Connected Particles | 连接粒子

📚 Connected Particles | 连接粒子

In Edexcel A Level Mechanics, connected particles are modelled as two or more objects joined by a light inextensible string, a rod, or a coupling. These problems test Newton’s laws, force diagrams, and the ability to resolve forces along the direction of motion. A clear, systematic approach is essential because the acceleration and tension in a connected system are determined by the combined effect of all external forces.

在 Edexcel A Level 力学中，连接粒子被建模为两个或更多由轻质不可伸长的绳子、杆或连接装置相连的物体。这类问题考查牛顿定律、受力图以及沿运动方向分解力的能力。清晰的系统方法至关重要，因为连接系统中的加速度和张力由所有外力的共同作用决定。

1. What Are Connected Particles? | 什么是连接粒子？

A connected particle system involves at least two bodies whose motions are linked by a string, rod, or coupling. Because the string is light and inextensible, the connected bodies move with the same speed and the same magnitude of acceleration while the string remains taut.

连接粒子系统涉及至少两个由绳子、杆或连接装置相连且运动相互关联的物体。由于绳子轻质且不可伸长，只要绳子保持张紧，相连物体就以相同的速率和相同的加速度大小运动。

Typical Edexcel problems include a particle on a table connected to a hanging particle, two particles over a smooth pulley, a particle on an inclined plane, and a person standing on a scale in a lift.

Edexcel 常见题型包括：桌面上的粒子与悬挂粒子相连、光滑滑轮两侧的两个粒子、斜面上的粒子，以及电梯中站在秤上的人。

2. Modelling Assumptions | 建模假设

The standard model makes several simplifying assumptions. These assumptions allow us to treat tension as constant and acceleration as common to all connected parts.

标准模型做出若干简化假设。这些假设使我们能够将张力视为恒定，并将加速度视为所有连接部分的共同加速度。

Assumption \| 假设	Consequence \| 结果
Light string	Tension is the same throughout the string
Inextensible string	All connected particles have the same acceleration
Smooth pulley	Tension is unchanged as it passes over the pulley
Particle model	Rotational effects and air resistance are ignored

These assumptions simplify the mathematics but must be stated or understood in exam solutions. If the string is modelled as light, its mass is negligible, so the forces acting on it must balance.

这些假设简化了数学处理，但在考试解答中必须说明或理解。如果绳子被建模为轻质，其质量可以忽略，因此作用在绳上的力必须平衡。

3. Newton’s Second Law in Connected Systems | 连接系统中的牛顿第二定律

Newton’s second law states that the resultant force acting on a particle is equal to the product of its mass and acceleration.

牛顿第二定律指出，作用在粒子上的合力等于其质量与加速度的乘积。

F = ma

In connected systems, you can apply F = ma to each particle separately, or to the whole system in the direction of motion. The whole-system method often removes the need to include internal tension forces.

在连接系统中，可以分别对每个粒子应用 F = ma，也可以沿运动方向对整个系统应用。整体法通常不需要考虑内部张力。

For a system of total mass M moving under an external resultant force F, the acceleration is given by a = F ÷ M. Internal forces such as tension cancel when the whole system is considered.

对于总质量为 M 的系统在外部合力 F 作用下运动，加速度为 a = F ÷ M。当考虑整个系统时，张力等内力相互抵消。

4. Tension in an Inextensible String | 不可伸长绳中的张力

Tension is the pulling force transmitted through a string, rope, or cable. In a light inextensible string, the tension has the same magnitude at both ends and throughout the string.

张力是通过绳子、绳索或缆绳传递的拉力。在轻质不可伸长的绳子中，张力在两端和整根绳上大小相同。

If two particles are connected by a taut string, each particle feels the tension pulling toward the string. Tension is not a fixed force; it adjusts according to the motion and external forces acting on the system.

如果两个粒子由张紧的绳子相连，每个粒子都会受到绳子方向相反的拉力。张力不是固定不变的力，它会根据系统的运动和外力进行调整。

You usually find tension by applying F = ma to one particle after the acceleration has been found. Do not assume tension is equal to the weight of any particle unless the system is in equilibrium.

通常在求出加速度后，通过对其中一个粒子应用 F = ma 来求张力。除非系统处于平衡状态，否则不要假设张力等于任何粒子的重力。

5. The Smooth Pulley System | 光滑滑轮系统

A classic problem is two particles of masses m₁ and m₂ connected by a light inextensible string passing over a smooth fixed pulley. If m₁ > m₂, the heavier particle accelerates downward and the lighter particle accelerates upward with the same acceleration.

