📚 IGCSE Physics: Calculating Work and Power | IGCSE物理：功与功率的计算
In IGCSE Physics (Edexcel), the concepts of work and power are fundamental to understanding how energy is transferred in mechanical systems. This article provides a comprehensive guide to calculating work done and power, complete with worked examples and examination tips tailored to the Edexcel specification.
在IGCSE物理（爱德思考试局）中，功和功率是理解机械系统中能量转移的核心概念。本文提供了功和功率计算的全面指南，包含典型例题和针对爱德思考纲的考试技巧。
1. What is Work Done? | 什么是功？
In physics, work is done when a force causes an object to move in the direction of the force. The key point is that work requires both a force and a displacement. If you push against a wall and it doesn't move, no work is done in the physics sense — even though you may feel tired!
在物理学中，当一个力使物体沿力的方向移动时，就说做了功。关键在于做功需要同时具备力和位移两个条件。如果你推一堵墙但墙没有移动，从物理学意义上讲就没有做功——即使你感到疲惫！
The equation for work done is:
Work Done (J) = Force (N) × Distance moved in the direction of the force (m)
W = F × s
Where W is measured in joules (J), F in newtons (N), and s in metres (m). One joule is defined as the work done when a force of one newton moves an object through a distance of one metre in the direction of the force.
其中W的单位是焦耳（J），F的单位是牛顿（N），s的单位是米（m）。一焦耳的定义是：一牛顿的力使物体沿力的方向移动一米所做的功。
2. The Units of Work and Energy | 功与能量的单位
Work and energy share the same unit — the joule (J). This is because doing work on an object transfers energy to it. The amount of energy transferred equals the amount of work done.
功和能量共享相同的单位——焦耳（J）。这是因为对物体做功就是将能量传递给物体。能量转移的量等于所做的功。
Common multiples and submultiples you should know for the Edexcel exam:
在爱德思考试中，你应该熟悉的常用倍数和分数单位：
| Unit | 单位 | Symbol | 符号 | Conversion | 换算 |
| Kilojoule | 千焦 | kJ | 1 kJ = 1000 J |
| Megajoule | 兆焦 | MJ | 1 MJ = 1,000,000 J |
| Kilowatt-hour | 千瓦时 | kWh | 1 kWh = 3.6 × 10⁶ J |
Beware: the kilowatt-hour (kWh) is a unit of energy, not power! It is commonly used by electricity companies to measure energy consumption.
注意：千瓦时（kWh）是能量单位，不是功率单位！电力公司通常用它来计量电能消耗。
3. Work Done Against Gravity | 克服重力做功
When you lift an object vertically, you do work against the gravitational force. The force you need to overcome is the weight of the object (mg). The distance is the vertical height (h) the object is raised.
当你垂直提起物体时，你在克服重力做功。你需要克服的力是物体的重量（mg）。距离是物体被提升的垂直高度（h）。
For lifting an object:
提升物体时：
Work Done = Weight × Vertical Height
W = m × g × h
Where m is mass in kilograms (kg), g is gravitational field strength (approximately 9.8 N/kg on Earth), and h is height in metres (m).
其中m是质量（单位：千克 kg），g是重力场强度（地球上约为9.8牛/千克 N/kg），h是高度（单位：米 m）。
Worked Example | 例题： om the floor onto a shelf that is 1.5 m high. Calculate the work done. (Take g = 9.8 N/kg)
例题： 一名学生将3千克的书包从地板提升到1.5米高的架子上。计算所做的功。（取g = 9.8牛/千克）
Solution | 解答： W = mgh = 3 × 9.8 × 1.5 = 44.1 J
W = 3 × 9.8 × 1.5 = 44.1 J
4. What is Power? | 什么是功率？
Power is the rate at which work is done — in other words, how quickly energy is transferred. Two students can both lift the same box to the same height, but the one who does it faster has a greater power output.
功率是做功的快慢——换句话说，就是能量转移的速度。两个学生都能把同一个箱子提到相同的高度，但做得更快的那位具有更大的功率输出。
The equation for power is:
功率的公式是：
Power (W) = Work Done (J) ÷ Time Taken (s)
P = W ÷ t
Power is measured in watts (W). One watt is equal to one joule per second (1 W = 1 J/s).
功率的单位是瓦特（W）。一瓦特等于每秒一焦耳（1瓦 = 1焦/秒）。
Worked Example | 例题： A crane lifts a load, doing 6000 J of work in 15 seconds. Calculate the power output of the crane.
