Work, Energy and Power | 功和能量 考点精讲

📚 Work, Energy and Power | 功和能量 考点精讲

In AQA A-Level Mathematics, the topic 'Work, Energy and Power' forms a bridge between forces and motion, offering a powerful set of tools for solving mechanics problems without directly resolving differential equations of motion. By understanding how work transfers energy and applying the principle of conservation of energy, you can model a wide range of physical situations, from sliding blocks to moving vehicles. Mastery of these concepts is essential for success in both the pure and applied sections of the examination, and it underpins many real-world engineering applications.

在AQA A-Level数学中，“功和能量”专题是力与运动之间的桥梁，为解决力学问题提供了一套无需直接求解运动微分方程的强大工具。通过理解功如何转移能量并应用能量守恒原理，你可以模拟从滑块运动到车辆行驶的各种物理情景。掌握这些概念对于在纯数学和应用数学部分取得成功至关重要，同时也是许多实际工程应用的基础。

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1. Definition of Work | 功的定义

ct through a displacement s is given by W = F s cosθ, where θ is the angle between the force vector and the displacement vector. Work is a scalar quantity, measured in joules (J); 1 J equals 1 newton metre (N m). Note that only the component of force parallel to displacement contributes to work.

在力学中，当力使其作用点沿力的方向移动时，力就做了功。恒力F使物体发生位移s所做的功W由公式W = F s cosθ给出，其中θ是力矢量与位移矢量之间的夹角。功是标量，单位为焦耳（J），1 J等于1牛顿米（N m）。注意，只有平行于位移的力分量才做功。

ransferred to the object. When the force opposes motion (90° < θ ≤ 180°), work is negative, as with friction. If the force is perpendicular to displacement (θ = 90°), no work is done — example, the normal reaction on an object sliding on a horizontal surface does zero work.

功可以是正值、负值或零。当力在位移方向上有分量（0° ≤ θ < 90°）时，功为正，能量传递给物体。当力阻碍运动（90° < θ ≤ 180°）时，功为负，例如摩擦力做负功。如果力垂直于位移（θ = 90°），则不做功——例如，在水平面上滑动的物体所受的法向反作用力做功为零。

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

ock being pulled along a rough horizontal surface by a force applied at an angle α to the horizontal. Here, the horizontal component F cosα does work over displacement s, while the vertical component affects the normal reaction and therefore friction, but does no work horizontally. Exam questions often require you to resolve forces and compute net work against resistance.

对于作用在沿直线运动的质点上的恒力，可以直接使用W = F s cosθ计算功。一个常见的场景是：一个物块被与水平方向成α角的力拉着沿粗糙水平面运动。此时，水平分量F cosα在位移s上做功，而垂直分量会影响法向反作用力从而影响摩擦力，但在水平方向上不做功。考试题目经常要求你分解力，并计算克服阻力所做的净功。

he weight mg, the work done to raise the object through a height h is W = mgh. This result is independent of the path taken as long as the motion is vertical, a fact that links directly to gravitational potential energy.

另一个典型例子是匀速竖直提升物体时克服重力所做的功。由于所施加的力等于重力mg，将物体提升高度h所做的功为W = mgh。只要运动是竖直的，这个结果与路径无关，这一事实直接与重力势能相关联。

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

he area under the force-displacement graph. Mathematically, if a force F(x) acts along the x-axis from x = a to x = b, the work done is the definite integral W = ∫ₐᵇ F(x) dx. In AQA examination contexts, this often involves a spring obeying Hooke's law or a force given as a function of position.

当作用在质点上的力随位移变化时，不能简单地用W = F s计算功。此时，功由力-位移图下的面积给出。从数学上看，如果一个力F(x)沿x轴从x = a作用到x = b，所做的功为定积分W = ∫ₐᵇ F(x) dx。在AQA考试中，这通常涉及遵循胡克定律的弹簧或作为位置函数给出的力。

xtension X is the area of a triangle under the F-x graph: W = ½ k X². Integration yields the same result: ∫₀ˣ k x dx = ½ k X². This work is stored as elastic potential energy.

例如，弹簧施加的力为F = k x，其中k为劲度系数，x为伸长量。将弹簧从自然长度拉伸到伸长量X所做的功是F-x图像下三角形的面积：W = ½ k X²。积分给出相同的结果：∫₀ˣ k x dx = ½ k X²。这部分功以弹性势能的形式储存起来。

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

The kinetic energy (KE) of a particle of mass m moving with speed v is given by:

KE = ½ m v²

Kinetic energy is also measured in joules and is always non-negative. The work-energy principle states that the net work done on a particle by all forces acting on it is equal to the change in its kinetic energy: W_net = ΔKE = KE_final - KE_initial.

