Work and Energy in IB Physics: Key Concepts and Exam Tips | IB物理功与能量考点精讲

📚 Work and Energy in IB Physics: Key Concepts and Exam Tips | IB物理功与能量考点精讲

In IB Physics, the topic of work and energy underlies much of mechanics and thermodynamics. Understanding how a force transfers energy, how energy is stored in different forms, and how the principle of conservation of energy governs every interaction is essential for both SL and HL students. This article breaks down the key syllabus points for Topic 2.3 – Work, Energy and Power – and provides exam-focused explanations, common pitfalls, and practical tips to master the concepts.

在IB物理中，功与能量是力学和热力学的重要基础。理解力如何传递能量、能量如何以不同形式储存，以及能量守恒原理如何支配每一次相互作用，对于SL和HL考生都至关重要。本文围绕大纲第2.3节“功、能量和功率”的核心考点，提供面向考试的解析、常见易错点和掌握概念的实用技巧。

1. Work: Definition and Calculation | 功的定义与计算

In physics, work is done when a force causes a displacement of an object in the direction of the force. The formal definition is W = F s cosθ, where F is the magnitude of the constant force, s is the magnitude of the displacement, and θ is the angle between the force and the displacement vectors. The unit of work is the joule (J), equivalent to one newton-metre (N m).

在物理学中，当一个力使物体沿力的方向发生位移时，该力做了功。正式定义为W = F s cosθ，其中F是恒力的大小，s是位移的大小，θ是力与位移矢量之间的夹角。功的单位是焦耳(J)，等于1牛·米(N·m)。

If the force is not constant or the path is curved, the work done can be found by calculating the area under a force–displacement graph. For IB exams, it is vital to remember that only the component of the force parallel to the displacement does work. A force perpendicular to the motion, such as the centripetal force in uniform circular motion, does zero work.

如果力不是恒力或运动轨迹为曲线，所做的功可以通过计算力–位移图下的面积求得。在IB考试中，必须牢记只有平行于位移的分力做功。垂直于运动方向的力，例如匀速圆周运动中的向心力，做功为零。

2. Positive, Negative, and Zero Work | 正功、负功与零功

Work can be positive, negative, or zero depending on the angle θ. When the force has a component in the same direction as the displacement (0° ≤ θ < 90°), work is positive and the force adds energy to the system. When the force has a component opposite to the displacement (90° < θ ≤ 180°), work is negative and the force removes energy from the system – for example, kinetic friction always does negative work. When θ = 90°, cos 90° = 0, so no work is done.

功的正负或零取决于角度θ。当力在位移方向上的分量与位移同向时（0° ≤ θ < 90°），功为正，力向系统输入能量。当力在位移方向上的分量与位移反向时（90° < θ ≤ 180°），功为负，力从系统取走能量——例如，动摩擦力总是做负功。当θ = 90°时，cos 90° = 0，因此不做功。

A common misconception is that carrying a heavy box horizontally at constant velocity does work. However, the upward normal force and the gravitational force are both perpendicular to the displacement, so the net work done by these forces is zero. Energy is consumed by the person’s muscles, but no mechanical work is done on the box in the physics sense.

一个常见误解是，水平匀速搬运重箱做了功。但实际上，向上的支持力和重力均与位移垂直，因此这些力做的净功为零。人体肌肉消耗了能量，但从物理学的机械功角度来看，并没有对箱子做功。

3. Kinetic Energy and the Work-Energy Theorem | 动能与动能定理

Kinetic energy (E_k) is the energy an object possesses due to its motion. It is given by the formula:

E_k = ½ m v²

其中 m 是物体的质量，v 是其速度。动能的单位同样是焦耳(J)。

The work-energy theorem states that the net work done on an object equals the change in its kinetic energy:

W_net = ΔE_k = E_kf – E_ki

This provides a powerful tool for solving problems without needing to know details of the acceleration or time. For example, to find the speed of a block sliding down a frictionless incline, one can simply equate the work done by gravity to the gain in kinetic energy.