例题： 一台起重机在15秒内做了6000焦耳的功。计算起重机的功率输出。
Solution | 解答： P = W ÷ t = 6000 ÷ 15 = 400 W
P = 6000 ÷ 15 = 400 W
5. Alternative Power Formula | 功率的另一种公式
Since work done W = F × s, we can substitute this into the power equation:
由于做功W = F × s，我们可以将其代入功率公式：
P = (F × s) ÷ t
Since speed v = s ÷ t, this simplifies to:
由于速度v = s ÷ t，这可以简化为：
Power = Force × Speed
P = F × v
This version is especially useful when dealing with vehicles moving at constant speed. For example, if a car experiences a resistive force of 800 N and travels at a constant speed of 25 m/s:
这个版本在处理以恒定速度运动的车辆时特别有用。例如，如果一辆汽车受到800牛的阻力并以25米/秒的恒定速度行驶：
P = 800 × 25 = 20,000 W = 20 kW
This tells us the engine must supply 20 kW of power to maintain this speed against the resistive forces.
这告诉我们发动机必须提供20千瓦的功率来克服阻力并维持这个速度。
6. Efficiency and Power | 效率与功率
In real-world machines, not all input energy is converted into useful output energy. Some energy is always lost, usually as heat and sound. Efficiency compares useful output to total input, and it applies to both energy and power.
在现实世界的机器中，并非所有输入能量都能转化为有用的输出能量。总会有一些能量损失，通常以热和声音的形式。效率比较的是有用输出与总输入的关系，它同时适用于能量和功率。
Efficiency = (Useful Output Energy ÷ Total Input Energy) × 100%
Or equivalently, in terms of power:
或者等效地用功率表示：
Efficiency = (Useful Output Power ÷ Total Input Power) × 100%
Worked Example | 例题： An electric motor is supplied with 500 W of electrical power. It produces 350 W of useful mechanical power. Calculate its efficiency.
例题： 一台电动机被输入500瓦的电功率。它产生350瓦的有用机械功率。计算其效率。
Solution | 解答： Efficiency = (350 ÷ 500) × 100% = 70%
效率 = (350 ÷ 500) × 100% = 70%
For the Edexcel exam, remember that efficiency has no units — it is expressed as a percentage.
在爱德思考试中，记住效率没有单位——它以百分比表示。
7. Work Done Against Friction | 克服摩擦力做功
Friction is a common force in everyday mechanics. When an object slides across a surface, work must be done to overcome the frictional force. This work is usually converted into thermal energy.
摩擦力是日常力学中常见的力。当物体在表面上滑动时，必须做功来克服摩擦力。这部分功通常转化为热能。
The equation remains the same:
公式保持不变：
Work Done = Frictional Force × Distance Slid
Worked Example | 例题： A wooden block is pushed across a floor with a frictional force of 12 N. If the block moves 4 m, how much work is done against friction?
例题： 一个木块在摩擦力为12牛的情况下被推过地板。如果木块移动了4米，克服摩擦力做了多少功？
Solution | 解答： W = F × s = 12 × 4 = 48 J
W = 12 × 4 = 48 J
8. Force-Distance Graphs | 力-距离图像
Edexcel exams often test your ability to interpret force-distance graphs. Work done can be determined from a graph of force against distance:
爱德思考卷经常测试你解读力-距离图像的能力。功可以通过力随距离变化的图像来确定：
Work Done = Area under a Force-Distance Graph
a changing force, you may need to count squares or divide the area into geometric shapes.
对于恒定力，图像是一条水平线，面积就是一个矩形（F × s）。对于变化的力，你可能需要数方格或将面积分成几何形状来计算。
This graphical method is particularly important in the IGCSE syllabus because it connects visual information with numerical calculations — a key skill assessed in Paper 2.
这种图像法在IGCSE考纲中特别重要，因为它将视觉信息与数值计算联系起来——这是试卷二考查的关键技能。
9. Kinetic Energy and Work | 动能与功的关系
There is a beautiful connection between work and kinetic energy. When a net force does work on an object, the object's kinetic energy changes by exactly the amount of work done.
功与动能之间有着美妙的联系。当合力对物体做功时，物体的动能恰好变化了所做的功的量。
The kinetic energy formula you must know:
你必须掌握的动能公式：
Kinetic Energy = ½ × mass × speed²
KE = ½mv²
This is the Work-Energy Theorem in its simplest form:
这是功-能定理的最简形式：
Work Done = Change in Kinetic Energy
Worked Example | 例题： A car of mass 1000 kg accelerates from 0 to 20 m/s. Calculate the work done on the car.