经典题型是质量分别为 m₁ 和 m₂ 的两个粒子，由一根绕过光滑定滑轮的轻质不可伸长绳子相连。如果 m₁ > m₂，较重粒子向下加速，较轻粒子以相同加速度向上加速。

a = (m₁ − m₂)g ÷ (m₁ + m₂)

The tension in the string is the same on both sides of the pulley and can be found from either particle’s equation of motion.

绳子两侧的张力相同，可以由任一个粒子的运动方程求出。

T = 2m₁m₂g ÷ (m₁ + m₂)

Always define the positive direction before writing equations. A common approach is to take the direction of motion of each particle as positive.

在列方程之前一定要先规定正方向。通常做法是取每个粒子的运动方向为正方向。

6. Horizontal Table and Hanging Particle | 水平桌面与悬挂粒子

Consider a particle of mass M on a smooth horizontal table, connected by a string passing over a smooth pulley at the edge to a hanging particle of mass m. The system accelerates because the weight of the hanging particle pulls the entire arrangement.

考虑质量为 M 的粒子放在光滑水平桌面上，通过绕过桌边光滑滑轮的绳子与质量为 m 的悬挂粒子相连。由于悬挂粒子的重力拉动整个装置，系统产生加速度。

For the hanging particle: mg − T = ma. For the particle on the table: T = Ma. Adding these equations gives the acceleration.

对悬挂粒子：mg − T = ma。对桌面上粒子：T = Ma。将两式相加可求出加速度。

a = mg ÷ (M + m)

The tension is then T = Ma, which gives:

然后张力为 T = Ma，即：

T = Mmg ÷ (M + m)

If the table is rough, include friction as a force opposing the motion of the table particle. Friction is calculated using F = μR, where R is the normal reaction.

如果桌面粗糙，需要将摩擦力作为阻碍桌面粒子运动的力。摩擦力用 F = μR 计算，其中 R 是法向反作用力。

7. Inclined Plane Systems | 斜面系统

Connected particles on inclined planes require resolving forces parallel and perpendicular to the slope. The weight component along the plane is mg sin θ, and the component perpendicular to the plane is mg cos θ.

斜面上的连接粒子需要沿斜面方向和垂直于斜面方向分解力。重力沿斜面的分量为 mg sin θ，垂直斜面的分量为 mg cos θ。

Weight component down slope = mg sin θ

When two particles are connected over a pulley with one or both on slopes, write separate equations of motion for each particle, taking care with the signs of the weight components.

当两个粒子通过滑轮连接，其中一个或两个都在斜面上时，要分别列出每个粒子的运动方程，并注意重力分量的正负号。

For a particle of mass m₁ on a smooth slope of angle θ and a hanging mass m₂, the acceleration can be found from:

对于质量为 m₁ 的粒子放在倾角为 θ 的光滑斜面上，另一个质量 m₂ 悬挂，加速度可由下式求得：

a = (m₂g − m₁g sin θ) ÷ (m₁ + m₂)

The direction of acceleration depends on which side of the equation is larger. If m₁ sin θ > m₂, the mass on the slope accelerates down the plane.

加速度方向取决于等式哪一侧更大。如果 m₁ sin θ > m₂，斜面上的质量将沿斜面向下加速。

8. Lift and Scale Problems | 电梯与秤问题

A person standing on a scale in a lift is another common connected-particle application. The scale reading is the normal reaction force, also called apparent weight.

站在电梯秤上的人是另一种常见的连接粒子应用。秤的读数是法向反作用力，也称为视重。

When the lift accelerates upward with acceleration a, the equation of motion for the person of mass m is:

当电梯以加速度 a 向上加速时，质量为 m 的人的运动方程为：

N − mg = ma ⇒ N = m(g + a)

When the lift accelerates downward, the normal reaction decreases:

当电梯向下加速时，法向反作用力减小：

mg − N = ma ⇒ N = m(g − a)

If the lift is in free fall, a = g, so N = 0 and the person experiences apparent weightlessness. If the lift moves at constant speed, acceleration is zero, so N = mg.

如果电梯自由下落，a = g，则 N = 0，人会感到失重。如果电梯匀速运动，加速度为零，因此 N = mg。

9. Step-by-Step Strategy | 分步解题策略

A consistent strategy helps avoid sign errors and lost marks in connected particle questions. Follow these steps for most Edexcel mechanics problems.

