Moments: Turning Forces, Equilibrium and Applications | 力矩：转动力、平衡与应用

📚 Moments: Turning Forces, Equilibrium and Applications | 力矩：转动力、平衡与应用

In A-Level Mechanics, a force can either translate an object or rotate it about a pivot. The rotating capability of a force is called its moment. This article covers the definition, calculation, equilibrium conditions, couples, centre of gravity, experimental verification and common exam techniques required by the Edexcel specification.

在 A-Level 力学中，力既可以使物体平动，也可以使物体绕支点转动。力使物体转动的能力称为力矩。本文涵盖 Edexcel 考试大纲所要求的力矩定义、计算、平衡条件、力偶、重心、实验验证和常见考试技巧。

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1. Definition of Moment | 力矩的定义

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. This distance is often called the lever arm or moment arm.

力对某点的力矩等于力乘以该点到力作用线的垂直距离。这个距离通常称为力臂或力矩臂。

M = F × d⊥

The SI unit of moment is the newton metre (N m). Although a newton metre has the same base units as a joule, a moment is not energy; it is a measure of turning effect, so the unit N m is not written as joules.

力矩的 SI 单位是牛顿米（N m）。虽然牛顿米与焦耳具有相同的基本单位，但力矩不是能量；它是转动效应的度量，因此单位 N m 不写作焦耳。

• M = moment, measured in newton metres (N m)
• F = force, measured in newtons (N)
• d⊥ = perpendicular distance from pivot to the line of action, measured in metres (m)

M 表示力矩，单位为牛顿米（N m）；F 表示力，单位为牛顿（N）；d⊥ 表示支点到力作用线的垂直距离，单位为米（m）。

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2. Calculating Moments | 力矩的计算

When the force acts perpendicular to the line joining the pivot to the point of application, the moment is simply M = F × d. When the force acts at an angle θ to the lever arm, only the perpendicular component F sin θ contributes to the moment.

当力垂直于支点与作用点连线时，力矩为 M = F × d。当力与力臂成 θ 角时，只有垂直分量 F sin θ 产生力矩。

M = F × d × sin θ

In exam questions, always draw the perpendicular distance from the pivot to the line of action. If the given distance is not perpendicular, resolve the force or the distance to avoid using the wrong length.

在考试题中，务必画出从支点到力作用线的垂直距离。如果给定的距离不是垂直距离，则需分解力或距离，避免使用错误长度。

Quantity	Symbol	Unit
Moment	M	N m
Force	F	N
Perpendicular distance	d⊥	m
Angle between F and lever arm	θ	degree

The table above summarises the quantities used in moment calculations. Always convert distances to metres and forces to newtons before substituting into the formula.

上表总结了力矩计算中使用的物理量。在代入公式之前，务必先将距离换算为米，将力换算为牛顿。

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3. Principle of Moments | 力矩原理

The principle of moments states that for an object to be in rotational equilibrium, the sum of all clockwise moments about any pivot must equal the sum of all anticlockwise moments about that same pivot.

力矩原理指出，物体处于转动平衡时，对任意支点的所有顺时针力矩之和必须等于所有逆时针力矩之和。

ΣM(clockwise) = ΣM(anticlockwise)

This principle is extremely powerful because it allows unknown forces or distances to be calculated without needing to know every reaction force, provided the pivot is chosen carefully.

这个原理非常有用，因为只要谨慎选择支点，就可以在不需知道每个反作用力的情况下计算未知力或距离。

For example, if a see-saw is balanced, the product of weight and distance on one side equals the product of weight and distance on the other side, measured about the central pivot.

例如，当跷跷板平衡时，一侧的重量与距离的乘积等于另一侧重量与距离的乘积，均以中心支点取矩。

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4. Conditions for Equilibrium | 平衡条件

A rigid body is in static equilibrium when two independent conditions are satisfied: the resultant force in any direction is zero, and the resultant moment about any point is zero.

刚体处于静力平衡时，必须满足两个独立条件：任意方向上的合力为零，且对任意点的合力矩为零。

ΣF = 0 and ΣM = 0

For coplanar forces, this usually means resolving forces horizontally and vertically, and taking moments about a convenient pivot. If only forces are being found, one condition may be enough; but a complete equilibrium problem uses both.

对于共面力系，通常需要水平和垂直方向分解力，并对方便选择的支点取矩。如果只求力，一个条件可能足够；但完整的平衡问题需要同时使用两个条件。

• Draw all forces acting on the body.
• Resolve forces into horizontal and vertical components if necessary.
• Take moments about a pivot that eliminates an unknown reaction.
• Solve the resulting equations simultaneously.

画出作用在物体上的所有力；必要时将力分解为水平和垂直分量；对可以消去未知反力的支点取矩；最后联立求解方程。

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5. Couples and Torque | 力偶与扭矩

A couple is a pair of equal and opposite parallel forces whose lines of action do not meet. A couple produces pure rotation with no resultant linear force. Its moment is the product of one force and the perpendicular distance between the two lines of action.

力偶是一对大小相等、方向相反、作用线不重合的平行力。力偶产生纯转动而没有净线性力。其力矩等于其中一个力乘以两条作用线之间的垂直距离。

C = F × d

Couples are important in steering wheels, taps and electric motors. Because the net force is zero, a couple has the same moment about every point in the plane.

力偶在方向盘、水龙头和电动机中很重要。由于净力为零，力偶对平面内任意点的力矩都相同。

The term torque is often used interchangeably with moment, especially when referring to a couple or a rotating shaft. In Edexcel questions, torque and moment of a couple are calculated in the same way.

“扭矩”一词常与力矩互换使用，尤其是在指力偶或旋转轴时。在 Edexcel 考题中，扭矩和力偶矩的计算方法相同

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