📚 Work and Energy for IB & WJEC Physics | IB & WJEC 物理功与能量考点精讲
Understanding work and energy is fundamental to mastering mechanics in both IB Physics and WJEC specifications. This article provides a comprehensive revision guide covering key concepts, formulas, and exam tips that you need to know for success. We will explore how work is defined, how energy is transformed, and how these principles apply to problem solving in a clear, bilingual format.
理解功与能量是掌握 IB 物理和 WJEC 考试中力学知识的基础。本文提供全面的复习指南，涵盖你需要掌握的关键概念、公式和应试技巧，助你取得成功。我们将以清晰的双语形式，探讨功的定义、能量的转化方式，以及这些原理如何应用于解题。
1. Work Done by a Constant Force | 恒力做的功
In physics, work is defined as the product of the force component in the direction of displacement and the magnitude of the displacement. When a constant force F acts on an object causing a displacement d, the work done W is given by W = F d cos θ, where θ is the angle between the force and displacement vectors. The unit of work is the joule (J).
在物理学中，功定义为力在位移方向上的分量与位移大小的乘积。当一个恒力 F 作用在物体上使其发生位移 d 时，所做的功 W 由 W = F d cos θ 给出，其中 θ 是力与位移矢量之间的夹角。功的单位是焦耳 (J)。
If the force is parallel to the displacement (θ = 0°), then cos 0° = 1 and W = F d. If the force is perpendicular (θ = 90°), no work is done because cos 90° = 0. This explains why the normal force or centripetal force does no work in many situations.
如果力平行于位移 (θ = 0°)，则 cos 0° = 1，W = F d。如果力垂直于位移 (θ = 90°)，则不做功，因为 cos 90° = 0。这就解释了为什么法向力或向心力在许多情况下不做功。
W = F d cos θ
2. Energy: The Capacity to Do Work | 能量：做功的能力
Energy is a scalar quantity that measures the ability of a system to perform work. It exists in many forms – kinetic, potential, thermal, chemical, nuclear, and more. In mechanics, we focus on kinetic energy and potential energy, which together form mechanical energy. Energy is always conserved; it can be transferred or transformed but never created or destroyed.
能量是一个标量，衡量系统做功的能力。它以多种形式存在——动能、势能、热能、化学能、核能等。在力学中，我们主要关注动能和势能，它们合称为机械能。能量总是守恒的；它可以转移或转化，但永远不会凭空产生或消失。
In IB and WJEC exams, you must be comfortable converting between energy descriptions and linking work and energy changes. For instance, when a net force does work on an object, the kinetic energy of the object changes by exactly that amount.
在 IB 和 WJEC 考试中，你需要熟练地在能量描述之间进行转换，并将功与能量变化联系起来。例如，当合外力对物体做功时，物体的动能变化量恰好等于所做的功。
3. Kinetic Energy (KE) | 动能
Kinetic energy is the energy an object possesses due to its motion. For an object of mass m moving with speed v, the kinetic energy is K = ½ m v². This formula applies to translational motion and is derived from the work done to accelerate a particle from rest. Kinetic energy is always positive and depends on the square of the speed.
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，其动能为 K = ½ m v²。该公式适用于平动，并且可以从将质点从静止加速所做的功推导出来。动能总是正值，并与速度的平方成正比。
Note that if the speed doubles, the kinetic energy quadruples. When solving problems, always use SI units: mass in kg, speed in m s⁻¹, and energy in J.
请注意，如果速度加倍，动能将变为原来的四倍。解题时务必使用国际单位制：质量用千克，速度用米每秒，能量用焦耳。
K = ½ m v²
4. Gravitational Potential Energy (GPE) | 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 change in GPE when an object of mass m is raised by a height Δh is ΔU = m g Δh, where g is the gravitational field strength (≈ 9.8 N kg⁻¹ on Earth). You can set a reference level where U = 0; only changes in GPE have physical significance.
重力势能是物体因在重力场中的位置而储存的能量。在地球表面附近，当质量为 m 的物体被抬升 Δh 高度时，重力势能的变化量为 ΔU = m g Δh，其中 g 是重力场强度（地球上约 9.8 N kg⁻¹）。你可以设定 U = 0 的参考面；只有重力势能的变化才有物理意义。
In many problems, the loss in GPE equals the gain in KE, provided air resistance and other non‑conservative forces are negligible. This is a direct consequence of the conservation of mechanical energy.
