Work and Energy Core Concepts | IB & CIE 物理功与能量考点精讲

📚 Work and Energy Core Concepts | IB & CIE 物理功与能量考点精讲

Work and energy are fundamental concepts in both IB and CIE Physics, forming the backbone of mechanics and problem-solving. Understanding the relationship between force, displacement, and energy transfer is crucial for mastering topics ranging from simple machines to particle interactions. This revision guide covers key definitions, equations, graphical analysis, and common pitfalls to help you excel in your exams.

功与能量是IB与CIE物理中的基础概念，构成了力学和问题解决的核心。理解力、位移和能量转移之间的关系对于掌握从简单机械到粒子相互作用的各种主题至关重要。本考点精讲涵盖了关键定义、方程、图像分析以及常见误区，帮助你在考试中脱颖而出。

1. Definition of Work | 功的定义

In physics, work is done when a force causes a displacement of an object in the direction of the force. Work is a scalar quantity, even though force and displacement are vectors. The SI unit of work is the joule (J), where 1 J = 1 N m.

在物理学中，当力使物体沿力的方向发生位移时，就做了功。功是标量，尽管力和位移都是矢量。功的国际单位是焦耳 (J)，1 J = 1 N m。

If a constant force F acts on an object that moves a distance d in a direction making an angle θ with the force, the work done W is given by:

如果一个恒力 F 作用在物体上，物体沿与力成 θ 角的方向移动了距离 d，则所做的功 W 为：

W = F d cosθ

When θ = 0°, cos0° = 1, so maximum positive work is done. When θ = 90°, cos90° = 0, no work is done. When θ = 180°, cos180° = -1, work is negative; energy is taken from the object.

当 θ = 0° 时，cos0° = 1，做最大正功。当 θ = 90° 时，cos90° = 0，不做功。当 θ = 180° 时，cos180° = -1，做功为负；能量从物体中移出。

2. Work Done by a Constant Force | 恒力做功的计算

For a constant force parallel to the displacement, work simplifies to W = F d. This occurs, for example, when lifting a mass vertically or pushing a box along a frictionless surface. Always ensure you use the component of the force in the direction of motion.

对于与位移平行的恒力，功简化为 W = F d。例如，竖直提升重物或在光滑表面上推动箱子。务必使用沿运动方向的力的分量。

Consider a block being pulled by a rope at an angle of 30° above the horizontal. If the tension is 50 N and the block moves 4 m, the work done is W = (50 N)(4 m) cos30° = 200 × 0.866 = 173.2 J. Only the horizontal component contributes.

考虑一个被与水平成 30° 角的绳子拉动的物块。若张力为 50 N，物块移动 4 m，则做功 W = (50 N)(4 m) cos30° = 200 × 0.866 = 173.2 J。只有水平分量做功。

Negative work means the force opposes the displacement. Friction acting opposite to motion always does negative work, converting mechanical energy into thermal energy.

负功意味着力与位移方向相反。与运动方向相反的摩擦力总是做负功，将机械能转化为内能。

3. Work as the Area Under a Force-Displacement Graph | 力-位移图像下的面积

When a force varies with position, the work done is found by calculating the area under the force vs. displacement graph. For a spring obeying Hooke’s law, F = k x, the graph is a straight line through the origin. The work done in stretching the spring from 0 to extension x is the triangular area:

当力随位置变化时，通过计算力-位移图像下的面积来求功。对于遵循胡克定律 F = k x 的弹簧，图像是通过原点的直线。将弹簧从 0 拉伸至伸长量 x 所做功为三角形面积：

W = ½ k x²

This is equal to the elastic potential energy stored. For arbitrary varying forces, you can estimate work by counting squares under a graph or, if the force function is known, using integration: W = ∫ F dx. Both IB and CIE syllabi expect you to interpret such graphs.

这等于储存的弹性势能。对于任意变力，可以通过图像下的方格计数估算功，或者如果力函数已知，使用积分 W = ∫ F dx。IB和CIE考纲都要求能解读此类图像。

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

Kinetic energy (Eₖ) is the energy an object possesses due to its motion. It is defined by:

动能 (Eₖ) 是物体因运动而具有的能量。其定义为：

Eₖ = ½ m v²

where m is mass (kg) and v is speed (m/s). The work-energy theorem states that the net work done on an object equals its change in kinetic energy:

其中 m 为质量 (kg)，v 为速率 (m/s)。动能定理指出，物体所受合力做的净功等于其动能的变化：

W_net = ΔEₖ = Eₖ_final – Eₖ_initial

This theorem is extremely powerful. For example, if a 2 kg block accelerates from 3 m/s to 7 m/s, the net work done is ½×2×(7² – 3²) = 1 × (49 – 9) = 40 J. You don’t need to know the individual forces if you can compute the change in speed.

