generates angular acceleration.
力偶的定义特征包括："平衡力"（合力为零）、"平行但不共线的作用线"以及"纯转动"。与单力既可能产生平动又可能绕定点转动不同，力偶只产生角加速度。
Because the net force is zero, a couple cannot change the translational velocity of an object's centre of mass. This property makes couples extremely useful in rotating objects like taps, dials, and valves without displacing them.
由于合力为零，力偶无法改变物体质心的平动速度。这一特性使力偶在旋转水龙头、旋钮和阀门等物体时非常有用，不必担心物体位置移动。
3. Definition of the Torque of a Couple | 力偶力矩的定义
The torque (or moment) of a couple is defined as the product of one of the forces and the perpendicular distance between their lines of action. If each force has magnitude F and the perpendicular separation is d, then:
力偶的力矩（或力偶矩）定义为其中一个力的大小与两力作用线之间垂直距离的乘积。若每个力大小为 F，垂直间距为 d，则：
Torque of a couple = F × d
This torque is measured in newton metres (N m). It is a scalar-like quantity in many calculations, but it has an associated direction perpendicular to the plane of rotation.
力偶矩的单位是牛·米（N m）。在许多计算中它表现为标量，但具有垂直于转动平面的方向。
It is important to note that d is not the distance between the points of application of the forces unless those points lie on a common perpendicular to the force lines. It is always the shortest distance between the two parallel lines of action.
必须注意，d 并不是力的作用点之间的距离，除非这两点恰好位于力作用线的公垂线上。d 始终是两平行作用线之间的最短距离。
4. The Crucial Role of Perpendicular Distance d | 垂直距离 d 的关键作用
wo forces on a rigid body, the torque remains F × d regardless of the shape of the object, as long as d is measured perpendicularly to the force lines.
很多学生容易混淆力实际作用点之间的距离与垂直距离 d。对于刚体上的力偶，无论物体形状如何，只要 d 是沿力线垂线测量的距离，力偶矩始终为 F × d。
ated vertically by 0.3 m, then d = 0.3 m, even if the plate is 0.5 m wide. The horizontal width is irrelevant.
设想一块矩形板在顶部和底部边缘受两个水平力。如果力水平作用但它们的力线在竖直方向相距 0.3 m，那么 d = 0.3 m，即使板宽 0.5 m。水平宽度无关紧要。
In the steering wheel example, the forces are tangential to the wheel rim, and d equals the diameter of the wheel. Not the rim length, but the straight-line distance between the two parallel tangent lines (which equals the diameter).
在方向盘例子中，力沿轮缘切向，d 等于方向盘的直径。不是轮缘弧长，而是两条平行切线之间的距离（即直径）。
5. Why the Resultant Force is Zero but Torque is Non-zero | 为何合力为零而力矩不为零
f we take any arbitrary pivot, the moments of the two forces add up to F × d in magnitude.
考虑一个大小为 F 向右的力作用在某点，另一个等大的力在垂直距离为 d 的另一点向左作用。取任意支点，这两个力的力矩大小之和总是 F × d。
× d about that pivot. The total turning effect is F × d, confirming a non-zero torque despite zero net force.
例如，将支点恰好选在其中一个力的作用线上。那么该力贡献的力矩为零，而另一个力对该支点的力矩为 F × d。总转动效应为 F × d，证实合力为零但力矩不为零。
This simple thought experiment shows that a couple always creates a pure rotation, and its torque magnitude does not depend on the chosen reference point.
这个简单的思想实验表明，力偶总是产生纯转动，且其力矩大小与所选的参考点无关。
6. Calculating the Torque: A Worked Example | 力偶力矩计算示例
A door handle is turned by applying a force of 12 N with each hand in opposite directions. The handles are 0.40 m apart, and the forces act tangentially, i.e., perpendicular to the line joining the handles. Find the torque of the couple.