在许多问题中，若空气阻力和其他非保守力可以忽略，重力势能的减少量等于动能的增加量。这是机械能守恒的直接结果。
ΔU = m g Δh
5. The 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: W_net = ΔK. This principle is extremely powerful because it relates the motion of an object to the forces acting on it without needing to know the details of intermediate velocities or accelerations.
功能定理指出，作用在物体上的合外力所做的功等于其动能的变化量：W_net = ΔK。这一原理非常强大，因为它将物体的运动与其所受的力联系起来，无需知道中间速度或加速度的细节。
Mathematically, if an object accelerates from velocity v₁ to v₂ under a net force, the work done is W_net = ½ m v₂² – ½ m v₁². This equation works for both straight-line and curved paths, provided you use the net force component along the displacement.
在数学上，如果物体在合外力作用下速度从 v₁ 加速到 v₂，则所做的功为 W_net = ½ m v₂² – ½ m v₁²。该方程既适用于直线路径，也适用于曲线路径，只要使用沿位移方向的合外力分量即可。
W_net = ½ m v₂² – ½ m v₁²
6. Conservation of Mechanical Energy | 机械能守恒
When only conservative forces (such as gravity or spring forces) do work, the total mechanical energy E = K + U remains constant. This is the principle of conservation of mechanical energy. For a falling object, the sum of kinetic and gravitational potential energy is constant: ½ m v² + m g h = constant.
当只有保守力（如重力或弹簧力）做功时，总机械能 E = K + U 保持不变。这就是机械能守恒定律。对于一个下落物体，动能与重力势能之和保持不变：½ m v² + m g h = 常数。
In exam questions, you will often use this principle to find the speed of an object at a certain height, or the maximum compression of a spring. Remember that if non‑conservative forces like friction exist, mechanical energy is not conserved; some energy is converted into thermal energy.
在考题中，你会经常利用这一原理来求物体在某一高度的速度，或者弹簧的最大压缩量。请记住，如果存在摩擦力等非保守力，机械能并不守恒；部分能量会转化为热能。
7. Power and Its Calculation | 功率及其计算
Power is the rate at which work is done or energy is transferred. The average power P is given by P = W / t, where W is the work done over time t. The SI unit of power is the watt (W), equivalent to J s⁻¹. For a constant force acting on an object moving with velocity v, instantaneous power can also be expressed as P = F v cos θ.
功率是做功或能量转移的速率。平均功率 P 由 P = W / t 给出，其中 W 是在时间 t 内所做的功。功率的国际单位是瓦特 (W)，相当于焦耳每秒 (J s⁻¹)。对于以速度 v 运动的物体上作用的恒力，瞬时功率也可以用 P = F v cos θ 表示。
This is particularly useful when dealing with engines or muscles providing a force to overcome resistance. For example, a car climbing a hill at constant speed works against gravity and friction; its engine power can be found by multiplying the resistive force by its speed.
在处理发动机或肌肉提供力克服阻力的情况时，该公式特别有用。例如，一辆汽车以恒定速度爬坡时需克服重力和摩擦力；其发动机功率可以用阻力乘以速度求得。
P = W / t and P = F v cos θ
8. Efficiency of Energy Transfers | 能量转换的效率
Efficiency measures how well a device converts input energy into useful output energy. It is defined as the ratio of useful output power or energy to total input, often expressed as a percentage: η = (useful output / total input) × 100%. No real machine is 100% efficient because of unavoidable energy losses, often due to friction, heat, sound, or light.
效率衡量设备将输入能量转化为有用输出能量的程度。它定义为有用的输出功率或能量与总输入之比，通常用百分比表示：η = (有用输出 / 总输入) × 100%。真实的机器不可能达到 100% 的效率，因为不可避免地存在能量损耗，通常来自摩擦、热、声或光。
In calculations, you may be asked to find the total input power required for a given useful output or to determine energy wasted. Always identify the useful energy pathway and the pathways of dissipation.
在计算中，你可能需要求给定有用输出所需的总输入功率，或确定浪费的能量。始终要明确哪些是有效的能量路径，哪些是耗散的路径。
9. Work Done by a Variable Force | 变力做的功
When the force acting on an object is not constant, the work done is found by calculating the area under the force–displacement graph. For a spring obeying Hooke’s law, F = k x, the force varies linearly with extension, so the work done to stretch or compress the spring is the area of a triangle: W = ½ k x².
