📚 6 Forces in Nature | 自然界中的力
In Edexcel A-Level Mathematics, mechanics is the study of forces and motion. Chapter 6 introduces forces that appear in nature: weight, normal reaction, tension, friction, air resistance, thrust and buoyancy. Understanding how to resolve forces and apply Newton’s laws is essential for modelling real-world situations accurately.
在爱德思 A-Level 数学中，力学研究力与运动。第 6 章介绍自然界中出现的力：重力、法向反作用力、张力、摩擦力、空气阻力、推力和浮力。理解如何分解力并应用牛顿定律是建立现实情境模型的关键。
1. Forces as Vectors | 力是矢量
A force is a vector quantity: it has both magnitude and direction. The SI unit of force is the newton (N). When several forces act on a particle, we add them as vectors. This can be done by drawing a scale diagram or by resolving into perpendicular components, usually horizontal and vertical, or parallel and perpendicular to a slope.
力是矢量：既有大小又有方向。国际单位制中力的单位是牛顿（N）。当多个力作用在一个质点上时，我们用矢量加法合成。可以通过画比例图或将力分解为互相垂直的分量来计算，通常沿水平与竖直方向，或沿斜面方向与垂直斜面方向。
For perpendicular forces of magnitudes F₁ and F₂, the resultant magnitude is given by:
对于大小为 F₁ 和 F₂ 的互相垂直的力，合力大小为：
F_resultant = √(F₁² + F₂²)
The direction θ relative to F₁ is found using tan θ = F₂ / F₁.
方向 θ 相对于 F₁ 通过 tan θ = F₂ / F₁ 求得。
2. Common Forces in Nature | 自然界中的常见力
In mechanics problems, we simplify real forces into a small set of model forces. These include weight, normal reaction, tension, friction, air resistance, thrust and buoyancy. Each has a standard direction and symbol, which helps when drawing free-body diagrams.
在力学问题中，我们将真实力简化为一小类模型力。这些力包括重力、法向反作用力、张力、摩擦力、空气阻力、推力和浮力。每种力都有标准的方向和符号，有助于画受力分析图。
| Force / 力 | Symbol / 符号 | Direction / 方向 |
|---|---|---|
| Weight / 重力 | W | Vertically downward / 竖直向下 |
| Normal reaction / 法向反作用力 | R | Perpendicular to contact surface / 垂直接触面 |
| Tension / 张力 | T | Along string or rope / 沿绳或索方向 |
| Friction / 摩擦力 | F | Parallel to surface, opposing sliding / 平行于接触面，阻碍滑动 |
| Air resistance / 空气阻力 | R_d | Opposite to velocity / 与速度方向相反 |
| Thrust / 推力 | P | Direction of motion / 运动方向 |
These model forces allow us to apply Newton’s laws without needing to consider atomic-level interactions.
这些模型力使我们能够应用牛顿定律，而无需考虑原子层面的相互作用。
3. Weight and Gravitational Force | 重力与万有引力
Weight is the force of gravity acting on a mass. Near the Earth’s surface, the gravitational field strength g is approximately 9.8 m s⁻². Weight is calculated as mass times g. Mass is a scalar quantity measured in kilograms, while weight is a vector measured in newtons.
重力是作用在质量上的万有引力。在地球表面附近，重力场强度 g 约为 9.8 m s⁻²。重力等于质量乘以 g。质量是标量，单位是千克，而重力是矢量，单位是牛顿。
W = m g
For example, a 3 kg object has weight W = 3 × 9.8 = 29.4 N. Weight always acts vertically downward, regardless of the shape of the surface the object is on.
例如，一个 3 kg 的物体所受重力为 W = 3 × 9.8 = 29.4 N。无论物体所在表面形状如何，重力始终竖直向下。
4. Normal Reaction and Tension | 法向反作用力与张力
The normal reaction R is the contact force exerted by a surface on an object. It is perpendicular to the surface, not necessarily vertical. For a block resting on a horizontal table with no other vertical forces, R balances the weight:
法向反作用力 R 是表面对物体施加的接触力。它垂直于表面，不一定是竖直方向。对于静止在水平桌面上且无其他竖直力的物块，R 与重力平衡：
R = m g
If an upward pulling force P at angle θ is applied, the vertical component P sin θ reduces the normal reaction:
若施加一个与水平方向成 θ 角的向上拉力 P，其竖直分量 P sin θ 会减小法向反作用力：
R = m g − P sin θ
Tension arises in strings, ropes and cables. For a light, inextensible string, the tension is the same throughout, and it acts along the string.
