Work & Power | 功与功率

📚 Work & Power | 功与功率

In physics, “work” has a very specific meaning that is different from everyday language. Work is done when a force causes an object to move in the direction of the force. Power is the rate at which work is done. This article covers the essential concepts, equations, units, and worked examples for the Edexcel IGCSE Physics syllabus.

在物理学中，”功”的含义与日常用语截然不同。当一个力使物体沿力的方向移动时，就说这个力做了功。功率则是做功的快慢。本文涵盖Edexcel IGCSE物理考纲中的核心概念、公式、单位及例题。

1. What is Work? | 什么是功

Work is defined as the product of the force applied on an object and the distance moved by the object in the direction of the force. For work to be done, there must be a force and there must be displacement in the direction of that force.

功定义为：作用在物体上的力与物体沿力的方向移动距离的乘积。要产生功，必须存在力，并且物体必须沿力的方向发生位移。

The equation for work is:

功的计算公式为：

Work = Force × Distance (in the direction of the force)

功 = 力 × 距离（沿力的方向）

In symbol form:

用符号表示：

W = F × s

W = work done in joules (J) | 功，单位焦耳（J）
F = force in newtons (N) | 力，单位牛顿（N）
s = distance in metres (m) | 距离，单位米（m）

One joule of work is done when a force of one newton moves an object one metre in the direction of the force. 1 J = 1 N·m.

1焦耳的功是指1牛顿的力使物体沿力的方向移动1米所做的功。1 J = 1 N·m。

2. Key Condition: Direction Matters | 关键条件：方向的重要性

Only the component of force in the direction of motion does work. If you push a wall and it does not move, no work is done in the physics sense, even if you feel tired. Similarly, if you carry a bag horizontally, the upward force from your hand does no work on the bag because the force is perpendicular to the displacement.

只有沿运动方向的力分量才做功。如果你推墙而墙不动，那么在物理学意义上你没有做功，即使你感到疲倦。同样地，如果你水平提着包行走，手对包向上的力并不做功，因为力与位移垂直。

For a force applied at an angle θ to the direction of motion:

当力与运动方向成角度θ时：

W = F × s × cos θ

When θ = 0°, cos θ = 1, and W = F × s. When θ = 90°, cos θ = 0, and no work is done.

当θ = 0°时，cos θ = 1，W = F × s。当θ = 90°时，cos θ = 0，不做功。

3. Units of Work and Energy | 功与能量的单位

Work and energy are measured in the same unit: the joule (J). This is because work is a transfer of energy. When work is done on an object, energy is transferred to that object. When an object does work, it transfers energy away.

功和能量使用相同的单位：焦耳（J）。这是因为功是能量的转移。当对物体做功时，能量被转移到该物体上；当物体对外做功时，它向外转移能量。

Common prefixes used:

常见的单位前缀：

Prefix 前缀	Value 数值	Example 示例
kilo (k) 千	10³	1 kJ = 1000 J
mega (M) 兆	10⁶	1 MJ = 1,000,000 J

4. What is Power? | 什么是功率

Power is the rate at which work is done or the rate at which energy is transferred. It tells us how quickly work is done, not how much total work is done. A more powerful machine does the same amount of work in less time.

功率是做功的快慢，即单位时间内所做的功或转移的能量。它表示做功有多快，而不是总共做了多少功。功率更大的机器能在更短时间内完成相同的功。

The equation for power is:

功率的计算公式为：

Power = Work done ÷ Time taken

功率 = 所做的功 ÷ 所用时间

In symbols:

用符号表示：

P = W ÷ t

P = power in watts (W) | 功率，单位瓦特（W）
W = work done in joules (J) | 功，单位焦耳（J）
t = time in seconds (s) | 时间，单位秒（s）

One watt is equal to one joule per second: 1 W = 1 J/s. Larger units like kilowatt (kW) and megawatt (MW) are also commonly used.