当作用在物体上的力不是恒力时，所做的功可以通过计算力–位移图下的面积求得。对于遵循胡克定律 F = k x 的弹簧，力随伸长量线性变化，因此拉伸或压缩弹簧所做的功等于三角形的面积：W = ½ k x²。
More generally, the work done by a variable force is given by the integral ∫ F(x) dx, but at IB and WJEC level, you will mainly use the area method on graphs. Make sure you can interpret a force–displacement graph correctly, counting squares or using geometric formulas.
更一般地，变力做的功由积分 ∫ F(x) dx 给出，但在 IB 和 WJEC 等级，你主要使用图形下的面积法。确保你能正确解读力–位移图，通过数格子或使用几何公式计算面积。
W = ½ k x² (for a spring)
10. Elastic Potential Energy and Hooke’s Law | 弹性势能与胡克定律
Elastic potential energy is the energy stored in a stretched or compressed elastic object, such as a spring. Hooke’s law states that the force needed to extend or compress a spring is proportional to the displacement from its natural length: F = k x, where k is the spring constant. The elastic potential energy stored in the spring is E_e = ½ k x², which equals the work done to deform it.
弹性势能是储存在被拉伸或压缩的弹性物体（如弹簧）中的能量。胡克定律指出，拉伸或压缩弹簧所需的力与偏离自然长度的位移成正比：F = k x，其中 k 是劲度系数。弹簧储存的弹性势能为 E_e = ½ k x²，等于使其形变所做的功。
Remember that the spring constant k is a measure of stiffness and is constant only within the elastic limit. Beyond that limit, permanent deformation occurs and Hooke’s law no longer applies.
请记住，劲度系数 k 是刚度的量度，且仅在弹性限度内为常数。超过该限度，会发生永久形变，胡克定律不再适用。
F = k x and E_e = ½ k x²
11. Non-Conservative Forces and Energy Dissipation | 非保守力与能量耗散
Non‑conservative forces, such as friction, air resistance, and applied forces that cause heating, dissipate energy from the system. When these forces act, mechanical energy is not conserved; the work done by non‑conservative forces equals the change in mechanical energy: W_nc = ΔK + ΔU.
非保守力，如摩擦力、空气阻力以及导致生热的外力，会耗散系统的能量。当这些力作用时，机械能不守恒；非保守力所做的功等于机械能的变化量：W_nc = ΔK + ΔU。
In many exam scenarios, you will be asked to calculate the amount of energy converted to thermal energy due to friction. Use the energy balance equation: initial total energy = final total energy + work done against friction. Understanding this helps you solve inclined plane or braking distance problems.
在许多考试情景中，你会被要求计算因摩擦而转化为热能的能量。利用能量平衡方程：初始总能量 = 最终总能量 + 克服摩擦力做的功。理解这一点有助于你解决斜面问题或刹车距离问题。
12. Solving Work and Energy Problems: Top Tips | 功与能量问题的解题技巧
First, always draw a clear diagram and identify the forces, displacements, and energy forms involved. Determine whether conservative or non‑conservative forces are at play. If only conservative forces do work, use conservation of mechanical energy; if friction or external forces are present, include work done against them.
首先，一定要画出清晰的示意图，并确定所涉及的力、位移和能量形式。判断是保守力还是非保守力在起作用。如果只有保守力做功，使用机械能守恒；如果存在摩擦力或外力，则要纳入克服它们所做的功。
Second, choose a convenient zero level for potential energy (often the lowest point or starting point). Express kinetic and potential energies in terms of the given variables, and then set up equations linking initial and final states. Substitute values carefully, keeping units consistent.
其次，选择一个方便的势能零点（通常是最低点或起始点）。用给定的变量表达动能和势能，然后建立联系初态和末态的方程。仔细代入数值，保持单位统一。
Finally, check whether the question asks for a scalar (energy) or a vector quantity. Energy and work are scalars, so directions may only matter in terms of the sign of work done (positive when energy is added, negative when removed). Practise past paper questions to gain confidence.
最后，检查题目要求的是标量（能量）还是矢量。能量和功是标量，所以方向只在功的正负号上有意义（能量增加时功为正，减少时为负）。通过练习历年真题来增强信心。
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