📚 KS3 CIE Moments Worksheet | KS3 CIE 力矩练习与讲解
In KS3 Science, a moment is the turning effect of a force. This worksheet explains how to calculate moments, use the principle of moments, and avoid the common errors that lose marks. Every idea is given in English and Chinese, with worked examples and practice questions you can try straight away.
在 KS3 科学中，力矩是力产生的转动效果。本练习讲解如何计算力矩、使用力矩原理，并避免那些容易丢分的常见错误。每个知识点都提供中英对照，并配有例题与可以立即动手尝试的练习题。
1. What is a Moment? | 什么是力矩？
A moment is the turning effect of a force acting on an object around a pivot. When you push a door open, tighten a nut with a spanner, or sit on a see-saw, you are using moments. The bigger the moment, the more easily the object turns.
力矩是力作用在物体上时绕支点产生的转动效果。当你推开门、用扳手拧螺母或坐在跷跷板上时，你都在使用力矩。力矩越大，物体就越容易被转动。
The size of a moment depends on two things: the size of the force, and the perpendicular distance from the pivot to the line of action of the force.
力矩的大小取决于两个因素：力的大小，以及从支点到力的作用线的垂直距离。
Three key words are used again and again in this topic, so learn them now.
这个主题里有三个关键词会反复出现，现在就记住它们。
- Pivot - the fixed point that the object turns around. 支点——物体绕其转动的固定点。
- Line of action - the imaginary line that a force acts along. 作用线——力沿其作用的假想线。
- Perpendicular distance - the shortest distance from the pivot to the line of action, measured at 90°. 垂直距离——从支点到作用线的最短距离，按 90° 测量。
A moment is not a force. A force is a push or a pull measured in newtons; a moment is a turning effect measured in newton metres.
力矩不是力。力是用牛顿测量的推或拉；力矩是用牛顿米测量的转动效果。
2. The Moment Equation | 力矩公式
You can calculate a moment using this equation:
你可以使用以下公式计算力矩：
moment = force × perpendicular distance from the pivot
M = F × d
In this equation, M stands for moment, F stands for force, and d stands for the perpendicular distance from the pivot to the line of action of the force.
在这个公式中，M 表示力矩，F 表示力，d 表示从支点到力的作用线的垂直距离。
Always check that the distance is measured at a right angle to the direction of the force. This is the perpendicular distance. If the distance is measured along the object but the force does not act at 90° to it, the calculation will give the wrong answer.
一定要检查距离是与力的方向成直角测量的。这就是垂直距离。如果距离是沿着物体测量的，而力并非与它成 90°，计算就会得出错误答案。
÷ d and d = M ÷ F. In KS3 papers you are usually asked to find the moment or the distance.
公式可以变形：M = F × d，所以 F = M ÷ d，d = M ÷ F。在 KS3 试卷中，通常要求你求力矩或距离。
3. Units and Symbols | 单位与符号
Force is measured in newtons (N) and distance in metres (m), so a moment is measured in newton metres (N m). You may also see it written as N·m; both mean exactly the same thing.
力的单位是牛顿（N），距离的单位是米（m），所以力矩的单位是牛顿米（N m）。你也可能看到写成 N·m，两者含义完全相同。
metres must be divided by 100 to turn it into metres: 25 cm = 0.25 m, and 120 cm = 1.20 m.
计算前先统一单位。如果距离以厘米给出，必须先除以 100 换算成米：25 cm = 0.25 m，120 cm = 1.20 m。
A moment is not energy, even though both can be written as N m. Energy is transferred when a force moves an object along its own line of action, while a moment only turns an object. Never write J for a moment in an exam.
力矩不是能量，尽管两者都可以写成 N m。当力沿着自己的作用线使物体移动时，能量才会被传递；而力矩只是使物体转动。考试中绝不要用 J 来写力矩。
- Moment: N m (newton metre). 力矩：N m（牛顿米）。
- Force: N (newton). 力：N（牛顿）。
- Distance: m (metre). 距离：m（米）。
4. Perpendicular Distance Explained | 垂直距离详解
The perpendicular distance is the shortest distance from the pivot to the line of action. Imagine extending the direction of the force into a long straight line, then drawing the shortest line from the pivot that meets it at 90°.
垂直距离是从支点到作用线的最短距离。想象把力的方向延长成一条长直线，然后从支点画一条与它成 90° 的最短直线。
If a force pulls at an angle, the perpendicular distance becomes smaller, so the moment becomes smaller too, even if the size of the force stays the same.
如果力斜着拉，垂直距离就会变小，因此即使力的大小不变，力矩也会随之变小。
A quick check: the perpendicular distance can never be longer than the distance from the pivot to the point where the force is applied. If your value is bigger, you have used the wrong distance.
