Doing Work, Transferring Energy | 做功与能量转移

📚 Doing Work, Transferring Energy | 做功与能量转移

In CIE A-Level Physics, the concept of work is central to understanding how energy is transferred between different stores. When a force acts on an object and causes displacement, work is done, and this process changes the energy content of the object or system. Mastering the links between work, force, displacement and energy is essential for solving both qualitative and quantitative problems in mechanics.

在 CIE A-Level 物理中，功的概念是理解能量如何在不同储存库之间转移的核心。当力作用在物体上并使其发生位移时，就做了功，这个过程改变了物体或系统的能量。掌握功、力、位移和能量之间的联系，对解决力学中的定性和定量问题都至关重要。

1. Work as a Means of Energy Transfer | 功作为能量转移的方式

Energy can be stored in many forms, such as kinetic energy, gravitational potential energy, elastic potential energy and internal energy. Doing work is one of the most important mechanical processes by which energy is transferred between these stores. When a force moves its point of application, work is done; this work transfers energy from one object or system to another.

能量可以多种形式储存，例如动能、重力势能、弹性势能和内能。做功是能量在这些储存库之间转移的最重要机械过程之一。当力使其作用点发生位移时，就做了功；这个功把能量从一个物体或系统转移到另一个物体或系统。

For example, when you lift a book upwards, you do work against gravity and energy is transferred from your muscles to the book’s gravitational potential energy store. Similarly, when a car engine does work on a car, chemical energy from the fuel is transferred to the car’s kinetic energy. In all such cases, the amount of work done quantifies the amount of energy transferred.

例如，当你向上举起一本书时，你对重力做功，能量从你的肌肉转移到书的重力势能储存。同样，当汽车发动机对汽车做功时，燃料中的化学能转化为汽车的动能。在所有这些情况中，做功的多少定量地反映了能量转移的多少。

2. Defining Work in Physics | 物理学中功的定义

In physics, work has a precise definition that is different from everyday usage. Work is done by a force when the point of application of that force moves. For a constant force, the work done is given by:

在物理学中，功有严格的定义，与日常用语不同。当力的作用点发生位移时，力就对物体做功。对于恒力，所做的功由下式给出：

W = Fs cos θ

Here, F is the magnitude of the force, s is the distance moved by the point of application, and θ is the angle between the force vector and the displacement vector. Work is a scalar quantity, so it has no direction, but it can be positive, zero or negative depending on the angle θ.

其中，F 是力的大小，s 是力的作用点移动的距离，θ 是力矢量与位移矢量之间的夹角。功是标量，没有方向，但根据夹角 θ 的不同，功可以为正、为零或为负。

The SI unit of work is the joule (J). One joule is defined as the work done when a force of one newton moves its point of application through a distance of one metre in the direction of the force: 1 J = 1 N m. Energy and work share the same unit because work is a measure of transferred energy.

功的国际单位是焦耳 (J)。1 焦耳定义为 1 牛的力使作用点沿力的方向移动 1 米所做的功：1 J = 1 N m。能量和功使用相同的单位，因为功是转移能量的量度。

3. Work Done by a Constant Force | 恒力做功

When the force is constant and acts along the same line as the displacement, the calculation simplifies. If the force and displacement are in the same direction, θ = 0°, so cos θ = 1 and W = F s. If the force is perpendicular to the displacement, θ = 90°, so cos θ = 0 and no work is done by that force. If the force opposes the displacement, θ = 180°, so cos θ = −1 and the work done is negative.

当力是恒力且与位移在同一直线上时，计算可以简化。如果力与位移方向相同，θ = 0°，则 cos θ = 1，W = Fs。如果力与位移垂直，θ = 90°，则 cos θ = 0，该力不做功。如果力与位移方向相反，θ = 180°，则 cos θ = −1，力做负功。

The sign of work indicates the direction of energy transfer. Positive work means energy is transferred to the object by the force, negative work means energy is transferred from the object, and zero work means no energy is transferred by that particular force. The table below summarises these cases.

功的正负表示能量转移的方向。正功表示力把能量传递给物体，负功表示能量从物体转移出去，零功表示该力没有发生能量转移。下表总结了这些情况。

Angle θ	Work done by force	Energy transfer	Example
θ = 0°	W = F s	To the object	Engine accelerating a car
θ = 90°	W = 0	None by this force	Normal contact force on a sliding block
θ = 180°	W = −F s	From the object	Friction slowing a box

Common exam questions ask you to identify whether a particular force does work. For example, a vertical lifting force does positive work when raising a bag, while the gravitational force does negative work because it acts downwards while the displacement is upwards. The normal force from a surface does zero work on an object moving horizontally because it is perpendicular to the displacement.

