AS Mathematics: Work and Energy | AS 数学：功和能量 考点精讲

📚 AS Mathematics: Work and Energy | AS 数学：功和能量 考点精讲

In AS Mathematics (Mechanics), the concepts of work and energy provide powerful tools for solving dynamics problems without having to analyse every instant of motion using kinematic equations. Understanding how forces do work and how energy is transferred between kinetic and potential forms simplifies many real-world situations, such as vehicles moving up slopes or objects falling under gravity.

在 AS 数学（力学）中，功和能量的概念为解决动力学问题提供了强大的工具，无需使用运动学方程分析运动的每一瞬间。理解力如何做功以及能量如何在动能和势能之间转移，可以简化许多现实情境，例如车辆爬坡或物体在重力作用下下落。

1. Definition of Work | 功的定义

Work is done when a force moves its point of application in the direction of the force. For a constant force F and displacement s in the same direction, the work done is given by:

当力的作用点沿力的方向发生位移时，力就做了功。对于方向相同的恒力 F 和位移 s，所做功由下式给出：

Work = F × s

If the force is at an angle θ to the displacement, the work done is the product of the component of the force in the direction of displacement and the displacement:

如果力与位移成角度 θ，则所做的功是力在位移方向上的分量与位移的乘积：

W = F s cos θ

The unit of work is the joule (J). 1 J = 1 N m.

功的单位是焦耳（J）。1 J = 1 N·m。

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

When a constant force acts along a straight line, the work done is simply the product of the force, displacement and the cosine of the angle between them. For example, a person pulling a sledge with a rope at an angle does work equal to the horizontal component of the tension multiplied by the distance moved.

当恒力沿直线作用时，所做的功简化为力、位移以及它们之间夹角的余弦的乘积。例如，一个人用绳子斜拉雪橇所做的功等于绳张力的水平分量乘以移动的距离。

Work can be positive, negative or zero:

• Positive work: force and displacement in the same general direction (0° ≤ θ < 90°). E.g. lifting a box.
• Negative work: force opposes motion (90° < θ ≤ 180°). E.g. friction or air resistance.
• Zero work: force perpendicular to displacement (θ = 90°). E.g. normal reaction.

功可以是正、负或零：

• 正功：力与位移大致同向（0° ≤ θ < 90°），如提起箱子。
• 负功：力阻碍运动（90° < θ ≤ 180°），如摩擦力或空气阻力。
• 零功：力垂直于位移（θ = 90°），如法向反作用力。

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

For a force that varies with position, the work done can be found from the area under a force–distance graph. If the force-distance relationship is linear, the work done equals the area of a trapezium; for a curve, integral calculus is required, though at AS level this is often evaluated by counting squares or using given geometric areas.

对于随位置变化的力，所做的功可以通过力—距离图下的面积求得。如果力与距离呈线性关系，功等于梯形的面积；对于曲线，需要积分运算，不过在 AS 阶段通常通过数方格或利用给定几何面积来计算。

In one dimension, if the force applied in the direction of motion is F(x), then work done from x₁ to x₂ is the definite integral:

在一维情况下，如果运动方向上施加的力为 F(x)，则从 x₁ 到 x₂ 所做的功为定积分：

W = ∫ F(x) dx from x₁ to x₂

This concept is used, for instance, to calculate the work done in stretching a spring or in resistances proportional to speed.

例如，该概念用于计算拉伸弹簧或与速度成正比的阻力所做的功。

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4. Kinetic Energy | 动能

The kinetic energy (KE) of an object of mass m moving at speed v is:

质量为 m 的物体以速度 v 运动时的动能（KE）为：

KE = ½ m v²

Kinetic energy is a scalar quantity, always positive, and its unit is also the joule (J). It represents the energy an object possesses due to its motion.

动能是标量，恒为正值，单位也是焦耳（J）。它表示物体因运动而具有的能量。

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5. Gravitational Potential Energy | 重力势能

The gravitational potential energy (GPE) gained by an object of mass m when lifted through a vertical height h in a uniform gravitational field is:

在均匀重力场中，质量为 m 的物体被提升垂直高度 h 时所增加的重力势能（GPE）为：

ΔGPE = m g h

where g is the acceleration due to gravity (9.8 m s⁻² or 10 m s⁻² as specified). The change in GPE depends only on the vertical displacement, not the path taken.

其中 g 为重力加速度（取 9.8 m s⁻² 或指定的 10 m s⁻²）。重力势能的变化仅取决于竖直位移，而与路径无关。

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6. The Work-Energy Principle | 功-能原理

The work-energy principle states that the net work done by all external forces acting on a particle is equal to the change in its kinetic energy:

功-能原理指出，作用在质点上的所有外力所做的净功等于其动能的改变量：

W_net = ΔKE = KE_final – KE_initial

This principle links the concepts of work and kinetic energy directly and can be used to find speeds, distances or forces without using kinematic equations of motion.

该原理直接联系了功与动能，可用于求解速度、距离或力，而无需使用运动学方程。

When more than one force does work, the work-energy principle incorporates the work of all forces, including gravity, friction and driving forces. For example, a car braking to rest: the work done by friction equals the decrease in kinetic energy.

当多个力做功时，功-能原理包含所有力的功，包括重力、摩擦力和驱动力。例如，汽车刹车至静止：摩擦力所做的功等于动能的减少量。

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7. Conservation of Mechanical Energy | 机械能守恒

If only conservative forces (such as gravity) do work (i.e. no friction, air resistance or external driving forces), the total mechanical energy (KE + GPE) remains constant:

如果只有保守力（如重力）做功（即没有摩擦、空气阻力或其他外部驱动力），则总机械能（动能+重力势能）保持不变：

KE₁ + GPE₁ = KE₂ + GPE₂

This is a special case of the work-energy principle and is extremely useful for problems involving objects sliding down smooth slopes, pendulums,

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