A-Level WJEC Physics: Work and Energy | A-Level WJEC 物理：功与能量 考点精讲

📚 A-Level WJEC Physics: Work and Energy | A-Level WJEC 物理：功与能量 考点精讲

Work and energy are among the most fundamental concepts in A-Level WJEC Physics. They provide a powerful framework for understanding how forces cause motion, how energy is stored and transferred, and how the principle of conservation of energy governs all physical processes. Mastering these ideas is essential not only for solving mechanics problems but also for tackling topics like electricity, thermal physics, and waves.

功与能量是 A-Level WJEC 物理中最基础的概念之一。它们构建了一个强大的框架，帮助我们理解力如何引起运动，能量如何被储存和转移，以及能量守恒定律如何支配所有物理过程。掌握这些概念对于解决力学问题以及攻克电学、热物理和波动等章节都至关重要。

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

In WJEC physics, work is defined as the product of the force and the displacement in the direction of the force. For a constant force F acting at an angle θ to the displacement s, the work done is W = F s cosθ. When the force is parallel to the displacement, θ = 0° and cosθ = 1, so the formula becomes W = F s. If the force is perpendicular to the displacement (θ = 90°), no work is done because cos90° = 0.

在 WJEC 物理中，功定义为力与在力的方向上的位移的乘积。对于一个与位移 s 成角度 θ 的恒力 F，做功为 W = F s cosθ。当力与位移平行时，θ = 0°，cosθ = 1，此时公式简化为 W = F s。如果力垂直于位移 (θ = 90°)，则不做功，因为 cos90° = 0。

W = F s cosθ

Work is a scalar quantity measured in joules (J). One joule is the work done when a force of one newton moves an object one metre in the direction of the force. It is important to remember that only the component of the force along the displacement contributes to work. The displacement must be the actual distance moved while the force is being applied.

功是一个标量，单位为焦耳 (J)。一焦耳是指一牛顿的力使物体沿力的方向移动一米所做的功。需要牢记的是，只有力沿位移方向的分量才对功有贡献。位移必须是在力作用期间实际移动的距离。

2. Work Done as Energy Transfer | 功是能量转移的量度

Work is a measure of energy transfer. Whenever work is done on an object, energy is transferred from one form to another or from one place to another. For example, when you lift a book, you do work against gravity, and chemical energy from your muscles is converted into gravitational potential energy. If a force does positive work on a body, the body gains energy; if the force does negative work (e.g., friction), the body loses energy.

功是能量转移的量度。每当对一个物体做功时，能量就从一种形式转移到另一种形式，或从一个地方转移到另一个地方。例如，当你举起一本书时，你克服重力做功，肌肉中的化学能转化为重力势能。如果一个力做正功，该物体就获得能量；如果力做负功（例如摩擦力），物体就损失能量。

This relationship is central to the work–energy principle: the net work done on an object equals the change in its kinetic energy. WJEC exam questions frequently ask you to link work and energy changes. Always state clearly what work is being done and where the energy is going.

这种关系是功能原理的核心：作用在物体上的净功等于其动能的变化量。WJEC 考试题目经常要求你联系功与能量的变化。始终要清楚地说明谁在做功以及能量去了哪里。

3. Kinetic Energy and the Work–Energy Theorem | 动能与功能定理

Kinetic energy (Eₖ) is the energy possessed by an object due to its motion. For an object of mass m moving at speed v, kinetic energy is given by:

动能 (Eₖ) 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，动能由下式给出：

Eₖ = ½ m v²

The work–energy theorem states that the net work done on an object is equal to the change in its kinetic energy: Wₙₑₜ = ΔEₖ = Eₖ,₂ − Eₖ,₁. This powerful result lets you calculate speed changes without having to deal with acceleration and time directly. In WJEC papers, you will often be asked to apply this theorem to a body sliding down an incline or being pulled by a variable force.

