Connected Bodies: System Motion and Interaction Forces | 连接体问题：系统运动与相互作用力

📚 Connected Bodies: System Motion and Interaction Forces | 连接体问题：系统运动与相互作用力

Connected-body problems are a fundamental class of mechanics questions that require us to analyse two or more objects moving together. These problems test your understanding of Newton’s laws, force diagrams, and the relation between the motion of a system and the forces exchanged between its parts.

连接体问题是力学中的一类基础题目，要求我们分析两个或多个物体共同运动的情形。这类问题考查对牛顿定律、受力图以及系统运动与其各部分之间相互作用力关系的理解。

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1. Introduction to Connected-Body Problems | 连接体问题概述

In a connected-body system, objects may be joined by a string, a rod, a spring, or direct contact. The connecting elements transmit forces between the objects, and these forces are internal to the whole system.

在连接体系统中，物体之间可以通过绳子、杆、弹簧或直接接触而连接。连接元件在物体之间传递力，而这些力对整个系统而言是内力。

The key idea is that, even when objects are connected, they usually have the same acceleration magnitude along the direction of the connection. This constraint greatly simplifies the analysis.

关键思想是，即使物体是连接在一起的，它们沿连接方向的加速度大小通常相同。这一约束大大简化了分析。

We shall examine the two complementary approaches: treating the whole system as one body, and isolating each body separately.

我们将研究两种互补的方法：把整个系统看作一个整体，以及分别隔离每个物体。

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2. Newton’s Laws for the System | 系统的牛顿定律

Newton’s second law applied to the whole system states that the net external force acting on the system equals the total mass times the acceleration of the centre of mass.

对系统整体应用牛顿第二定律时，合外力等于系统总质量乘以质心的加速度。

∑F = Ma

Here M is the total mass of the system and a is the acceleration of its centre of mass.

其中 M 是系统的总质量，a 是系统质心的加速度。

Internal forces, such as the tension in a string between two blocks, occur in action-reaction pairs. Their vector sum is zero, so they do not affect the motion of the centre of mass.

内力，例如两个木块之间绳子中的张力，以作用力和反作用力成对出现。它们的矢量和为零，因此不影响质心的运动。

This is why the whole-system method can find the acceleration of the system without knowing the internal forces.

这就是为什么整体法可以在不知道内力的情况下求出系统的加速度。

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3. The Whole-System Method | 整体法

Step 1: Choose a positive direction along which the system accelerates. Step 2: Identify all external forces acting on the system. Step 3: Apply Newton’s second law to the total mass.

第一步：选定系统加速的方向为正方向。第二步：确定作用在系统上的所有外力。第三步：对总质量应用牛顿第二定律。

Example: two blocks of masses m₁ and m₂ on a smooth horizontal table are pulled by a horizontal force F applied to m₁. The acceleration is

例：在光滑水平面上，质量为 m₁ 和 m₂ 的两个木块，通过水平拉力 F 作用在 m₁ 上。加速度为

a = F / (m₁ + m₂)

Notice that the tension between the blocks does not appear. The external force F is the only horizontal external force.

注意木块之间的张力没有出现。在水平方向上，外力只有 F。

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4. The Isolation Method | 隔离法

To find internal forces, we isolate one object at a time and draw its free-body diagram. Each isolated body obeys Newton’s second law independently.

为了求出内力，我们一次隔离一个物体并画出它的受力图。每个隔离体独立地满足牛顿第二定律。

For the two-block example above, for block m₁: F – T = m₁a. For block m₂: T = m₂a.

对于上面的两个木块，对 m₁ 有：F – T = m₁a；对 m₂ 有：T = m₂a。

Combining these equations gives the same acceleration as the whole-system method, and also gives T = m₂F / (m₁ + m₂).

联立这些方程可得到与整体法相同的加速度，并且还得到 T = m₂F / (m₁ + m₂)。

T = m₂F / (m₁ + m₂)

This shows that isolation is necessary when an internal force itself is required.

这表明，当题目要求某个内力本身时，隔离法是必要的。

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5. Relationship Between Accelerations | 加速度之间的关系

For an ideal string that does not stretch, the magnitudes of the accelerations of all objects connected by the string are equal along the string direction.

对于不能伸长的理想绳子，由该绳连接的所有物体沿绳方向的加速度大小相等。

In a simple Atwood machine, two

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