Torque of a Couple | 力偶的转矩计算

📚 Torque of a Couple | 力偶的转矩计算

The concept of a couple is fundamental in rotational mechanics. In A-Level Physics, understanding how to calculate the torque produced by a couple is essential for solving problems involving rotation. This article explores the definition, formula, and applications of the torque of a couple, with worked examples tailored to the CIE syllabus.

力偶是旋转力学中的基本概念。在 A-Level 物理中，理解如何计算力偶产生的转矩对于解决涉及旋转的问题至关重要。本文深入探讨力偶转矩的定义、公式及应用，并结合 CIE 考纲提供详细的例题讲解。

1. What Is a Couple? | 什么是力偶？

A couple consists of two equal and opposite forces that are parallel to each other and act along different lines of action. Because the two forces are equal in magnitude and opposite in direction, the net force acting on the body is zero. However, since their lines of action do not coincide, they produce a rotational effect.

力偶由两个大小相等、方向相反且彼此平行的力组成，这两个力沿不同的作用线作用于物体。由于两个力大小相等、方向相反，物体所受的合力为零。然而，由于力的作用线不重合，它们会产生旋转效果。

For example, when you turn a steering wheel, you apply two equal forces in opposite directions — one hand pulls down while the other pushes up. These two forces form a couple.

例如，当你转动方向盘时，两只手施加两个大小相等、方向相反的力——一只手向下拉，另一只手向上推。这两个力构成一个力偶。

2. The Torque of a Couple | 力偶的转矩

The torque (or moment) of a couple is defined as the product of one of the forces and the perpendicular distance between the two forces. It is also known as the moment of the couple.

力偶的转矩（或力矩）定义为其中一个力与两个力之间的垂直距离的乘积，也称为力偶矩。

T = F × d

where T is the torque (in N m), F is the magnitude of one of the forces (in N), and d is the perpendicular distance between the two forces (in m).

其中 T 为转矩（单位 N m），F 为其中一个力的大小（单位 N），d 为两个力之间的垂直距离（单位 m）。

It is important to note that d must be measured as the perpendicular distance between the lines of action of the two forces, not the distance along the surface of the body.

需要注意，d 必须测量为两个力作用线之间的垂直距离，而不是沿物体表面的距离。

3. Key Characteristics of a Couple | 力偶的关键特性

A couple has several distinctive characteristics that set it apart from a single force:

力偶具有若干区别于单个力的显著特性：

The resultant force is zero, so there is no net translational acceleration; the body undergoes pure rotation.

合力为零，因此没有净平动加速度；物体处于纯旋转状态。
The torque of a couple is independent of the point about which moments are taken. You can choose any pivot position and still obtain the same torque value.

力偶的转矩与取矩点的位置无关。无论选择哪个支点，所得的转矩值都相同。
The torque of a couple remains unchanged regardless of where the couple is applied on a rigid body.

力偶作用于刚体上的位置无论如何改变，其转矩始终不变。

Because the net force is zero, a couple cannot be replaced by a single force acting at the centre of mass; it must be analysed separately from translational forces.

由于合力为零，力偶不能被作用于质心的单个力所替代；必须将其与平动力分开分析。

4. Direction of the Torque | 转矩的方向

The direction of the torque produced by a couple is given by the right-hand rule. If the fingers of your right hand curl in the direction of rotation caused by the couple, your thumb points along the direction of the torque vector.

力偶产生的转矩方向由右手定则确定。如果你右手的四指沿着力偶引起的旋转方向弯曲，那么拇指所指的方向即为转矩矢量的方向。

In two-dimensional problems, we often simply state whether the torque is clockwise (CW) or anticlockwise (ACW). By convention, a clockwise torque is taken as negative and an anticlockwise torque as positive, although the sign convention may vary between problems.

在二维问题中，我们通常只说明转矩是顺时针（CW）还是逆时针（ACW）。按照惯例，顺时针转矩取负值，逆时针转矩取正值，但不同题目中的符号约定可能有所不同。

If the two forces cause the body to rotate anticlockwise, the torque is positive; if they cause clockwise rotation, the torque is negative.

如果两个力使物体逆时针旋转，则转矩为正；若使物体顺时针旋转，则转矩为负。

5. Torque of a Couple vs Moment of a Single Force | 力偶矩与单力力矩的区别

Many students confuse the torque of a couple with the moment of a single force. The moment of a single force is calculated as M = F × r, where r is the perpendicular distance from the pivot to the line of action of the force. This moment depends on the choice of pivot.

许多学生容易将力偶的转矩与单个力的力矩混淆。单个力的力矩计算公式为 M = F × r，其中 r 是从支点到力的作用线的垂直距离。该力矩依赖于支点的选择。

In contrast, the torque of a couple is independent of the pivot point. This is a key difference tested in CIE examinations. For a couple, no matter where you take moments, the result is always T = F × d.

相比之下，力偶的转矩与支点位置无关。这是 CIE 考试中经常考查的关键区别。对于力偶，无论在哪里取矩，结果始终为 T = F × d。

Moreover, a single force applied at the centre of mass of a body produces only translation. To produce pure rotation, a body must be acted upon by a couple. This distinction is frequently tested in multiple-choice questions.

此外，作用于物体质心的单个力只能产生平动。要使物体产生纯旋转，必须有力偶作用。这一区别经常出现在选择题中。

6. Applications of Couples | 力偶的应用

Couples appear in many everyday situations and engineering applications. The following table lists some common examples:

力偶出现在许多日常和工程场景中。下表列举了一些常见实例：

Application 应用	Description 说明
Steering wheel 方向盘	Two hands apply equal and opposite forces to turn the wheel. 双手施加等大反向的力转动方向盘。
Screwdriver 螺丝刀	The handle is turned by a couple to rotate the screw. 手柄通过力偶转动螺丝。
Propeller 螺旋桨	The engine applies a couple to rotate the propeller blades. 发动机施加力偶使螺旋桨叶片旋转。
Door handle 门把手	Pushing and pulling on opposite sides creates a couple. 在相对两侧推拉形成力偶。

In each case, the magnitude of the torque depends on the force applied and the perpendicular distance between the forces. For a steering wheel of diameter 0.5 m, if each hand applies a force of 40 N, the torque is T = 40 × 0.5 = 20 N m.

在上述每种

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Torque of a Couple | 力偶的转矩计算