张力出现在绳子、绳索和缆绳中。对于轻质且不可伸长的绳，绳中各处张力相同，方向沿绳的方向。
5. Friction and the Coefficient of Friction | 摩擦力与摩擦系数
Friction opposes relative motion between two surfaces in contact. If an object is moving, friction acts opposite to the direction of motion. If an object is stationary, static friction can be any value up to a maximum. The maximum possible friction force is:
摩擦力阻碍两接触面之间的相对运动。若物体在运动，摩擦力与运动方向相反。若物体静止，静摩擦力可为不超过最大值的任意值。最大可能摩擦力为：
F_max = μ R
Here μ is the coefficient of friction, a dimensionless constant depending on the two surfaces. For static equilibrium, friction satisfies F ≤ μ R. Once sliding begins, friction is usually modelled as F = μ R.
其中 μ 是摩擦系数，一个由两接触面决定的无量纲常数。对于静力平衡，摩擦力满足 F ≤ μ R。一旦开始滑动，摩擦力通常按 F = μ R 建模。
| Surfaces / 表面 | Typical μ / 典型 μ |
|---|---|
| Rubber on concrete / 橡胶与混凝土 | 0.7 − 1.0 |
| Wood on wood / 木头与木头 | 0.4 |
| Ice on ice / 冰与冰 | 0.05 |
In nature, friction generates heat. This is one way a small brush fire can start, but in mechanics we focus on the force model rather than the heat.
在自然界中，摩擦会产生热量。这是灌木丛火灾可能发生的一种方式，但在力学中我们关注的是力的模型而不是热量。
6. Resolving Forces on an Inclined Plane | 斜面受力分解
When an object of mass m rests on a slope inclined at angle θ to the horizontal, its weight mg acts vertically downward. Resolving perpendicular and parallel to the slope gives:
当质量为 m 的物体静止在倾角为 θ 的斜面上时，其重力 mg 竖直向下。将重力沿垂直斜面和平行斜面方向分解可得：
R = m g cos θ
Down-slope component = m g sin θ
If the object is in equilibrium, friction balances the down-slope component, so F = m g sin θ. If the object is on the point of sliding down, limiting friction applies and tan θ = μ.
若物体处于平衡状态，摩擦力与沿斜面向下的分量平衡，因此 F = m g sin θ。若物体恰好要沿斜面向下滑动，则极限摩擦力适用，且 tan θ = μ。
7. Newton’s Laws of Motion | 牛顿运动定律
Newton’s first law states that if the resultant force on an object is zero, the object remains at rest or moves with constant velocity. Newton’s second law relates resultant force, mass and acceleration:
牛顿第一定律指出，若物体所受合力为零，物体保持静止或匀速直线运动。牛顿第二定律将合力、质量和加速度联系起来：
F_resultant = m a
Newton’s third law states that for every action there is an equal and opposite reaction. These action-reaction forces act on different bodies, so they never cancel each other on the same body.
牛顿第三定律指出，每一个作用力都有一个大小相等、方向相反的反作用力。这对作用力与反作用力作用在不同物体上，因此它们从不在同一物体上相互抵消。
For example, a 10 kg object accelerating at 2 m s⁻² has resultant force F = 10 × 2 = 20 N. This resultant can be provided by a slightly larger driving force than the total resistive force.
例如，一个 10 kg 的物体以 2 m s⁻² 加速，所受合力为 F = 10 × 2 = 20 N。这个合力可以由略大于总阻力的驱动力提供。
8. Equilibrium and Resultant Force | 平衡与合力
An object is in equilibrium when the vector sum of all forces acting on it is zero. This means both the horizontal and vertical components, or any two perpendicular directions
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