Connected Particles | 连接粒子问题

📚 Connected Particles | 连接粒子问题

Connected particles problems are a family of mechanics questions in Edexcel A-Level Mathematics that involve two or more particles joined by strings, often passing over smooth pulleys. The central idea is that the tension in a light, inextensible string and the shared acceleration create a link between the motion of each particle.

连接粒子问题是 Edexcel A-Level 数学中力学部分的一类经典题型，涉及由绳子连接并通过光滑滑轮的两个或多个粒子。核心思想是：轻质不可伸长绳子中的张力以及粒子间共同的加速度，将各个粒子的运动联系在一起。

---

1. Fundamental Assumptions | 基本假设

Every connected particles problem rests on a set of standard assumptions. The string is modelled as light (zero mass) and inextensible (it does not stretch). This means tension is the same at every point of the string, and both particles move with the same magnitude of acceleration.

每个连接粒子问题都基于一组标准假设。绳子被建模为轻质（质量为零）且不可伸长（不会拉伸）。这意味着绳子各点的张力相同，且两个粒子以相同大小的加速度运动。

• Light string: tension is uniform throughout | 轻绳：张力处处相等
• Inextensible: equal acceleration for both particles | 不可伸长：两个粒子加速度大小相等
• Smooth pulley: no friction at the pulley, tension passes over unaltered | 光滑滑轮：滑轮处无摩擦，张力不变地传递
• Particles: treated as point masses | 粒子：视为质点

---

2. Newton’s Second Law Applied | 牛顿第二定律的应用

For a single particle, Newton’s second law states that the resultant force on the particle equals mass times acceleration: F = ma. For connected particles, we apply this law separately to each particle along its direction of motion.

对于单个粒子，牛顿第二定律指出：粒子所受合外力等于质量乘以加速度，即 F = ma。对于连接粒子，我们沿每个粒子的运动方向分别应用该定律。

F = ma

When solving problems, it is essential to choose a positive direction first. Typically, the direction of expected motion is taken as positive, and forces opposing that direction are written with a negative sign.

在解题时，必须先选定正方向。通常取预期的运动方向为正，与该方向相反的力则冠以负号。

---

3. The Atwood Machine: Two Hanging Particles | 阿特伍德机：两个悬挂粒子

The simplest connected-particles system is the Atwood machine: two particles of masses m₁ and m₂ hang from a light string passing over a smooth fixed pulley. Assuming m₁ > m₂, the heavier particle accelerates downwards and the lighter one accelerates upwards.

最简单的连接粒子系统是阿特伍德机：质量分别为 m₁ 和 m₂ 的两个粒子悬挂在绕过光滑定滑轮的轻绳两端。假设 m₁ > m₂，则较重的粒子向下加速，较轻的粒子向上加速。

For the heavier particle, taking downwards as positive:

对较重的粒子，取向下为正方向：

m₁g − T = m₁a

For the lighter particle, taking upwards as positive:

对较轻的粒子，取向上为正方向：

T − m₂g = m₂a

Adding the two equations eliminates T and gives the acceleration. Alternatively, treating the whole system as a single object of mass m₁ + m₂, the net driving force is (m₁ − m₂)g.

将两式相加消去 T 即可得到加速度。或者，将整个系统视为质量为 m₁ + m₂ 的整体，净驱动力为 (m₁ − m₂)g。

a = (m₁ − m₂)g / (m₁ + m₂)

Once a is known, substitute back into either equation to find the tension:

求出 a 后，代回任一方程即可求得张力：

T = 2m₁m₂g / (m₁ + m₂)

---

4. Particle on a Smooth Horizontal Table | 光滑水平桌面上的粒子

A common variation places one particle on a smooth horizontal table, connected by a string that passes over a pulley at the edge of the table to a second particle hanging vertically. The hanging particle accelerates downwards, pulling the table particle horizontally.

一个常见的变形是：一个粒子放在光滑水平桌面上，绳子绕过桌边滑轮与一个竖直悬挂的粒子相连。悬挂的粒子向下加速，拉动桌面上的粒子水平运动。

For the hanging particle (mass m), taking downwards as positive: mg − T = ma.

对悬挂粒子（质量 m），取向下为正：mg − T = ma。

For the table particle (mass M), taking the direction of motion (towards the pulley) as positive: T = Ma.

对桌面上的粒子（质量 M），取运动方向（朝向滑轮）为正：T = Ma。

Substituting and solving gives:

代入求解得：

a = mg / (M + m), T = Mmg / (M + m)

Notice the structure: the acceleration is the “driving force” divided by the “total mass” of the whole system.

注意其结构：加速度等于”驱动力”除以整个系统的”总质量”。

---

5. Rough Table and Friction | 粗糙桌面与摩擦力

If the table is rough, a frictional force acts on the table particle opposing its motion. For a particle of mass M on a rough horizontal surface, the normal reaction is N = Mg, and the limiting friction is F = μN = μMg.

若桌面粗糙，则桌面上的粒子会受到与运动方向相反的摩擦力。对于质量 M 的粒子，水平面上的法向反力 N = Mg，最大静摩擦力 F = μN = μMg。

Assuming motion occurs (so kinetic friction applies), Newton’s second law for the table particle becomes:

假设运动发生（此时为动摩擦力），桌面上粒子的牛顿第二定律为：

T − μMg = Ma

For the hanging particle, still: mg − T = ma.

悬挂粒子仍然满足：mg − T = ma。

Adding the equations yields:

相加得：

a = g(m − μM) / (M + m)

For motion to occur at all, we must have m > μM. Otherwise the system remains at rest with the friction balancing the tension.

要使系统运动，必须有 m > μM。否则系统保持静止，摩擦力与张力平衡。

---

6. Inclined Plane Systems | 斜面系统

Particles may also lie on smooth or rough inclined planes. The key step is resolving the weight into components parallel and perpendicular to the plane: mg sinθ down the plane and mg cosθ perpendicular to the plane.

粒子也可能位于光滑或粗糙的斜面上。关键步骤是将重力分解为平行于斜面和垂直于斜面的分量：沿斜面向下的 mg sinθ 和垂直于斜面的 mg cosθ。

For a particle of mass M on a rough plane inclined at angle θ, connected via a pulley at the top to a hanging particle of mass m, the equations depend on the direction of motion. Assuming the hanging particle is heavy enough to pull the plane particle up the slope:

对于粗糙斜面上质量为 M 的粒子，斜面倾角为 θ，通过斜面顶端滑轮与质量为 m 的悬挂粒子相连，方程的形式取决于运动方向。假设悬挂粒子足够重，能将斜面上的粒子沿斜面向上拉动：

For the particle on the plane, up the slope is positive:

对斜面上的粒子，沿斜面向上为正：

T − Mg sinθ − μMg cosθ = Ma

For the hanging particle, downwards is positive:

对

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）