Work Done and Energy Transfer | 做功与能量转移

📚 Work Done and Energy Transfer | 做功与能量转移

In everyday language, ‘work’ means effort, duty, or employment. In physics, however, work has a precise meaning: it is the process by which energy is transferred from one body or system to another by a force acting over a distance. This article explains the concept of work, the forms of energy involved in mechanical systems, and how energy transfers are quantified in the CIE A-Level Physics syllabus.

在日常语言中，“work”意味着努力、职责或职业。但在物理学中，功有精确的定义：它是力使物体沿力的方向移动一段距离时，能量从一个物体或系统转移到另一个物体或系统的过程。本文将解释功的概念、机械系统中涉及的能量形式，以及如何量化能量转移，以对应 CIE A-Level 物理考纲。

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1. The Definition of Work | 功的定义

Work is done when a force acts on an object and causes a displacement in the direction of the force. The amount of work done is equal to the product of the magnitude of the force and the distance moved along the line of action of the force.

功是在力作用于物体并使物体沿力的方向发生位移时完成的。功的大小等于力的大小与物体沿力的作用线移动距离的乘积。

For a constant force applied along the direction of motion, work done W is given by:

对于沿运动方向施加的恒力，功 W 由下式给出：

W = F × d

where W is measured in joules (J), F is the force in newtons (N), and d is the displacement in metres (m). One joule is defined as the work done when a force of one newton moves an object one metre in the direction of the force.

其中 W 的单位是焦耳（J），F 是力，单位为牛顿（N），d 是位移，单位为米（m）。1 焦耳定义为 1 牛顿的力使物体沿力的方向移动 1 米所做的功。

Work is a scalar quantity. Even though force and displacement are vectors, their scalar product gives a scalar result. This means work has magnitude but no direction.

功是标量。尽管力和位移都是矢量，但它们的标量积产生标量结果。这意味着功只有大小，没有方向。

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2. Work Done by a Constant Force at an Angle | 恒力与做功角度

In many real situations, the force is not parallel to the displacement. For example, a person pulls a suitcase with a strap at an angle θ to the horizontal. Only the component of the force in the direction of the displacement does useful work.

在许多真实情境中，力并不平行于位移。例如，一个人用与水平方向成 θ 角的带子拉行李箱。只有力沿位移方向的分量做有用功。

When a constant force F acts at an angle θ to the direction of displacement d, the work done is:

当恒力 F 与位移 d 方向成 θ 角时，做功为：

W = F d cos θ

This equation is the scalar product of the force and displacement vectors. If cos θ is positive, the force has a component in the direction of motion and work is positive, meaning energy is transferred to the object. If cos θ is negative, the force opposes the motion, work is negative, and energy is transferred away from the object.

该公式是力和位移矢量的标量积。若 cos θ 为正，力在运动方向上有分量，功为正，能量转移给物体；若 cos θ 为负，力阻碍运动，功为负，能量从物体转移出去。

• θ = 0°: W = Fd, the force acts exactly in the direction of motion, maximum positive work.

  θ = 0°：W = Fd，力完全沿运动方向，做最大正功。

• θ = 90°: W = 0, the force is perpendicular to motion, such as the tension in a string when an object moves in a horizontal circle. No work is done because energy is not transferred along the direction of motion.

  θ = 90°：W = 0，力与运动垂直，例如物体在水平圆周运动时绳子的张力。因为没有沿运动方向转移能量，所以不做功。

• θ = 180°: W = −Fd, the force directly opposes motion, such as kinetic friction acting on a sliding block. Negative work removes kinetic energy from the block.

  θ = 180°：W = −Fd，力直接阻碍运动，例如作用在滑动物块上的动摩擦力。负功从物块中移除动能。

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3. Work Done by a Variable Force | 变力做功

When the force is not constant, the equation W = Fd cos θ cannot be used directly. Instead, the work done by a variable force is found from the area under a force–displacement graph.

当力不是恒定值时，不能直接使用 W = Fd cos θ。变力做功可通过力–位移图像下的面积求得。

For a force that varies with position, divide the displacement into many small intervals. In each interval the force is approximately constant, so the small amount of work is δW = F δd. The total work is the sum of all these small contributions, which is the area under the F–d graph.

