Work and Energy in IB and OCR Physics: Key Points Explained | IB OCR 物理：功与能量考点精讲

📚 Work and Energy in IB and OCR Physics: Key Points Explained | IB OCR 物理：功与能量考点精讲

Work and energy form the backbone of classical mechanics, linking forces to the motion they produce. In both IB and OCR A Level Physics, mastering these concepts is essential for solving problems ranging from simple inclined planes to complex energy transformations. This article breaks down the key ideas, equations, and common pitfalls you need to know for your examinations.

功与能量是经典力学的基石，将力与其产生的运动联系起来。在 IB 和 OCR A Level 物理中，掌握这些概念对于解决从简单斜面到复杂能量转换的各种问题至关重要。本文详细讲解了您考试所需的关键概念、方程和常见误区。

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. It is a scalar quantity measured in joules (J). The work done by a constant force F acting at an angle θ to the displacement s is given by W = F s cos θ. If the force and displacement are parallel, cos θ = 1 and work is simply F s.

在物理学中，当力使物体沿力的方向发生位移时，就说力做了功。功是标量，单位为焦耳（J）。恒力 F 与位移 s 夹角为 θ 时所做的功为 W = F s cos θ。如果力与位移平行，cos θ = 1，功简化为 F s。

W = F s cos θ

When θ = 90°, cos 90° = 0, so no work is done. For example, the centripetal force acting on an object in uniform circular motion does zero work because the force is always perpendicular to the instantaneous velocity (and thus to the displacement).

当 θ = 90° 时，cos 90° = 0，不做功。例如，作匀速圆周运动的物体所受的向心力不做功，因为力始终垂直于瞬时速度（从而垂直于位移）。

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

Work can be positive, negative, or zero. A force does positive work when it has a component in the direction of motion (e.g., a pushing force that speeds up a cart). Negative work is done when the force opposes motion (e.g., friction or a braking force). The sign is determined by the cosine of the angle between F and s.

功可以是正功、负功或零。当力在运动方向有分量时（例如推车使其加速的力），力做正功。当力阻碍运动时（例如摩擦力或制动力），力做负功。正负由 F 与 s 夹角的余弦决定。

To compute the total work done by several forces, you can either find the net force and then calculate the work it does, or calculate the work done by each force separately and sum them algebraically. Both approaches give the same result.

要计算多个力所做的总功，可以先求合力再计算合力做的功，也可以分别计算每个力做的功再代数相加。两种方法结果相同。

3. 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 is a direct consequence of Newton’s second law and kinematic equations, and it holds even when forces are not constant, as long as we consider the net work.

功-能定理指出，物体所受合外力做的净功等于其动能的变化量：W_net = ΔK。这是牛顿第二定律和运动学方程的直接推论，即使力不是恒力，只要考虑净功，该定理依然成立。

W_net = ½ m v₂² – ½ m v₁²

This theorem is extremely useful for solving problems where forces vary or where only speeds and not forces are given. For instance, when a car accelerates on a horizontal road, the net work of the driving force and resistive forces equals the increase in kinetic energy.

该定理在力变化或只知速度未知力的问题中极其有用。例如，汽车在水平路面上加速时，驱动力与阻力的净功等于动能的增加量。

4. Kinetic Energy | 动能

Kinetic energy (K or KE) is the energy an object possesses due to its motion. For an object of mass m moving at speed v, it is defined as K = ½ m v². It is always positive or zero, and its SI unit is the joule (J).

动能（K 或 KE）是物体由于运动而具有的能量。质量为 m、速度为 v 的物体的动能定义为 K = ½ m v²。动能总是非负的，SI 单位是焦耳（J）。

K = ½ m v²

Note that kinetic energy depends on the square of speed, so doubling the speed quadruples the energy. This is why high-speed collisions are far more destructive and why braking distances increase dramatically with speed.

请注意动能与速度的平方成正比，因此速度加倍时能量变为四倍。这就是高速碰撞破坏性远更严重、制动距离随速度急剧增加的原因。

5. Gravitational Potential Energy | 重力势能

Gravitational potential energy (GPE) is the energy an object stores due to its position in a gravitational field. Near Earth’s surface, where g is approximately constant, the change in GPE when an object of mass m is raised by a vertical height Δh is ΔU_g = m g Δh.

重力势能（GPE）是物体因在引力场中的位置而储存的能量。在地面附近 g 近似恒定的情况下，质量为 m 的物体竖直升高 Δh 时，重力势能的变化量为 ΔU_g = m g Δh。

ΔU_g = m g Δh

The work done by gravity is the negative of the change in GPE: W_gravity = –ΔU_g. When an object falls, gravity does positive work and GPE decreases. The zero of GPE can be chosen arbitrarily; only changes in GPE are physically meaningful.

重力做的功等于重力势能变化量的负值：W_gravity = –ΔU_g。物体下落时，重力做正功，重力势能减少。重力势能的零点可以任意选择；只有重力势能的变化才有物理意义。

6. Elastic Potential Energy | 弹性势能

Elastic potential energy is stored in deformed elastic objects, such as springs. For a spring obeying Hooke’s law (F = –k x), the elastic potential energy when stretched or compressed by an amount x from its natural length is U_s = ½ k x². Here k is the spring constant and x is the extension or compression.

弹性势能储存在发生弹性形变的物体中，例如弹簧。对于遵从胡克定律（F = –k x）的弹簧，当它从原长被拉伸或压缩了 x 时，弹性势能为 U_s = ½ k x²。其中 k 是劲度系数，x 是形变量。

U_s = ½ k x²

The work done by the spring force is also equal to –ΔU_s. In many problems, you will consider the conversion between elastic potential energy, kinetic energy and gravitational potential energy, for example in a mass-spring system or a bungee jump.

