Work, Energy and Power | 功、能与功率

📚 Work, Energy and Power | 功、能与功率

In physics, the concepts of work, energy and power are fundamental to understanding how forces cause motion and how systems evolve. Work quantifies the action of a force over a displacement, energy represents the capacity to do work, and power measures the rate at which work is performed. Mastering these ideas allows us to analyse mechanical systems, from simple machines to complex engines, and forms a cornerstone of both classical mechanics and thermal physics.

在物理学中，功、能与功率的概念是理解力如何引起运动以及系统如何演化的基础。功量化了力在位移上的作用，能量代表做功的能力，而功率则衡量做功的快慢。掌握这些概念使我们能够分析从简单机械到复杂发动机的各种力学系统，它们构成了经典力学和热物理学的基石。

1. The Concept of Work | 功的概念

In everyday language, ‘work’ may mean any physical or mental effort, but in physics it has a precise meaning. Work is done on an object when a force acts on the object and causes a displacement in the direction of the force. If you push a wall and it does not move, no work is done on the wall, regardless of how much effort you exert. Work, therefore, requires both a force and movement.

在日常语言中，“做功”可能指任何体力或脑力的付出，但在物理学中它有精确的定义。当一个力作用在物体上并使物体沿力的方向发生位移时，我们就说力对物体做了功。如果你推一堵墙而墙不动，无论你用了多大力气，都没有对墙做功。因此，功既需要力，也需要移动。

2. Calculating Work Done | 计算所做的功

For a constant force F acting at an angle θ to the displacement d, the work done W is defined as the product of the force component in the direction of the displacement and the displacement itself. The mathematical expression is:

对于一个与位移 d 成角度 θ 的恒力 F，所做的功 W 定义为力在位移方向上的分量与位移的乘积。数学表达式为：

W = F d cos θ

When the force is parallel to the displacement (θ = 0°), cos θ = 1 and the formula simplifies to W = F d. The SI unit of work is the joule (J), where 1 J is the work done when a force of 1 newton displaces an object by 1 metre in the direction of the force.

当力与位移平行时（θ = 0°），cos θ = 1，公式简化为 W = F d。功的国际单位是焦耳（J），1 J 等于1牛顿的力使物体沿力的方向移动1米所做的功。

3. Energy and Its Forms | 能量及其形式

Energy is often described as the ability to do work. It is a scalar quantity measured in joules (J). Energy can exist in many forms, including kinetic energy (energy of motion), gravitational potential energy (energy due to position in a gravitational field), elastic potential energy (energy stored in deformed objects), thermal energy, chemical energy, and many others. In mechanics, we mainly focus on kinetic and potential energies.

能量通常被描述为做功的能力。它是一个标量，单位为焦耳（J）。能量有多种形式，包括动能（运动的能量）、重力势能（因在引力场中的位置而具有的能量）、弹性势能（储存在形变物体中的能量）、热能、化学能等等。在力学中，我们主要关注动能和势能。

4. Kinetic Energy | 动能

Kinetic energy (KE or Eₖ) is the energy an object possesses due to its motion. For an object of mass m moving at speed v, the kinetic energy is given by:

动能（KE 或 Eₖ）是物体由于运动而具有的能量。对于质量为 m、速度为 v 的物体，其动能由下式给出：

KE = ½ m v²

This relationship shows that kinetic energy depends on the square of the speed; doubling the speed quadruples the kinetic energy. To derive this, we can consider the work done by a net force accelerating an object from rest: the work W = F d = m a d, and using v² = 2 a d, we obtain W = ½ m v², which equals the gain in kinetic energy.

这个关系表明动能与速度的平方成正比；速度翻倍，动能变为原来的四倍。为推导该公式，可以考虑合外力将物体从静止加速所做的功：功 W = F d = m a d，再利用 v² = 2 a d，得到 W = ½ m v²，它等于物体获得的动能。

5. Gravitational Potential Energy | 重力势能

Gravitational potential energy (GPE or Eₚ) is the energy stored in an object because of its height above a reference level in a gravitational field. Near the Earth’s surface, where the gravitational field is approximately uniform, the change in gravitational potential energy when raising an object of mass m by a vertical height h is:

重力势能（GPE 或 Eₚ）是物体由于在引力场中相对于参考平面的高度而储存的能量。在地球表面附近，引力场近似均匀，当把质量为 m 的物体竖直升高 h 时，重力势能的变化为：

ΔEₚ = m g h

Here, g is the acceleration due to gravity (≈ 9.81 m s⁻²). The work done against gravity when lifting an object slowly at constant speed is stored as gravitational potential energy. The zero level of potential energy can be chosen arbitrarily; only changes in potential energy have physical significance.

