Work and Energy in IB Edexcel Physics | IB Edexcel 物理：功与能量考点精讲

📚 Work and Energy in IB Edexcel Physics | IB Edexcel 物理：功与能量考点精讲

Work and energy are central pillars of classical mechanics, linking force, motion, and the capacity to do work. In the IB Edexcel Physics syllabus, a thorough understanding of work, kinetic energy, potential energy and the principle of conservation of energy is essential for solving a wide range of problems – from simple inclined planes to complex systems involving springs and circular motion. This article revisits every key concept, formula and typical exam pitfall, helping you build a rock‑solid foundation.

功和能量是经典力学的核心支柱，它将力、运动和做功的能力紧密相连。在 IB Edexcel 物理课程中，透彻理解功、动能、势能以及能量守恒原理对于解决从简单的斜面到涉及弹簧和圆周运动的复杂系统等各种问题至关重要。本文回顾每一个关键概念、公式和常见考试陷阱，帮助你建立扎实的基础。

1. Definition of Work | 功的定义

Work is done when a force causes a displacement of the point of application in the direction of the force. For a constant force F acting at an angle θ to a straight displacement s, the work done is W = F s cos θ. If the force and displacement are in the same direction, θ = 0° and W = F s. If the force is perpendicular to the displacement, θ = 90° and no work is done because cos 90° = 0. Work is a scalar quantity; its SI unit is the joule (J), where 1 J = 1 N m.

当一个力使其作用点沿力的方向发生位移时，就做了功。对于与直线位移 s 成角度 θ 的恒力 F，所做的功为 W = F s cos θ。如果力与位移同向，θ = 0°，则 W = F s。如果力垂直于位移，θ = 90°，由于 cos 90° = 0，不做功。功是标量；其国际单位是焦耳 (J)，1 J = 1 N m。

It is crucial to identify the component of force that acts along the displacement. When pushing a lawnmower, only the horizontal component of the applied force does work on the mower; the vertical component does no work on the horizontal displacement. The area under a force–distance graph also represents the work done, a principle that extends to variable forces.

关键是找出沿位移方向的力的分量。推割草机时，只有施加力的水平分量对割草机做功；垂直分量对水平位移不做功。力–距离图下的面积也代表所做的功，这一原理可以推广到变力的情况。

W = F s cos θ

2. Work Done by a Variable Force | 变力做功

When the force is not constant, the work done is found by integrating the force with respect to displacement. For a one‑dimensional motion, W = ∫ F(x) dx from the initial position x_i to the final position x_f. Graphically, this is the exact area under the force–displacement curve. In many IB questions, a graph of F against x is provided, and you are required to count squares or calculate the area of simple geometric shapes to determine the work done.

当力不是恒力时，做功通过对力关于位移积分求得。对于一维运动，W = ∫ F(x) dx，从初始位置 x_i 到最终位置 x_f。从图形上看，这就是力–位移曲线下的精确面积。在许多 IB 考题中，会提供 F 随 x 变化的图像，要求你通过数格子或计算简单几何图形的面积来确定所做的功。

A common example is the stretching of a spring obeying Hooke’s law, F = k x, where k is the spring constant. The work done in stretching the spring from extension 0 to e is the area of a triangle: W = ½ k e². This result is stored as the elastic potential energy.

一个常见的例子是遵循胡克定律 F = k x 的弹簧的拉伸，其中 k 是弹簧常数。将弹簧从伸长量 0 拉伸到 e 所做的功是三角形的面积：W = ½ k e²。这个结果被储存为弹性势能。

W = ∫ F dx = ½ k e² (for a spring)

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

Kinetic energy (E_k) is the energy an object possesses due to its motion. For a body of mass m moving with speed v, E_k = ½ m v². The net work done on an object by all the forces acting on it equals the change in its kinetic energy: W_net = ΔE_k = E_k,final – E_k,initial. This statement is the work–energy theorem and is a direct consequence of Newton’s second law.