一致的解题策略有助于避免符号错误和失分。在大多数 Edexcel 力学问题中，可遵循以下步骤。

Draw a clear diagram showing all forces acting on each particle.
Define the positive direction for each particle’s motion.
Write down F = ma for each particle separately, or write one equation for the whole system.
Solve the equations simultaneously to find acceleration and tension.
Check that the direction and magnitude of your answers are physically reasonable.

画出清晰的受力图，标出每个粒子所受的全部力。规定每个粒子运动的正方向。分别对每个粒子列出 F = ma，或对整个系统列一个方程。联立方程求解加速度和张力。检查答案的方向和大小是否合理。

For whole-system equations, use the total external force in the direction of motion and the total mass. This is often faster for finding acceleration.

使用整体法时，用运动方向上的总外力和总质量。这通常能更快求出加速度。

10. Common Mistakes | 常见错误

One common mistake is assuming the tension equals the weight of one of the particles. This is only true when the system is at rest or moving at constant speed.

一个常见错误是假设张力等于某个粒子的重力。只有当系统静止或匀速运动时，这才是正确的。

Another error is using the wrong sign for acceleration when particles move in opposite directions. Define positive as the direction of motion for each particle and keep signs consistent.

另一个错误是粒子向相反方向运动时加速度符号使用不当。应把每个粒子的运动方向定义为正，并保持符号一致。

Students also sometimes forget that a rough surface introduces friction, which is proportional to the normal reaction and opposes motion. Always check whether the surface is smooth or rough.

学生有时还会忘记粗糙表面会产生摩擦力，摩擦力与法向反作用力成正比并阻碍运动。务必检查表面是光滑还是粗糙。

Finally, do not mix whole-system and individual-particle equations without care. Internal tension cancels in the whole system but must be included when considering a single particle.

最后，不要随意混用整体法和单个粒子的方程。在整体法中内张力相互抵消，但在考虑单个粒子时必须纳入。

11. Worked Example | 例题

A particle of mass 3 kg rests on a smooth horizontal table. It is connected by a light inextensible string passing over a smooth pulley at the edge of the table to a particle of mass 2 kg hanging freely. Find the acceleration of the system and the tension in the string.

一个质量为 3 kg 的粒子静止在光滑水平桌面上。它通过一根绕过桌边光滑滑轮的轻质不可伸长绳子，与一个质量为 2 kg 的自由悬挂粒子相连。求系统的加速度和绳中的张力。

Take g = 9.8 m s⁻². For the 2 kg hanging mass, using downward as positive:

取 g = 9.8 m s⁻²。对 2 kg 的悬挂质量，以向下为正：

2g − T = 2a

For the 3 kg table mass, using the direction of motion as positive:

对 3 kg 的桌面质量，以运动方向为正：

T = 3a

Substitute T = 3a into the first equation:

将 T = 3a 代入第一个方程：

2g − 3a = 2a ⇒ 2g = 5a ⇒ a = 2g ÷ 5 = 3.92 m s⁻²

The tension is:

张力为：

T = 3 × 3.92 = 11.76 N

The system accelerates at 3.92 m s⁻² and the string tension is 11.76 N. This is less than the hanging weight of 19.6 N, confirming that the hanging particle accelerates downward.

系统以 3.92 m s⁻² 加速，绳中张力为 11.76 N。这小于悬挂物的重力 19.6 N，说明悬挂粒子正在向下加速。

12. Exam Tips | 考试技巧

In Edexcel exams, always state modelling assumptions clearly, such as ‘light inextensible string’ and ‘smooth pulley’. These phrases justify why tension is constant and acceleration is common.

在 Edexcel 考试中，务必清楚说明建模假设，例如“轻质不可伸长的绳子”和“光滑滑轮”。这些表述说明为什么张力恒定且加速度相同。

Set out your working vertically, with one equation per line and a clear positive direction for each particle. This makes your method easier to follow and earns method marks even if a numerical slip occurs.

解题过程要竖排书写，每个粒子一行方程，并明确正方向。这样方法清晰，即使数值计算失误也能获得方法分。

Always check the units and whether the acceleration is less than g where expected. If the system includes a hanging mass, acceleration should be positive and less than 9.8 m s⁻² unless additional forces act.

务必检查单位，以及加速度是否符合预期且小于 g。如果系统包含悬挂质量，除非有其他外力作用，加速度应为正值且小于 9.8 m s⁻²。

Finally, revise both the whole-system method and the individual-particle method. The whole-system method is faster for acceleration, while the individual-particle method is needed for tension.

最后，要同时复习整体法和单个粒子法。整体法求加速度更快，而求张力需要用到单个粒子法。

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Connected Particles | 连接粒子