这里 g 是重力加速度（≈ 9.81 m s⁻²）。缓慢、匀速提升物体时克服重力所做的功被储存为重力势能。势能的零水平可以任意选择；只有势能的变化才有物理意义。

6. Elastic Potential Energy | 弹性势能

Elastic potential energy is the energy stored in elastic materials when they are stretched or compressed. For an ideal spring obeying Hooke’s law, the force required to extend or compress the spring by a displacement x is F = k x, where k is the spring constant. The work done to change the spring’s length from its equilibrium position to x is stored as elastic potential energy:

弹性势能是弹性材料被拉伸或压缩时储存的能量。对于遵守胡克定律的理想弹簧，使弹簧产生位移 x 所需的力为 F = k x，其中 k 是弹簧常数。将弹簧从平衡位置改变到 x 所做的功以弹性势能的形式储存起来：

Eₑ = ½ k x²

This formula arises from the fact that the force varies linearly with displacement; the work done is the area under the force–displacement graph, which forms a triangle of base x and height k x.

这个公式源于力随位移线性变化的事实；所做的功等于力-位移图下的面积，该图形成一个底为 x、高为 k x 的三角形。

7. The Work–Energy Theorem | 功-能定理

The work–energy theorem is a central linking concept: the net work done on an object by all external forces equals the change in its kinetic energy. Expressed mathematically:

功-能定理是一个核心的连接性概念：所有外力对物体所做的净功等于物体动能的变化。用数学式表示为：

W_net = ΔKE = ½ m v_f² − ½ m v_i²

This theorem allows us to analyse motion without directly dealing with acceleration and time. For example, when a car brakes to a stop, the negative work done by friction reduces the kinetic energy to zero. The theorem holds regardless of the path taken, as long as we consider all forces acting on the object.

这一定理使我们能够在不直接处理加速度和时间的情况下分析运动。例如，当汽车刹车停下时，摩擦力所做的负功使动能减小到零。无论路径如何，只要考虑作用在物体上的所有力，该定理都成立。

8. Conservation of Mechanical Energy | 机械能守恒

When only conservative forces (such as gravity and elastic spring forces) do work on a system, the total mechanical energy — the sum of kinetic energy and all relevant potential energies — remains constant. That is:

当只有保守力（如重力和弹簧弹力）对系统做功时，系统的总机械能——动能与所有相关势能之和——保持恒定。即：

KE_i + PE_i = KE_f + PE_f

In the presence of non‑conservative forces like friction, mechanical energy is not conserved; it is transformed into thermal energy or other forms. However, the broader principle of conservation of energy states that the total energy of an isolated system is always conserved.

当存在如摩擦力等非保守力时，机械能不守恒，它转化为热能或其他形式的能量。然而，更广泛的能量守恒定律指出，孤立系统的总能量总是守恒的。

9. Power | 功率

Power is defined as the rate at which work is done or energy is transferred. The average power P_avg over a time interval Δt is:

功率定义为做功或能量传递的速率。在一段时间 Δt 内的平均功率 P_avg 为：

P_avg = ΔW / Δt

The SI unit of power is the watt (W), where 1 W = 1 J s⁻¹. For a constant force moving an object at steady speed v in the direction of the force, power can also be expressed as P = F v. This relation is particularly useful for calculating the power output of engines and motors.

功率的国际单位是瓦特（W），1 W = 1 J s⁻¹。当恒力使物体沿力的方向以稳定速度 v 运动时，功率也可表示为 P = F v。这一关系在计算引擎和电动机的输出功率时非常有用。

10. Efficiency | 效率

Efficiency (η) measures how well a device converts input energy into useful output energy. It is defined as the ratio of useful work or energy output to the total energy input, usually expressed as a percentage:

效率（η）衡量设备将输入能量转化为有用输出能量的能力。它定义为有用的功或能量输出与总能量输入之比，通常以百分比表示：

η = (E_useful / E_input) × 100%

Because some energy is always dissipated as heat due to friction or resistance, efficiency is always less than 1 (or less than 100%). Improving efficiency is a key goal in engineering and energy management, such as designing low‑friction engines and optimising electrical circuits.

由于摩擦或电阻等原因总会有一部分能量以热能的形式耗散，效率总是小于1（或小于100%）。提高效率是工程和能源管理中的关键目标，例如设计低摩擦引擎和优化电路。

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Work, Energy and Power | 功、能与功率