常见的考题要求你判断某个力是否做功。例如，竖直向上的提升力在提起包时做正功，而重力做负功，因为它的方向向下而位移向上。水平面上的支持力对水平运动的物体不做功，因为它与位移垂直。

4. Work Done by a Varying Force | 变力做功

In many real situations, the force is not constant. For example, the force needed to stretch a spring increases as the extension increases. When a force varies in magnitude or direction, you cannot simply use W = Fs cos θ with a single value of F. Instead, the work done is found from the area under a force–distance graph.

在许多实际情况中，力不是恒定的。例如，拉伸弹簧所需的力随着伸长量的增大而增大。当力的大小或方向变化时，不能简单地用一个固定的 F 值代入 W = Fs cos θ。此时，做功可由力-位移图像下的面积求出。

If a force increases uniformly from zero to a maximum value F over a displacement x, the average force is F/2. Therefore, the work done is:

如果力从零均匀增加到最大值 F，位移为 x，则平均力为 F/2。因此，所做的功为：

W = ½ Fx

For an ideal spring obeying Hooke’s law, the force is F = kx, where k is the spring constant. The work done in stretching the spring from zero extension to extension x is stored as elastic potential energy:

对于遵守胡克定律的理想弹簧，力为 F = kx，其中 k 是弹簧劲度系数。将弹簧从零伸长量拉伸到 x 所做的功储存为弹性势能：

W = ½ kx²

This result explains why the work done by a varying force is generally not half of the maximum force times displacement in all cases; the area method must be used unless the force–distance relationship is simple and linear.

这个结果说明了为什么变力做功一般不能简单地用最大力乘以位移的一半来计算；除非力-位移关系是简单且线性的，否则必须使用面积法。

5. Force–Distance Graphs and Work | 力-位移图像与功

CIE exam questions frequently ask you to calculate work done from a force–distance graph. The area under the graph between two positions represents the total work done by the force over that displacement. This is true whether the force is constant or varying, because work is the integral of force with respect to displacement.

CIE 考试题目经常要求你根据力-位移图像计算做功。图像下两个位置之间的面积表示该力在这段位移上所做的总功。无论力是恒定的还是变化的，这一点都成立，因为功是力对位移的积分。

When the graph lies above the distance axis, the area represents positive work. When the graph lies below the axis, the area represents negative work. In calculations, you should subtract any area below the axis from the area above the axis to find the net work done.

当图像位于距离轴上方时，面积代表正功。当图像位于轴下方时，面积代表负功。在计算中，你应从轴上方面积中减去轴下方面积，以求得净功。

For a graph made of straight lines, you can divide the area into rectangles and triangles. For a curved graph, you may need to estimate the area by counting squares, especially in structured questions where the grid is given.

对于由直线构成的图像，你可以把面积分成矩形和三角形。对于曲线图像，你可能需要像网格题那样通过数格来估算面积，特别是在给出方格纸的结构化试题中。

6. Work and Kinetic Energy | 功与动能

The work–energy principle states that the net work done on an object equals its change in kinetic energy. This is one of the most powerful tools in mechanics because it allows you to relate the total force acting on an object to its change in speed without separately analysing the motion.

功能原理指出，作用在物体上的合外力所做的净功等于其动能的变化量。这是力学中最有力的工具之一，因为你能将作用在物体上的总力与其速度变化联系起来，而无需单独分析运动过程。

W_net = ΔEₖ = ½ mv² − ½ mu²

Here, m is the mass of the object, u is the initial speed, v is the final speed, and W_net is the total work done by all forces acting on the object. If the net work is positive, the kinetic energy increases; if it is negative, the kinetic energy decreases.

其中，m 是物体质量，u 是初速度，v 是末速度，W_net 是作用在物体上所有力所做的总功。如果净功为正，动能增加；如果净功为负，动能减少。

For constant acceleration, this principle can be derived from the equation v² = u² + 2as. Rearranging gives as = (v² − u²)/2. Multiplying both sides by mass m gives m a s = ½ mv² − ½ mu². Since F_net = ma, the left side is the net work F_net s, giving the work–energy principle.