对于随位置变化的力，将位移分成许多微小间隔。在每个间隔内力近似恒定，因此微功 δW = F δd。总功是所有微小贡献之和，也就是 F–d 图像下的面积。

If the graph is a straight line passing through the origin, the force increases uniformly with displacement. The work done is the area of a triangle:

若图像是过原点的直线，则力随位移均匀增加。功等于三角形的面积：

W = ½ × F_max × d

This situation occurs when stretching a spring obeying Hooke’s law: F = kd, where k is the spring constant. The work done in extending the spring from 0 to d is:

这种情况出现在拉伸符合胡克定律的弹簧时：F = kd，其中 k 是劲度系数。将弹簧从 0 拉伸到 d 所做的功为：

W = ½ k d²

This energy is stored in the spring as elastic potential energy. The spring constant k has units N m⁻¹, and d is the extension in metres.

这部分能量以弹性势能的形式储存在弹簧中。劲度系数 k 的单位是 N m⁻¹，d 是伸长量，单位为米。

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4. Energy and Its Main Forms | 能量及其主要形式

Energy is defined as the capacity to do work. An object or system that has energy is able to bring about a change, such as moving an object, heating a substance, or emitting radiation. Energy is a scalar quantity and is measured in joules, the same unit as work.

能量定义为做功的能力。具有能量的物体或系统能够引起变化，例如移动物体、加热物质或发射辐射。能量是标量，单位是焦耳，与功的单位相同。

The main forms of energy in A-Level mechanics are kinetic energy, gravitational potential energy, elastic potential energy, chemical energy, thermal energy, electrical energy, and nuclear energy. Energy can change from one form to another, but the total energy of an isolated system is conserved.

A-Level 力学中的主要能量形式有动能、重力势能、弹性势能、化学能、热能、电能和核能。能量可以从一种形式转变为另一种形式，但孤立系统的总能量保持不变。

Form | 形式	Description | 描述
Kinetic energy | 动能	Energy due to motion of an object | 物体由于运动而具有的能量
Gravitational potential energy | 重力势能	Energy stored due to height in a gravitational field | 在重力场中由于高度而储存的能量
Elastic potential energy | 弹性势能	Energy stored in a deformed spring or material | 在形变的弹簧或材料中储存的能量
Thermal energy | 热能	Energy associated with the random motion of particles | 与粒子无规则运动相关的能量
Chemical energy | 化学能	Energy stored in bonds between atoms and molecules | 储存在原子和分子化学键中的能量

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5. Kinetic Energy and the Work–Energy Principle | 动能与动能定理

Kinetic energy is the energy an object possesses because of its motion. For an object of mass m moving with speed v, the kinetic energy Eₖ is:

动能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，动能 Eₖ 为：

Eₖ = ½ m v²

This equation is valid for speeds much less than the speed of light. Because v is squared, doubling the speed increases the kinetic energy by a factor of four.

该公式适用于远小于光速的速度。由于 v 是二次方，速度加倍时动能增大为原来的四倍。

The work–energy principle states that the total work done on an object is equal to its change in kinetic energy:

动能定理指出，对物体所做的总功等于物体动能的变化量：

W_total = ΔEₖ = ½ m v² − ½ m u²

where u is the initial speed and v is the final speed. This principle is a direct consequence of Newton’s second law. If the net force acting on a body causes it to accelerate over a distance, the work done by that force appears as an increase in kinetic energy.

其中 u 是初速度，v 是末速度。动能定理是牛顿第二定律的直接推论。若作用在物体上的合力使其在位移上加速，则合力做的功表现为动能的增加。

A common exam question asks you to find the stopping distance of a car. If a constant braking force F acts opposite to motion, the work done is −Fd. Setting −Fd = 0 − ½ m v² gives d = m v² / 2F. Doubling the speed quadruples the stopping distance, which is why speeding is so dangerous.