例题： 一辆质量为1000千克的汽车从0加速到20米/秒。计算对汽车所做的功。
Solution | 解答： ce initial speed = 0 m/s). Final KE = ½ × 1000 × 20² = ½ × 1000 × 400 = 200,000 J = 200 kJ
W = ΔKE = ½mv² ﹣ ½mu² = 200,000 ﹣ 0 = 200,000 J
10. Gravitational Potential Energy vs Work | 重力势能与功
The gain in gravitational potential energy (GPE) when lifting an object is exactly equal to the work done against gravity:
提升物体时重力势能（GPE）的增加恰好等于克服重力所做的功：
GPE = mgh = Work Done Against Gravity
This emphasises the principle of energy conservation — the work put in is stored as potential energy, which can later be converted back into kinetic energy (for example, when the object falls).
这凸显了能量守恒原理——输入的功被储存为势能，之后可以转化为动能（例如，当物体下落时）。
For the Edexcel exam, you should be comfortable converting between these energy forms in calculation questions. A typical question might involve a ball rolling down a ramp: the GPE lost equals the KE gained (ignoring friction).
在爱德思考试中，你应该能熟练地在这些能量形式之间进行换算计算。一个典型问题可能涉及球从斜坡上滚下：损失的重力势能等于获得的动能（忽略摩擦力）。
11. Common Exam Mistakes and Tips | 常见考试错误与技巧
Students frequently make the following mistakes in IGCSE Physics exams on this topic:
学生在IGCSE物理考试中在这个主题上经常犯以下错误：
Mixing up units: Remember that W stands for both work (joules) and watts. Check carefully whether the question uses J or W/N.
混淆单位：记住W既代表功（焦耳）也代表瓦特。仔细检查题目中使用的是J还是W/N。
Not converting units: If distance is given in kilometres (km), convert to metres (m) before substituting into the formula. Similarly, convert grams to kilograms.
不进行单位换算：如果距离给出的是千米（km），代入公式前要转换为米（m）。同样地，克要转换为千克。
Forgetting the direction condition: Work is only done when the displacement is in the direction of the force. A force perpendicular to motion (like a centripetal force) does no work.
忘记方向条件：只有当位移方向与力的方向相同时才做功。垂直于运动方向的力（如向心力）不做功。
Using the wrong value for g: s, g = 9.8 N/kg. Do not round to 10 unless the question explicitly instructs you to do so.
使用错误的g值：在爱德思IGCSE物理中，g = 9.8牛/千克。除非题目明确要求，否则不要四舍五入为10。
Confusing power and energy: Power is the rate of energy transfer. A high-power device transfers energy quickly, but total energy depends on how long it runs.
混淆功率和能量：功率是能量转移的速率。高功率设备能够快速转移能量，但总能量取决于运行时间。
12. Practice Questions | 练习题
Test your understanding with these Edexcel-style questions:
用这些爱德思风格的问题来测试你的理解：
Question 1 | 题目1： airs 12 m high in 20 seconds. Calculate the work done and the power output. (g = 9.8 N/kg)
题目1： 一名50千克的学生在20秒内爬了高12米的楼梯。计算所做的功和功率输出。（g = 9.8牛/千克）
Answer | 答案： W = mgh = 50 × 9.8 × 12 = 5880 J. P = W ÷ t = 5880 ÷ 20 = 294 W
W = 50 × 9.8 × 12 = 5880 J; P = 5880 ÷ 20 = 294 W
Question 2 | 题目2： An engine provides a driving force of 1500 N to move a car at a constant speed of 30 m/s. Calculate the power developed by the engine.
题目2： 一台发动机提供1500牛的驱动力使汽车以30米/秒的恒定速度行驶。计算发动机产生的功率。
Answer | 答案： P = F × v = 1500 × 30 = 45,000 W = 45 kW
P = 1500 × 30 = 45,000 W = 45 kW
Question 3 | 题目3： lifts 200 kg of water through a height of 5 m in 10 seconds. What is the power of the pump? (g = 9.8 N/kg)
题目3： 一台水泵在10秒内将200千克的水提升5米。水泵的功率是多少？（g = 9.8牛/千克）
Answer | 答案： W = mgh = 200 × 9.8 × 5 = 9800 J. P = 9800 ÷ 10 = 980 W
W = 200 × 9.8 × 5 = 9800 J; P = 9800 ÷ 10 = 980 W
Summary | 总结
mgh for lifting objects against gravity. Always check your units, consider the direction of force relative to displacement, and verify whether the question is about work or power.
功和功率本质上是关于能量转移的。记住核心公式：W = F × s用于计算功，P = W ÷ t用于计算功率。从这些可以推导出P = F × v用于运动物体，W = mgh用于克服重力提升物体。始终检查单位，考虑力的方向相对于位移的方向，并确认题目问的是功还是功率。
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