动能 (E_k) 是物体由于运动而具有的能量。对于质量为 m、运动速度为 v 的物体，E_k = ½ m v²。作用在物体上所有力的合力所做的净功等于其动能的变化：W_net = ΔE_k = E_k,末 – E_k,初。这一表述就是功能定理，是牛顿第二定律的直接结果。

The theorem allows you to find the speed of an object without directly solving the equations of motion, as long as you can compute the net work. For example, the work done by gravity on a falling object increases its kinetic energy, while the work done by friction dissipates mechanical energy, appearing as heat.

该定理使你可以无需直接求解运动方程就能求出物体的速度，只要你能计算出净功。例如，重力对下落物体做的功增加了其动能，而摩擦力做的功耗散了机械能，表现为热量。

E_k = ½ m v² W_net = ΔE_k

4. Gravitational Potential Energy | 重力势能

Gravitational potential energy (E_p) near the Earth’s surface is defined as E_p = m g h, where h is the height above a chosen reference level, m is the mass and g is the gravitational field strength (9.81 m s⁻² on Earth). The work done by gravity when an object moves vertically from height h₁ to h₂ is W_g = –ΔE_p = –(m g h₂ – m g h₁) = m g (h₁ – h₂). If the object falls, gravity does positive work and E_p decreases.

地球表面附近的重力势能 (E_p) 定义为 E_p = m g h，其中 h 是所选参考水平面以上的高度，m 是质量，g 是重力场强度（地球上为 9.81 m s⁻²）。当物体从高度 h₁ 竖直移动到 h₂ 时，重力做的功为 W_g = –ΔE_p = –(m g h₂ – m g h₁) = m g (h₁ – h₂)。如果物体下落，重力做正功，E_p 减少。

E_p is a relative quantity: only differences in gravitational potential energy are physically meaningful. You must always specify the reference level (often the lowest point in the problem). In problems involving inclined planes, the change in height Δh = s sin θ is used, where s is the distance along the slope.

E_p 是一个相对量：只有重力势能的变化才有物理意义。你必须始终指定参考水平面（通常是问题中的最低点）。在涉及斜面的问题中，使用高度变化 Δh = s sin θ，其中 s 是沿斜面的距离。

E_p = m g h

5. Elastic Potential Energy | 弹性势能

Elastic potential energy is stored in deformed elastic objects such as springs. For a spring that obeys Hooke’s law (F = k x), the elastic potential energy when stretched or compressed by an amount x is E_e = ½ k x². This is always positive, as x² ensures that energy is stored regardless of the direction of displacement. The spring constant k has units N m⁻¹ and measures the stiffness of the spring.

弹性势能储存在发生形变的弹性物体中，例如弹簧。对于遵循胡克定律 (F = k x) 的弹簧，当被拉伸或压缩量 x 时，弹性势能为 E_e = ½ k x²。这个值总是正的，因为 x² 确保无论位移方向如何都储存能量。弹簧常数 k 的单位是 N m⁻¹，它衡量弹簧的刚度。

The work done to stretch a spring from extension x₁ to x₂ is ΔE_e = ½ k (x₂² – x₁²). In an oscillating mass–spring system, energy continuously converts between kinetic energy and elastic potential energy, with the total mechanical energy remaining constant if no dissipative forces are present.

将弹簧从伸长量 x₁ 拉伸到 x₂ 所做的功为 ΔE_e = ½ k (x₂² – x₁²)。在振动的质量–弹簧系统中，能量不断地在动能和弹性势能之间转换，如果没有耗散力存在，总机械能保持不变。

E_e = ½ k x²

6. Principle of Conservation of Mechanical Energy | 机械能守恒原理

When only conservative forces (such as gravity and ideal spring forces) do work on a system, the total mechanical energy (kinetic + potential) remains constant. That is, E_total = E_k + E_p + E_e = constant. This principle lets us equate the energy at two different points in the motion without worrying about the intermediate accelerations and velocities.