该定理非常强大。例如，一个 2 kg 的物块从 3 m/s 加速到 7 m/s，净功为 ½×2×(7² – 3²) = 1 × (49 – 9) = 40 J。如果能够计算速度变化，就不需要知道每个力。

5. Gravitational Potential Energy | 重力势能

Gravitational potential energy (Eₚ) 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 by a vertical height Δh is:

重力势能 (Eₚ) 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，质量为 m 的物体被垂直提升高度 Δh 时，重力势能的变化为：

ΔEₚ = m g Δh

The zero of potential energy can be chosen arbitrarily. Often the ground or the lowest point in a problem is taken as the reference level. In more advanced IB problems, you may use the universal formula Eₚ = -G M m / r, but near the surface, m g Δh is sufficient and is common in both IB and CIE exams.

势能的零点可以任意选择。通常将地面或问题中的最低点选为参考水平面。在一些更深入的IB题目中，可能会用到万有引力势能公式 Eₚ = -G M m / r，但在表面附近用 m g Δh 就足够了，在IB和CIE考试中都很常见。

6. Elastic Potential Energy | 弹性势能

Elastic potential energy is stored when a spring or any elastic object is deformed. For an ideal spring obeying Hooke’s law F = k x, the energy stored when stretched or compressed by displacement x from its natural length is:

弹性势能储存在弹簧或任何弹性物体发生形变时。对于遵循胡克定律 F = k x 的理想弹簧，从原长拉伸或压缩位移 x 时储存的能量为：

Eₑ = ½ k x²

This is the same as the work done to deform the spring. The spring constant k measures stiffness (N/m). The formula applies equally to compression and extension. In problems, you must ensure x is measured from the equilibrium position.

这与使弹簧形变所做的功相同。劲度系数 k 衡量弹簧的刚度 (N/m)。该公式对压缩和拉伸同样适用。解题时务必确保 x 是从平衡位置测量的位移。

7. Principle of Conservation of Energy | 能量守恒原理

The total energy of an isolated system remains constant. Energy can be transformed from one form to another, but it cannot be created or destroyed. In mechanical systems without friction or air resistance, the sum of kinetic and potential energies stays constant.

孤立系统的总能量保持不变。能量可以从一种形式转化为另一种形式，但不会凭空产生或消失。在没有摩擦或空气阻力的机械系统中，动能和势能的总和保持恒定。

For a freely falling object, as it loses height, gravitational potential energy converts into kinetic energy: ½ m v² + m g h = constant. In a pendulum, energy continuously oscillates between kinetic and gravitational potential. In real systems, some energy is dissipated as heat due to resistive forces, so total mechanical energy decreases, but total energy including thermal energy is still conserved.

对于自由落体，随着高度降低，重力势能转化为动能：½ m v² + m g h = 常数。在单摆中，能量在动能和重力势能之间不断转换。在真实系统中，由于阻力一些能量以热的形式耗散，因此机械能减少，但包括内能在内的总能量依然守恒。

8. Power and Efficiency | 功率与效率

Power is the rate at which work is done or energy is transferred. The SI unit is the watt (W), where 1 W = 1 J/s. Average power P = W / t, and instantaneous power can be expressed as:

功率是做功或能量传递的速率。国际单位是瓦特 (W)，1 W = 1 J/s。平均功率 P = W / t，瞬时功率可表示为：

P = F v cosθ

where v is the instantaneous velocity. When force and velocity are in the same direction, this simplifies to P = F v. This is particularly useful when determining the power output of an engine moving at constant speed against a resistive force.

其中 v 是瞬时速度。当力与速度同向时，可简化为 P = F v。这在确定以恒定速度抵抗阻力运动的发动机输出功率时特别有用。

Efficiency η (eta) of a device is the ratio of useful output power (or energy) to total input power (or energy):

设备的效率 η 是有用输出功率（或能量）与总输入功率（或能量）之比：

η = (useful output / total input) × 100%

No real machine is 100% efficient because of losses such as friction, sound, and heat. Exam questions often ask you to calculate the energy wasted.

任何真实机器的效率都不可能达到 100%，因为存在摩擦、声音和热等损失。考题常要求你计算浪费的能量。

9. Energy in Collisions | 碰撞中的能量

In collisions, momentum is always conserved, but kinetic energy is not always conserved. There are three main types:

在碰撞中，动量总是守恒的，但动能未必守恒。主要有三种类型：

Type	Momentum	Kinetic Energy	Example
Elastic	Conserved	Conserved	Ideal gas molecule collisions
Inelastic	Conserved	Not conserved	Most everyday collisions
Perfectly inelastic	Conserved	Not conserved; bodies stick together	Bullet embedding in a block

In an elastic collision, both momentum and kinetic energy are conserved. For two objects of masses m₁ and m₂, you can use the equations for final velocities derived from these conservation laws. In IB and CIE, you may be required to determine whether a collision is elastic by comparing total kinetic energy before and after.