动能定理指出，合力对物体做的功等于物体动能的变化量：W_net = ΔE_k = E_kf – E_ki。这提供了一个无需详细知道加速度或时间的强大解题工具。例如，求一个滑块沿光滑斜面下滑的速度时，可以直接将重力做的功等于动能的增加量。

4. Gravitational Potential Energy | 重力势能

Gravitational potential energy (E_p) is the energy stored in an object due to its position in a gravitational field. In a uniform field near Earth’s surface, the change in gravitational potential energy is:

ΔE_p = m g Δh

其中 Δh 是高度的变化，g 为重力场强度(约9.81 N/kg)。通常选取一个方便的参考水平面为零势能面，因为只有势能的变化量具有物理意义。

The work done against gravity is stored as potential energy. When an object is lifted at constant speed, the lifting force does positive work while gravity does equal negative work, so the net work is zero but the gravitational potential energy increases. In IB problems, it is crucial to define the reference level clearly when applying conservation laws.

克服重力所做的功以重力势能的形式储存起来。当物体匀速上升时，举力做正功，重力做等量负功，净功为零，但重力势能增加了。在IB问题中，应用守恒定律时必须明确定义参考水平面。

5. Elastic Potential Energy | 弹性势能

Elastic potential energy is the energy stored in an object when it is deformed elastically, such as a spring or rubber band. For an ideal spring obeying Hooke’s law (F = k x, where k is the spring constant and x is the extension or compression from equilibrium), the elastic potential energy is:

E_PE = ½ k x²

注意弹性势能总是非负的，且无论拉伸还是压缩，只要形变量相同，储存的能量相同。这可以通过力–位移图下的面积来推导，因为线性力的平均力为½ k x，位移为x。

In IB exams, you may be asked to find the energy stored in a spring that is stretched by a hanging mass, or to calculate the maximum speed of an oscillating system. Always check whether the spring is ideal and whether the limit of proportionality is exceeded. Energy conversions involving springs are often combined with gravitational potential energy to test conservation of mechanical energy.

在IB考试中，可能会要求计算悬挂重物拉伸弹簧所储存的能量，或求振荡系统的最大速度。务必检查弹簧是否为理想弹簧，是否超出比例极限。涉及弹簧的能量转换常与重力势能结合，以考查机械能守恒。

6. Conservation of Mechanical Energy | 机械能守恒

When only conservative forces (gravity, elastic spring force, electrostatic force) do work on a system, the total mechanical energy (E_m = E_k + E_p) remains constant. That is,

E_initial = E_final

or more explicitly, E_ki + E_pi = E_kf + E_pf. This principle simplifies many problems involving roller coasters, pendulums, and projectiles.

或者更明确地表示为 E_ki + E_pi = E_kf + E_pf。这个原理简化了许多涉及过山车、摆和抛体的问题。

However, the condition must be checked carefully. If non-conservative forces (e.g., friction, air resistance) are present, mechanical energy is no longer conserved – some of it is converted into thermal energy or sound. The IB often tests your ability to identify whether a system is isolated and whether only conservative forces act.

然而，必须仔细检验条件。如果存在非保守力（如摩擦力、空气阻力），机械能就不再守恒——部分机械能转化为内能或声能。IB经常考查学生辨别系统是否孤立、是否只有保守力做功的能力。

7. Non-Conservative Forces and Energy Dissipation | 非保守力与能量耗散

The work done by non-conservative forces, such as friction or drag, changes the total mechanical energy of a system. The relationship is given by:

W_nc = ΔE_m = (E_kf + E_pf) – (E_ki + E_pi)

其中 W_nc 是非保守力所做的总功。若摩擦做负功，系统的机械能减少，这部分“损失”的能量实际上转化为热能和周围环境的内能。能量本身从未被毁灭，只是从有用的机械形式转变为分散的形式。

This leads to the more general principle of conservation of energy: energy cannot be created or destroyed, only transferred or transformed. IB students should be prepared to discuss energy transformations using Sankey diagrams or bar charts, showing the flow of energy from input to useful output and wasted energy.