质量为m、速度为v的质点的动能（KE）由下式给出：

KE = ½ m v²

动能也以焦耳为单位，并且始终为非负值。动能定理指出，作用在质点上的所有力所做的净功等于其动能的变化量：W_net = ΔKE = KE_final - KE_initial。

This principle is extremely useful when forces vary or when direct application of Newton's second law involves complex integration. For example, a car accelerating on a horizontal road: the engine provides a driving force, while air resistance and friction do negative work. The net work equals the gain in kinetic energy from u to v. Always remember to include all forces acting on the particle when calculating net work.

该定理在力发生变化或直接应用牛顿第二定律需要复杂积分时极为有用。例如，一辆汽车在水平路面上加速：发动机提供驱动力，而空气阻力和摩擦力做负功。净功等于动能从u到v的增加量。在计算净功时，务必记住将所有作用在质点上的力都考虑进去。

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5. Gravitational Potential Energy | 重力势能

Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field. Near the Earth's surface, where the gravitational field is uniform, the change in GPE when a mass m is raised through a vertical height Δh is:

ΔGPE = m g Δh

significance. In many problems, you can set the lowest point of the motion as the reference level where GPE = 0.

重力势能（GPE）是物体因其在引力场中的位置而具有的能量。在地球表面附近，引力场是均匀的，当质量为m的物体被竖直提升高度Δh时，重力势能的变化量为：

ΔGPE = m g Δh

其中g为重力加速度（9.8 m s⁻²）。势能的零点可以任意选定；只有GPE的差值才具有物理意义。在许多问题中，你可以将运动的最低点设为参考水平面，令该处GPE = 0。

When a particle moves along a smooth slope, the change in GPE depends only on the vertical height descended, not on the length of the slope. This makes energy methods particularly convenient for inclined plane problems: the loss in GPE is m g h, where h is the vertical drop, and this loss converts into kinetic energy if no other forces do work.

当质点沿光滑斜面运动时，GPE的变化只取决于下降的竖直高度，而与斜面长度无关。这使得能量方法在斜面问题中特别方便：GPE的减少量为m g h，其中h为竖直落差，如果没有其他力做功，这部分减少量会转化为动能。

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6. Elastic Potential Energy | 弹性势能

string is deformed, it stores elastic potential energy (EPE). For a material obeying Hooke' onstant. The EPE stored is the work done in producing the deformation from its natural length:

EPE = ½ k x²

This formula applies whether the spring is stretched or compressed, as the force and displacement are always in opposite directions, but the work done on the spring is positive. The EPE is always non-negative and is measured in joules.

当弹簧或弹性绳发生形变时，会储存弹性势能（EPE）。对于遵循胡克定律的材料，张力或推力T与伸长量或压缩量x成正比：T = k x，其中k为劲度系数。储存的EPE等于使弹簧从其自然长度形变过程中所做的功：

EPE = ½ k x²

无论弹簧是拉伸还是压缩，该公式均适用，因为力和位移总是反向，但对弹簧做的功为正。EPE始终为非负值，单位为焦耳。

In AQA mechanics problems, you often encounter a particle attached to a spring on a smooth surface, or a falling mass compressing a spring. At maximum compression or extension, the particle's kinetic energy is zero, and the initial energy (GPE or KE) has all been converted to EPE, provided there are no energy losses. Setting up an energy conservation equation with these terms is a standard solution method.

在AQA力学问题中，你经常会遇到质点系着弹簧在光滑面上运动，或者下落物体压缩弹簧的情境。在最大压缩量或伸长量处，质点的动能为零，如果无能量损失，初始能量（GPE或KE）已全部转化为EPE。建立包含这些项的能量守恒方程是一种标准的求解方法。

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7. Conservation of Mechanical Energy | 机械能守恒

rticle, the sum of its kinetic energy and potential energies remains constant: KE + GPE + EPE = constant. This provides a powerful shortcut for analysing motion between two points without calculating accelerations or time.

机械能守恒原理指出，如果只有保守力（如重力和理想弹簧力）对质点做功，那么其动能与势能之和保持恒定：KE + GPE + EPE = 常数。这为分析两点之间的运动提供了强大的捷径，无需计算加速度或时间。

ation equating the total mechanical energy. For example, a pendulum bob swinging from its highest point (all GPE) to its lowest point (all KE) satisfies m g h = ½ m v², from which the speed can be found. Always check that no non-conservative forces, such as friction or air resistance, are doing work; if they are present, the mechanical energy is not conserved and the work-energy principle with the inclusion of work done by these forces must be used.

应用此原理时，确定初态和末态，为重力势能选取方便的零点，并写出总机械能相等的方程。例如，一个摆锤从最高点（全为GPE）摆动到最低点（全为KE）满足m g h = ½ m v²，由此可求速度。始终检查是否有非保守力（如摩擦力或空气阻力）做功；如果存在这些力，则机械能不守恒，必须使用包含这些力做功的动能定理。

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

the path taken and cannot be expressed as a change in a simple potential function. When such forces act on a particle, mechanical energy is dissipated, usually as heat or sound. In these cases, the work-energy principle becomes: W_conservative + W_non-conservative = ΔKE.