在弹性碰撞中，动量和动能都守恒。对于质量为 m₁ 和 m₂ 的两个物体，可利用守恒定律推导末速度方程。IB和CIE可能要求你通过比较碰撞前后的总动能来判断碰撞是否为弹性碰撞。

10. Key Graphs and Problem-Solving | 关键图像与解题策略

Force–displacement graphs: area represents work done. For a spring, the area is a triangle; for friction, it could be a rectangle. Always check whether the force is constant or varying.

力–位移图像：面积代表所做的功。对于弹簧，面积为三角形；对于摩擦力，可能是矩形。务必检查力是恒力还是变力。

Kinetic energy versus displacement graphs can be used to infer net force. The gradient of an Eₖ vs. x graph gives the net force acting on the object (since W = F d and W = ΔEₖ). Power–time graphs show total energy transferred as the area under the curve.

动能–位移图像可用于推断合力。Eₖ-x 图像的斜率表示作用在物体上的合力（因为 W = F d 且 W = ΔEₖ）。功率–时间图像中曲线下的面积表示转移的总能量。

When solving problems, use the following systematic approach: (1) Identify the system and energy forms present. (2) Draw a free-body diagram if forces are involved. (3) Apply the work-energy theorem or conservation of energy. (4) For collisions, check momentum first, then check kinetic energy to classify collision type.

解题时，采用以下系统方法：(1) 确定系统及存在的能量形式。(2) 如果涉及力，画出受力示意图。(3) 应用动能定理或能量守恒定律。(4) 对于碰撞，先检查动量，再检查动能以分类碰撞类型。

11. Common Errors and Misconceptions | 常见错误与误区

Confusing work with torque: Both use unit N m, but work is scalar and torque is vector. Never say work is done when holding a heavy object stationary — there is no displacement, so W = 0.

混淆功与力矩：两者单位都用 N m，但功是标量，力矩是矢量。手持重物静止不动时并没有做功——因为没有位移，所以 W = 0。

Forgetting the cosθ factor: Many students use W = F d regardless of angle. Always resolve the force component parallel to motion. Similarly, in power P = F v, ensure F is the component in the direction of velocity.

忘记 cosθ 因子：许多学生不论角度如何都使用 W = F d。始终要将力沿运动方向分解。同样，在 P = F v 中，确保 F 是速度方向上的分量。

Misapplying energy conservation: In a non-isolated system, mechanical energy may decrease, but total energy including heat, sound, etc., is conserved. Do not blindly presume kinetic + potential energy is constant if friction is present.

误用能量守恒：在非孤立系统中，机械能可能减少，但包括热、声音等在内的总能量是守恒的。如果存在摩擦，不要盲目假定动能与势能之和保持不变。

Using wrong reference level for gravitational potential: Only changes in potential energy matter, but if you are using Eₚ = mgh, be consistent with the zero height. In spring problems, ensure x is measured from the spring’s unstretched length.

重力势能参考水平选错：只有势能的变化才有意义，但如果使用 Eₚ = mgh，要保持零高度一致。在弹簧问题中，确保 x 是从弹簧原长测量的位移。

12. Summary Checklist | 总结检查表

Before the exam, ensure you can do the following:

考试前，确保你能做到以下几点：

Define work and state W = F d cosθ; identify conditions for positive, negative, and zero work.

定义功并写出 W = F d cosθ；判断正功、负功和零功的条件。
Calculate work from a force–displacement graph by finding area.

通过求面积计算力–位移图像中的功。
State and apply the work-energy theorem: W_net = ΔEₖ.

陈述并应用动能定理：W_net = ΔEₖ。
Compute kinetic energy Eₖ = ½ m v² and potential energies (gravitational and elastic).

计算动能 Eₖ = ½ m v² 以及势能（重力和弹性）。
Apply conservation of energy to problems involving motion, springs, and collisions.

运用能量守恒解决涉及运动、弹簧和碰撞的问题。
Calculate power as P = W/t or P = F v, and efficiency η.

用 P = W/t 或 P = F v 计算功率，并计算效率 η。
Distinguish between elastic, inelastic, and perfectly inelastic collisions using kinetic energy.

利用动能区分弹性、非弹性和完全非弹性碰撞。

Master these concepts, and you will have a solid foundation for tackling energy-related questions in both IB and CIE Physics examinations.

掌握这些概念，你将拥有扎实的基础，从容应对IB和CIE物理考试中与能量相关的题目。

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Work and Energy Core Concepts | IB & CIE 物理功与能量考点精讲