The Torque of a Couple | 力偶矩

📚 The Torque of a Couple | 力偶矩

Rotational motion in A-Level Physics often involves more than one force. When two equal and opposite forces act along parallel but separate lines, they form a couple, and the turning effect of this pair is called the torque of a couple. Understanding this concept is essential for analysing steering wheels, electric motors, and many mechanical systems.

在 A-Level 物理中，转动往往涉及不止一个力。当两个大小相等、方向相反的力沿着平行但不重合的直线作用时，它们构成一个力偶，这一对力的转动效果称为力偶矩。理解这一概念对分析方向盘、电动机以及许多机械系统至关重要。

1. From Torque to a Couple | 从力矩到力偶

Torque, or moment of a force, measures how effectively a single force can rotate an object about a pivot. It is defined as the product of the force and the perpendicular distance from the pivot to the line of action of that force.

力矩（也叫力的矩）用于度量单个力使物体绕支点转动的作用效果。它定义为力与该支点到力作用线的垂直距离的乘积。

A couple is different: it is not about one force acting at a distance from a pivot. Instead, it is a pair of forces that work together to produce a pure rotation, with no resultant linear force on the body.

力偶则不同：它并不是一个力相对于支点产生力矩。力偶是由两个力共同作用产生纯转动，物体整体不受到净平动力的组合。

2. Defining a Couple | 力偶的定义

A couple is a pair of forces that are equal in magnitude, opposite in direction, parallel, and do not share the same line of action.

力偶是一对大小相等、方向相反、平行且不共线的力。

Because the two forces are equal and opposite, their vector sum is zero. This means the object experiences no resultant linear force, so its centre of mass does not accelerate in a straight line.

由于这两个力大小相等、方向相反，它们的矢量和为零。这意味着物体不受净平动力，因此其质心不会发生直线加速。

However, because the forces do not act along the same line, their turning effects do not cancel. This gives rise to a net torque, which is the torque of the couple.

但由于这两个力不沿同一条直线作用，它们的转动效果并不会相互抵消，于是产生一个净力矩，这就是力偶矩。

3. Key Conditions for a Couple | 力偶的关键条件

For two forces to qualify as a couple, three conditions must be satisfied:

要使两个力构成力偶，必须满足以下三个条件：

Equal magnitude: Both forces have the same size, so |F₁| = |F₂| = F.
中文：大小相等，两力大小相同，即 |F₁| = |F₂| = F。
Opposite direction: The forces act along parallel lines but in opposite senses.
中文：方向相反，两力沿平行线作用但指向相反。
Parallel but non-collinear: The lines of action are separated by a perpendicular distance d.
中文：平行但不共线，两力作用线之间相隔垂直距离 d。

If the lines of action of the two forces coincide, their moments about any point cancel exactly, so there is no net torque. Such a pair is not a couple.

如果两个力的作用线重合，它们对任意点的力矩会完全相互抵消，因此没有净力矩。这样的一对力不称为力偶。

4. Torque of a Couple: The Core Formula | 力偶矩：核心公式

For a couple made of two forces, each of magnitude F, with a perpendicular separation d between their lines of action, the magnitude of the torque is given by:

对于由两个力组成的力偶，设每个力的大小为 F，两力作用线之间的垂直距离为 d，则力偶矩的大小为：

M = F × d

Here, M is the torque of the couple, also called the moment of the couple. The SI unit is the newton metre, written as N m.

其中 M 是力偶矩，也叫力偶的力矩。它的 SI 单位是牛米，写作 N m。

It is important to note that d is the perpendicular distance between the two forces themselves, not the distance from a pivot to one of the forces. This is a common source of confusion.

需要特别注意的是，d 是两个力自身之间的垂直距离，而不是支点到其中一个力的距离。这是常见的混淆点。

5. Why the Distance Is Perpendicular | 为什么距离取垂直距离

When a force acts at an angle to a line, only the perpendicular component of the distance contributes to the moment. For a couple, the effective separation is always measured perpendicular to the two parallel forces.

当一个力与某条直线成一定角度时，只有距离的垂直分量才会对力矩有贡献。对于力偶，有效距离始终是沿垂直于两个平行力的方向测量的。

If the physical separation along a rod is s and the forces act at an angle θ to the rod, then the perpendicular distance between the forces is:

如果沿一根杆的实际间距为 s，且力与杆的夹角为 θ，那么两个力之间的垂直距离为：

d = s sin θ

This relationship appears regularly in CIE A-Level questions, especially when a couple acts on a coil in a magnetic field or on a lever at an angle.

这一关系式在 CIE A-Level 考试中经常出现，尤其是当力偶作用于磁场中的线圈或成一定角度的杠杆时。

6. Direction of the Couple Torque | 力偶矩的方向

Torque is a vector quantity, and its direction can be described as clockwise or anticlockwise. For calculations, you must choose one direction as positive, usually anticlockwise.

力矩是矢量，其方向可以描述为顺时针或逆时针。在计算中，必须规定一个方向为正，通常取逆时针为正。

For a couple, the torque vector is perpendicular to the plane containing the two forces, following the right-hand rule. At A-Level, it is usually sufficient to state whether the torque is clockwise or anticlockwise.

