📚 Doing work, transferring energy | 做功与能量传递
In physics, the concept of work provides a precise way to describe how forces cause changes in energy. Whenever a force acts on an object and causes displacement, energy is transferred from one system to another. This article explores the definition of work, its calculation for constant and variable forces, the relationship between work and kinetic energy, the potential energy stored in gravitational and elastic systems, the distinction between conservative and non‑conservative forces, the principle of conservation of mechanical energy, and the related quantities of power and efficiency. All explanations are aligned with the CIE A‑Level Physics syllabus and use clear, exam‑focused language.
在物理学中，功的概念提供了一种精确的方式来描述力如何引起能量的变化。每当力作用于物体并使其发生位移时，能量就会从一个系统转移到另一个系统。本文探讨了功的定义，恒力与变力做功的计算方法，功与动能之间的关系，储存在重力场和弹性系统中的势能，保守力与非保守力的区别，机械能守恒定律，以及相关的功率与效率概念。所有内容均紧扣 CIE A‑Level 物理考试大纲，并使用清晰、面向考试的语言进行表述。
1. Work Defined as Force × Distance | 功定义为力与位移的乘积
Work is done when a force causes an object to move in the direction of the force. The amount of work done, W, is equal to the product of the force component parallel to the displacement and the magnitude of the displacement. Work is a scalar quantity, even though force and displacement are vectors, and it is measured in joules (J). One joule is defined as the work done by a force of one newton moving an object through one metre in the direction of the force.
当一个力使物体沿力的方向运动时，我们就说力做了功。功的大小 W 等于力沿位移方向的分量与位移大小的乘积。尽管力和位移都是矢量，功却是标量，其单位是焦耳（J）。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。
The most general expression for work done by a constant force is:
恒力做功的最一般表达式为：
W = F d cos θ
If the force and displacement are in the same direction, cos θ = 1 and W = F d. If they are perpendicular, cos θ = 0 and no work is done.
其中 F 是力的大小，d 是位移的大小，θ 是力矢量与位移矢量之间的夹角。如果力与位移方向相同，cos θ = 1，则 W = F d；如果二者相互垂直，cos θ = 0，则不做功。
2. Work Done by a Constant Force | 恒力所做的功
When a constant force acts on an object that moves in a straight line, the work done is simply the product of the force component along the line of motion and the distance moved. For example, when a person pulls a sled with a rope at an angle to the horizontal, only the horizontal component of the tension does work on the sled. The vertical component, which is balanced by the normal reaction and weight, does not contribute to the transfer of energy as far as horizontal motion is concerned.
当恒力作用在沿直线运动的物体上时，所做的功就是力沿运动方向的分量与移动距离的乘积。例如，一个人用绳子以一定角度拉雪橇时，只有拉力的水平分量对雪橇做功。竖直分量被法向反作用力和重力平衡，并不参与水平方向上的能量传递。
Positive work increases the object’s mechanical energy; negative work decreases it. If a force opposes the motion, such as friction, the work done by that force is negative, often converting mechanical energy into thermal energy.
正功增加物体的机械能，负功减少机械能。如果力阻碍运动，例如摩擦力，所做的功为负，通常会把机械能转化为热能。
3. Work Done by a Variable Force | 变力所做的功
Not all forces are constant. When a force varies in magnitude or direction as the object moves, the work done cannot be found simply by multiplying force by displacement. Instead, we consider the area under a force–displacement graph. For a variable force applied along the direction of motion, the work done is equal to the area between the force curve and the displacement axis, taking account of signs.
并非所有的力都是恒定的。当力的大小或方向随着物体的运动而变化时，我们不能简单地用力乘以位移来求功。此时，我们需要考虑力–位移图下的面积。对于沿运动方向施加的变力，所做的功等于力曲线与位移轴之间所围成的面积，并需考虑正负号。
In many A‑Level problems, the force may change linearly with extension, such as in a spring obeying Hooke’s law. The work done in stretching such a spring from extension zero to x is the area of a triangle under the force–extension graph:
在许多 A‑Level 问题中，力可能随伸长量成线性变化，例如满足胡克定律的弹簧。将这种弹簧从伸长量为零拉至 x 的过程中，力–伸长量图下的面积是一个三角形，所做的功为：
W = ½ F x = ½ k x²
This result is important for calculating the elastic potential energy stored in a deformed spring.
