Work and Energy | 功与能量

📚 Work and Energy | 功与能量

In IGCSE Physics, the concepts of work and energy form the foundation for understanding how forces cause changes in motion and how energy is transferred between different forms. This article systematically explains the key ideas, formulas, and applications you need to master for the CIE examination. Read on to build a strong grasp of work, kinetic energy, potential energy, power, and the principle of conservation of energy.

在IGCSE物理中，功与能量的概念是理解力如何引起运动变化以及能量如何在各种形式之间转化的基础。本文系统讲解CIE考试中必须掌握的核心概念、公式及其应用。继续阅读，扎实掌握功、动能、势能、功率和能量守恒原理。

1. What is Work? | 什么是功？

In physics, work is done when a force causes an object to move in the direction of the force. If there is no movement, or the movement is perpendicular to the force, no work is done in the scientific sense.

在物理学中，当一个力使物体沿力的方向发生位移时，就做了功。如果没有移动，或者移动方向与力垂直，则从科学意义上讲没有做功。

Work is a scalar quantity, measured in joules (J). One joule is the work done when a force of 1 newton moves an object 1 metre in the direction of the force.

功是一个标量，单位为焦耳（J）。1焦耳就是1牛顿的力使物体沿力的方向移动1米所做的功。

The mathematical relationship is expressed as:

其数学关系表达为：

W = F × d

where W is work (J), F is force (N), and d is distance moved in the direction of the force (m).

其中W表示功（J），F表示力（N），d表示沿力方向移动的距离（m）。

2. Calculating Work Done at an Angle | 计算力与位移成角度时的功

When the force is applied at an angle θ to the displacement, only the component of the force in the direction of motion does work. The formula becomes:

当力与位移方向成θ角时，只有力在运动方向上的分量做功。公式变为：

W = F × d × cos θ

Note: In the IGCSE CIE syllabus, most questions involve forces parallel to the displacement, but you should be aware that perpendicular forces (cos 90° = 0) do zero work.

注意：在IGCSE CIE大纲中，大多数题目涉及力与位移平行，但你需要知道垂直力（cos 90°=0）做功为零。

For example, when you carry a heavy bag horizontally, the upward force you apply is perpendicular to the motion, so you do no work on the bag in the physics sense.

例如，当你水平提着重物时，你施加的向上的力与运动方向垂直，因此在物理意义上你没有对包做功。

3. Energy – The Capacity to Do Work | 能量——做功的本领

Energy is defined as the ability or capacity to do work. It is also measured in joules (J). Energy exists in many forms: kinetic, gravitational potential, elastic potential, thermal, chemical, nuclear, and more.

能量定义为做功的能力或本领。它的单位也是焦耳（J）。能量有多种形式：动能、重力势能、弹性势能、热能、化学能、核能等等。

Energy can be transferred from one form to another, but it is never created or destroyed. This is the principle of conservation of energy.

能量可以从一种形式转化为另一种形式，但永远不会被创造或消灭。这就是能量守恒原理。

4. Kinetic Energy (Eₖ) | 动能 (Eₖ)

Kinetic energy is the energy an object possesses due to its motion. Any moving object has kinetic energy. The formula is:

动能是物体由于运动而具有的能量。任何运动的物体都具有动能。其公式为：

Eₖ = ½ m v²

where m is mass (kg) and v is speed (m/s). Notice that kinetic energy depends on the square of the speed, so doubling the speed quadruples the kinetic energy.

其中m是质量（kg），v是速度（m/s）。注意动能取决于速度的平方，因此速度加倍将使动能变为原来的四倍。

In IGCSE problems, you often need to calculate the kinetic energy of a car, a runner, or an object just before it hits the ground. Always remember to square the speed first, then multiply by half the mass.

在IGCSE题目中，经常需要计算汽车、跑步者或物体落地前的动能。务必先对速度进行平方，再乘以质量的一半。

5. Gravitational Potential Energy (Eₚ) | 重力势能 (Eₚ)

Gravitational potential energy is the energy stored in an object because of its position in a gravitational field. The higher the object is above a reference level, the greater its GPE.

重力势能是由于物体在重力场中的位置而储存的能量。物体离参考水平面越高，其重力势能越大。

The formula for changes in gravitational potential energy near the Earth’s surface is:

在地球表面附近，重力势能的变化公式为：

ΔEₚ = m g Δh

where m is mass (kg), g is gravitational field strength (9.8 N/kg or approximately 10 N/kg on Earth), and Δh is the change in vertical height (m).

其中m是质量（kg），g是重力场强度（在地球上为9.8 N/kg或近似取10 N/kg），Δh是高度的变化（m）。

When an object falls, GPE is converted into kinetic energy. If air resistance is negligible, the loss in GPE equals the gain in kinetic energy.

物体下落时，重力势能转化为动能。如果空气阻力可以忽略，则重力势能的减少量等于动能的增加量。

6. Elastic Potential Energy (Strain Energy) | 弹性势能（应变能）

Elastic potential energy is stored in objects that are stretched, compressed, or bent, and can return to their original shape. Typical examples include springs and rubber bands.

弹性势能储存在被拉伸、压缩或弯曲并能恢复原状的物体中。典型例子包括弹簧和橡皮筋。

For a spring obeying Hooke’s Law (extension proportional to load), the elastic potential energy stored is given by:

对于遵守胡克定律（伸长量与负载成正比）的弹簧，储存的弹性势能为：

Eₑ = ½ F x or Eₑ = ½ k x²

where F is the applied force (N), x is the extension or compression (m), and k is the spring constant (N/m).

其中F是施加的力（N），x是伸长量或压缩量（m），k是弹簧常数（N/m）。

IGCSE questions may ask you to calculate the energy stored in a spring from a force-extension graph; the area under the graph gives the work done and hence the elastic energy stored.

