📚 Pulleys | 滑轮
In Edexcel A Level Mechanics, pulley problems are a core application of Newton's second law. A pulley allows a string to change the direction of a force, so two or more particles can be connected and move with linked motion. To solve these problems, students must combine modelling assumptions, force diagrams, and simultaneous equations.
在 Edexcel A Level 力学中，滑轮问题是牛顿第二定律的核心应用。滑轮可以改变绳子中力的方向，使两个或多个物体连接起来并产生联动运动。解决这类问题需要综合运用建模假设、受力分析和联立方程。
1. The Standard Pulley Model | 标准滑轮模型
In most A Level questions, a smooth pulley is fixed at the edge of a table or at the top of a vertical system. A light inextensible string passes over the pulley, with particles attached to each end. The pulley only changes the direction of the tension force; it does not change its magnitude.
在大多数 A Level 题目中，光滑滑轮固定在桌边或竖直系统顶端。一条轻质且不可伸长的绳子跨过滑轮，两端各系一个质点。滑轮只改变绳子张力的方向，不改变其大小。
You should always begin by drawing a clear diagram showing the pulley, the string, the masses, and the directions of motion. Label the tension T on both sides of the string and choose a positive direction for each particle.
解题时应首先画出清晰的示意图，标出滑轮、绳子、质量以及运动方向。在绳子两侧标出张力 T，并为每个质点选定正方向。
- Masses are often labelled m₁ and m₂.
- 质量通常标记为 m₁ 和 m₂。
- The string is treated as having no mass and no stretching.
- 绳子被视为没有质量且不可伸长。
- The pulley is smooth, so there is no friction between the string and the pulley.
- 滑轮是光滑的，因此绳子与滑轮之间没有摩擦。
2. Modelling Assumptions | 建模假设
Pulley problems rely on three standard modelling assumptions. These assumptions simplify the real situation and allow the use of constant acceleration equations and Newton's second law.
滑轮问题依赖于三个标准的建模假设。这些假设简化了实际情况，使我们可以使用匀加速运动方程和牛顿第二定律。
| Assumption | Consequence | 中文解释 |
| Light string | Tension is the same throughout the string. | 轻绳：张力在绳中处处相同。 |
| Inextensible string | Connected particles have the same acceleration. | 不可伸长的绳：相连质点的加速度相同。 |
| Smooth pulley | The pulley does not change the tension magnitude. | 光滑滑轮：滑轮不改变张力大小。 |
If a question does not state that the string is light or the pulley is smooth, you should still assume these conditions unless a rough pulley or a heavy string is specified.
如果题目没有明确说明绳子是轻的或滑轮是光滑的，除非题目特别说明滑轮粗糙或绳子有质量，否则仍应默认这些条件成立。
3. Tension in a Light String | 轻绳中的张力
Tension is the pulling force transmitted through a string. In a light inextensible string passing over a smooth pulley, the tension can be treated as the same on both sides of the pulley. This is because a light string has no mass, so no resultant force is needed to accelerate the string itself.
张力是通过绳子传递的拉力。在跨过光滑滑轮的轻质不可伸长绳子中，绳子两边的张力可以视为相等。这是因为轻绳没有质量，因此不需要合力来加速绳子本身。
When you draw force diagrams, show the tension T pulling away from each particle along the string. Tension always pulls, never pushes. The direction of T on each mass depends on which end of the string is attached to that mass.
画受力图时，应标出张力 T 沿着绳子方向拉离每个质点。张力总是拉力，而不是推力。每个质量所受的张力的方向取决于绳子连接的是哪一端。
Tleft = Tright = T
左侧张力 = 右侧张力 = T
4. Setting Up Equations of Motion | 建立运动方程
To solve a pulley problem, apply Newton's second law to each particle separately. Start by choosing a positive direction for each particle, usually the direction in which it accelerates.
