Connected Particles | 连接粒子系统

📚 Connected Particles | 连接粒子系统

In A-Level Mechanics, connected particles refer to systems where two or more objects are linked by a light, inextensible string, a rod, or are in contact, causing their motions to be interdependent. The analysis of such systems relies on Newton's laws of motion and the concept of tension, and forms a core part of the Edexcel Mechanics syllabus.

在A-Level力学中，连接粒子系统指的是两个或多个物体通过轻绳、轻杆或直接接触而相互连接，使得它们的运动相互依赖。分析这类系统需要运用牛顿运动定律和张力的概念，是爱德思力学考纲的核心内容。

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1. Key Assumptions in Connected Particle Problems | 连接粒子问题的关键假设

When solving connected particle problems, we rely on several key modelling assumptions to simplify the analysis while keeping the physics accurate.

在解决连接粒子问题时，我们依赖几个关键的建模假设来简化分析，同时保持物理的准确性。

1. Light string: The string is considered to have negligible mass. This means the tension is uniform throughout its length and we do not need to account for any additional inertia from the string itself.

1. 轻绳：绳的质量被视为可忽略不计。这意味着绳上各处的张力是均匀的，我们不需要考虑绳本身带来的额外惯性。

2. Inextensible string: The string does not stretch under tension. Therefore, any acceleration of one attached particle is immediately transmitted to the other, so all connected particles move with the same magnitude of acceleration.

2. 不可伸长的绳：绳在拉力作用下不会伸长。因此，一个连接粒子的加速度会立即传递给另一个，使得所有连接粒子的加速度大小相同。

3. Smooth pulley: The pulley has no friction at its axle and its own mass is negligible. This ensures the tension is the same on both sides of the pulley.

3. 光滑滑轮：滑轮轴处无摩擦，且滑轮自身的质量可忽略。这样就保证了滑轮两侧的张力相等。

4. Air resistance is ignored unless otherwise stated.

4. 除非另有说明，空气阻力忽略不计。

5. Strings remain taut throughout the motion, so tension never falls to zero (unless the problem specifically asks for a slack condition).

5. 在整个运动过程中绳子保持绷紧，因此张力不会降为零（除非题目特别要求松弛条件）。

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2. System Approach vs. Isolation Method | 整体法与隔离法

Two fundamental methods are used to solve connected particle problems: the system (or whole-system) approach and the isolation (or free-body) method. Understanding when to use each can greatly simplify your working.

解决连接粒子问题有两种基本方法：整体法（系统法）和隔离法（受力分析法）。理解何时使用每种方法可以大大简化你的解题过程。

Aspect	System Approach	Isolation Method
Description	Treat all connected particles as a single object; internal tension forces cancel.	Consider each particle separately, showing all forces acting on it.
When to use	Best for finding the common acceleration directly.	Necessary for finding tension in the string or normal reaction forces.
Advantage	Reduces the number of unknowns; equation is often simpler.	Gives detailed internal forces.

整体法：将所有连接粒子视为单一物体；内部张力相互抵消。
隔离法：单独考虑每个粒子，显示作用在其上的所有力。
最佳用途：整体法适合直接求出共同的加速度；隔离法对求解绳的张力和法向反作用力必不可少。
优势：整体法减少未知数数量，方程通常更简单；隔离法提供详细的内部力。

The system approach uses the fact that internal forces (like tension) appear in equal and opposite pairs and sum to zero. Therefore the net force on the whole system is just the vector sum of external forces, equated to total mass times acceleration.

整体法利用了这样一个事实：内力（如张力）成对出现且大小相等方向相反，其总和为零。因此，整个系统的净力仅为外力的矢量和，等于总质量乘以加速度。

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3. Tension in a Light Inextensible String | 轻绳中的张力

Tension is the pulling force exerted along the entire length of a string, acting equally on both connected objects. For a light, inextensible string passing over a smooth pulley, the tension is constant and the string merely transmits the force.

张力是沿绳子整个长度施加的拉力，等量地作用在两个连接的物体上。对于绕过光滑滑轮的轻质不可伸长绳，张力处处相等，绳子仅传递力。

Newton's third law explains that if the string pulls particle A with force T to the right, then particle A pulls the string with force T to the left. The same pair of forces exists at the other end. Consequently, in equations of motion, tension appears with opposite signs when writing the equations for two connected bodies.

牛顿第三定律解释了如果绳用向右的力T拉粒子A，那么粒子A用向左的力T拉绳。在绳的另一端也存在同样的一对力。因此，在运动方程中，当为两个连接的物体写方程时，张力会带着相反的符号出现。

For example, if a horizontal force pulls particle A and particle B is pulled by the string, the equations are: F - T = m₁a and T = m₂a.

例如，如果一个水平力拉粒子A，而粒子B被绳拉动，方程就是：F - T = m₁a 和 T = m₂a。

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4. Two Particles on a Horizontal Surface | 水平面上的两个连接粒子

Imagine two particles of masses m₁ and m₂ connected by a light inextensible string on a smooth horizontal table. A horizontal force F is applied to m₁, pulling the system.

