📚 Work, Energy and Power | 功、能量和功率
Work, energy and power are foundational concepts in mechanics that describe how forces cause motion and transfer capacity to do work. Understanding their precise definitions and the relationships between them is essential for solving physical problems and analysing real‑world systems.
功、能量和功率是力学中的基础概念，用于描述力如何引起运动并传递做功的能力。准确理解它们的定义及其相互关系，是解决物理问题、分析实际系统所必需的。
1. Work Done by a Constant Force | 恒力做功
In physics, work is done when a force moves an object through a distance in the direction of the force. For a constant force F acting at an angle θ to the displacement s, the work done is the product of the force component along the displacement and the magnitude of the displacement.
在物理学中，当力使物体沿力的方向移动一段距离时就做了功。对于恒力F与位移s成角度θ的情况，所做的功等于力在位移方向的分量与位移大小的乘积。
W = F s cos θ
re 1 J = 1 N m. When the force is parallel to the displacement (θ = 0), work simplifies to W = F s. If the force is perpendicular (θ = 90°), no work is done because cos 90° = 0.
功是标量，其国际单位为焦耳（J），1 J = 1 N m。当力与位移平行时（θ = 0），功简化为W = F s；若力与位移垂直（θ = 90°），则因cos 90° = 0而不做功。
2. Work Done by a Varying Force | 变力做功
ply by multiplying force and displacement. Instead, the work equals the area under a force– F = k x) from 0 to an extension x is ½ k x², which is the area of the triangle under the F–x line.
当施加的力随位置变化时，不能简单地用力与位移相乘来计算功。此时，功等于力–位移图下的面积。例如，将遵循胡克定律（F = k x）的弹簧从原长拉伸至伸长量x，所做的功为½ k x²，即F–x图线下三角形的面积。
This area method is powerful: it can be used for any varying force, as long as the graph is available or can be integrated.
这种面积法功能强大：只要有图线或可积分，就可用于任何变力做功的情况。
3. Energy and its Forms | 能量及其形式
Energy is defined as the capacity to do work. In mechanics, the two most important forms are kinetic energy (energy possessed by a moving object) and potential energy (energy stored due to position or configuration). Other forms include thermal, chemical and nuclear energy, but in pure mechanics problems we often focus on mechanical energy.
能量被定义为做功的能力。在力学中，最重要的两种形式是动能（运动物体具有的能量）和势能（因位置或形态而储存的能量）。其他形式还包括热能、化学能和核能，但在纯力学问题中我们通常只关注机械能。
4. Kinetic Energy Derivation | 动能公式推导
Kinetic energy (KE) is the energy a body has by virtue of its motion. For an object of mass m moving with speed v, the kinetic energy is given by KE = ½ m v². This expression can be derived by considering the net work done to accelerate the object from rest.
动能是物体因运动而具有的能量。质量为m、速度为v的物体的动能为KE = ½ m v²。这一表达式可通过考虑合外力将物体从静止加速所做的功推导出来。
Using a constant net force F over a displacement s: W = F s = m a s. From the kinematic relation v² = u² + 2 a s gives W = m a × (v² / (2a)) = ½ m v². Thus the work done equals the kinetic energy gained.
利用恒定的合外力F和位移s：W = F s = m a s。由运动学关系v² = u² + 2 a s，取u = 0得s = v² / (2a)。代入得W = m a × (v² / (2a)) = ½ m v²。因此，所做的功等于获得的动能。
5. Gravitational Potential Energy | 重力势能
Gravitational potential energy (GPE) is the energy stored in an object due to its height above a chosen reference level. Near the Earth’s surface, when an object of mass m is lifted through a vertical height h, the change in GPE is ΔGPE = m g h, where g is the acceleration of free fall (≈ 9.81 m s⁻²).
重力势能（GPE）是由于物体相对于选定参考平面的高度而储存的能量。在地球表面附近，将质量为m的物体垂直提升高度h时，重力势能的变化量为ΔGPE = m g h，其中g为自由落体加速度（≈ 9.81 m s⁻²）。
Gravitational force is conservative, meaning the work done against gravity depends only on the change in vertical height, not on the path taken. This property allows us to define a unique potential energy at any height.
重力是保守力，意味着克服重力做的功只取决于垂直高度的变化，与路径无关。这一性质使我们能够定义任一高度处的唯一势能值。
6. Principle of Conservation of Energy | 能量守恒定律
The principle of conservation of energy states that energy cannot be created or destroyed; it can only be transformed from one form to another or transferred between bodies. In a closed system, the total energy remains constant. For mechanical systems involving only conservative forces (gravity, elastic spring force), the sum of kinetic and potential energies is conserved.
能量守恒定律指出，能量既不能创造也不能消灭，只能从一种形式转化为另一种形式，或在物体间传递。在封闭系统中，总能量保持不变。对于仅涉及保守力（重力、弹性弹簧力）的机械系统，动能与势能之和守恒。
This conservation is expressed as: KE₁ + PE₁ = KE₂ + PE₂. It provides a shortcut for solving problems where force analysis would be complicated.
