📚 Work and Energy: Key Exam Points for IB and AQA Physics | 功与能量考点精讲
Work and energy are foundational concepts that bridge mechanics, thermal physics, and even modern physics. In both IB and AQA A‑level syllabi, you are expected to understand how energy transforms, how to calculate work done by forces, and how to apply conservation principles to solve complex problems. Mastering this topic means you can confidently tackle anything from simple inclined planes to roller coaster loops and nuclear collisions.
功与能量是连接力学、热物理乃至现代物理的基础概念。在 IB 和 AQA A‑level 课程中，你需要理解能量如何转化，如何计算力所做的功，并应用守恒原理解决复杂问题。掌握这个主题意味着你能够自信地应对从简单斜面到过山车圆周运动乃至核碰撞的各种题目。
1. Definition of Work | 功的定义
In physics, work is done when a force causes an object to move in the direction of the force. It is a scalar quantity, measured in joules (J). For a constant force F acting on an object that undergoes a displacement s, the work W is defined as the product of the component of the force along the displacement and the magnitude of the displacement.
在物理学中，当力使物体沿力的方向发生位移时，我们就说力做了功。功是标量，单位为焦耳（J）。对于作用在物体上的恒力 F 和发生的位移 s，功 W 定义为力沿位移方向的分量与位移大小的乘积。
The general equation is:
一般公式为：
W = F s cos θ
cos θ = 1, and W = F s. If they are perpendicular, θ = 90°, cos θ = 0, and no work is done — a classic example being the normal force on a block sliding along a horizontal surface.
其中 θ 是力矢量与位移矢量之间的夹角。若力和位移方向相同，θ = 0°，cos θ = 1，则 W = F s。若二者垂直，θ = 90°，cos θ = 0，则不做功——一个典型例子是物块在水平面上滑动时法向力不做功。
Work can be positive or negative. When the force has a component in the direction of motion, it does positive work and transfers energy to the object. When it opposes motion, it does negative work and removes energy from the object, often appearing as thermal energy due to friction.
功可以是正也可以是负。当力在运动方向上有分量时，做正功并将能量传递给物体。当力与运动方向相反时，做负功并从物体中带走能量，通常表现为因摩擦而产生的内能。
2. Calculating Work Done by a Constant Force | 恒力做功的计算
For an object pulled along a rough surface by a rope at an angle, you must resolve the applied force into horizontal and vertical components. Only the horizontal component (F cos θ) does work if the motion is horizontal. The vertical component affects the normal contact force and therefore the friction, but does not directly contribute to the work done on the object's horizontal displacement.
当一个物体被一根与水平方向成角度的绳子沿粗糙表面拉动时，必须将该拉力分解为水平和竖直分量。如果运动是水平的，只有水平分量（F cos θ）做功。竖直分量会影响法向接触力进而影响摩擦力，但并不会直接对物体的水平位移做功。
Common exam scenarios include:
常见的考试情景包括：
- A box pulled up a frictionless incline — the work done by the pulling force equals the change in gravitational potential energy plus any change in kinetic energy.
- 一个箱子沿无摩擦斜面拉上去——拉力所做的功等于重力势能的增加加上动能的变化。
- Pushing a lawnmower — the applied force has a downward component that increases friction.
- 推割草机——施加的力具有向下的分量，会增加摩擦力。
- A car engine providing a driving force — the work done by the driving force equals the force multiplied by the distance travelled, provided the force is constant.
- 汽车发动机提供驱动力——如果力是恒定的，驱动力所做的功等于力乘以行驶的距离。
Always draw a free‑body diagram and identify which forces actually perform work. Forces like tension and engine thrust do work, while the normal force and the weight component perpendicular to motion usually do zero work.
一定要画隔离体受力图，并识别哪些力实际上在做功。像张力和发动机推力等会做功，而法向力和垂直于运动方向的重力分量通常不做功。
3. Work Done by a Variable Force and Graphical Methods | 变力做功与图像法
t, you cannot simply multiply F by s. The work done is found by calculating the area under a force–displacement graph. For a spring obeying Hooke' arly with extension, and the work done to stretch it from 0 to x is the area of a triangle, giving the familiar elastic potential energy formula.