一个门把手用双手各施加 12 N 的力向相反方向转动。两个把手相距 0.40 m，力沿切向作用，即垂直于两把手连线。求力偶力矩。
parallel force lines is exactly the handle separation = 0.40 m. Each force F = 12 N. Thus:
两平行力线之间的垂直距离 d 恰好等于手柄间距 0.40 m。每个力 F = 12 N。因此：
Torque = 12 N × 0.40 m = 4.8 N m
he couple torque already accounts for both forces. The moment of one force about the other's line of action gives F × d, which is the total torque of the pair.
注意我们并没有乘以 2；力偶力矩已计入两个力的贡献。一个力对另一力作用线的力矩为 F × d，这就是力偶的总力矩。
If the forces were not perpendicular to the separation, we would need to find the perpendicular distance between their parallel lines of action using trigonometry.
如果力不垂直于间隔，就需要用三角学求出两平行力线之间的垂直距离。
7. Independence from the Choice of Pivot | 力偶力矩与转动轴的无关性
One remarkable property of a couple is that its torque is a 'free vector': it is the same about any point in space. To prove this, take a couple with forces F at A and -F at B, where the vector AB is not parallel to F. The moment about an arbitrary point O is rOA × F + rOB × (-F) = (rOA - rOB) × F = BA × F.
力偶的一个显著特性是其力矩是一个"自由矢量"：对空间任意一点的力矩都相同。证明如下：设力偶中力 F 作用于 A，-F 作用于 B，矢量 AB 不与 F 平行。关于任意点 O 的力矩为 rOA × F + rOB × (-F) = (rOA - rOB) × F = BA × F。
where d is the perpendicular distance between the lines of action. The result depends only on the couple, not on O.
由于 BA 是从 B 到 A 的矢量，其大小乘以 F 垂直于 BA 的分量恰好等于 F × d，其中 d 是力线间的垂直距离。这一结果只取决于力偶本身，与 O 无关。
This is why in mechanics problems you can sum the torques of all couples acting on a body without worrying about a common pivot; each couple contributes an independent torque value.
这就是为什么在力学问题中你可以将作用在物体上所有力偶的力矩直接相加，而不必担心统一取矩点；每个力偶都贡献一个独立的力矩值。
8. Vector Nature of the Torque of a Couple | 力偶力矩的矢量性质
Although we often treat the torque of a couple as a scalar in two-dimensional problems, it is fundamentally a vector perpendicular to the plane containing the two forces. The direction is given by the right-hand grip rule: if the fingers curl in the sense of rotation the couple tends to produce, the thumb points in the direction of the torque vector.
虽然在二维问题中我们常把力偶力矩当作标量处理，但它本质上是一个垂直于力偶平面的矢量。方向由右手螺旋定则确定：四指沿力偶产生的转动方向弯曲，拇指所指即为力矩矢量方向。
For a couple acting in the xy-plane, the torque vector points either in the +z or -z direction. Positive torque is usually associated with anticlockwise rotation (palm rule), but CIE often uses the convention that anticlockwise moments are positive.
对于作用在 xy 平面内的力偶，力矩矢量指向 +z 或 -z 方向。通常逆时针转动对应正力矩（右手螺旋定则），CIE 一般规定逆时针力矩为正。
F. Its magnitude is rF sinθ, where θ is the angle between r and F, and its direction is perpendicular to both r and F.
用矢量表示，若 r 是一条力线到另一条力线的位置矢量，则力偶力矩 = r × F。其大小为 rF sinθ，θ 为 r 与 F 的夹角，方向垂直于 r 和 F 所在平面。
9. Work Done by a Couple | 力偶做的功
oduct of the torque and the angular displacement. This is analogous to work done by a linear force: W = F × distance for translation; for rotation, W = τ × θ.
当力偶作用于物体使其转动一个角位移 θ（以弧度计）时，力偶做的总功等于力矩与角位移的乘积。这与平动中力做功类似：平动中 W = F × 距离；转动中 W = τ × θ。
/2)θ. Summing for both forces gives F d θ = τ θ. This elegant result shows that the couple's torque directly links to energy transferred in rotation.
推导：力偶中的每个力沿半径为 d/2 的弧运动，因此每个力做功 F × (d/2)θ。两力做功之和为 F d θ = τ θ。这一简洁的结果表明力偶力矩与转动中传递的能量直接相关。
ngular speed ω is P = τ ω. This is widely used in analysing motors and rotating machinery.