IGCSE题目可能会要求根据力-伸长图计算弹簧中储存的能量；图下方的面积表示所做的功，也就是储存的弹性能量。

7. The Relationship Between Work and Energy | 功与能量之间的关系

When work is done on an object, energy is transferred to that object. Conversely, when an object does work, it transfers energy away. This is often summarised as:

对一个物体做功时，能量就传递给该物体。反之，当物体对外做功时，它传递出能量。这通常概括为：

Work done = Energy transferred

For instance, when you lift a book, you do work against gravity and the book gains gravitational potential energy. When you push a trolley and it speeds up, the work done is converted into kinetic energy.

例如，当你举起一本书时，你克服重力做功，书获得重力势能。当你推动手推车并使其加速时，所做的功转化为动能。

In mechanics problems, the work-energy principle states that the net work done on an object equals its change in kinetic energy.

在力学问题中，功–能原理指出，对物体所做的净功等于其动能的变化量。

8. Conservation of Energy | 能量守恒

The law of conservation of energy states that energy cannot be created or destroyed, only transferred or converted from one form to another. The total energy of an isolated system remains constant.

能量守恒定律指出，能量不能被创造或消灭，只能从一种形式传递或转化为另一种形式。孤立系统的总能量保持不变。

In a pendulum, energy continuously changes between gravitational potential energy (at highest points) and kinetic energy (at lowest point). If there is no friction, the total mechanical energy stays the same.

在单摆中，能量在重力势能（最高点）和动能（最低点）之间不断转换。如果没有摩擦，总机械能保持不变。

A common IGCSE question is to describe energy changes: e.g., a ball thrown upwards: KE → GPE at the top; then GPE → KE as it falls back down. Some energy is often transferred to thermal energy due to air resistance.

IGCSE中常见的问题是描述能量转化：例如，上抛的球：动能→最高点的重力势能；下落时重力势能→动能。由于空气阻力，部分能量通常转化为热能。

9. Power – The Rate of Doing Work | 功率——做功的快慢

Power is defined as the rate at which work is done or the rate at which energy is transferred. It is a scalar quantity measured in watts (W), where 1 W = 1 J/s.

功率定义为单位时间内做功的多少或能量传递的快慢。它是标量，单位是瓦特（W），1 W = 1 J/s。

The formula for power is:

功率的公式为：

P = W / t or P = ΔE / t

where P is power (W), W is work done (J) or ΔE is energy transferred (J), and t is time taken (s).

其中P是功率（W），W是做的功（J）或ΔE是传递的能量（J），t是所用时间（s）。

Another useful form arises when a constant force moves something at constant speed: P = F × v, where v is velocity. This helps compare engine power, for example.

当恒力使物体匀速运动时，还有一个有用的形式：P = F × v，其中v是速度。例如，这可用于比较发动机功率。

10. Efficiency of Energy Transfers | 能量转化的效率

In real systems, not all input energy is converted into useful output energy. Some energy is always dissipated, often as thermal energy, due to friction, air resistance, or electrical heating.

在真实系统中，并非所有输入能量都转化为有用的输出能量。由于摩擦、空气阻力或电热效应，总有一部分能量被耗散，通常是热能。

Efficiency is the ratio of useful output energy (or power) to total input energy (or power), expressed as a percentage:

效率是有用输出能量（或功率）与总输入能量（或功率）之比，用百分比表示：

Efficiency = (Useful output energy / Total input energy) × 100%

No machine can be 100% efficient because of energy dissipation. In IGCSE, you will be expected to calculate efficiency or identify ways to increase it, such as lubrication to reduce friction.

由于能量耗散，没有任何机器可以达到100%的效率。在IGCSE中，你需要计算效率或提出提高效率的方法，例如通过润滑减少摩擦。

11. Worked Example – Falling Object | 例题——落体问题

A stone of mass 0.5 kg is dropped from a height of 20 m. Calculate (a) its initial GPE, (b) its speed just before hitting the ground (take g = 10 N/kg).

一块质量为0.5 kg的石头从20 m高处静止下落。计算：(a) 初始重力势能；(b) 落地速度（取g = 10 N/kg）。

(a) GPE = m g h = 0.5 × 10 × 20 = 100 J.

(b) Assuming no air resistance, GPE → KE: 100 = ½ m v² → 100 = ½ × 0.5 × v² → v² = 400 → v = 20 m/s.

(b) 假设无空气阻力，重力势能全部转化为动能：100 = ½ × 0.5 × v² → v² = 400 → v = 20 m/s.

12. Key Points Summary | 重点总结

Here is a quick checklist of the essential equations and concepts for your IGCSE revision:

以下是IGCSE复习时必须掌握的公式和概念的快速清单：

Work done: W = F d (force parallel to displacement) | 功：W = F d（力与位移平行）
Kinetic energy: Eₖ = ½ m v² | 动能：Eₖ = ½ m v²
Gravitational potential energy: ΔEₚ = m g Δh | 重力势能：ΔEₚ = m g Δh
Elastic potential energy: Eₑ = ½ F x = ½ k x² | 弹性势能：Eₑ = ½ F x = ½ k x²
Conservation of energy: total energy constant in isolated system | 能量守恒：孤立系统总能量恒定
Power: P = W/t = E/t | 功率：P = W/t = E/t
Efficiency: (Useful output / total input) × 100% | 效率：（有用输出/总输入）×100%

Always remember to use SI units: mass in kg, distance in m, time in s, force in N, and energy in J. Practice energy conversion problems and graph interpretation to strengthen your understanding.

始终记住使用国际单位制：质量用kg，距离用m，时间用s，力用N，能量用J。多练习能量转化问题和图像分析以加深理解。

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Work and Energy | 功与能量