Moments and Rotational Equilibrium in IGCSE Physics | IGCSE物理：力矩与物体的转动平衡

📚 Moments and Rotational Equilibrium in IGCSE Physics | IGCSE物理：力矩与物体的转动平衡

In IGCSE Physics, forces are not only responsible for making objects move in straight lines — they can also cause objects to rotate. The turning effect of a force is called a moment, and understanding how moments combine to produce rotational equilibrium is essential for solving many exam problems involving levers, beams, and balanced objects.

在 IGCSE 物理中，力不仅仅能使物体沿直线运动——它们还能使物体转动。力对物体产生的转动效果称为力矩。理解力矩如何组合并达到转动平衡，是解答涉及杠杆、横梁和平衡物体等许多考题的关键。

1. What Is a Moment? | 什么是力矩？

A moment (sometimes called the moment of a force) is the turning effect that a force produces around a pivot or fulcrum. Everyday examples include pushing a door handle, using a spanner to turn a nut, or applying force to a seesaw.

力矩（有时称为力的力矩）是力围绕支点或转轴所产生的转动效果。日常生活中的例子包括推门把手、用扳手拧螺母、或者对跷跷板施力。

The size of the moment depends on two factors: the size of the force applied, and the perpendicular distance from the pivot to the line of action of that force. A larger force or a larger distance produces a greater turning effect.

力矩的大小取决于两个因素：所施加的力的大小，以及从支点到该力的作用线的垂直距离。力越大或距离越大，产生的转动效果就越显著。

2. The Formula: Moment = Force × Perpendicular Distance | 力矩的计算公式

For a force F acting at a perpendicular distance d from the pivot, the moment M is calculated using the following equation:

对于作用在距支点垂直距离 d 处的力 F，力矩 M 可用以下公式计算：

M = F × d

In this equation, F is measured in newtons (N), d is measured in metres (m), and the moment M is measured in newton-metres (N m).

在此公式中，F 以牛顿（N）为单位，d 以米（m）为单位，力矩 M 以牛米（N m）为单位。

It is important to remember that d is not simply the distance from the pivot to where the force is applied — it must be the perpendicular distance, i.e. the shortest distance between the pivot and the line of action of the force.

务必记住，d 并不是支点到力的作用点之间的任意距离——它必须是垂直距离，即支点到力的作用线之间的最短距离。

3. The Perpendicular Distance (Lever Arm) | 力臂

The perpendicular distance between the pivot and the line of action of the force is often called the lever arm. To interpret this correctly, imagine extending the line of action of the force indefinitely in both directions; the lever arm is the perpendicular distance from the pivot to this extended line.

支点到力的作用线之间的垂直距离通常称为力臂。为了正确理解这一点，可以想象将力的作用线向两端无限延伸；力臂即支点到这条延伸线之间的垂直距离。

When the force is applied at an angle, the lever arm is shorter than the actual distance from the pivot to the point of application. For example, if you push a door at its edge at an angle of 30° to the door surface, only the component of force perpendicular to the door contributes to rotation.

当力以一定角度施加时，力臂比从支点到施力点的实际距离更短。例如，如果你以与门面成 30° 的角度推门的边缘，只有垂直于门的力的分量才产生转动效果。

lever arm = perpendicular distance from pivot to line of action

力臂 = 从支点到力的作用线的垂直距离

4. Units of Moment | 力矩的单位

The unit of moment is the newton-metre (N m). Although newton-metre has the same base units as the joule (J), they are physically different quantities: a joule measures energy or work, while a newton-metre measures the turning effect of a force. In exam questions, always write the unit as N m, not J.

力矩的单位是牛米（N m）。虽然牛米与焦耳（J）具有相同的基本单位，但它们是物理意义不同的量：焦耳衡量能量或做功，而牛米衡量力的转动效果。在考题中，务必写成 N m，不可写成 J。

Quantity	Symbol	Unit
Force	F	newton (N)
Perpendicular distance	d	metre (m)
Moment	M	newton-metre (N m)

In the Edexcel IGCSE syllabus, you may also see the symbol Mᶜ used for the moment of a force in structured calculations — this simply stands for the calculated moment and does not change the formula.

在 Edexcel IGCSE 考纲中，你也可能会看到符号 Mᶜ 用于结构化的力矩计算——这只是代表计算所得的力矩，并不会改变公式本身。

5. Direction of Rotation and Sign Convention | 转动方向与正负约定

A moment can turn an object either clockwise or anticlockwise around the pivot. In calculations, it is convenient to label one direction as positive and the other as negative. By convention, when analysing rotational equilibrium, clockwise moments are usually taken as negative and anticlockwise moments as positive — but the key rule is to be consistent.