这一结果对计算变形弹簧中储存的弹性势能非常重要。
4. Work–Energy Theorem | 功能定理
The work–energy theorem states that the net work done on an object is equal to the change in its kinetic energy. Mathematically:
功能定理指出：合力对物体所做的功等于物体动能的变化量。数学表达式为：
Wₙₑₜ = ΔK = ½ m v² − ½ m u²
where m is the mass, u is the initial speed, and v is the final speed. This theorem is a direct consequence of Newton’s second law and the definition of work, and it provides a powerful tool for solving problems without having to consider the detailed time variation of forces.
其中 m 为质量，u 为初速度，v 为末速度。该定理是牛顿第二定律和功的定义的直接推论，它提供了一个无需考虑力随时间变化细节就能解决问题的有力工具。
When the net work is positive, the kinetic energy increases; when it is negative, the kinetic energy decreases. All forms of work—gravitational, elastic, applied forces, friction—contribute to the net work, and therefore to the change in kinetic energy.
合力做正功时，动能增加；做负功时，动能减少。所有形式的功——重力功、弹性功、外力功、摩擦功——都包含在合力功中，都会影响动能的变化。
5. Gravitational Potential Energy | 重力势能
Gravitational potential energy is the energy stored in an object by virtue of its position in a gravitational field. Near the Earth’s surface, the change in gravitational potential energy when an object of mass m is raised through a vertical height Δh is given by:
重力势能是物体因在引力场中的位置而储存的能量。在地球表面附近，质量为 m 的物体竖直升高 Δh 时，重力势能的变化量为：
ΔU = m g Δh
where g is the acceleration of free fall. The work done against gravity in lifting the object is stored as an increase in potential energy. When the object falls, gravity does positive work and the potential energy decreases, being converted into kinetic energy.
其中 g 为自由落体加速度。克服重力举起物体所做的功以势能增加的形式储存起来。当物体下落时，重力做正功，势能减少，转化为动能。
ded h is measured from that reference level. The choice of reference does not affect the change in potential energy.
我们通常会定义一个零势能参考面，并将高度 h 处的势能表示为 U = mgh（前提是 h 从该参考面量起）。参考面的选择不影响势能的变化量。
6. Kinetic Energy | 动能
Kinetic energy is the energy an object possesses due to its motion. For a rigid body of mass m moving with speed v, the kinetic energy is:
动能是物体因运动而具有的能量。对于质量为 m、速度为 v 的刚体，其动能为：
K = ½ m v²
Kinetic energy is always positive (or zero) and is a scalar. Doubling the speed quadruples the kinetic energy, which has important safety implications in road transport because the work needed to stop a vehicle is directly related to its kinetic energy.
动能恒为正值（或零），且是一个标量。速度加倍会使动能增加为原来的四倍，这在交通领域具有重要的安全意义，因为使车辆停下来所需的功直接与其动能有关。
In many A‑Level problems, the transfer between kinetic and potential energy is the central theme. For a falling object in the absence of air resistance, the loss in gravitational potential energy equals the gain in kinetic energy.
在许多 A‑Level 问题中，动能与势能之间的转换是核心主题。对于不考虑空气阻力的自由落体，减少的重力势能等于增加的动能。
7. Conservative and Non‑conservative Forces | 保守力与非保守力
A conservative force is one for which the work done in moving an object between two points is independent of the path taken. Gravitational force and the elastic force of an ideal spring are examples of conservative forces. For such forces, a potential energy function can be defined, and the work done by the force is equal to the negative of the change in potential energy.
保守力是指，将物体在两点之间移动时，所做的功与所经路径无关的力。重力和理想弹簧的弹力都是保守力的例子。对这类力，可以定义一个势能函数，而且力所做的功等于势能减少量。
Non‑conservative forces, such as friction and air resistance, cause the mechanical energy of a system to change into forms that are not recoverable as mechanical work, typically thermal energy. The work done by non‑conservative forces depends on the path, and no useful potential energy function can be introduced for them.
非保守力，如摩擦力和空气阻力，会使系统的机械能转化为无法作为机械功回收的形式，通常是热能。非保守力所做的功依赖于路径，我们无法为它们引入有意义的势能函数。
8. Conservation of Mechanical Energy | 机械能守恒
The principle of conservation of mechanical energy states that if only conservative forces do work within a system, the total mechanical energy—the sum of kinetic and potential energy—remains constant:
机械能守恒定律指出：如果在一个系统内只有保守力做功，那么总机械能——动能与势能之和——保持恒定：
K₁ + U₁ = K₂ + U₂
This principle allows us to relate speeds and positions without calculating the work done by internal conservative forces. For example, a pendulum swinging without air resistance converts gravitational potential energy into kinetic energy and back, with the total remaining the same at all points.
