📚 Work Done and Energy Transfer | 功与能量转移
l ideas in physics: work done = force × distance moved in the direction of the force. At A‑ d the principle of conservation of energy. This article bridges the GCSE foundation with A‑Level depth, equipping you with the understanding needed for Mechanics and beyond.
在Edexcel联合科学大纲中，知识点1.40引入了物理学最基本的概念之一：功 = 力 × 沿力方向移动的距离。在A‑Level阶段，这一概念进一步拓展到变力、动能与势能转化以及能量守恒原理。本文从GCSE基础过渡到A‑Level深度，帮助你掌握力学及更高层次所需的理解。
1. Work Done by a Constant Force | 恒力做功
The work done W by a constant force F acting on an object is defined as the product of the force and the displacement s in the direction of the force. The scalar formula is W = F s cos θ rallel to the displacement (θ = 0°), cos 0 = 1, giving the familiar GCSE relation W = F s.
恒力F对物体所做的功W定义为力与沿力方向的位移s的乘积。标量公式为 W = F s cos θ，其中θ是力矢量与位移矢量之间的夹角。当力平行于位移时（θ = 0°），cos 0 = 1，就得到了GCSE中熟悉的W = F s。
he force does no work. This explains why the normal reaction force on a sliding block or the centripetal force on an object moving in a circle does zero work, despite acting over a distance.
如果力垂直于位移（θ = 90°），cos 90° = 0，则该力不做功。这就解释了为什么斜面上滑块所受的法向反作用力，或者物体做圆周运动时所受的向心力，虽然作用了一段距离，却做功为零。
2. Work Done as Energy Transfer | 做功即能量转移
away. The SI unit of work and energy is the joule (J), where 1 J = 1 N m. In mechanics, the terms ‘work done’ and ‘energy transferred’ are interchangeable.
功是能量转移的量度。对物体做功时，能量传递给物体；物体对外做功时，能量从物体传出。功和能的国际单位是焦耳 (J)，1 J = 1 N m。在力学中，“做功”与“能量转移”这两个术语可以互换使用。
The table below summarises common units and quantities that appear in A‑Level work–energy problems.
下表总结了A‑Level功能问题中常见的单位和物理量。
| Quantity | Symbol | Unit |
|---|---|---|
| Work / Energy / Heat | W, E, Q | joule (J) |
| Force | F | newton (N) |
| Distance / Displacement | s, d, x | metre (m) |
| Power | P | watt (W) |
3. Work Done on an Inclined Plane | 斜面上的功
g sin θ, where sin θ = h/L. The minimum pulling force needed is F = mg sin θ. The work done by this force is W = F × L = (mg sin θ) × L = mgh. This elegant result shows that the work done against gravity depends only on the vertical height gained, not on the slope length.
考虑一个重量为mg的物块沿光滑斜面匀速上移，斜面长度为L、竖直高度为h。重力沿斜面向下的分量为 mg sin θ，其中 sin θ = h/L。所需的最小拉力为 F = mg sin θ。这个力所做的功为 W = F × L = (mg sin θ) × L = mgh。这一简洁的结果表明，克服重力所做的功只取决于升高的竖直高度，与斜面的长度无关。
sipated as heat. The work done against friction is f × L, where f is the frictional force.
如果存在摩擦，总功会更大，因为还必须额外用力克服摩擦力，且部分能量以热的形式耗散。克服摩擦力所做的功为 f × L，其中f是摩擦力。
4. Work Done by a Variable Force | 变力做功
ulated simply by multiplying force by distance. Instead, you find the area under the force– ’s law, the force varies linearly with extension: F = k x, where k is the spring constant.
当力不恒定时，不能简单地用力乘以位移来计算功。此时应通过力–位移图下的面积来求解。对于遵循胡克定律的弹簧，力随伸长量线性变化：F = k x，其中k为劲度系数。
W = ½ k x²
al energy stored in a stretched or compressed spring. The area of the triangle under the F–x graph from 0 to x is indeed ½ × base × height = ½ × x × kx = ½ k x².
该表达式代表拉伸或压缩弹簧中储存的弹性势能。F–x 图中从0到x这一段下的三角形面积正是 ½ × 底 × 高 = ½ × x × kx = ½ k x²。
work done is the definite integral ∫ F(x) dx between the initial and final positions. At A‑ ou will either use the area method for linear graphs or be given the average force for non‑linear cases.