力矩可以使物体绕支点顺时针或逆时针转动。在计算中，约定一个方向为正、另一个方向为负是比较方便的。按惯例，在分析转动平衡时，通常将顺时针力矩取为负值、逆时针力矩取为正值——但关键是保持一致。

To determine the direction of a moment, imagine applying the force and ask: would this turn the object clockwise or anticlockwise about the pivot? Do not guess — trace the motion mentally or with a diagram.

要判断力矩的方向，可以想象施加该力，并思考：这个力会使物体绕支点顺时针还是逆时针转动？不要凭直觉猜测——请在脑海中追踪运动或借助图示判断。

clockwise moment: M = −F × d | anticlockwise moment: M = +F × d

顺时针力矩：M = −F × d | 逆时针力矩：M = +F × d

6. Moving a Force along Its Line of Action | 沿力的作用线平移力

A special property of moments is that if a force is shifted along its line of action (without rotating it), the moment it produces about a given pivot remains unchanged. This is because the perpendicular distance from the pivot to the line of action stays the same.

力矩的一个特殊性质是：如果力沿其作用线平移（不旋转方向），则它对给定支点所产生的力矩保持不变。这是因为支点到作用线的垂直距离始终保持不变。

This idea is very useful in problem solving. For example, the weight of a uniform rod acts at its centre, regardless of how the rod is positioned. When drawing a free-body diagram for rotational problems, you can treat the entire weight of an object as acting at its centre of gravity.

这一思路在解题中非常有用。例如，均匀杆的重力作用在其中心，无论杆如何放置都如此。在绘制转动问题的受力图时，可以将整个物体的重力视为作用在其重心上。

7. The Principle of Moments (First Law) | 力矩原理（第一定律）

When an object is in rotational equilibrium, the total clockwise moment about any pivot equals the total anticlockwise moment about that same pivot. This statement is called the principle of moments.

当物体处于转动平衡状态时，绕任意支点的顺时针力矩之总和等于绕同一支点的逆时针力矩之总和。这一表述称为力矩原理。

sum of clockwise moments = sum of anticlockwise moments

顺时针力矩之和 = 逆时针力矩之和

This principle allows us to solve for unknown forces or distances in a balanced system. For example, if a beam is balanced on a pivot and we know three of the four quantities (forces and distances), we can always find the fourth.

这一原理使我们能够求解平衡系统中的未知力或未知距离。例如，如果一根横梁在支点上保持平衡，并且我们已知四个量（力和距离）中的三个，就总能求出第四个量。

The first law of moments applies only to rotation. For an object to be completely at rest, it must also satisfy the condition that all external forces balance in both the vertical and horizontal directions.

力矩第一定律仅适用于转动。要使物体完全静止，还必须满足所有外力在竖直和水平方向上都平衡的条件。

8. Complete Conditions for Equilibrium | 完整的平衡条件

For a rigid object to be in complete equilibrium (both translational and rotational), two conditions must be satisfied simultaneously:

要使刚体处于完全平衡状态（既平动平衡又转动平衡），必须同时满足两个条件：

First condition (translational): the vector sum of all external forces acting on the object is zero, i.e. the resultant force in any direction is zero.

第一条件（平动平衡）：物体受到的所有外力的矢量和为零，即任意方向上的合力为零。
Second condition (rotational): the sum of all clockwise moments equals the sum of all anticlockwise moments about any chosen pivot.

第二条件（转动平衡）：绕任意选定支点，所有顺时针力矩之和等于所有逆时针力矩之和。

This two-part condition is sometimes referred to in textbooks as the first and second laws of moments. In Edexcel IGCSE questions, you may be asked to verify equilibrium by checking both the forces and the moments separately.

这一双重要件有时在教科书中被称为力矩第一定律和第二定律。在 Edexcel IGCSE 考题中，你可能会被要求分别检查力与力矩来验证平衡状态。

9. Worked Example: Balancing a Uniform Rod | 计算示例：平衡的均匀杆

A uniform rod AB has length 5.0 m and mass 2.0 kg. It is pivoted at a point O which is 2.0 m from end A. A 5.0 kg mass is hung at a point 1.5 m from O on the side of A. What downward force F must be applied at end B to keep the rod horizontal?

一根均匀杆 AB 长 5.0 m，质量为 2.0 kg。支点 O 距 A 端 2.0 m。一个 5.0 kg 的物体悬挂在距 O 点 1.5 m 的 A 侧。为使杆保持水平，需在 B 端施加多大的竖直向下力 F？

Take g = 10 N/kg. First, identify all forces and their distances from O.