功能定理指出，作用在物体上的净功等于其动能的变化量：Wₙₑₜ = ΔEₖ = Eₖ,₂ − Eₖ,₁。这一强大的结论使你在无需直接处理加速度和时间的情况下就能计算速度变化。在 WJEC 试卷中，经常会要求你将该定理应用于沿斜面下滑或受变力拉动的物体。

Remember that the net work includes work done by all forces – applied forces, gravity, friction, and normal reaction (which often does no work because it is perpendicular to motion).

请记住，净功包括所有力所做的功——作用力、重力、摩擦力和法向反作用力（通常不做功，因为它垂直于运动）。

4. Gravitational Potential Energy | 重力势能

Gravitational potential energy (Eₚ) is the energy an object possesses due to its position in a gravitational field. Near the Earth’s surface, where the gravitational field strength g is approximately constant at 9.81 N kg⁻¹, the change in gravitational potential energy when an object of mass m is raised through a vertical height Δh is:

重力势能 (Eₚ) 是物体因其在重力场中的位置而具有的能量。在地球表面附近，重力场强度 g 近似为常数 9.81 N kg⁻¹，质量为 m 的物体被举高竖直高度 Δh 时，重力势能的变化量为：

ΔEₚ = m g Δh

If you take a reference level where Eₚ = 0, then the potential energy at height h is Eₚ = m g h. In WJEC problems, you must be careful to use the vertical component of displacement when calculating work done against gravity. If an object is lifted along a slope, only the vertical rise matters for the change in Eₚ.

若取某参考面使 Eₚ = 0，则高度 h 处的势能为 Eₚ = m g h。在 WJEC 问题中，计算克服重力做功时必须使用位移的竖直分量。如果物体沿斜面被抬高，只有竖直升高部分才影响 Eₚ 的改变。

The gravitational potential energy gained equals the work done against gravity, provided no other forces (like friction) are present. This principle is frequently used to determine speeds at the bottom of a fall or heights reached by projectiles.

如果没有其他力（如摩擦力）存在，增加的重力势能等于克服重力所做的功。此原理常被用来求下落到底部的速度或抛体所能达到的高度。

5. Hooke’s Law and Elastic Potential Energy | 胡克定律与弹性势能

Many WJEC questions involve springs or elastic materials obeying Hooke’s Law: the extension x of a spring is directly proportional to the applied force F, as long as the elastic limit is not exceeded. Mathematically, F = k x, where k is the spring constant (N m⁻¹). The elastic potential energy stored in a stretched or compressed spring is:

许多 WJEC 试题会涉及遵循胡克定律的弹簧或弹性材料：只要不超过弹性极限，弹簧的伸长量 x 与施加的力 F 成正比。数学表达式为 F = k x，其中 k 是劲度系数 (N m⁻¹)。储存在被拉伸或压缩的弹簧中的弹性势能为：

Eₑ = ½ F x = ½ k x²

This formula arises because the average force needed to stretch the spring from 0 to x is ½ F. It is a common exam point that the work done in extending a spring is not simply F × x, but rather the area under the force–extension graph, which is a triangle for Hookean materials. As the spring is extended, the force is not constant, so work done = average force × extension.

该公式的由来是因为将弹簧从 0 拉伸至 x 所需的平均力为 ½ F。常见的考点是，拉伸弹簧所做的功并非简单的 F × x，而是力-伸长量图下的面积，对于遵循胡克定律的材料，该区域是一个三角形。由于弹簧伸长过程中力并非恒力，所以功 = 平均力 × 伸长量。

When you solve problems involving springs, remember that elastic potential energy is a scalar quantity that can be fully converted into kinetic energy or gravitational potential energy in the absence of dissipative forces.

当你解决涉及弹簧的问题时，要记住弹性势能是一个标量，在没有耗散力的情况下可以完全转化为动能或重力势能。

6. Conservation of Mechanical Energy | 机械能守恒

The principle of conservation of energy states that energy cannot be created or destroyed, only transferred or converted from one form to another. In an isolated system where only conservative forces (like gravity or spring forces) do work, the total mechanical energy (Eₖ + Eₚ + Eₑ) remains constant.