摩擦力、空气阻力以及非弹性绳中的张力都是非保守力的例子。这些力所做的功取决于所经过的路径，无法表示为简单势能函数的变化。当这些力作用在质点上时，机械能被耗散，通常转化为热能或声能。在这些情况下，动能定理变为：W_conservative + W_non-conservative = ΔKE。

A common task is to calculate the work done against a constant friction force F_r acting over a distance d: work done by friction = -F_r rvative forces = final energy. For instance, a block sliding down a rough slope: m g h - F_r s = ½ m v², where s is the distance along the slope.

一个常见的任务是计算克服在距离d上作用的恒定摩擦力Fr所做的功：摩擦力做的功 = -F_r d。这个负功减少了总机械能。建立形如：初始能量 + 非保守力做的功 = 最终能量 的方程。例如，一个物块沿粗糙斜面滑下：m g h - F_r s = ½ m v²，其中s为沿斜面的距离。

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9. Power | 功率

Power is the rate at which work is done. The average power P_av over a time interval Δt is P_av object in the direction of the force. Power is measured in watts (W), with 1 W = 1 J s⁻¹.

功率是做功的速率。在一段时间Δt内的平均功率P_av为P_av = ΔW / Δt，其中ΔW为所做的功。瞬时功率P可表示为P = F v，其中F为驱动力，v为物体在力的方向上的速度。功率的单位是瓦特（W），1 W = 1 J s⁻¹。

F that exactly balances the total resistance R, so the net force is zero and the power output is P = F v = R v. If the vehicle moves up an incline, the resistance includes a component of weight down the slope: R + m g sinθ acts against motion, and the engine power must overcome this to maintain speed.

典型的AQA问题涉及车辆以恒定速度克服阻力行驶。在最大速度下，发动机提供的驱动力F恰好平衡总阻力R，因此净力为零，功率输出为P = F v = R v。如果车辆沿斜坡向上行驶，阻力包括沿斜面的重力分量：R + m g sinθ阻碍运动，发动机功率必须克服该阻力以维持速度。

n engine works at a constant power, the driving force decreases as speed increases. This relationship often appears in problems requiring you to find acceleration when power and resistance are known: F_drive = P / v, then apply Newton's second law.

当加速度不恒定时，必须谨慎使用P = F v，但如果发动机以恒定功率工作，驱动力会随着速度的增加而减小。这种关系经常出现在已知功率和阻力、需要求加速度的问题中：F_drive = P / v，然后应用牛顿第二定律。

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10. Efficiency | 效率

Efficiency measures how effectively a system converts input energy or power into useful output. It is defined as:

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

or equivalently for power: Efficiency = (useful power output / total power input) × 100% es due to friction, heat, and sound, real machines always have an efficiency less than 100%.

效率衡量系统将输入能量或功率转化为有用输出的有效程度。其定义为：

效率 = (有用的能量输出 / 总的能量输入) × 100%

对于功率也等价：效率 = (有用功率输出 / 总功率输入) × 100%。由于摩擦、热和声音造成的能量损失，实际机器的效率总是小于100%。

In exam questions, you may be given the total power generated by an engine and the rate at which it does work against resistance, and asked to find the efficiency. For example, a car engine produces 50 kW of power, but 15 kW is lost to internal friction and heat; the useful power driving the wheels is 35 kW, giving an efficiency of 70%. Always express the ratio using the same units.

在考试题中，可能会给出发动机产生的总功率及其克服阻力做功的速率，要求计算效率。例如，一辆汽车发动机产生50 kW的功率，但有15 kW损失在内摩擦和发热上；驱动车轮的有用功率为35 kW，效率为70%。始终使用相同单位表示该比率。

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11. Typical Exam-Style Problems | 典型考题剖析

A classic AQA problem combines several energy concepts. Consider a block of mass 2 kg released from rest at the top of a smooth curved track of vertical height 5 m. At the bottom, it slides onto a rough horizontal surface and compresses a spring of stiffness 200 N m⁻¹. The coefficient of friction on the horizontal part is 0.3 and the distance from the foot of the track to the spring is 1.5 m. Find the maximum compression of the spring.

一个经典的AQA问题综合了多个能量概念。考虑一个质量为2 kg的物块，从竖直高度为5 m的光滑弧形轨道顶端由静止释放。在底部，它滑上粗糙水平面并压缩一个劲度系数为200 N m⁻¹的弹簧。水平面上的摩擦系数为0.3，轨道底部到弹簧的距离为1.5 m。求弹簧的最大压缩量。

smooth track: the loss in GPE is entirely converted into KE at the bottom: m g h = ½ m v² ⇒ v = √2 g h = √2 × 9.8 × 5 ⁻¹. Next, sliding on the rough surface before touching the spring: the friction force is F_r s the mechanical energy. The remaining kinetic energy just before hitting the spring is KE' = ½ m v² - F_r 89.18 J. Then, as the block compresses the spring by x, this remaining energy is converted into EPE plus further work against friction over the compression distance x: ½ k x² + F_r x = 89.18. Substituting values: 100 x² + 5.88 x

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