想象两个质量分别为m₁和m₂的粒子，在光滑水平桌面上由一根轻质不可伸长绳连接。一个水平力F施加在m₁上，拉动整个系统。

System approach: The whole system accelerates as one. The only external horizontal force is F. So:

整体法：整个系统作为一个整体加速。唯一的外加水平力是F。因此：

F = (m₁ + m₂) a

Thus the common acceleration is a = F / (m₁ + m₂).

因此共同的加速度为 a = F / (m₁ + m₂)。

Isolation method: To find the tension T, consider m₂ alone. The only horizontal force on m₂ is the tension, so:

隔离法：为了求张力T，单独考虑m₂。作用在m₂上的水平力只有张力，所以：

T = m₂ a

Substituting the acceleration gives T = (m₂ / (m₁ + m₂)) × F.

代入加速度得到 T = (m₂ / (m₁ + m₂)) × F。

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5. Connected Particles on an Inclined Plane | 斜面上的连接粒子

A common configuration consists of a mass m₁ on a smooth inclined plane (angle θ to the horizontal), connected by a light string over a pulley to a hanging mass m₂.

一种常见结构是：质量m₁放在光滑斜面上（与水平面的夹角为θ），通过绕过滑轮的轻绳连接到一个悬挂质量m₂。

Assuming m₂ descends and m₁ moves up the slope, we write the equations of motion for each mass.

假设m₂下降而m₁沿斜面向上运动，我们为每个质量写运动方程。

For m₁ (up the slope positive): T - m₁g sin θ = m₁a

对于m₁（沿斜面向上为正）：T - m₁g sin θ = m₁a

For m₂ (downwards positive): m₂g - T = m₂a

对于m₂（向下为正）：m₂g - T = m₂a

Solving simultaneously gives the acceleration and tension:

联立求解得到加速度和张力：

a = g (m₂ - m₁ sin θ) / (m₁ + m₂)

T = m₁m₂g (1 + sin θ) / (m₁ + m₂)

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6. Pulley Systems – The Atwood Machine | 滑轮系统——阿特伍德机

The classic Atwood machine consists of two masses m₁ and m₂ (with m₂ > m₁) suspended vertically over a smooth pulley. The string is light and inextensible.

经典的阿特伍德机由两个质量m₁和m₂（其中m₂ > m₁）通过光滑滑轮垂直悬挂组成。绳子是轻质且不可伸长的。

Motion equations with upward positive for m₁ and downward positive for m₂:

运动方程，对m₁取向上为正，对m₂取向下为正：

T - m₁g = m₁a

m₂g - T = m₂a

Adding the equations eliminates T and gives the acceleration of the system:

将两个方程相加消去T，得到系统的加速度：

a = g (m₂ - m₁) / (m₁ + m₂)

The tension in the string can then be found as:

然后可以求出绳中的张力：

T = 2 m₁ m₂ g / (m₁ + m₂)

Notice that if the masses are equal, a = 0 and T = mg, as expected for static equilibrium.

注意，如果两个质量相等，a = 0，T = mg，这符合静力平衡的预期。

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7. One Particle on a Table and One Hanging Vertically | 一物在桌，另一物悬挂

Consider a mass m₁ resting on a smooth horizontal table, connected by a light string passing over a smooth pulley at the edge to a hanging mass m₂. The system is released from rest.

考虑一个质量m₁静止在光滑水平桌上，通过一根绕过桌边光滑滑轮的轻绳连接到一个悬挂质量m₂。系统由静止释放。

Since the table is smooth, no friction acts on m₁. The tension pulls m₁ horizontally:

因为桌面光滑，m₁上无摩擦力。张力水平拉m₁：

T = m₁ a

For the hanging mass m₂, taking downward as positive:

对于悬挂质量m₂，取向下为正：

m₂ g - T = m₂ a

Solving yields:

解得：

a = m₂ g / (m₁ + m₂)

and T = m₁ m₂ g / (m₁ + m₂).

且 T = m₁ m₂ g / (m₁ + m₂)。

tion of m₁, where R = m₁g. The equation for m₁ becomes T - f = m₁a, altering both a and T.

如果有摩擦力，摩擦力 f = μR 会阻碍m₁的运动，其中 R = m₁g。此时m₁的方程变为 T - f = m₁a，从而改变了a和T。

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8. Systems with Friction | 含有摩擦力的系统

When a surface is rough, we must include the frictional force. For a particle on a horizontal rough surface, friction is given by F_f sually mg) and μ is the coefficient of friction. The direction of friction opposes the motion.