一个常见的考题是求汽车的制动距离。若恒定制动力 F 与运动方向相反，则做功为 −Fd。由 −Fd = 0 − ½ m v² 可得 d = m v² / 2F。速度加倍会使制动距离变为原来的四倍，这就是超速危险的原因。

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6. Gravitational Potential Energy | 重力势能

Gravitational potential energy is the energy stored in an object due to its position in a gravitational field. Near the Earth’s surface, the gravitational field strength g is approximately constant, so the gravitational potential energy Eₚ of an object of mass m at height h above a reference level is:

重力势能是物体由于在重力场中的位置而储存的能量。在地球表面附近，重力场强度 g 近似恒定，因此质量为 m 的物体在参考水平面以上高度 h 处的重力势能 Eₚ 为：

Eₚ = m g h

The reference level can be chosen arbitrarily because only changes in gravitational potential energy are physically meaningful. If an object moves downwards, h decreases, the gravitational potential energy decreases, and the lost energy may become kinetic energy.

参考水平面可以任意选取，因为只有重力势能的变化才具有物理意义。若物体向下运动，h 减小，重力势能减小，损失的能量可能转化为动能。

When an object is lifted vertically at constant speed by an external force, the external force does positive work equal to mgh, and the gravitational potential energy increases by mgh. If the object is lowered at constant speed, the external force does negative work, and the gravitational potential energy decreases.

当物体以恒定速度被外力竖直提起时，外力做正功，大小等于 mgh，重力势能增加 mgh。若物体以恒定速度被放下，外力做负功，重力势能减小。

In projectile motion, the total mechanical energy of the object is conserved if air resistance is negligible. At its highest point, the vertical speed is zero, so the kinetic energy is minimum and the gravitational potential energy is maximum.

在抛体运动中，若忽略空气阻力，物体的总机械能守恒。在最高点，竖直速度为零，因此动能最小，重力势能最大。

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7. Elastic Potential Energy | 弹性势能

Elastic potential energy is stored when an elastic material, such as a spring, is stretched or compressed. For an ideal spring obeying Hooke’s law, the force required to produce an extension d is F = kd, where k is the spring constant.

弹性势能是在弹簧等弹性材料被拉伸或压缩时储存的。对于符合胡克定律的理想弹簧，产生伸长量 d 所需的外力为 F = kd，其中 k 是劲度系数。

The work done in stretching the spring from no extension to a final extension d is equal to the area under the F–d graph. Because the force increases linearly, the area is a triangle, giving:

将弹簧从无伸长拉伸到最终伸长量 d 所做的功等于 F–d 图像下的面积。由于力线性增加，面积为三角形，因此：

Eₑ = ½ k d²

where Eₑ is the elastic potential energy stored in the spring. The same equation applies to a compression if the spring remains within its elastic limit.

其中 Eₑ 是弹簧中储存的弹性势能。只要弹簧不超出弹性限度，该公式同样适用于压缩。

If a spring-mass system is set into vertical oscillation, energy continuously changes between elastic potential energy, gravitational potential energy, and kinetic energy. At the maximum displacement, all energy is stored in the spring; at the equilibrium position, the speed is greatest and kinetic energy is maximum.

若弹簧质量系统做竖直振动，能量不断在弹性势能、重力势能和动能之间转化。在最大位移处，所有能量都储存在弹簧中；在平衡位置，速度最大，动能最大。

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8. Power and the Rate of Energy Transfer | 功率与能量转移速率

Power is defined as the rate at which work is done, or the rate at which energy is transferred. The average power P is:

功率定义为做功的速率，或能量转移的速率。平均功率 P 为：

P = W / t

where W is the work done in joules and t is the time taken in seconds. The unit of power is the watt (W), where 1 W = 1 J s⁻¹.

其中 W 是以焦耳为单位的功，t 是以秒为单位的时间。功率的单位是瓦特（W），1 W = 1 J s⁻¹。

For an object moving with constant velocity v while a constant force F acts in the direction of motion, the instantaneous power is:

当物体以恒定速度 v 运动，且恒力 F 沿运动方向作用时，瞬时功率为：

P = F v

This equation is obtained by substituting W = Fd into P = W/t and noting that d/t = v. It is particularly useful for vehicle problems: the engine provides a driving force, and at maximum speed the driving power equals the power dissipated against resistive forces.