当施加的力不是恒力时，不能简单地将 F 与 s 相乘。变力所做的功通过计算力—位移图像下的面积求得。对于遵循胡克定律的弹簧，F = k x，力随伸长量线性增大，将其从 0 拉伸至 x 所做的功等于一个三角形的面积，从而得出熟悉的弹性势能公式。
The work done by a variable force is:
变力所做的功为：
W = Area under F–x graph
In IB and AQA exams, you may be asked to estimate the area by counting squares or using trapezoidal rules on a given graph. This skill also applies to finding the work done by a gas during expansion, where the area under a pressure–volume (p–V) diagram represents the work done by the gas.
在 IB 和 AQA 考试中，可能要求你通过数格点或利用梯形法则估算给定图像下的面积。这一技能也适用于求气体膨胀时所做的功，压强—体积（p–V）图下的面积即表示气体对外做的功。
For a spring, work done = average force × displacement = (½ Fmax) × x = ½ k x², consistent with the stored elastic potential energy.
对于弹簧，所做的功 = 平均力 × 位移 = (½ Fmax) × x = ½ k x²，与储存的弹性势能一致。
4. Kinetic Energy and the Work–Kinetic Energy Theorem | 动能与动能定理
Kinetic energy (K or Eₖ) is the energy possessed by an object due to its motion. It is given by:
动能（K 或 Eₖ）是物体由于运动而具有的能量，其表达式为：
Eₖ = ½ m v²
where m is the mass in kilograms, v is the speed in metres per second, and the unit is the joule. Kinetic energy is always positive and depends on the square of speed, so doubling the speed quadruples the kinetic energy.
其中 m 是以千克为单位的质量，v 是以米每秒为单位的速度，单位为焦耳。动能总是正值，并且与速度的平方成正比，因此速度加倍会使动能变为原来的四倍。
The work–kinetic energy theorem states that the net work done on an object equals its change in kinetic energy:
动能定理指出，对物体做的净功等于其动能的变化量：
Wnet = ΔEₖ = ½ m v² − ½ m v₀²
This theorem is extremely powerful because it links the total effect of all forces (including friction) directly to the change in speed, avoiding the need to consider acceleration and time in many problems. In both IB and AQA, you must be able to apply it to situations like cars braking, bullets penetrating barriers, and objects sliding down rough slopes.
这一定理非常强大，因为它将全部力（包括摩擦）的净效果直接与速度变化联系起来，在许多问题中无需考虑加速度和时间。在 IB 和 AQA 课程中，你必须能够将它应用于汽车刹车、子弹穿透障碍物以及物体沿粗糙斜面下滑等情景。
5. Gravitational Potential Energy | 重力势能
Gravitational potential energy (Eₚ or Ug) is the energy stored in an object due to 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 height h is:
重力势能（Eₚ 或 Ug）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，将质量为 m 的物体竖直升高 h 时，重力势能的变化量为：
ΔEₚ = m g h
nt of the motion or at a convenient reference level. Only changes in potential energy have physical significance.
重力势能的零点可以任意选取；你可以将运动的最低点或方便的参考面设为 h = 0。只有势能的变化才有物理意义。
When an object falls freely, gravitational potential energy is transformed into kinetic energy. In the absence of air resistance, mechanical energy is conserved: m g htop ½ m v²_bottom. This provides a quick way to find speed without using kinematic equations.
当物体自由下落时，重力势能转化为动能。若没有空气阻力，机械能守恒：m g htop = ½ m v²_bottom。这提供了一种无需使用运动学公式即可求速度的快捷方法。
For inclined planes, the change in height matters, not the distance along the slope. So the work done against gravity when moving an object up a frictionless ramp is always mgh, regardless of the ramp's length.
在斜面上，起决定作用的是高度的变化，而不是沿斜面的距离。因此，将一个物体沿无摩擦斜面向上移动时克服重力做的功总是 mgh，与斜面长度无关。
6. Elastic Potential Energy | 弹性势能
n objects that can be stretched or compressed, such as springs and rubber bands. For an ideal spring obeying Hooke' ergy stored when the spring is stretched or compressed by an amount x from its natural length is:
弹性势能储存在能够被拉伸或压缩的物体中，例如弹簧和橡皮筋。对于遵循胡克定律的理想弹簧，F = k x，当弹簧从原长被拉伸或压缩 x 时，储存的弹性势能为：
E_elastic = ½ k x²
Here k is the spring constant (N m⁻¹) and x is the extension or compression (m). This energy can be fully recovered when the spring returns to its equilibrium position, provided no energy is lost as heat.