取 g = 10 N/kg。首先确定所有力及其到 O 点的距离。

The weight of the rod acts at its centre. Since the rod is uniform, the centre is at 2.5 m from A, which is 0.5 m to the right of O. This produces a clockwise moment.

杆的重力作用在其中心。由于杆是均匀的，中心距 A 端 2.5 m，即位于 O 点右侧 0.5 m 处。这产生一个顺时针力矩。

The 5.0 kg mass exerts a downward force of 5.0 × 10 = 50 N, at a distance of 1.5 m to the left of O. This produces an anticlockwise moment.

5.0 kg 的物体施加向下的力 5.0 × 10 = 50 N，位于 O 点左侧 1.5 m 处。这产生一个逆时针力矩。

The unknown force F acts at end B, which is 3.0 m to the right of O. It produces a clockwise moment.

未知力 F 作用在 B 端，位于 O 点右侧 3.0 m 处。它产生一个顺时针力矩。

Force	Force (N)	Distance from O (m)	Direction
Weight of rod	20	0.5	Clockwise
5.0 kg mass	50	1.5	Anticlockwise
Force F at end B	F	3.0	Clockwise

Using the principle of moments:

利用力矩原理：

Anticlockwise moment = Clockwise moments

逆时针力矩 = 顺时针力矩之和

50 × 1.5 = 20 × 0.5 + F × 3.0

Solving:

求解：

75 = 10 + 3F → 3F = 65 → F = 21.7 N

So a downward force of approximately 21.7 N must be applied at end B for the rod to remain horizontal.

因此，为使杆保持水平，需在 B 端施加约 21.7 N 的竖直向下的力。

10. Centre of Gravity and Stability | 重心与稳定性

The centre of gravity of an object is the point through which its entire weight appears to act. For a uniform object, the centre of gravity is at its geometric centre. For irregularly shaped objects, the centre of gravity can be found experimentally by hanging the object from different points and drawing vertical lines from each suspension point; the intersection of these lines gives the centre of gravity.

物体的重心是物体全部重力似乎作用通过的那一点。对于均匀物体，重心位于其几何中心。对于形状不规则的物体，可以通过实验方法找到重心：从不同点悬挂物体，并从每个悬挂点画出竖直线；这些线的交点即为重心。

The position of the centre of gravity determines whether an object is stable or topples over. An object will topple if the vertical line through its centre of gravity falls outside its base of support. The wider the base and the lower the centre of gravity, the more stable the object.

重心的位置决定了物体是稳定还是翻倒。如果通过重心的竖直线落在物体的支撑面之外，物体就会翻倒。支撑面越宽、重心越低，物体就越稳定。

For example, a double-decker bus is made stable by placing heavy luggage compartments at the bottom, lowering the centre of gravity. Racing cars are designed low and wide for the same reason.

例如，双层巴士通过将沉重的行李舱设置在底部来降低重心，从而使其更加稳定。赛车设计得又低又宽也是出于同样的原因。

11. Everyday Applications of Moments | 力矩在日常生活中的应用

Moments are at work in countless everyday tools. A lever such as a crowbar allows a small effort to move a large load because the effort arm is longer than the load arm. The mechanical advantage of a lever is given by the ratio of the load arm to the effort arm.

力矩广泛应用于无数日常工具中。撬棍等杠杆允许用较小的力移动较大的重物，因为施力臂比阻力臂更长。杠杆的机械优势等于阻力臂与施力臂之比。

Other common examples include:

其他常见示例包括：

A door handle is placed far from the hinges to maximise the distance d, making it easier to rotate the door about its hinge.

门把手设置在远离铰链的位置，以增大距离 d，使门更容易绕铰链转动。
A spanner has a long handle for the same reason — increasing the lever arm increases the moment for a given force.

扳手具有长手柄也是出于同样的原因——增大力臂可以在相同作用力下增大力矩。
A wheelbarrow uses the principle of moments to lift heavy loads: the wheel acts as the pivot, and the handles provide a long lever arm for the applied effort.

独轮手推车运用力矩原理来抬起重物：轮子作为支点，手柄为所施加的力提供较长的力臂。

In each of these examples, the fundamental physics reduces to one simple relationship: moment equals force times perpendicular distance. By adjusting either factor, the turning effect can be increased or decreased as needed.

在以上每一个例子中，基本的物理都归结为一个简单的关系式：力矩等于力乘以垂直距离。通过调整任意一个因素，转动效果都可以按需增大或减小。

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Moments and Rotational Equilibrium in IGCSE Physics | IGCSE物理：力矩与物体的转动平衡