该公式通过将 W = Fd 代入 P = W/t，并注意到 d/t = v 而得出。它特别适用于车辆问题：发动机提供驱动力，在最高速度时，发动机的输出功率等于克服阻力所消耗的功率。

In an electric motor, the input electrical power is P = VI, where V is the voltage and I is the current. Part of this input power is converted to useful mechanical power, and the rest is transferred to the surroundings as thermal energy due to friction and resistance in the wires.

在电动机中，输入的电功率为 P = VI，其中 V 是电压，I 是电流。输入功率的一部分转换为有用的机械功率，其余部分由于摩擦和导线电阻而以热能形式转移到周围环境。

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9. Efficiency of Energy Transfers | 能量转移效率

In any real energy transfer, not all input energy is converted into useful output energy. Some energy is always dissipated, often as thermal energy, sound, or vibration. Efficiency measures how much of the total input energy is usefully transferred.

在任何真实的能量转移中，并非所有输入能量都转化为有用的输出能量。总有一些能量被耗散，通常以热能、声音或振动形式存在。效率衡量输入能量中有多少被有效转移。

Efficiency is defined as the ratio of useful output energy to total input energy, often expressed as a percentage:

效率定义为有用输出能量与总输入能量的比值，通常以百分比表示：

efficiency = (useful output energy / total input energy) × 100%

Since power is the rate of energy transfer, efficiency can also be expressed in terms of useful output power and total input power:

由于功率是能量转移的速率，效率也可以用有用输出功率与总输入功率之比表示：

efficiency = (useful output power / total input power) × 100%

In an ideal machine, energy would be perfectly transformed and efficiency would be 100%. In practice, friction between moving parts and electrical resistance convert some energy to thermal energy, which is usually wasted because it is spread too thinly to be recovered.

在理想的机械中，能量会被完美转化，效率为 100%。实际上，运动部件之间的摩擦和电阻会把部分能量转化为热能，而这些热能通常分布过于分散而无法回收，因此被浪费。

When analysing a system, it is helpful to draw an energy flow diagram showing the input energy, useful output, and dissipated energy. The total input energy must equal the sum of useful output and dissipated energy, in agreement with the principle of conservation of energy.

分析系统时，可以画出能量流动图，标出输入能量、有用输出和耗散能量。总输入能量必须等于有用输出与耗散能量之和，这符合能量守恒原理。

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10. Conservation of Energy in Mechanical Systems | 机械系统中的能量守恒

The law of conservation of energy states that energy cannot be created or destroyed, only transferred from one form to another. In a closed system, the total energy before a process equals the total energy after the process.

能量守恒定律指出，能量不能被创造或消灭，只能从一种形式转化为另一种形式。在封闭系统中，过程前后的总能量相等。

In mechanics, the sum of kinetic energy and potential energy is called mechanical energy. When only conservative forces, such as gravity and ideal spring forces, do work, mechanical energy is conserved:

在力学中，动能与势能之和称为机械能。当只有保守力，如重力和理想弹簧力做功时，机械能守恒：

KE_initial + PE_initial = KE_final + PE_final

Returning to the gravitational context, for an object falling from rest through a height h, its gravitational potential energy mgh is converted entirely into kinetic energy:

回到重力情境，对于从静止下落高度 h 的物体，其重力势能 mgh 完全转化为动能：

½ m v² = m g h

Cancelling m gives v = √(2gh). Notice that the final speed is independent of the mass of the object. This result is often tested in exam questions involving falling objects and roller coasters.

约去 m 得到 v = √(2gh)。注意最终速度与物体质量无关。这个结果经常在涉及下落物体和过山车的考题中出现。

When non-conservative forces such as friction or air resistance are present, mechanical energy is not conserved. The work done by these forces equals the loss of mechanical energy. For example, a skydiver who reaches terminal velocity has constant kinetic energy but continues to lose gravitational potential energy; the lost energy is transferred to thermal energy in the air and the surrounding environment.

当存在摩擦力或空气阻力等非保守力时，机械能不守恒。这些力所做的功等于机械能的损失。例如，达到收尾速度的跳伞运动员动能不变，但持续损失重力势能；损失的能量被转移到空气和周围环境中的热能。

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