1瓦特等于1焦耳每秒：1 W = 1 J/s。常用的较大单位还有千瓦（kW）和兆瓦（MW）。

5. Alternative Power Equation | 功率的另一种表达形式

Since work W = F × s, we can substitute this into the power equation:

因为功 W = F × s，我们可以将其代入功率公式：

P = (F × s) ÷ t = F × (s ÷ t) = F × v

Since s ÷ t is the average velocity v, power can also be calculated as force multiplied by velocity. This is especially useful for vehicles and machines that move at constant speed.

由于 s ÷ t 等于平均速度 v，因此功率也可以用力乘以速度来计算。这对于匀速运动的车辆和机械特别有用。

P = F × v

P = power in watts (W) | 功率，单位瓦特（W）
F = force in newtons (N) | 力，单位牛顿（N）
v = average speed in metres per second (m/s) | 平均速度，单位米每秒（m/s）

For example, if a car engine provides a driving force of 1500 N while the car moves at 20 m/s, the power output is 1500 × 20 = 30,000 W = 30 kW.

例如，如果汽车发动机提供1500 N的驱动力，而汽车以20 m/s的速度行驶，则输出功率为1500 × 20 = 30,000 W = 30 kW。

6. Work Done Against Gravity | 克服重力做功

When an object is lifted vertically, work is done against the gravitational force. The force needed to lift an object at constant speed equals its weight, which is mg, where m is mass and g is the gravitational field strength (approximately 9.8 N/kg on Earth).

当物体被垂直提升时，需要克服重力做功。匀速提升物体所需的力等于物体的重量，即 mg，其中 m 是质量，g 是重力场强度（地球表面约为9.8 N/kg）。

The work done in lifting an object to a height h is:

将物体提升到高度 h 所做的功为：

W = m × g × h

This work is stored as gravitational potential energy in the object, and it can be recovered when the object falls. This is a key idea linking work and energy.

这些功以重力势能的形式储存在物体中，当物体下落时可以重新释放。这是联系功与能量的关键概念。

Example: A 2 kg book is lifted from the floor to a shelf 1.5 m high. Taking g = 9.8 N/kg, the work done is W = 2 × 9.8 × 1.5 = 29.4 J. The book now has 29.4 J of gravitational potential energy.

例：将一本2 kg的书从地板提升到1.5 m高的书架上。取 g = 9.8 N/kg，所做的功为 W = 2 × 9.8 × 1.5 = 29.4 J。这本书现在具有29.4 J的重力势能。

7. Work-Energy Principle | 功能原理

The work-energy principle states that the work done on an object equals the change in its kinetic energy. If an object speeds up, work is done on it. If it slows down, it does work on something else.

功能原理指出：对物体所做的功等于物体动能的变化量。如果物体加速，则对它做了正功；如果物体减速，则它对外做了功。

When a constant force F acts on an object of mass m over a distance s, causing it to accelerate from velocity u to velocity v, the work done is:

当恒力 F 作用在质量为 m 的物体上，经过距离 s，使其速度从 u 加速到 v 时，所做的功为：

W = F × s = ½ × m × v² − ½ × m × u²

This shows that work done equals the change in kinetic energy. The term ½mv² is the kinetic energy of the object.

这表明所做的功等于动能的变化量。½mv² 项就是物体的动能。

8. Efficiency and Power | 效率与功率

Not all energy input to a machine is converted into useful output. Some energy is always wasted, often as heat and sound due to friction. Efficiency measures how much of the input energy or power is converted into useful output.

并非所有输入机器的能量都能转化为有用的输出。总有一部分能量被浪费，通常以热和声音的形式耗散于摩擦。效率衡量的是输入能量或功率中有多少被转化为有用的输出。

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

效率 = （有用能量输出 ÷ 总能量输入）× 100%

Since power is energy per unit time, efficiency can also be calculated using power:

由于功率是单位时间的能量，效率也可以用功率来计算：

Efficiency = (Useful power output ÷ Total power input) × 100%

效率 = （有用功率输出 ÷ 总功率输入）× 100%

Efficiency has no units and is often expressed as a percentage. A machine with 100% efficiency is impossible in practice due to friction and other energy losses.