解决滑轮问题时，需要分别对每个质点应用牛顿第二定律。首先要为每个质点选定正方向，通常选取其加速度方向为正方向。
For a particle of mass m, the equation of motion is:
对于质量为 m 的质点，运动方程为：
F = ma
合力 = 质量 × 加速度
Write the resultant force as the sum of all forces acting along the chosen positive direction. For example, if upward is positive for one mass, then tension is positive and weight is negative.
把合力写成沿所选正方向的所有力的代数和。例如，如果某个质量以向上为正方向，那么张力为正，重力为负。
Because the string is inextensible, both particles have the same magnitude of acceleration a. This is the key link between the two equations.
由于绳子不可伸长，两个质点的加速度大小 a 相同。这是联系两个方程的关键。
5. Two Masses Hanging Vertically | 两个竖直悬挂的质量
The classic pulley problem involves two masses m₁ and m₂ hanging vertically on either side of a fixed smooth pulley. If m₂ is greater than m₁, then m₂ accelerates downwards and m₁ accelerates upwards.
经典滑轮问题是两个质量 m₁ 和 m₂ 竖直悬挂在固定光滑滑轮两侧。如果 m₂ 大于 m₁，则 m₂ 向下加速，m₁ 向上加速。
For mass m₁, taking upward as positive:
对于质量 m₁，取向上为正方向：
T − m₁g = m₁a
For mass m₂, taking downward as positive:
对于质量 m₂，取向下为正方向：
m₂g − T = m₂a
Adding the two equations eliminates T and gives:
将两个方程相加消去 T，得到：
a = (m₂ − m₁)g / (m₁ + m₂)
Substituting this acceleration back into either equation gives the tension:
将加速度代回任意一个方程，可求出张力：
T = 2m₁m₂g / (m₁ + m₂)
These two results are very useful for checking your working. The tension is always less than the weight of the heavier mass but greater than the weight of the lighter mass.
这两个结果非常便于检查计算过程。张力总是小于较重物体的重力，但大于较轻物体的重力。
6. Worked Example | 例题解析
ing. Find the acceleration of the system and the tension in the string. Take g = 9.8 m/s².
一条轻质不可伸长的绳子跨过光滑滑轮，两端分别系有 3 kg 和 5 kg 的质量。求系统的加速度和绳中的张力。取 g = 9.8 m/s²。
Let m₁ = 3 kg and m₂ = 5 kg. Since m₂ is heavier, m₂ moves downwards and m₁ moves upwards.
设 m₁ = 3 kg，m₂ = 5 kg。由于 m₂ 更重，m₂ 向下运动，m₁ 向上运动。
Using the standard formula for acceleration:
使用标准加速度公式：
a = (5 − 3) × 9.8 / (5 + 3) = 19.6 / 8 = 2.45 m/s²
For the tension, use the equation for the 3 kg mass:
求张力时，对 3 kg 的质量列方程：
T − 3g = 3a
T = 3 × 9.8 + 3 × 2.45 = 29.4 + 7.35 = 36.75 N
You can check this by using the 5 kg mass equation:
你也可以用 5 kg 质量的方程进行检验：
5g − T = 5a
T = 49 − 12.25 = 36.75 N
Both equations give the same tension, which confirms the solution is consistent.
两个方程给出的张力相同，这证明解答是一致的。
7. One Mass on a Smooth Horizontal Table | 光滑水平桌面上的连接体
In this type of question, a mass m₁ rests on a smooth horizontal table and is connected by a light inextensible string passing over a smooth pulley at the edge of the table to a hanging mass m₂.
在这类题目中，质量 m₁ 静止在光滑水平桌面上，通过一条跨过桌边光滑滑轮的轻质不可伸长绳子与悬挂的质量 m₂ 相连。
For the mass on the table, the only horizontal force is the tension T, so:
对于桌面上的质量，水平方向只受张力 T 作用，因此：
T = m₁a
For the hanging mass, taking downward as positive:
对于悬挂的质量，取向下为正方向：
m₂g − T = m₂a
Adding the two equations gives:
将两个方程相加，得到：
a = m₂g / (m₁ + m₂)
The tension is then:
张力为：
T = m₁m₂g / (m₁ + m₂)
Notice that the acceleration is smaller than g because the table mass must also be accelerated. If the table is rough, you would need to include a friction force acting against the motion.