这里 k 是劲度系数（N m⁻¹），x 是伸长量或压缩量（m）。只要没有能量以热的形式耗散，弹簧恢复到平衡位置时，这部分能量可以完全回收。
c potential energy, giving ½ m v² = ½ k x²_max. This enables you to find the maximum compression without needing to track the force‑time history.
在涉及弹簧振子或与弹簧发生碰撞的问题中，常将碰撞瞬间的动能与最大弹性势能相等，得到 ½ m v² = ½ k x²_max。这样一来，无需追踪力随时间的变化就能求出最大压缩量。
Note that the force–extension graph is a straight line through the origin, so the work done (area) naturally yields the ½ factor. Always check whether the spring is within its elastic limit; beyond that, Hooke's law no longer applies.
注意，力—伸长量图像是一条过原点的直线，因此所做的功（面积）自然会产生因子 ½。务必检查弹簧是否在弹性限度内；超出限度胡克定律不再适用。
7. Conservation of Mechanical Energy | 机械能守恒
The principle of conservation of mechanical energy states that if only conservative forces (such as gravity and spring forces) do work, the total mechanical energy (Eₖ + Eₚ) of a system remains constant:
机械能守恒定律指出，如果只有保守力（如重力和弹簧力）做功，系统的总机械能（Eₖ + Eₚ）保持不变：
Eₖ₁ + Eₚ₁ = Eₖ₂ + Eₚ₂
However, when non‑conservative forces like friction, air resistance, or applied forces do work, mechanical energy is not conserved — it is transformed into other forms such as thermal energy or sound. In these cases, the work done by non‑conservative forces equals the change in mechanical energy:
然而，当有摩擦力、空气阻力或外加力等非保守力做功时，机械能不守恒——它会转化为热能或声能等其它形式。此时，非保守力所做的功等于机械能的变化：
Wnc = ΔEₖ + ΔEₚ
IB exams frequently ask you to discuss energy transformations in contexts like a bouncing ball, a pendulum with air resistance, or a roller coaster with braking sections. AQA questions similarly expect you to apply energy principles alongside Newton's laws. Be precise: identify the system, list the energy stores at the start and end, and account for any work done by external forces or dissipation.
IB 考试经常要求你讨论诸如弹跳的球、受空气阻力影响的摆锤或带有制动段的过山车等情境中的能量转化。AQA 考题同样期望你在结合牛顿定律的同时应用能量原理。务必精确：明确所研究的系统，列出始末状态的能量储存，并考虑外力做功或耗散的影响。
8. Power and Efficiency | 功率与效率
Power is the rate at which work is done or energy is transferred. The average power is given by:
功率是做功或能量转移的速率。平均功率由下式给出：
P = W / t = ΔE / t
where W is work done in joules, t is time in seconds, and the unit of power is the watt (W), equivalent to J s⁻¹. For a constant force F acting on an object moving with constant speed v in the direction of the force, the instantaneous power can be expressed as:
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），相当于 J s⁻¹。若物体在恒力 F 作用下沿力的方向以恒定速度 v 运动，则瞬时功率可表示为：
P = F v
This relation is particularly useful for vehicles: the driving force exerted by the engine times the car's velocity gives the power output at that instant. To accelerate against resistive forces, the engine must supply power to increase kinetic energy and overcome drag.
这一关系对车辆问题特别有用：发动机的驱动力乘以车的速度即得出该时刻的输出功率。为对抗阻力进行加速时，发动机必须提供功率以增加动能并克服阻力。
Efficiency measures how much of the input energy or power is converted into useful output. It is defined as:
效率衡量输入能量或功率有多大比例转化为有用输出。其定义为：
Efficiency = (useful output energy / total input energy) × 100%
No real machine can be 100% efficient because of friction, heat losses, and sound. In exam problems, you may be asked to calculate the efficiency of an electric motor lifting a load, or the wasted power in a transmission system.