当只有保守力（如重力和理想的弹簧力）对系统做功时，总机械能（动能 + 势能）保持不变。即 E_total = E_k + E_p + E_e = 常数。这条原理使我们能够将运动中两个不同点的能量等同起来，而无需担心中间的加速度和速度。

For example, a pendulum bob at its highest point has maximum gravitational potential energy and zero kinetic energy; at the lowest point, E_p is minimum and E_k is maximum. Using ½ m v² = m g h_max gives the speed at the bottom. However, if non‑conservative forces like friction or air resistance do work, the total mechanical energy changes and we must use the work–energy theorem including the work done by these forces.

例如，摆锤在最高点具有最大的重力势能和零动能；在最低点，E_p 最小而 E_k 最大。利用 ½ m v² = m g h_max 可求出底部的速度。然而，如果像摩擦力或空气阻力这样的非保守力做功，总机械能就会改变，我们必须使用包含这些力所做功的功能定理。

E_k1 + E_p1 = E_k2 + E_p2 (no friction)

7. Power | 功率

Power is the rate at which work is done or energy is transferred. The average power is P_avg = ΔW / Δt, and instantaneous power is P = dW/dt. The SI unit of power is the watt (W), where 1 W = 1 J s⁻¹. For a constant force F acting on an object moving with velocity v, the power delivered is P = F v cos θ, with θ the angle between F and v.

功率是做功或能量转移的速率。平均功率为 P_avg = ΔW / Δt，瞬时功率为 P = dW/dt。功率的国际单位是瓦特 (W)，1 W = 1 J s⁻¹。对于一个恒力 F 作用在速度为 v 的物体上，传递的功率为 P = F v cos θ，其中 θ 是 F 与 v 之间的夹角。

This relationship is useful when analysing engines: a car moving at constant speed up a hill needs an engine power P = F_tot v, where F_tot is the total resistive force plus the component of weight down the incline. As the velocity increases, a given engine power produces less driving force, explaining why vehicles have a top speed.

这一关系在分析发动机时非常有用：一辆以恒定速度上坡的汽车需要的发动机功率为 P = F_tot v，其中 F_tot 是总阻力加上沿斜面的重力分量。随着速度增加，给定的发动机功率产生的驱动力就会减小，这就解释了为什么车辆有最高速度。

P = ΔW / Δt = F v cos θ

8. Efficiency | 效率

Efficiency measures how much of the input energy or power is converted to useful work. It is defined as η = (useful work output / energy input) × 100%, or η = (useful power output / power input) × 100%. Efficiency is always less than 1 (or 100%) due to energy dissipated as heat, sound or other non‑useful forms.

效率衡量输入能量或功率中有多少转化为有用的功。它定义为 η = (有用功输出 / 能量输入) × 100%，或 η = (有用功率输出 / 功率输入) × 100%。效率总是小于 1（或 100%），因为能量会以热量、声音或其他无用的形式耗散。

In practical systems, such as electric motors or pulleys, you may be asked to calculate efficiency from the measured work done against gravity and the work input. The energy lost is mostly due to friction. The concept of efficiency reinforces the first law of thermodynamics – energy is conserved, but its “usefulness” degrades.

在实际系统中，比如电动机或滑轮，你可能需要根据测得的克服重力所做的功和输入的功来计算效率。能量损失主要来自摩擦。效率的概念强化了热力学第一定律——能量是守恒的，但其“有用性”会降低。

η = (useful output / input) × 100%

9. Interpreting Energy Bar Charts and Sankey Diagrams | 能量柱状图和桑基图的解读

Energy bar charts (LOL diagrams) are visual tools that track the distribution of energy between kinetic, gravitational potential, elastic potential, and internal (thermal) energy at different stages of a process. Each type of energy is represented by a bar, and the total height remains constant if no energy enters or leaves the system.

能量柱状图（LOL 图）是一种可视化工具，用于跟踪过程中不同阶段动能、重力势能、弹性势能和内能（热能）之间的能量分布。每种能量用一个柱子表示，如果没有能量进入或离开系统，总高度保持不变。

Sankey diagrams show energy transfers as flow arrows, with the thickness of the arrow proportional to the amount of energy. They clearly illustrate how input energy splits into useful output and wasted energy. IB exam questions often ask you to deduce the efficiency from a Sankey diagram or to identify which energy transformation is missing from a given description.