对于恒力矩，以角速度 ω 转动的力偶输出的功率为 P = τ ω。这被广泛用于分析电机和旋转机械。
10. Real-world Examples of Couples | 力偶的实际应用
Couples are everywhere in daily life. Turning a tap requires two fingers applying forces on opposite sides of the handle – a clear couple. The magnetic forces on opposite sides of a current-carrying coil in a uniform magnetic field also form a couple, which is the operating principle of a DC motor.
力偶在日常生活中随处可见。转动水龙头需要两根手指在把手两侧施加力——一个清晰的力偶。均匀磁场中通电线圈左右两侧所受的磁力也构成一个力偶，这是直流电动机的工作原理。
When you use a screwdriver, your hand applies a couple to the handle: the palm and fingers push in opposite directions separated by the handle diameter. Similarly, in a bicycle pedal, the downward force on one pedal and the upward pull on the other (or vice versa) create a couple that turns the crank.
使用螺丝刀时，手对手柄施加了一个力偶：手掌和手指沿手柄直径方向施加相反的作用力。类似地，自行车踏板上一脚向下踩、另一脚向上提（或反之）形成力偶驱动曲柄转动。
Even the steering wheel of a car is designed to be operated by two forces forming a couple, making it easier to control the vehicle's direction without shifting the driver's body.
汽车方向盘的设计也是通过双手施加力偶来操作，使得在控制车辆方向时驾驶者身体不易移位。
11. Common Misunderstandings and Pitfalls | 常见误解与易错点
Misconception: 'The torque of a couple is 2F × d.' one force, the other gives F × d; the first contributes zero, giving total F × d, not 2Fd.
误解："力偶力矩等于 2F × d"。请记住，定义已经结合了两个力的效应。如果对其中一个力取矩，另一个力产生 F × d；第一个力贡献为零，总力矩为 F × d，非 2Fd。
Another error is using the distance between points of application rather than the perpendicular distance between lines of action. Always draw parallel lines through the force vectors and measure the perpendicular gap.
另一个错误是使用力作用点之间的距离而非力作用线之间的垂直距离。务必画出通过力矢量的平行线并测量其垂直间距。
Some students think a couple can cause translation if the object is not perfectly rigid. In real materials, couples may cause slight deformation, but the net translational force remains zero according to Newton's laws. The principle remains valid in rigid-body mechanics.
有些学生认为若物体非完美刚体，力偶可能引起平动。实际材料中力偶可能导致微小形变，但根据牛顿定律，净平动力始终为零。刚体力学中这一原理仍然成立。
Finally, forgetting that torque is a vector can lead to sign errors when combining multiple couples. Assign positive to anticlockwise and negative to clockwise, as per CIE convention, and sum algebraically.
最后，忘记力矩是矢量，组合多个力偶时可能导致符号错误。按照CIE惯例，逆时针为正、顺时针为负，然后代数和。
12. Key Points Summary | 重点总结
d, where d is the perpendicular distance between the lines of action. Its SI unit is N m.
力偶由两个大小相等、方向相反、作用线平行但不共线的力构成。力偶的力矩为 F × d，d 为力线间垂直距离。SI单位是 N m。
The resultant force is zero, so the couple causes only rotation. The torque is independent of the pivot point and acts as a free vector perpendicular to the plane of the forces.
合力为零，力偶只产生转动。力偶力矩与取矩点无关，作为自由矢量垂直于力偶平面。
r displacement (rad). The direction of the torque follows the right-hand grip rule: anticlockwise positive.
力偶做的功 = 力矩 × 角位移（弧度）。力矩方向遵循右手螺旋定则：逆时针为正。
Mastering couples is vital for solving rigid-body equilibrium problems where both linear and rotational equilibrium must be satisfied. In any static problem, the sum of all forces must be zero and the sum of all torques (including those from couples) must also be zero.
掌握力偶对于解决刚体平衡问题至关重要，这类问题中需要同时满足平动平衡和转动平衡。在任何静力问题中，合力必须为零，合力矩（包括力偶的力矩）也必须为零。
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