一般来说，对于任意变力F(x)，功等于初、末位置间 F(x) dx 的定积分∫。在A‑Level阶段，对于线性图像可以直接使用面积法，对于非线性情形通常会给出平均力。
5. Kinetic Energy and the Work–Energy Theorem | 动能与功能定理
The kinetic energy Eₖ of an object of mass m moving at speed v is given by:
质量为m、速度为v的物体的动能Eₖ由下式给出：
Eₖ = ½ m v²
The work–energy theorem states that the net work done by all forces acting on an object equals the change in its kinetic energy: Wnet = ΔEₖ. In equation form, for an object accelerating from speed u to v:
功能定理指出，作用在物体上的所有力所做的净功等于其动能的变化：Wnet = ΔEₖ。对于从速度u加速到v的物体，方程形式为：
Wnet = ½ m v² − ½ m u²
This theorem is extremely powerful in A‑ ere d is the stopping distance. This shows that stopping distance is proportional to the square of the initial speed.
该定理在A‑Level力学中非常实用。例如，在制动距离问题中，初始动能转化为克服摩擦力所做的功：½ m u² = F d，其中d为制动距离。这表明制动距离与初速度的平方成正比。
6. Gravitational Potential Energy | 重力势能
When an object is lifted near the Earth’s surface, the work done against gravity increases its gravitational potential energy Eₚ. For a vertical rise Δh:
当物体在地表附近被举高时，克服重力所做的功增加了它的重力势能Eₚ。对于竖直升高Δh：
ΔEₚ = m g Δh
where g is the gravitational field strength, taken as 9.81 N kg⁻¹ (or m s⁻²) in most A‑Level calculations. The zero level of potential energy is arbitrary; only changes in Eₚ are physically meaningful.
其中g为重力场强度，大多数A‑Level计算中取9.81 N kg⁻¹（或 m s⁻²）。势能的零水平面可以任意选取，只有势能的变化ΔEₚ才具有物理意义。
In an ideal pendulum or free‑fall situation, the loss in Eₚ equals the gain in Eₖ, assuming no air resistance. This is the foundation of energy conservation problems.
在理想单摆或自由落体情境中，假设没有空气阻力，重力势能的减少等于动能的增加。这是能量守恒问题的基础。
7. Conservation of Mechanical Energy | 机械能守恒
In a system where only conservative forces (such as gravity or ideal spring forces) do work, the total mechanical energy remains constant:
在只有保守力（如重力或理想弹力）做功的系统中，总机械能保持不变：
Etotal = Eₖ + Eₚ = constant
ass falling from rest at height h, the energy transformation is ½ m v² = m g h, giving v = √2 g h. This result is independent of the mass, a key demonstration of Galileo’s principle.
对于从高度h静止下落的物体，能量转化为 ½ m v² = m g h，得出 v = √2 g h。该结果与质量无关，是伽利略原理的重要体现。
When non‑ ction, W_friction, must be included in the energy balance: initial energy = final energy + W_friction.
当存在摩擦力等非保守力时，机械能不守恒，而是转化为热能。克服摩擦力所做的功W_friction必须纳入能量平衡：初始能量 = 末能量 + W_friction。
8. Power: Rate of Doing Work | 功率：做功的快慢
Power P is the rate at which work is done or energy is transferred:
功率P是做功或能量转移的速率：
P = W / t = ΔE / t
The unit of power is the watt (W), where 1 W = 1 J s⁻¹. In many A‑ constant force F propelling an object at constant velocity v, the power output is P = F v.
功率的单位是瓦特（W），1 W = 1 J s⁻¹。在许多A‑Level问题中，尤其是涉及车辆的问题，会用到恒力F推动物体以恒定速度v运动时的输出功率公式 P = F v。
ider the instantaneous power P = F v cos θ. This ties back to the definition of work done.
注意：如果速度不恒定，或者力以某一角度作用，必须考虑瞬时功率 P = F v cos θ。这又回归到功的定义。
9. Efficiency | 效率
No practical machine can convert all input energy into useful output energy. Efficiency is defined as:
任何实际机械都无法将全部输入能量转化为有用的输出能量。效率的定义为：
Efficiency = (useful output energy / total input energy) × 100%
expressed in terms of power: Efficiency = (useful output power / total input power) × 100%. In A‑ cy of a motor lifting a load or the overall efficiency of a power station from given data.
效率也可用功率表示：效率 = (有用输出功率 / 总输入功率) × 100%。在A‑Level考试中，你可能需要根据给定数据求电动机提升重物时的效率，或发电站的整体效率。
Heating effects due to friction, air resistance, and electrical resistance are the main reasons for energy loss. Always account for these when calculating the work that must be supplied to a real system.
摩擦、空气阻力和电阻引起的热效应是能量损失的主要原因。在计算实际系统必须提供的功时，一定要把这些因素考虑进去。
10. Practical Examples and Exam-Style Problems | 实例与考试题型
Let us examine two classic A‑Level scenarios that integrate work, energy, and power.
让我们研究两个经典的A‑Level情境，它们将功、能和功率融会贯通。
Car braking: A car of mass 120
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