📚 Energy Changes: Work, Energy Stores and Conservation | 能量变化：功、能量储存与守恒
Energy is a central idea in CIE A-Level Physics. The principle of conservation of energy states that energy cannot be created or destroyed, only transferred or transformed between different stores. This article reviews the key concepts, equations, graphs and exam-style applications of energy changes, from the definition of work to internal energy and the first law of thermodynamics.
能量是 CIE A-Level 物理的核心概念。能量守恒定律指出，能量不能被创造或消灭，只能在不同的储存库之间转移或转化。本文回顾能量变化的关键概念、公式、图像以及考试常见应用，内容从功的定义一直延伸到内能与热力学第一定律。
1. Work and Energy Transfer | 功与能量转移
In mechanics, work is done whenever a force moves its point of application. Work is a scalar quantity, but it can be positive, negative or zero depending on the direction of the force relative to the displacement.
在力学中，只要力使其作用点发生移动，就做了功。功是标量，但根据力相对于位移的方向，功可以是正值、负值或零。
For a constant force F acting at an angle θ to a displacement s, the work done W is:
对于与位移 s 成 θ 角的恒力 F，所做的功 W 为：
W = F s cos θ
oes no work at all. If θ = 180°, the force opposes the motion and does negative work, which means energy is taken from the system.
有三种情况值得牢记。若 θ = 0°，力与位移平行，做最大正功；若 θ = 90°，cos θ = 0，力完全不做功；若 θ = 180°，力与运动方向相反，做负功，表示能量从系统中被取走。
- The normal force on a box sliding along a horizontal floor does no work, because it is perpendicular to the displacement.
- 沿水平地面滑动的箱子所受的支持力不做功，因为它与位移垂直。
- Friction does negative work, converting kinetic energy into internal (thermal) energy of the surfaces.
- 摩擦做负功，把动能转化为接触面的内能（热能）。
- A crane lifting a load at constant velocity: the cable tension does positive work mgΔh, gravity does negative work −mgΔh, and the net work is zero.
- 起重机匀速吊起重物：缆绳拉力做正功 mgΔh，重力做负功 −mgΔh，净功为零。
2. The SI Unit and the Joule | 国际单位与焦耳
The SI unit of work and of every form of energy is the joule (J). One joule is the work done when a force of one newton moves its point of application one metre in the direction of the force.
功以及一切形式能量的国际单位都是焦耳（J）。1 焦耳就是 1 牛顿的力使其作用点在力的方向上移动 1 米所做的功。
1 J = 1 N m = 1 kg m² s⁻²
Because energy and work share the same unit, a joule can also be described as the transfer of one joule of energy by any pathway: mechanical, electrical, thermal or radiative. In calculations you must always convert to SI base units first: grams to kilograms, centimetres to metres, minutes to seconds.
由于能量与功的单位相同，焦耳也可以描述为通过任何途径（力学、电学、热学或辐射）转移 1 焦耳的能量。计算时必须先换算成国际单位制基本单位：克换千克、厘米换米、分钟换秒。
lamp transfers 60 J each second, and burning 1 kg of petrol releases roughly 4.5 × 10⁷ J.
记住一些典型数量级有助于检查答案：举起一个苹果约获得 1 J，60 W 的灯泡每秒转移 60 J，燃烧 1 kg 汽油约释放 4.5 × 10⁷ J。
3. Energy Stores and Transfer Pathways | 能量的储存与转移途径
Modern syllabuses describe energy as being held in stores and moved along pathways. Energy is never 'used up'; it is only transferred from one store to another, or dissipated to the surroundings.
现代考纲把能量描述为储存在不同的"储存库"中，并沿特定路径转移。能量永远不会被"用掉"，它只是从一个储存库转移到另一个储存库，或耗散到周围环境。
| Store 储存库 | Description 说明 | Typical example 典型例子 |
| Kinetic 动能 | due to motion 由运动具有 | moving car 行驶的汽车 |
| Gravitational 重力势能 | due to height in a field 由场中的高度决定 | raised water 高处的水 |
| Elastic 弹性势能 | due to stretching or compression 由形变决定 | stretched spring 拉伸的弹簧 |
| Chemical 化学能 | stored in bonds 储存在化学键中 | food, fuel 食物、燃料 |
| Internal 内能 | random kinetic and potential energy of particles 粒子无规则动能与势能 | hot water 热水 |
| Nuclear 核能 | stored in nuclei 储存在原子核中 | reactor fuel 反应堆燃料 |
The main transfer pathways are mechanical work (a force moving through a distance), electrical work (charge moving through a potential difference), heating (because of a temperature difference) and radiation (electromagnetic or sound waves).
主要的转移途径有：机械功（力通过一段位移）、电功（电荷通过电势差）、加热（由于温度差）以及辐射（电磁波或声波）。
4. Kinetic Energy and the Work-Energy Theorem | 动能与动能定理
The kinetic energy of a body of mass m moving at speed v is:
质量为 m、速率为 v 的物体的动能为：
Eₖ = ½ m v²
Kinetic energy is a scalar and is always positive or zero. Because it depends on v², doubling the speed quadruples the kinetic energy; this is why braking distance increases so rapidly with speed.
