📚 Work, Energy and Power | 功、能量与功率
In A-Level OCR Physics, the topic of work, energy and power is fundamental to understanding mechanics. You need to master calculations of work done, energy transfers, conservation of energy, power, and efficiency, and apply these concepts to real-world problems.
在A-Level OCR物理中，功、能量和功率是理解力学的核心。你需要掌握功的计算、能量转换、能量守恒、功率和效率，并能将这些概念应用于实际问题。
1. Work: Definition and Calculation | 功：定义与计算
Work is done when a force moves an object in the direction of the force. For a constant force F acting at an angle θ to the displacement s, the work done W is given by:
当一个力使物体沿力的方向移动时，力做了功。对于恒力F与位移s成角度θ，做的功W由下式给出：
W = F s cosθ
The SI unit of work is the joule (J), equivalent to 1 N m.
功的国际单位是焦耳（J），等于1 N·m。
If the force is parallel to the displacement, θ = 0, so cosθ = 1, and W = F s. When the force is perpendicular (θ = 90°), no work is done, as cos90° = 0. For example, the gravitational force does no work on a satellite in a circular orbit because the force is perpendicular to the velocity.
如果力与位移平行，θ = 0，cosθ = 1，功W = F s。当力与位移垂直（θ = 90°）时，不做功，因为cos90° = 0。例如，引力对圆轨道上的卫星不做功，因为力与速度垂直。
Work can be positive (force and displacement in same direction) or negative (opposite direction, e.g. friction). Negative work means the force takes energy away from the object.
功可以是正功（力与位移同向）或负功（反向，如摩擦力）。负功意味着该力从物体带走能量。
2. Work Done by a Variable Force: Force-Distance Graphs | 变力做功：力-位移图像
When the force is not constant, work done can be found from the area under a force–distance graph. If force F is plotted against displacement x, the work done equals the area between the graph and the x-axis.
当力不是恒力时，做功可以从力-位移图像下的面积求得。如果力F对位移x作图，所做的功等于图线与x轴之间的面积。
For a spring obeying Hooke’s law, F = kx, the graph is a straight line through the origin. The work done in stretching the spring from 0 to extension x is the area of a triangle:
对于满足胡克定律的弹簧，F = kx，图像是一条过原点的直线。将弹簧从0拉伸至伸长x所做的功是三角形的面积：
W = ½ F x = ½ k x2
This energy is stored as elastic potential energy.
这份能量储存为弹性势能。
In general, for any force–distance graph, counting squares or using integration gives the work done.
一般而言，对于任何力-位移图像，数格子或使用积分可以求出做功。
3. Kinetic Energy | 动能
Kinetic energy (Ek) is the energy an object possesses due to its motion. For an object of mass m moving at speed v,
动能（Ek）是物体由于运动而具有的能量。对于质量为m、速度为v的物体，
Ek = ½ m v2
Kinetic energy is a scalar quantity, and its unit is the joule.
动能是标量，单位是焦耳。
The work–energy theorem (derived from Newton’s second law) states that the net work done on an object equals its change in kinetic energy:
功能定理（从牛顿第二定律导出）指出，作用在物体上的合外力做的功等于其动能变化量：
Wnet = ΔEk = ½ m vf2 – ½ m vi2
Always be careful: Wnet is the work done by all forces acting on the object, including friction and gravity.
注意：Wnet 是作用在物体上所有力（包括摩擦力和重力）所做的总功。
4. 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, where g is constant, the change in Ep when an object of mass m is lifted through a vertical height Δh is:
重力势能（Ep）是物体因其在引力场中的位置而储存的能量。在地表附近，g恒定，质量为m的物体升高竖直高度Δh时，重力势能的变化为：
ΔEp = mgΔh
The zero of potential energy can be chosen arbitrarily, but only changes in Ep have physical meaning. When an object falls, it loses gravitational potential energy and gains kinetic energy (if no air resistance).
势能的零点可以任意选择，但只有Ep的变化具有物理意义。物体下落时，失去重力势能并获得动能（如果没有空气阻力）。
In OCR problems, often you equate loss in Ep to gain in Ek or use conservation of energy.
在OCR题目中，经常将Ep的减少等同于Ek的增加，或利用能量守恒。
5. Elastic Potential Energy (Extension) | 弹性势能（扩展）
Elastic potential energy is stored in stretched or compressed elastic objects. For a spring with spring constant k extended or compressed by x from its natural length, provided the elastic limit is not exceeded, the energy stored is:
弹性势能储存在被拉伸或压缩的弹性物体中。对于劲度系数为k的弹簧，从原长拉伸或压缩x（未超出弹性极限），储存的能量为：
Ee = ½ k x2
This result comes from the work done to extend the spring, as shown in the force–distance graph. The area under the F–x graph is ½ Fmax x = ½ (kx) x = ½ k x2.
该结果源自拉伸弹簧所做的功，如力-位移图像所示。F-x图像下面积为½ Fmax x = ½ (kx) x = ½ k x2。
Energy can be transferred to and from elastic potential energy, e.g. in a mass–spring system, energy oscillates between kinetic and elastic potential energy.
能量可以与弹性势能相互转换，例如在弹簧振子中，能量在动能和弹性势能之间振荡。
6. The Work–Energy Principle | 功能原理
The work–energy principle is a key concept: the total work done by all forces acting on a body equals the change in its kinetic energy. This includes work done by conservative forces (like gravity) and non‑conservative forces (like friction).
功能原理是一个关键概念：作用在物体上的所有力所做的总功等于其动能的变化量。这包括保守力（如重力）和非保守力（如摩擦力）所做的功。
Mathematically, Wtotal = ΔEk. When dealing with gravitational force, you can include mgΔh as part of Wtotal or move it to the other side as ΔEp. The conservation of mechanical energy (Ek + Ep = constant) applies only when no external non‑conservative forces do work.
数学表达为Wtotal = ΔEk。在涉及重力时，你可以把mgΔh包含在Wtotal中，或者将其移到另一边作为ΔEp。机械能守恒（Ek + Ep = 恒量）仅当没有外部的非保守力做功时才成立。
For example, a block sliding down a rough incline: Wgravity + Wfriction = ΔEk. Since friction does negative work, some energy is dissipated as heat.
例如，物块沿粗糙斜面下滑：重力功 + 摩擦力功 = ΔEk。由于摩擦力做负功，部分能量以热能形式耗散。
7. Conservation of Mechanical Energy
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