由于摩擦、热量散失和声音，任何真实机器的效率都不可能达到 100%。考试题目中可能要求你计算电动机提升重物时的效率，或传动系统中损失的功率。
9. Common Misconceptions and Exam Tips | 常见误区与应试技巧
Many students confuse work with effort. Holding a heavy book stationary does zero mechanical work because there is no displacement, even though you feel tired. The metabolic energy consumed inside your body is not accounted for by the physical definition of work.
许多学生将“做功”与“用力”混淆。静止地托着一本厚重书本所做的机械功为零，因为没有位移，尽管你感觉疲劳。你身体内部消耗的代谢能并不符合物理学中功的定义。
Another common error is forgetting the cos θ factor. If a force is perpendicular to the displacement, it does no work. For circular motion at constant speed, the centripetal force is always perpendicular to velocity, so it does zero work and cannot change the kinetic energy.
另一个常见错误是忘记 cos θ 因子。如果一个力与位移垂直，它就不做功。在匀速圆周运动中，向心力始终与速度垂直，因此不做功，不会改变动能。
s often misapply conservation of energy when friction is present. Always check whether non‑ clude the work they do (usually negative) in your energy equation. Writing W_friction = ΔEmech is safer than assuming Etotal is constant.
学生常常在有摩擦时错误地应用能量守恒。一定要检查是否存在非保守力，若存在，则应在能量方程中包含它们所做的功（通常为负）。写出 W_friction = ΔEmech 比假定总能量恒定更为妥当。
rms of work per unit time. Also, remember that the watt is a compound unit: 1 W = 1 J s⁻¹.
在功率计算中，只有力和速度方向相同且恒定时才能用 P = Fv 乘以平均速度。对于加速物体，应从单位时间做功的角度思考。还要记住瓦特是复合单位：1 W = 1 J s⁻¹。
Finally, always convert all quantities to SI units: mass in kg, distance in m, time in s, force in N. Using grams or centimetres will lead to inconsistent energy values unless carefully converted.
最后，务必将所有量化为国际单位制：质量用 kg，距离用 m，时间用 s，力用 N。使用克或厘米会导致能量数值不一致，除非经过仔细换算。
10. Linking Work and Energy Across the Syllabus | 功与能量在课程中的横向联系
W/q. In thermal physics, the internal energy of a system changes via heat and work (First Law of Thermodynamics). In AQA, similar connections appear when discussing the efficiency of heat engines or the kinetic energy of gas molecules.
功与能量原理远不止于力学。在 IB 物理中，你利用电场做功来定义电势差：V = W/q。在热物理中，系统的内能通过热量和功发生变化（热力学第一定律）。在 AQA 中，讨论热机效率或气体分子的动能时也会出现类似的联系。
asses of nuclei are often given in atomic mass units, and the energy released in reactions is calculated using Einstein's mass‑ is is a profound extension of the idea that energy is conserved and can be stored as mass.
在学习核物理时，原子核的质量常以原子质量单位给出，反应释放的能量则通过爱因斯坦的质能方程 E = m c² 计算。这是能量守恒并可以质量形式储存这一思想的深刻延伸。
Understanding work and energy provides a unifying framework. Whether you are analyzing a satellite's orbit, a capacitor's stored energy, or the photoelectric effect, the ability to account for energy inputs, outputs, and transformations is essential. IB and AQA exams reward your skill at tracing energy pathways and applying the correct formula in unfamiliar contexts.
理解功与能量提供了一个统一的分析框架。无论你是在分析卫星轨道、电容器的储能还是光电效应，能够核算能量的输入、输出和转化都是至关重要的。IB 和 AQA 考试会奖励你在陌生情境中追踪能量路径并正确运用公式的能力。
As a final check, always verify that your energy calculations respect the law of conservation of energy: what goes in must either be stored, used to do work, or dissipated. A clear energy flow diagram can often simplify the most challenging multi‑step problems.
作为最终检查，始终要保证你的能量计算满足能量守恒定律：输入的能量必然被储存、做有用功或耗散掉。一张清晰的能量流向图常常能简化最难的多步骤问题。
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