弹簧弹力做的功也等于 –ΔU_s。在许多问题中，需要考虑弹性势能、动能和重力势能之间的转换，例如在弹簧振子或蹦极跳中。

7. Conservation of Mechanical Energy | 机械能守恒

When only conservative forces (such as gravity and spring force) do work on a system, the total mechanical energy E = K + U remains constant. This principle allows you to relate speeds and positions without directly using forces and accelerations.

当只有保守力（如重力和弹簧弹力）对系统做功时，总机械能 E = K + U 保持不变。这一原理使您可以在不直接使用力和加速度的情况下建立速度与位置的关系。

K₁ + U₁ = K₂ + U₂

However, if non-conservative forces like friction or air resistance are present, mechanical energy is not conserved. Some energy is transformed into internal energy (heat, sound), but total energy of an isolated system always remains conserved.

然而，如果存在摩擦力或空气阻力等非保守力，机械能就不守恒。部分能量转化为内能（热、声），但孤立系统的总能量始终守恒。

8. Power | 功率

Power is the rate at which work is done or energy is transferred. The average power P_av = W / Δt, and the instantaneous power can be expressed as P = F v cos θ, where F is the applied force and v is the velocity of the object at that instant.

功率是做功或能量传递的速率。平均功率 P_av = W / Δt，瞬时功率可表示为 P = F v cos θ，其中 F 是作用力，v 是物体在该时刻的速度。

P = W / Δt and P = F v cos θ

The SI unit of power is the watt (W), where 1 W = 1 J s⁻¹. In problems involving engines moving at constant speed, the power required to overcome resistive forces is often given by P = F v. The same equation helps explain why vehicles must downshift to exert larger force at low speeds.

功率的 SI 单位是瓦特（W），1 W = 1 J s⁻¹。在涉及发动机恒速运动的问题中，克服阻力所需的功率常由 P = F v 给出。该方程也有助于解释为什么车辆在低速时需降挡以提供更大的力。

9. Work Done by Non-Conservative Forces | 非保守力做功

Non-conservative forces, such as friction, air resistance and applied forces from muscles or engines, change the mechanical energy of a system. The work done by these forces equals the change in mechanical energy: W_nc = ΔE_mech = (K₂ + U₂) – (K₁ + U₁).

摩擦力、空气阻力以及肌肉或发动机的施加力等非保守力会改变系统的机械能。这些力做的功等于机械能的变化量：W_nc = ΔE_mech = (K₂ + U₂) – (K₁ + U₁)。

W_nc = ΔK + ΔU

When friction is present, W_nc is negative, indicating that mechanical energy dissipates as thermal energy. In contrast, an engine doing positive non-conservative work can increase mechanical energy, for instance when a lift raises a load or a car accelerates up a hill.

存在摩擦时，W_nc 为负，表明机械能耗散为热能。相反，发动机做正的非保守功可以增加机械能，例如电梯提升重物或汽车加速上坡时。

10. Efficiency | 效率

Efficiency measures how well a device converts energy input into useful energy output. It is defined as the ratio of useful output work (or power) to total input energy (or power): η = W_out / E_in. Efficiency is often expressed as a percentage.

效率衡量设备将输入能量转化为有用输出能量的能力。它定义为有用输出功（或功率）与总输入能量（或功率）之比：η = W_out / E_in。效率通常以百分比表示。

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

Real machines always have an efficiency less than 100% because of unavoidable energy losses, mainly due to friction and heat. Calculations of efficiency often appear in problems involving motors, pulleys and heat engines, linking physics to engineering contexts.

真实机器的效率始终低于 100%，因为不可避免地存在能量损失，主要源自摩擦和发热。效率计算经常出现在涉及电机、滑轮和热机的题目中，将物理学与工程实际联系起来。

11. Common Misconceptions and Exam Tips | 常见误区与应考建议

Many students confuse force with energy or think that a moving object must have a force acting on it. In reality, an object moving at constant velocity has no net force and its kinetic energy remains constant, but forces may still be doing zero net work.

许多学生混淆了力与能量，或认为运动的物体一定有其上作用的力。实际上，匀速运动的物体所受合外力为零，其动能保持不变，但各力仍可能做净功为零。

Always identify the system and all forces acting on it. Distinguish between conservative and non-conservative forces. When using energy conservation, check whether non-conservative forces are present; if they are, account for their work. In OCR exams, you may be asked to describe energy changes in qualitative terms, while IB often requires detailed quantitative calculations including diagrams.

务必明确所研究的系统及其所受全部力。区分保守力和非保守力。使用能量守恒时，检查是否有非保守力存在；若有，则需考虑它们做的功。OCR 考试可能会要求定性描述能量变化，而 IB 则常要求进行详细的定量计算并画图。

A crucial skill is working with the area under a force–distance graph, which gives the work done. For a varying force, the work equals the area under the F–s curve. This is directly testable in both syllabi.

一个关键技能是根据力-位移图下方的面积求功。对于变力，功等于 F–s 曲线下的面积。这在两个考纲中都是直接考查点。

Always include units: work and energy in joules (J), power in watts (W).

总是带上单位：功和能量的单位是焦耳（J），功率的单位是瓦特（W）。
When dealing with inclined planes, use correct components of weight and displacement.

处理斜面问题时，使用正确的重力分量和位移分量。
For spring problems, the energy stored depends on extension squared, not on the direction of extension.

在弹簧问题中，储存的能量取决于形变量的平方，与形变方向无关。

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Work and Energy in IB and OCR Physics: Key Points Explained | IB OCR 物理：功与能量 考点精讲