对于力偶，力矩矢量垂直于两个力所在的平面，方向遵循右手定则。在 A-Level 考试中，通常只需说明力矩是顺时针还是逆时针即可。

7. Independence from the Axis | 与转轴位置无关

One of the most important properties of a couple is that its torque is the same about any point. This is why a couple is sometimes called a pure torque or a free vector.

力偶最重要的性质之一是：它对任意点的力矩都相同。因此力偶有时被称为纯力矩或自由矢量。

To see this, suppose one force acts at a perpendicular distance x from a chosen origin O, and the other force acts at a distance x + d from the same point. The moments are Fx and F(x + d) in opposite directions. The net moment is:

为了理解这一点，假设一个力到所选原点 O 的垂直距离为 x，另一个力到同一点的距离为 x + d。两个力产生的力矩方向相反，分别为 Fx 和 F(x + d)。净力矩为：

M = F(x + d) − Fx = Fd

The position x cancels out, so the torque of the couple depends only on F and d. This means you can calculate the couple torque without needing to identify a pivot point.

位置 x 被抵消了，因此力偶矩只取决于 F 和 d。这意味着计算力偶矩时无需确定支点位置。

8. Couples in Equilibrium | 力偶的平衡

An object is in rotational equilibrium when the total clockwise torque equals the total anticlockwise torque about any point. A couple produces a net torque, so a body acted on by only one couple cannot be in equilibrium.

当绕任意点的顺时针力矩总和等于逆时针力矩总和时，物体处于转动平衡状态。力偶会产生净力矩，因此仅受一个力偶作用的物体不可能处于平衡状态。

A couple cannot be balanced by a single force, because a single force would create a resultant linear force. To balance a couple, an equal and opposite couple must be applied.

单个力无法平衡一个力偶，因为单个力会产生净平动力。要平衡一个力偶，必须施加一个大小相等、方向相反的力偶。

For example, when a driver holds a steering wheel steady, one hand pushes up and the other pushes down. The two hands form a couple, and any external opposing torque must come from another couple, such as the reaction of the steering column.

例如，当驾驶员稳住方向盘时，一只手向上推，另一只手向下推。双手形成力偶，任何外部反抗力矩都来自另一个力偶，例如转向柱的反作用力矩。

9. Multiple Couples and Resultant Torque | 多个力偶与合力矩

When several couples act on the same object in the same plane, their torques combine algebraically. You must include the signs for clockwise and anticlockwise directions.

当多个力偶在同一平面内作用于同一物体时，它们的力矩按代数量合成。计算时必须包括顺时针和逆时针方向的正负号。

If anticlockwise is taken as positive, the resultant torque is:

如果规定逆时针为正，则合力矩为：

τₜₒₜₐₗ = τ₁ + τ₂ + τ₃

where each torque includes its appropriate sign. If the resultant torque is zero, the object is in rotational equilibrium.

其中每一个力矩都带有相应的正负号。如果合力矩为零，则物体处于转动平衡状态。

10. Worked Examples | 例题解析

A driver applies two equal and opposite forces of 12 N at opposite ends of a steering wheel of diameter 0.30 m. Calculate the torque of the couple.

一名驾驶员在直径 0.30 m 的方向盘两端施加两个大小相等、方向相反的 12 N 力。计算力偶矩。

The perpendicular distance between the forces is the diameter of the wheel, d = 0.30 m. Using M = F × d:

两个力之间的垂直距离为方向盘的直径，d = 0.30 m。利用 M = F × d：

M = 12 × 0.30 = 3.6 N m

A second example: two forces of 8.0 N act at the ends of a rod of length 0.50 m. The forces make an angle of 30° with the rod. Find the torque of the couple.

第二个例子：两个 8.0 N 的力作用在一根长 0.50 m 的杆两端，力的方向与杆成 30° 角。求力偶矩。

The perpendicular separation is d = 0.50 sin 30° = 0.25 m. Therefore:

垂直距离为 d = 0.50 sin 30° = 0.25 m。因此：

M = 8.0 × 0.25 = 2.0 N m

11. Common Mistakes to Avoid | 常见错误

Using the distance from a pivot to one force. In a couple, there is no pivot; use the perpendicular distance between the two forces.
中文：错误地使用支点到某一个力的距离。力偶中没有支点，应使用两个力之间的垂直距离。
Adding the two forces together. The force F in the formula is the magnitude of one force only, not the sum of both.
中文：将两个力相加。公式中的 F 只是其中一个力的大小，而不是两个力的总和。
Treating collinear equal and opposite forces as a couple. If the lines of action are the same, the net torque is zero.
中文：把共线等大反向的力当作力偶。如果作用线相同，净力矩为零。
Forgetting to state direction. Torque has a direction, so always give clockwise or anticlockwise in your final answer.
中文：忘记说明方向。力矩有方向，因此最终答案中必须说明顺时针或逆时针。
Confusing distance along the object with perpendicular distance. Only the perpendicular separation contributes to the torque.
中文：混淆沿物体的距离与垂直距离。只有垂直距离才对力矩有贡献。

12. Summary and Key Takeaways | 小结与要点

The torque of a couple is a pure rotational effect produced by two equal, opposite, parallel forces whose lines of action do not coincide.

力偶矩是由两个大小相等、方向相反、平行且不共线的力产生的纯转动效果。

Its magnitude is given by M = F × d, where d is the perpendicular distance between the two forces. The torque is independent of

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The Torque of a Couple | 力偶矩