注意到加速度小于 g，因为桌面上的质量也需要被加速。如果桌面粗糙，还需要考虑与运动方向相反的摩擦力。
8. Pulley with an Inclined Plane | 斜面上的滑轮
A common exam variation places one mass on a smooth inclined plane while the other mass hangs vertically. The string lies parallel to the slope. The component of the weight acting down the slope is m₁g sin θ, where θ is the angle of inclination.
一种常见的考试变形是把一个质量放在光滑斜面上，另一个质量竖直悬挂。绳子与斜面平行。重力沿斜面向下的分量为 m₁g sin θ，其中 θ 为斜面倾角。
If m₂ is large enough, m₂ moves downwards and m₁ moves up the slope. Taking up the slope as positive for m₁ and downward as positive for m₂:
如果 m₂ 足够大，m₂ 向下运动，m₁ 沿斜面向上运动。对 m₁ 取沿斜面向上为正方向，对 m₂ 取向下为正方向：
T − m₁g sin θ = m₁a
m₂g − T = m₂a
Adding these equations gives:
将方程相加，得到：
a = (m₂g − m₁g sin θ) / (m₁ + m₂)
If m₁g sin θ is greater than m₂g, then the motion reverses: m₁ slides down the slope and m₂ rises. Always compare the driving forces before choosing the direction of acceleration.
如果 m₁g sin θ 大于 m₂g，则运动方向相反：m₁ 沿斜面下滑，m₂ 上升。在选择加速度方向之前，一定要先比较驱动力的大小。
9. Common Mistakes and Exam Tips | 常见错误与应试技巧
One common mistake is choosing different positive directions for the two masses without realising it. This can lead to incorrect signs and an impossible acceleration.
一个常见错误是在没有意识到的情况下为两个质量选择了不同的正方向。这可能导致符号错误以及不可能的加速度结果。
Another mistake is assuming the acceleration is equal to g or that the tension is equal to the weight of one mass. In a connected system, both acceleration and tension must be found from the simultaneous equations.
另一个错误是认为加速度等于 g，或认为张力等于某个质量的重力。在连接体系中，加速度和张力都必须通过联立方程求出。
Exam tips:
应试技巧：
- Draw a labelled force diagram for each particle.
- 为每个质点画出带标注的受力图。
- Choose and clearly state a positive direction for each mass.
- 为每个质量选择并明确说明正方向。
- Use F = ma in the direction of motion, not perpendicular to it.
- 沿运动方向使用 F = ma，而不是垂直于运动方向。
- Check that the acceleration is less than g for vertical systems.
- 检查竖直系统中加速度是否小于 g。
- Check the tension by substituting into the other equation.
- 将结果代入另一个方程检验张力。
10. Summary Checklist | 总结清单
Before you finish a pulley question, run through this checklist. It will help you avoid losing marks on both method and accuracy.
完成滑轮题目之前，请对照这份清单检查。它能帮助你避免在方法和计算准确性上失分。
- Assume the string is light, inextensible and the pulley is smooth unless told otherwise.
- 除非另有说明，假设绳子是轻质且不可伸长的，滑轮是光滑的。
- Mark the tension T as the same on both sides of the pulley.
- 标出滑轮两侧张力 T 相同。
- Give both particles the same acceleration a.
- 让两个质点具有相同的加速度 a。
- Choose a positive direction for each mass and stick to it.
- 为每个质量选择一个正方向并保持一致。
- Write an F = ma equation for each mass.
- 为每个质量写出 F = ma 方程。
- Solve the simultaneous equations for a and T.
- 联立求解 a 和 T。
- Check units and give final answers to an appropriate degree of accuracy.
- 检查单位，并以适当的精度给出最终答案。
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