利用这一定理，我们可以将速度与位置联系起来，而不必计算内部保守力所做的功。例如，一个没有空气阻力的摆，会在重力势能和动能之间来回转换，且在所有位置上总机械能保持不变。
When non‑conservative forces are present, the change in mechanical energy equals the work done by those forces:
当存在非保守力时，机械能的变化等于这些力所做的功：
Wₙₒₙ₋ₓₒₙₛₑᵣᵥ = Δ(K + U)
This generalisation is essential for solving realistic problems involving friction or driving forces.
这一推广对于解决涉及摩擦力或驱动力的实际问题至关重要。
9. Elastic Potential Energy | 弹性势能
ic potential energy. For a spring that obeys Hooke’s law, F = k x, where k is the spring constant, the stored energy for an extension x is:
当弹簧或任何弹性体在其比例限度内发生形变时，拉伸或压缩它所做的功以弹性势能的形式储存起来。对于符合胡克定律的弹簧，F = k x，其中 k 为劲度系数，伸长量为 x 时所储存的能量为：
Uₑ = ½ k x²
This energy can be recovered as kinetic energy when the spring returns to its natural length. Elastic potential energy is a form of mechanical potential energy, and when combined with kinetic energy and gravitational potential energy, it participates in the overall conservation of mechanical energy in elastic systems.
当弹簧恢复到原长时，这部分能量可以以动能的形式释放。弹性势能是机械势能的一种形式，与动能和重力势能一起，参与弹性系统总机械能的守恒。
In exam questions, students are often required to relate the loss in elastic potential energy to the gain in kinetic energy, or to calculate the work done against friction during the expansion of a spring.
在考试题中，学生常常需要将弹性势能的减少量与动能的增加量联系起来，或者计算弹簧伸长过程中克服摩擦力所做的功。
10. Power | 功率
Power is defined as the rate of doing work or the rate of energy transfer. The average power produced by a force over a time interval Δt is:
功率定义为做功的速率或能量传递的速率。一个力在时间间隔 Δt 内产生的平均功率为：
P = W / Δt
Power is a scalar quantity and its SI unit is the watt (W). One watt is equivalent to one joule per second.
功率是标量，其国际单位是瓦特（W）。一瓦特等于一焦耳每秒。
For an object moving at constant speed v under the action of a constant force F that acts in the direction of motion, the instantaneous power can also be expressed as:
对于在恒力 F 作用下以恒定速度 v 运动的物体，且力的方向与运动方向一致时，瞬时功率还可以表达为：
P = F v
This relationship is particularly useful when analysing the performance of engines: to maintain a constant speed against resistive forces, the engine must deliver a power equal to the product of the resistive force and the speed.
这一关系在分析发动机性能时特别有用：为了保持恒定速度以克服阻力，发动机必须提供等于阻力与速度乘积的功率。
11. Efficiency of Energy Transfer | 能量传递的效率
In any practical process, not all the input energy is converted into useful output energy; some is always dissipated as thermal energy due to friction, air resistance or electrical resistance. Efficiency is a measure of how well a system transfers energy into the desired form, and is defined as:
在任何实际过程中，输入的能量不可能全部转化为有用的输出能量，总有一部分因摩擦、空气阻力或电阻而耗散为热能。效率是衡量系统将能量转化为所需形式的有效程度的量度，其定义为：
Efficiency = (useful energy output / total energy input) × 100%
Efficiency can also be expressed in terms of power: useful output power divided by input power. Since no real device is perfectly efficient, the efficiency is always less than 100%. In examination questions, you may be asked to calculate the efficiency of a motor lifting a load, or to relate the energy dissipated to the work done against friction.
效率也可以用功率来表示：有用的输出功率除以输入功率。由于任何真实装置都不可能完美无摩擦，效率始终低于 100%。在考试问题中，你可能需要计算电动机提升重物时的效率，或者将耗散的能量与克服摩擦力所做的功联系起来。
A clear understanding of efficiency helps students appreciate energy conservation in real‑world systems: the energy “lost” is not destroyed, but transferred to the surroundings, raising the internal energy and temperature of the environment.
清晰理解效率有助于学生认识到真实系统中的能量守恒：“损失”的能量并没有被消灭，而是传递到了周围环境中，使环境的内能和温度升高。
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