动能是标量，恒为正值或零。由于它与 v² 成正比，速率加倍则动能变为四倍；这正是刹车距离随车速迅速增加的原因。
The work-energy theorem states that the net work done on a body equals its change in kinetic energy:
动能定理指出，作用在物体上的净功等于其动能的变化量：
Wnet = ΔEₖ = ½ m v² − ½ m u²
This result follows directly from Newton' Fs with F = ma and v² = u² + 2as, substituting s = (v² − u²) / (2a) gives W = ½mv² − ½mu².
该结果可直接由牛顿第二定律推出。由 W = Fs，令 F = ma，并用 v² = u² + 2as，代入 s = (v² − u²) / (2a) 即得 W = ½mv² − ½mu²。
Momentum links usefully to kinetic energy. Since p = mv:
动量与动能之间有很有用的联系。因为 p = mv，所以：
Eₖ = p² / (2m)
5. Gravitational Potential Energy | 重力势能
For a small change in height near the Earth's surface, the change in gravitational potential energy is:
在地球表面附近，高度有微小变化时，重力势能的变化为：
ΔEₚ = m g Δh
This expression is only valid when g is effectively constant, that is, for height changes that are small compared with the Earth's radius. The reference level for h is arbitrary: only changes in potential energy are physically meaningful, so you may choose any convenient zero.
该式仅在 g 可视为常量时成立，即高度变化远小于地球半径时。h 的参考面可以任意选取：只有势能的变化才有物理意义，因此可以选择任意方便的零点。
When a body falls freely, gravity does positive work mgh and the potential energy store decreases by the same amount while the kinetic energy store increases. When a body is raised at constant speed, the lifting force does work mgh and the gravitational store increases. The two descriptions are two ways of bookkeeping the same transfer.
物体自由下落时，重力做正功 mgh，重力势能库减少同样的数值，而动能库增加。物体被匀速提升时，提升力做功 mgh，重力势能库增加。两种描述只是对同一次转移的两种记账方式。
s and you must use the general expression Eₚ = −G M m / r, measured from infinity as zero.
对于大距离情形，例如卫星，简单的 mgh 公式不再适用，必须使用一般表达式 Eₚ = −G M m / r，并取无穷远处为零点。
6. Elastic Potential Energy in Springs | 弹簧的弹性势能
For a spring obeying Hooke's law, the extension x is proportional to the applied force F, with spring constant k:
对于遵守胡克定律的弹簧，伸长量 x 与所加力 F 成正比，比例系数为劲度系数 k：
F = k x
The energy stored in the stretched or compressed spring equals the area under the force-extension graph, which is a triangle of base x and height F:
拉伸或压缩的弹簧所储存的能量等于力–伸长量图像下的面积，即底为 x、高为 F 的三角形面积：
E = ½ F x = ½ k x²
Two warnings are essential in exams. First, Hooke' up to the limit of proportionality; beyond that point the graph curves and the area must be found graphically. Second, the unit of k is N m⁻¹, and x must be in metres.
考试中有两个关键提醒。第一，胡克定律以及 E = ½kx² 只在正比极限以内成立；超过该点图像变弯，必须用图像面积求能量。第二，k 的单位是 N m⁻¹，x 必须以米为单位。
.15)² = 2.25 J. When released, this 2.25 J may be transferred to a ball as kinetic energy.
例题：劲度系数 k = 200 N m⁻¹ 的弹簧被压缩 0.15 m，储存能量 E = ½ × 200 × (0.15)² = 2.25 J。释放后，这 2.25 J 可以转化为小球的动能。
7. The Principle of Conservation of Energy | 能量守恒定律
The principle of conservation of energy states that the total energy of a closed system remains constant; energy may be transferred between stores, but the total amount is unchanged. Energy is never created or destroyed.
能量守恒定律指出，封闭系统的总能量保持不变；能量可以在各储存库之间转移，但总量不变。能量既不能被创造，也不能被消灭。
In mechanics, the principle is usually applied in the form 'loss in one store equals gain in another'. For a frictionless pendulum or a free-falling object:
在力学中，该原理通常写作"一个储存库的减少量等于另一个储存库的增加量"。对于无摩擦的单摆或自由落体：
m g h = ½ m v² → v = √2 g h
Notice that the mass cancels, so a heavy ball and a light ball released from the same height reach the same speed. This is a common exam question, and the algebraic cancellation is usually worth a mark.
注意质量被约去，因此从同一高度释放的重球与轻球落地速率相同。这是常见考题，代数上的约去通常可以得一分。
When friction or air resistance is present, the mechanical energy is not conserved on its own, but the total energy still is: it reappears as internal energy, with a small temperature rise in the surfaces and the air.