能量守恒定律指出，能量既不能被创造也不能被消灭，只能从一种形式转化为另一种形式。在一个只有保守力（如重力或弹簧力）做功的孤立系统中，总机械能（Eₖ + Eₚ + Eₑ）保持不变。

Eₖ,₁ + Eₚ,₁ + Eₑ,₁ = Eₖ,₂ + Eₚ,₂ + Eₑ,₂

WJEC questions often present a scenario where a pendulum swings, a roller coaster moves, or a mass oscillates on a spring. You are expected to equate the total energy at two different positions to find unknown speeds or displacements. It is vital to choose a clear reference level for gravitational potential energy and to consistently include all forms of mechanical energy.

WJEC 试题常会给出一个场景，比如摆锤摆动、过山车运动或弹簧上的质量块振动。你需要通过使两个不同位置的总能量相等来求出未知速度或位移。至关重要的一点是，要选择一个明确的重力势能参考面，并且一致地包含所有形式的机械能。

If non-conservative forces such as friction or air resistance do work, the mechanical energy is not conserved; instead, some energy is transferred to thermal energy and the system heats up.

如果存在像摩擦力或空气阻力这样的非保守力做功，机械能便不再守恒；此时部分能量转化为内能，系统会升温。

7. Power | 功率

Power is defined as the rate of doing work or the rate of energy transfer. It is a scalar quantity with the unit watt (W), where 1 W = 1 J s⁻¹. The average power P when work W is done in a time interval t is:

功率定义为做功的速率或能量转移的速率。它是一个标量，单位是瓦特 (W)，1 W = 1 J s⁻¹。在时间间隔 t 内做功 W 时的平均功率 P 为：

P = W / t

For a constant force F acting on an object moving at constant speed v in the direction of the force, the instantaneous power can also be expressed as P = F v. This relation is particularly useful in problems involving vehicles moving at a steady speed against resistive forces. The WJEC syllabus expects you to be able to derive P = F v from the definitions of work and power.

对于一个作用于以恒定速度 v 沿力方向运动的物体上的恒力 F，瞬时功率也可表示为 P = F v。在处理车辆以恒定速度克服阻力运动的问题时，这一关系式很有用。WJEC 教学大纲要求你能够从功和功率的定义推导出 P = F v。

Be careful with units: force in newtons, speed in m s⁻¹, power in watts. Sometimes power is given in kilowatts (kW) and you must convert to watts.

注意单位：力以牛顿计，速度以 m s⁻¹ 计，功率以瓦特计。有时功率会以千瓦 (kW) 给出，你必须将其转换为瓦特。

8. Efficiency | 效率

Efficiency is a measure of how much useful energy or work output we get compared with the total energy input. It can be expressed as a ratio or as a percentage. In the WJEC specification, efficiency is given by:

效率是指我们获得的有用能量或有用功与总输入能量相比的量度。它可以表达为比值或百分比。在 WJEC 规范中，效率由下式给出：

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

Equivalently, you can use power: Efficiency = (useful output power / total input power) × 100%. No real machine is 100% efficient; there are always energy losses due to friction, air resistance, sound, and heat. Typical WJEC questions may ask you to calculate efficiency from energy values, or to explain ways to reduce energy waste.

等效地，你可以使用功率：效率 = (有用输出功率 / 总输入功率) × 100%。没有任何实际机器能达到 100% 的效率；总是存在因摩擦、空气阻力、声音和热量造成的能量损失。典型的 WJEC 问题可能会要求你根据能量值计算效率，或解释如何减少能量浪费。

Efficiency is a dimensionless quantity, but it is important to keep the percentage form when expressing final answers. Always state clearly what you consider ‘useful’ output in the context of the question.

效率是无量纲量，但在表达最终答案时保留百分比形式很重要。始终要清晰地说明，在该问题的情境中你将什么视为“有用”输出。

9. Energy and Non-Conservative Forces | 能量与非保守力

In the real world, non-conservative forces such as friction and drag are always present. These forces remove mechanical energy from a system and convert it into thermal energy (internal energy). The work done by a non-conservative force on an object is equal to the change in the object’s mechanical energy, and it is always path-dependent.