这导致了更普遍的能量守恒原理：能量不能被创造或销毁，只能被转移或转化。IB学生应能使用桑基图或能量棒图讨论能量转换，展示从输入能量到有用输出和浪费能量的流动过程。

8. Power | 功率

Power is defined as the rate of doing work or the rate of energy transfer. The average power is:

P = W / t or P = ΔE / t

For a force acting at constant velocity, the instantaneous power can also be expressed as P = F v cosθ, where v is the speed. The SI unit of power is the watt (W), where 1 W = 1 J/s.

对于恒速运动的受力，瞬时功率也可以表示为 P = F v cosθ，其中 v 是速率。功率的国际单位是瓦特(W)，1 W = 1 J/s。

A classic IB problem involves a car engine producing a constant power to overcome resistive forces. At maximum speed, the driving force equals the resistive force, and P = F v_max. Understanding the inverse relationship between force and velocity at constant power helps explain the gear system of vehicles.

一个经典的IB问题涉及汽车发动机以恒定功率克服阻力做功。当达到最大速度时，驱动力等于阻力，且 P = F v_max。理解恒定功率下力与速度的反比关系有助于解释车辆的齿轮系统。

9. Efficiency | 效率

Efficiency (η) is a measure of how much useful work or energy output is obtained from a process compared to the total energy input. It is usually expressed as a percentage:

η = (useful work output / total energy input) × 100% or η = (useful power output / total power input) × 100%

效率永远低于100%（如果不是理想过程），因为实际系统中总有一些能量耗散为热量。IB要求能将效率应用于机械、热机和发电站等情境。平时练习中，用水壶加热水或电动机提升重物是常见的效率计算试题。

When analyzing efficiency, always identify the energy destination that is considered ‘useful’. For an electric motor lifting a load, the useful output is the increase in gravitational potential energy. The input is the electrical energy supplied. The difference is lost mainly through heating in the windings and friction.

分析效率时，务必确定被视为“有用”的能量归宿。对于提升重物的电动机，有用输出是重力势能的增加量；输入是提供的电能；差值主要因绕组发热和摩擦而损失。

10. Common Problem-Solving Strategies | 典型问题分析

Many IB exam questions on work and energy require a systematic approach. First, define the system and identify all forces. For isolated systems with only conservative forces, apply conservation of mechanical energy. If friction or air drag is present, use the work-energy theorem including W_nc = ΔE_m.

许多IB功与能量考题需要系统性的解题步骤。首先，定义系统并识别所有力。对于只有保守力的孤立系统，应用机械能守恒；若存在摩擦或空气阻力，则使用包含 W_nc = ΔE_m 的动能定理。

For slope problems, resolve the weight into components parallel and perpendicular to the slope. The parallel component (mg sinθ) does work to increase kinetic energy, while the perpendicular component affects the normal force and thus the friction if present. In spring problems, combine the elastic potential energy ½ k x² with kinetic and gravitational terms, and carefully choose a reference for each.

处理斜面问题时，将重力分解为平行和垂直于斜面的分量。平行分量（mg sinθ）做功增加动能，垂直分量影响支持力进而影响摩擦力（若存在）。在弹簧问题中，将弹性势能 ½ k x² 与动能项和重力项结合，并谨慎选择各势能的参考点。

It is also important to verify units and to clearly state assumptions, such as g = 9.81 m/s², negligible air resistance, or ideal springs. For multi-step calculations, show the energy transformations step by step and indicate whether energy is being converted into thermal or other forms. The IB rewards clear reasoning and proper use of physics terminology.

验证单位、明确陈述假设（如 g = 9.81 m/s²、空气阻力忽略不计、理想弹簧）也至关重要。对于多步计算，应逐步展示能量转换过程，并指明能量是否转化为内能或其他形式。IB鼓励清晰的推理和正确使用物理术语。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Work and Energy in IB Physics: Key Concepts and Exam Tips | IB物理功与能量考点精讲