当存在摩擦或空气阻力时，机械能本身不守恒，但总能量仍然守恒：它以内能的形式重新出现，使接触面与空气的温度略微升高。
8. Efficiency and Dissipation | 效率与能量耗散
Efficiency compares the useful energy output with the total energy input, and is usually expressed as a percentage:
效率比较有用的能量输出与总能量输入，通常用百分数表示：
efficiency = (useful output / total input) × 100%
The same formula applies to power, because power is simply energy per unit time. A device can never have an efficiency greater than 100%, because that would violate the conservation of energy.
同样的公式也适用于功率，因为功率就是单位时间的能量。任何装置的效率都不可能大于 100%，否则将违反能量守恒。
- A filament lamp: about 5% of the input becomes light, the rest heats the surroundings.
- 白炽灯：约 5% 的输入转化为光，其余用于加热周围环境。
- A large electric motor: efficiencies of 90% or more are typical.
- 大型电动机：效率通常可达 90% 以上。
- A car engine: roughly 25% to 30% of the chemical energy becomes useful kinetic energy.
- 汽车发动机：大约 25% 到 30% 的化学能变为有用的动能。
Dissipated energy is not destroyed, but it is spread out into the random motion of many particles at low temperature, where it is much less useful. Reducing waste usually means reducing friction and unnecessary heating.
耗散的能量并没有被消灭，而是分散成许多粒子在较低温度下的无规则运动，因而用处小得多。减少浪费通常意味着减少摩擦和不必要的发热。
9. Power and the Rate of Energy Transfer | 功率与能量转移速率
Power is defined as the rate of transfer of energy, or the rate of doing work:
功率定义为能量转移的速率，即做功的速率：
P = ΔW / Δt
stant speed v under a driving force F in the same direction as its motion, the mechanical power is:
功率的国际单位是瓦特（W），1 W = 1 J s⁻¹。若物体在与其运动方向相同的驱动力 F 作用下以恒定速率 v 运动，则机械功率为：
P = F v
Domestic electricity is billed in kilowatt-hours rather than joules. One kilowatt-hour is the energy transferred by a 1 kW device in one hour:
家庭用电以千瓦时而非焦耳计费。1 千瓦时是 1 kW 的用电器工作 1 小时所转移的能量：
1 kW h = 1000 W × 3600 s = 3.6 × 10⁶ J
e kettle transfers 2.0 × 10³ × 180 = 3.6 × 10⁵ J, while the lamp transfers 100 × 3600 = 3.6 × 10⁵ J: the same energy at very different powers.
典型考题要求比较两种电器，例如使用 3 分钟的 2.0 kW 水壶与亮 60 分钟的 100 W 灯。水壶转移 2.0 × 10³ × 180 = 3.6 × 10⁵ J，灯转移 100 × 3600 = 3.6 × 10⁵ J：能量相同而功率差别很大。
10. Internal Energy and the First Law | 内能与热力学第一定律
The internal energy of a system is the sum of the random kinetic energy and the potential energy of all its particles. It depends on both the temperature and the state of the substance, so temperature and internal energy are not the same quantity.
系统的内能是其所有粒子的无规则动能与势能之和。它同时取决于温度和物质的状态，因此温度与内能并不是同一个量。
The first law of thermodynamics is a statement of conservation of energy for thermal processes:
热力学第一定律是能量守恒在热过程中的表述：
ΔU = q + w
= q − w with w defined as work done by the system; always state which convention you use.
在此约定下，q 是通过加热供给系统的能量，w 是外界对系统所做的功。有些教材写作 ΔU = q − w，其中 w 定义为系统对外做的功；答题时应说明所用的约定。
q = +500 J and w = −200 J, so ΔU = 500 − 200 = +300 J. The internal energy rises by 300 J.
例题：气体通过吸热获得 500 J 能量，同时对外做功 200 J。则 q = +500 J，w = −200 J，故 ΔU = 500 − 200 = +300 J，内能增加 300 J。
A useful check: heating a gas at constant volume changes only its internal energy, while heating at constant pressure also does work of expansion, so more energy is needed for the same temperature rise.
一个有用的检验：在定容条件下加热气体只改变其内能，而在定压条件下加热还要做膨胀功，因此升高同样温度需要更多能量。
11. Reading Energy Graphs | 能量图像的读法
Graphs carry several marks in this topic, and the routine is always the same: identify what the axes are, then ask whether the useful information is the gradient or the area.
本专题的图像题占分不少，方法总是相同的：先认清坐标轴，再判断有用信息来自斜率还是面积。
| Graph 图像 | Meaning of gradient 斜率的含义 | Meaning of area 面积的含义 |
| force against extension 力–伸长量 | spring constant k 劲度系数 k | elastic energy stored 弹性势能 |
| force against displacement 力–位移 | stiffness of interaction 相互作用刚度 | work done 所做的功 |
| power against time 功率–时间 | rate of change of power 功率变化率 | energy transferred 转移的能量 |
| energy against time 能量–时间 | power 功率 | <