在现实世界中，总是存在诸如摩擦力和阻力这样的非保守力。这些力从系统中带走机械能并将其转化为热力学能（内能）。非保守力对物体做的功等于该物体机械能的变化量，而且它总是与路径有关的。

Wₙc = ΔEₖ + ΔEₚ

For instance, when a block slides down a rough incline, the work done against friction is equal to the loss in total mechanical energy. The WJEC syllabus expects you to apply this extended work–energy principle, especially in questions where the final speed is lower than that predicted by energy conservation alone. Always include the work done against friction as a negative contribution to the total energy equation.

例如，当一个滑块沿粗糙斜面下滑时，克服摩擦力做的功等于总机械能的减少量。WJEC 教学大纲要求你运用这个扩展的功能原理，尤其是在那些最终速度低于仅由能量守恒所预测的速度的问题中。始终要将克服摩擦力做的功作为总能量方程的负贡献项包含进来。

When friction is present, the dissipated energy E = F_friction × d, where d is the distance over which the friction acts. Some energy may also be converted to sound, but these are usually negligible at this level.

当存在摩擦力时，耗散的能量 E = F_摩擦 × d，其中 d 是摩擦力作用的距离。部分能量也可能转化为声能，但在此阶段通常可忽略不计。

10. Common Misconceptions and Exam Tips | 常见误区与应试技巧

Students often confuse force and energy: a force does not possess energy; it is an agent that transfers energy. Another typical mistake is forgetting that work is only done when there is a displacement in the direction of the force. Holding a heavy object stationary might feel tiring, but no work is done on the object in the physics sense because there is no displacement.

学生常会混淆力和能量：力并不具有能量，它是转移能量的一种作用。另一个典型错误是忘记只有在沿力方向有位移时才做功。静止地搬着重物可能会让你感到疲劳，但从物理学意义上讲，并没有对物体做功，因为没有位移。

Always draw clear force and displacement diagrams. Label all forces, decide which ones do work, and use the correct angle in W = F s cosθ. When dealing with slopes, decompose the weight into components parallel and perpendicular to the incline to calculate work done by gravity correctly.

一定要画出清晰的力和位移示意图。标出所有力，判断哪些力做功，并在 W = F s cosθ 中使用正确的角度。处理斜面问题时，要将重力分解为平行和垂直于斜面的分量，以便正确计算重力做的功。

In energy conservation problems, carefully choose your zero reference for gravitational potential energy. The change in potential energy depends only on the vertical displacement, not on the path taken. Avoid double counting energy conversions: each energy ‘unit’ should only be accounted for once in your equation.

在能量守恒问题中，要仔细选择重力势能的零参考点。势能的变化只取决于竖直位移，与所经路径无关。避免重复计算能量转换：每一个能量“单位”在你的方程中只应计入一次。

Finally, always check the units. Energy and work are both in joules. When using P = F v, ensure v is in m s⁻¹, not km h⁻¹. Many WJEC past papers test this conversion. Practise extracting information from graphs, especially force–distance and force–extension graphs, because the area under these graphs represents work done or energy stored.

最后，始终要检查单位。能量和功的单位都是焦耳。使用 P = F v 时，确保 v 的单位为 m s⁻¹，而非 km h⁻¹。许多 WJEC 历年试卷都考查了这一换算。要练习从图像中提取信息，尤其是力-距离图像和力-伸长量图像，因为这些图像下的面积代表所做的功或储存的能量。

By methodically applying the principles of work and energy, you can solve a wide variety of problems with confidence. Remember that WJEC examiners value clear reasoning, correct use of formulas, and precise unit handling.

有条不紊地运用功与能量原理，你就能自信地解决各种问题。请记住，WJEC 考官看重清晰的推理过程、正确的公式运用和准确的单位处理。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）