对于匀加速运动，这一原理可由方程 v² = u² + 2as 推出。整理得 as = (v² − u²)/2。两边乘以质量 m 得 mas = ½ mv² − ½ mu²。由于 F_net = ma，左边就是净功 F_net s，从而得到功能原理。

7. Gravitational Potential Energy | 重力势能

When an object is lifted at constant speed, the upward force applied to it must balance the object’s weight. The work done by this lifting force is W = mgΔh, where m is the mass, g is the gravitational field strength, and Δh is the vertical height gained. This work is stored as gravitational potential energy:

当物体匀速上升时，施加在物体上的向上力必须与物体的重力平衡。这个提升力所做的功为 W = mgΔh，其中 m 是质量，g 是重力场强度，Δh 是增加的竖直高度。这个功以重力势能形式储存：

ΔEₚ = mgΔh

Strictly speaking, ΔEₚ is the change in gravitational potential energy relative to a chosen reference level. We usually choose the initial position as having zero potential energy, but any reference level can be used because only changes in potential energy are physically significant.

严格来说，ΔEₚ 是相对于选定参考水平面的重力势能变化量。我们通常把初始位置设为零势能点，但实际上可以使用任何参考水平面，因为只有势能的变化量才有物理意义。

If the object is released, this stored gravitational potential energy can be transferred to kinetic energy as the object falls. Assuming no air resistance, the loss in potential energy equals the gain in kinetic energy: mgΔh = ½ mv² − ½ mu².

如果释放物体，这些储存的重力势能在物体下落时转化为动能。假设没有空气阻力，重力势能的减少量等于动能的增加量：mgΔh = ½ mv² − ½ mu²。

8. Work Done Against Friction and Dissipation | 克服摩擦做功与能量耗散

Work done against friction or air resistance is different from work done against gravity. When a force overcomes friction, the energy is not stored as useful potential energy; instead, it is dissipated as internal energy, often called thermal energy or heat. This energy spreads into the surroundings and becomes difficult to recover for useful work.

克服摩擦或空气阻力做功不同于克服重力做功。当一个力克服摩擦时，能量不会以有用的势能形式储存，而是耗散为内能，通常称为热能或热量。这些能量扩散到周围环境中，很难再回收为有用的功。

For example, when a block slides along a rough surface and comes to rest, its kinetic energy is transferred to thermal energy by the negative work done by friction. The temperature of the block and surface may rise slightly. Because this energy is scattered and cannot be fully recovered, friction is described as a non-conservative force.

例如，当一个滑块在粗糙表面上滑动并最终停下时，摩擦做负功，把滑块和表面的温度略微升高，其动能转化为热能。由于这些能量被分散且无法完全回收，摩擦被描述为非保守力。

In energy calculations, you often write: initial energy = final energy + work done against friction. This form is useful when mechanical energy is not conserved, and it allows you to account for dissipated energy.

在能量计算中，你通常写成：初始能量 = 末态能量 + 克服摩擦做的功。当机械能不守恒时，这种形式很有用，它能帮助你计入耗散的能量。

9. Power and the Rate of Doing Work | 功率与做功速率

Power is defined as the rate at which work is done or energy is transferred. If a quantity of work ΔW is done in a time interval Δt, the average power is:

功率定义为做功或能量转移的速率。如果在时间间隔 Δt 内做了功 ΔW，则平均功率为：

P = ΔW / Δt

The SI unit of power is the watt (W), where 1 W = 1 J s⁻¹. In many mechanical systems, a constant force acts on an object moving at speed v in the direction of the force. The power delivered by the force is:

功率的国际单位是瓦特 (W)，1 W = 1 J s⁻¹。在许多机械系统中，恒力沿力的方向以速度 v 作用在运动物体上。该力提供的功率为：

P = Fv

This equation is especially useful for vehicles. For example, a car travelling at constant speed on a level road experiences resistive forces, such as air resistance and friction. The engine’s useful power output equals the driving force multiplied by the car’s speed. At a given power output, a vehicle can provide a larger driving force at lower speed, which explains why cars use lower gears for climbing steep hills.

这个公式对车辆问题尤其有用。例如，汽车在水平路面上匀速行驶时会受到空气阻力和摩擦等阻力。发动机的有用输出功率等于驱动力乘以车速。在给定输出功率下，车辆在较低速度时能提供更大的驱动力，这就解释了为什么汽车爬陡坡时要使用低档位。

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Doing Work, Transferring Energy | 做功与能量转移