该守恒关系可表示为：KE₁ + PE₁ = KE₂ + PE₂。它为求解那些受力分析会很复杂的问题提供了一条捷径。
7. Work–Energy Theorem | 功能定理
The work–energy theorem is a direct consequence of Newton’s second law and states that the net work done on an object equals the change in its kinetic energy: Wnet = ΔKE = ½ m v² – ½ m u². This theorem applies even when non‑conservative forces are present, provided we account for all forces.
功能定理是牛顿第二定律的直接推论，它指出：作用在物体上的合外力所做的功等于其动能的变化量：Wnet = ΔKE = ½ m v² – ½ m u²。只要考虑了所有的力，即使存在非保守力，该定理也成立。
It is especially useful for linking force, displacement and speed without explicitly calculating acceleration or time.
它特别适用于无需显式计算加速度或时间，就能将力、位移和速度联系起来的情形。
8. Mechanical Energy and Non‑conservative Forces | 机械能与非保守力
Mechanical energy is conserved only when all the forces doing work are conservative. If non‑conservative forces such as friction or air resistance do work, the total mechanical energy changes. The work done by non‑conservative forces equals the change in mechanical energy: Wnc = ΔKE + ΔPE.
只有当所有做功的力都是保守力时，机械能才守恒。如果摩擦力或空气阻力等非保守力做功，总机械能就会改变。非保守力做的功等于机械能的变化量：Wnc = ΔKE + ΔPE。
For example, a block sliding down a rough incline loses some gravitational potential energy to kinetic energy and the rest to thermal energy due to friction. This energy is not lost but dissipated, still conserving total energy overall.
例如，一个物块沿粗糙斜面下滑时，部分重力势能转化为动能，其余部分因摩擦转化为热能。这些能量并未消失，而是耗散了，总体上的总能量仍然守恒。
9. Power | 功率
Power measures the rate at which work is done or energy is transferred. Average power is defined as Pavg = ΔW / Δt (or ΔE / Δt). The SI unit is the watt (W), where 1 W = 1 J s⁻¹.
功率度量做功或传递能量的快慢。平均功率定义为Pavg = ΔW / Δt（或ΔE / Δt）。国际单位是瓦特（W），1 W = 1 J s⁻¹。
When a constant force F acts on an object moving at speed v in the direction of the force, the instantaneous power is P = F v. If the force is at an angle θ to the velocity, P = F v cos θ. This relation is often used in transport problems to find maximum speeds for a given engine power.
当恒力F作用在沿力的方向以速度v运动的物体上时，瞬时功率为P = F v。若力与速度成θ角，则P = F v cos θ。这个关系常用于交通工具问题，以求解给定发动机功率下的最大速度。
10. Efficiency | 效率
Efficiency quantifies how well a machine or process converts input energy into useful output work. It is expressed as a percentage:
效率量化了机器或过程将输入能量转化为有用输出功的程度，以百分数表示：
η = (useful output energy / total input energy) × 100%
Because of friction, sound and heat losses, real machines always have an efficiency less than 100%. In many A‑Level problems, the efficiency is given, and you must find the input or output power accordingly.
由于摩擦、声音和热量损失，实际机器的效率总是低于100%。在许多A‑Level问题中，会给出效率，要求你据此求输入或输出功率。
11. Applications and Problem‑Solving Strategies | 应用与解题策略
Work, energy and power principles are applied in a wide range of mechanics contexts: a car engine delivering power to overcome drag and climb slopes; a pendulum swinging between kinetic and potential extremes; or a projectile converting kinetic energy into gravitational potential energy.
功、能量和功率原理广泛应用于各种力学情境中：汽车发动机输出功率以克服阻力并爬坡；单摆在动能和势能极值之间摆动；抛体将动能转化为重力势能。
A typical problem‑solving approach involves identifying the system, writing down the initial and final energies (kinetic and potential), calculating the work done by any non‑conservative forces, and applying either conservation of energy or the work–energy theorem. Careful choice of reference level for potential energy simplifies calculations.
典型的解题方法包括：确定系统，写出初始和最终能量（动能和势能），计算所有非保守力做的功，然后应用能量守恒或功能定理。审慎选取势能的参考水平可简化计算。
12. Summary of Key Equations and Concepts | 关键方程与概念总结
The table below collects the most important relationships for revision. All symbols follow standard usage: m mass, v speed, h height, g gravitational field strength, F force, s displacement, t time.
下表汇总了最重要的关系式供复习。所有符号遵循标准用法：m 质量，v 速度，h 高度，g 重力场强度，F 力，s 位移，t 时间。
| Concept (English) | Equation / Relation | Translated notes |
|---|---|---|
| Work done by constant force | W = F s cos θ | 恒力做功，θ为力与位移的夹角 |
| Kinetic energy | KE = ½ m v² | 动能，½ m v² |
| Change in gravitational potential energy | ΔGPE = m g h | 重力势能变化，h为垂直高度差 |
| Work–energy theorem |
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