一个快速判断方法：垂直距离永远不可能大于从支点到力的作用点的距离。如果你的数值更大，说明你用错了距离。
| Situation 情形 | Perpendicular distance 垂直距离 |
| Force at 90° to the handle 力与手柄成 90° | d equals the distance along the handle d 等于沿手柄的距离 |
| Force acting at an angle 力斜着作用 | d is shorter; measure the shortest distance d 更短；测量最短距离 |
| Force passing through the pivot 力通过支点 | d = 0 m, so the moment is zero d = 0 m，所以力矩为零 |
5. Turning Moments Up or Down | 增大或减小力矩
To increase a moment you can increase the force, or increase the perpendicular distance. Everyday designs usually use the second method because it costs nothing extra.
要增大力矩，可以增大作用力，或者增大垂直距离。日常生活中的设计通常采用第二种方法，因为它不需要额外费力。
- Door handles are placed far from the hinge, so a small push gives a large moment. 门把手安装在远离合页的位置，因此轻轻一推就能产生较大的力矩。
- A long spanner gives a bigger moment for the same push than a short one. 相同的推力下，长扳手比短扳手产生更大的力矩。
- The wheelbarrow, the crowbar and the bottle opener are all levers that increase the moment of your hand. 独轮车、撬棍和开瓶器都是增大人手力矩的杠杆。
To decrease a moment, do the opposite: shorten the distance or reduce the force. A short spanner is used for small nuts so that they are not over-tightened.
要减小力矩则相反：缩短距离或减小力。小螺母用小扳手，就是为了避免拧得过紧。
A longer handle always makes the job feel easier, but it does not change the amount of work you do - it only changes how much force you need.
手柄越长，操作感觉越省力，但这并不会改变你做的功——它只改变你需要用多大的力。
6. The Principle of Moments | 力矩原理
When an object is balanced, the total clockwise moment about the pivot equals the total anticlockwise moment about the same pivot. This is called the principle of moments.
当物体保持平衡时，绕支点的顺时针力矩总和等于绕同一支点的逆时针力矩总和。这被称为力矩原理。
sum of clockwise moments = sum of anticlockwise moments
Σ Mclockwise = Σ Manticlockwise
Three details matter. First, both sides must be measured from the same pivot. Second, balanced means the object is not turning at all. Third, the weight of the beam or ruler can be ignored if the question says the beam is uniform and pivoted at its centre, because its own weight then has zero moment.
有三个细节很重要。第一，两侧都必须从同一个支点测量。第二，平衡意味着物体完全不转动。第三，如果题目说明横杆均匀且以中点为支点，就可以忽略横杆自身的重力，因为它自身重力的力矩为零。
If the two moments are not equal, the object will rotate towards the side with the larger moment.
如果两侧力矩不相等，物体就会向力矩较大的一侧转动。
7. Worked Example 1 - A Spanner | 例题一：扳手
A mechanic pushes the end of a spanner with a force of 40 N. The force acts at 90° to the spanner, 0.25 m from the centre of the nut. Calculate the moment.
一位技师用 40 N 的力推扳手末端。力与扳手成 90°，作用点距螺母中心 0.25 m。计算这个力矩。
bstitute the values, M = 40 N × 0.25 m. Step 3: give the answer with the unit, M = 10 N m.
第一步：写出公式，M = F × d。第二步：代入数值，M = 40 N × 0.25 m。第三步：写出带单位的答案，M = 10 N m。
roduced by half the force, because the distance is doubled. This is exactly why long spanners are useful.
现在把力改成 20 N，但推在最末端，距螺母 0.5 m。则 M = 20 N × 0.5 m = 10 N m。力减小一半，距离增大一倍，产生的力矩相同。这正是长扳手有用武之地的原因。
.1 m from the nut, the moment would be 40 × 0.1 = 4 N m - too small to loosen a tight nut.
如果同样的 40 N 只作用在距螺母 0.1 m 处，力矩只有 40 × 0.1 = 4 N m——小到无法拧松卡紧的螺母。
8. Worked Example 2 - A Balanced See-Saw | 例题二：平衡的跷跷板
A see-saw is balanced on a pivot. A boy with a weight of 300 N sits 1.2 m from the pivot on the left. A girl with a weight of 200 N sits on the right. How far from the pivot must she sit to balance the see-saw?
一个跷跷板以支点支撑并保持平衡。体重 300 N 的男孩坐在支点左侧 1.2 m 处。体重 200 N 的女孩坐在右侧。她应坐在距支点多远处才能使跷跷板平衡？
e girl produces a clockwise moment: 200 N × d. By the principle of moments, 360 = 200 × d.
男孩产生逆时针力矩：300 N × 1.2 m = 360 N m。女孩产生顺时针力矩：200 N × d。根据力矩原理，360 = 200 × d。
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