当接触面粗糙时，我们必须计入摩擦力。对在粗糙水平面上的粒子，摩擦力由 F_f = μR 给出，其中R是法向反作用力（通常为mg），μ是摩擦系数。摩擦力的方向与运动方向相反。

Consider m₁ on a horizontal rough table connected to a hanging m₂. The motion equations become:

考虑m₁在粗糙水平桌上，连接到悬挂的m₂。运动方程变为：

T - μ m₁ g = m₁ a (for m₁)

m₂ g - T = m₂ a (for m₂)

m₁ g = (m₁ + m₂) a, so a = g (m₂ - μ m₁) / (m₁ + m₂). The system only moves if m₂ > μ m₁.

将两式相加得到 m₂ g - μ m₁ g = (m₁ + m₂) a，因此 a = g (m₂ - μ m₁) / (m₁ + m₂)。只有当 m₂ > μ m₁ 时系统才会运动。

m₁ g cos θ. The equation for m₁ up the slope becomes T - m₁ g sin θ - μ m₁ g cos θ = m₁ a.

在粗糙斜面上，摩擦力沿斜面阻碍运动，大小为 μ R，其中 R = m₁ g cos θ。m₁沿斜面向上运动的方程变为 T - m₁ g sin θ - μ m₁ g cos θ = m₁ a。

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9. Lift or Elevator with Multiple Objects | 电梯中的连接物体

In lift problems, two or more objects may be stacked or connected inside an accelerating elevator. Although not connected by strings, they are connected by contact forces. Newton's second law applies to each object.

在电梯问题中，两个或多个物体可能在加速的电梯中叠放或连接。尽管它们不是用绳子连接的，但通过接触力相连。牛顿第二定律适用于每个物体。

Suppose a block of mass m₂ sits on top of mass m₁, and the system is in a lift accelerating upward at a m/s². The normal reaction between the floor and m₁, and between m₁ and m₂, can be found by considering the acceleration. For the combined system, the net force from the lift floor is (m₁+m₂)(g+a). Individual normal reactions follow from isolation of each mass.

假设一个质量为m₂的物块放在质量m₁的上面，且该系统在一个以a m/s²向上加速的电梯中。地板对m₁的正压力，以及m₁与m₂之间的正压力，可以通过考虑加速度求出。对于整体系统，电梯地板提供的净力为 (m₁+m₂)(g+a)。各别正压力则通过对每个质量隔离受力分析得出。

The key is that the acceleration is identical for all parts of the connected system, and contact forces adjust accordingly.

关键之处在于，系统的所有部分加速度相同，接触力会相应地自动调节。

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10. Limiting Tension and String Breaking | 极限张力与绳断裂

Sometimes a problem gives a maximum tension that the string can withstand before breaking. The string will break if the required tension exceeds this limit.

有时题目会给出绳子在断裂前能承受的最大张力。如果所需的张力超过这个极限，绳子就会断裂。

for the maximum possible acceleration or maximum hanging mass without breaking, set T = T_max and solve.

要检验绳子是否断裂，计算给定条件下的张力，并与断裂强度比较。如果题目要求找出不断裂条件下的最大加速度或最大悬挂质量，只需令 T = T_max 并求解。

For the Atwood machine, if T_max is given, you can find the limiting acceleration a_max = (T_max - m₁g) / m₁ (if m₁ is the lighter side) and then find the corresponding mass.

对于阿特伍德机，如果给出了T_max，你可以求出极限加速度 a_max = (T_max - m₁g) / m₁（如果m₁是较轻的一侧），然后求出相应的质量。

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11. Problem-Solving Strategy for Connected Particles | 连接粒子系统解题策略

Following a clear sequence of steps will help you tackle any connected particle question confidently.

遵循清晰的解题步骤将帮助你自信地处理任何连接粒子问题。

Step 1: Draw a clear diagram showing all forces for each particle. Define a positive direction for the entire system.

步骤1：画出清晰的受力图，标出每个粒子上的所有力。为整个系统定义一个正方向。

Step 2: Write down Newton's second law for the whole system (system approach) to find the common acceleration, if possible. This avoids handling tension at first.

步骤2：如果可能，写出整个系统的牛顿第二定律（整体法）以求出共同的加速度。这首先避免了处理张力。

Step 3: Then isolate each particle to write individual equations of motion. Use these to find the tension or normal reactions.

步骤3：然后隔离每个粒子，写出各自的运动方程。利用这些方程求出张力或法向反作用力。

Step 4: Solve the simultaneous equations. Remember acceleration is the same in magnitude for all connected parts.

步骤4：求解联立方程。记住所有连接部分的加速度大小相同。

Step 5: Check if friction or limiting tension conditions apply and substitute correctly.

步骤5：检查是否有摩擦力或极限张力条件，并正确代入。

Step 6: Interpret the results in context: does the system move in the assumed direction? If acceleration turns out negative, reverse the assumed motion and re-solve if necessary.

步骤6：结合实际情况解释结果：系统是否沿假设的方向运动？如果得出的加速度为负，则反转假设的运动方向，必要时重新求解。

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12. Summary and Key Takeaways | 总结与关键点

Connected particle problems always rely on a few central principles: (1) the string is light and inextensible, giving the same acceleration to all linked masses; (2) tension is the same throughout the string and is an

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