效率没有单位，通常用百分比表示。由于摩擦和其他能量损失，效率为100%的机器在实践中是不存在的。

9. Worked Example: Calculating Work | 例题：计算功

Question: A student pushes a box with a horizontal force of 50 N across a floor for a distance of 4 m. How much work is done?

题目：一名学生用50 N的水平力推动一个箱子在地板上移动了4 m。做了多少功？

Solution: Since the force is in the direction of motion, W = F × s = 50 N × 4 m = 200 J.

解答：由于力与运动方向一致，W = F × s = 50 N × 4 m = 200 J。

Follow-up: If the push takes 10 seconds, what is the power?

追问：如果推动过程用了10秒，功率是多少？

Solution: P = W ÷ t = 200 J ÷ 10 s = 20 W.

解答：P = W ÷ t = 200 J ÷ 10 s = 20 W。

10. Worked Example: Lifting a Load | 例题：提升重物

Question: A crane lifts a 500 kg steel beam vertically at constant speed to a height of 20 m. Calculate (a) the work done by the crane, and (b) the power output if the lift takes 40 seconds. Take g = 9.8 N/kg.

题目：一台起重机将500 kg的钢梁匀速垂直提升到20 m的高度。计算（a）起重机做的功，（b）若提升过程用时40秒，求输出功率。取 g = 9.8 N/kg。

Solution (a): The force needed equals the weight: F = m × g = 500 × 9.8 = 4900 N. Work done = F × s = 4900 × 20 = 98,000 J = 98 kJ.

解答（a）：所需力等于重量：F = m × g = 500 × 9.8 = 4900 N。功 = F × s = 4900 × 20 = 98,000 J = 98 kJ。

Solution (b): Power = Work ÷ Time = 98,000 J ÷ 40 s = 2450 W = 2.45 kW.

解答（b）：功率 = 功 ÷ 时间 = 98,000 J ÷ 40 s = 2450 W = 2.45 kW。

11. Common Mistakes and Exam Tips | 常见错误与考试提示

Forgetting direction: Only distance moved in the direction of the force counts. If force and motion are at an angle, use W = F × s × cos θ. | 忘记方向：只有沿力的方向移动的距离才算数。若力与运动方向有夹角，使用 W = F × s × cos θ。
Confusing work and power: Work is energy transferred (J); power is the rate of transfer (W). | 混淆功和功率：功是转移的能量（J）；功率是转移的速率（W）。
Using incorrect units: Always convert minutes to seconds, km to metres, and grams to kilograms before substituting into equations. | 使用错误的单位：代入公式前，务必把分钟换算为秒、千米换算为米、克换算为千克。
No displacement means no work: Holding an object still, no matter how heavy, involves no physics work. | 没有位移就没有功：无论物体多重，只要保持不动，就没有物理意义上的功。
Power = F × v: This form is useful when speed is constant and force is known. | 功率 = F × v：当速度恒定且力已知时，此公式非常有用。
Efficiency less than 100%: In calculations, efficiency is always less than 100% for real machines. | 效率小于100%：在计算中，真实机器的效率总是小于100%。

12. Quick Revision Summary | 快速复习总结

Quantity 物理量	Equation 公式	Unit 单位
Work 功	W = F × s	J (N·m)
Work against gravity 克服重力做功	W = m × g × h	J
Power 功率	P = W ÷ t	W (J/s)
Power (alternative) 功率（另一形式）	P = F × v	W
Efficiency 效率	(Useful output ÷ Total input) × 100%	%

Remember: Work is done when a force moves an object in the direction of the force. Power is the rate of doing work. Both are fundamental ideas that link mechanics, energy, and real-world machines.

记住：当力使物体沿力的方向移动时，就做了功。功率是做功的快慢。两者都是联系力学、能量和实际机械的基本概念。

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Work & Power | 功与功率