桑基图用箭头流表示能量传递，箭头的粗细与能量的多少成正比。它们清楚地展示了输入能量如何分为有用的输出和浪费的能量。IB 考试经常要求你根据桑基图推断效率，或者找出给定描述中缺少了哪种能量转化。

10. Work and Energy in Circular Motion | 圆周运动中的功和能量

An object moving in a circle at constant speed has a centripetal force directed towards the centre. Since the displacement at any instant is tangential and thus perpendicular to the centripetal force, no work is done by the centripetal force. This is why the speed – and hence the kinetic energy – remains constant even though the object accelerates.

以恒定速率做圆周运动的物体受到指向圆心的向心力。由于任一时刻的位移沿切线方向，因此与向心力垂直，向心力不做功。这就是为什么尽管物体在加速，其速率——因而动能——保持不变的原因。

If there is a tangential component of the net force (e.g., a car accelerating around a bend), work is done by that tangential component, causing a change in kinetic energy. In vertical circular motion, gravitational potential energy and kinetic energy interconvert, so speed varies. Use conservation of energy to find speeds at different points in the loop.

如果净力有切向分量（例如，汽车在弯道上加速），该切向分量就会做功，导致动能改变。在竖直圆周运动中，重力势能和动能相互转化，因此速率会变化。使用能量守恒可以求出环中不同点的速度。

11. Common Misconceptions and Exam Tips | 常见误区与考试技巧

Many students confuse work with effort: you can push against a wall and feel tired, but you have done zero work on the wall because its displacement is zero. Always check for displacement in the direction of the force. Another pitfall is thinking that potential energy is absolute; only changes matter, and you must consistently use the same reference level.

许多学生将功与所花的力气混淆：你可以推墙并感到累，但由于墙的位移为零，你对墙做的功为零。始终检查在力的方向上是否有位移。另一个陷阱是认为势能是绝对的；只有变化才有意义，而且你必须始终使用同一个参考水平面。

When using conservation of energy, clearly state your “zero level” at the beginning of the solution. For spring problems, make sure you distinguish between the extension of the spring and the height change. In power questions, check whether you need average or instantaneous power. Also, be comfortable converting between units like kJ, MJ and J, and remember that 1 kWh = 3.6 × 10⁶ J.

在使用能量守恒时，在解答开头清楚地说明你的“零点水平面”。对于弹簧问题，确保区分弹簧的伸长量和高度变化。在功率问题中，检查你需要的是平均功率还是瞬时功率。此外，要能熟练地在 kJ、MJ 和 J 等单位之间换算，并记住 1 kWh = 3.6 × 10⁶ J。

A reliable exam technique is to list all the energy forms present at two key positions, then write the conservation equation and solve for the unknown. Always double‑check that dissipated energy (due to friction) is subtracted or added correctly. When drawing graphs, label axes with quantities and units, and remember that the gradient of a W–s graph is force, and the area under a P–t graph is energy.

一个可靠的考试技巧是列出在两个关键位置存在的所有能量形式，然后写出守恒方程并求解未知量。始终反复检查因摩擦而耗散的能量是否被正确地减去或加上。画图时，用物理量和单位标注坐标轴，并记住 W–s 图的斜率是力，P–t 图下的面积是能量。

12. Summary of Essential Formulae | 核心公式总结

Work: W = F s cos θ
Kinetic energy: E_k = ½ m v²
Work–energy theorem: W_net = ΔE_k
Gravitational PE: E_p = m g h
Elastic PE: E_e = ½ k x²
Power: P = W / t = F v cos θ
Efficiency: η = (E_useful / E_input) × 100%

These equations form the backbone of energy analysis. Practice applying them to multi‑step problems involving ramps, pulleys, springs and collisions to build confidence for the exam.

这些方程构成了能量分析的支柱。通过将它们应用到涉及斜面、滑轮、弹簧和碰撞的多步骤问题中进行练习，以建立考试的信心。

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Work and Energy in IB Edexcel Physics | IB Edexcel 物理：功与能量 考点精讲