📚 A.3 Work, energy and power | A.3 功、能量和功率
Work, energy and power are foundational concepts in IB Physics, linking force and motion through the principle of energy conservation. In this topic, you will learn how to calculate work done by forces, distinguish between kinetic and potential energy, apply the work–energy theorem, and analyse power and efficiency in mechanical systems. A deep understanding of these ideas is essential for solving real-world physics problems and for later topics such as thermal physics and oscillations.
功、能量和功率是IB物理的基础概念，通过能量守恒原理将力与运动联系在一起。在本主题中，你将学习如何计算力所做的功、区分动能与势能、应用功-能定理，并分析机械系统中的功率与效率。深入理解这些概念对于解决现实世界的物理问题以及后续的热学、振动等章节至关重要。
1. Work and Its Definition | 功及其定义
In physics, work (W) is defined as the product of the force applied to an object and the displacement of the object in the direction of the force. If the force makes an angle θ with the displacement, only the component of force along the displacement does work. Mathematically:
W = F s cos θ
ent vector. Work is a scalar quantity and its SI unit is the joule (J), where 1 J = 1 N·m.
在物理学中，功（W）定义为作用在物体上的力与物体在力的方向上发生的位移的乘积。如果力与位移方向成角度θ，只有沿位移方向的分力做功。数学表达式为：
W = F s cos θ
其中F为恒力的大小，s为位移的大小，θ为力矢量与位移矢量之间的夹角。功是标量，其国际单位是焦耳（J），1 J = 1 N·m。
When the force has a component in the same direction as displacement, work is positive (energy transferred to the object). When the component is opposite to the displacement, work is negative (energy taken from the object). For example, friction does negative work.
当力在位移方向上有同向分量时，做正功（能量传递给物体）；当力在位移方向上有反向分量时，做负功（从物体中提取能量）。例如，摩擦力做负功。
2. Calculating Work Done by a Constant Force | 恒力做功的计算
e, when carrying a bag horizontally at constant speed, the vertical force does no work because there is no vertical displacement. Similarly, the normal force and centripetal force often do zero work as they are perpendicular to motion.
对于以一定角度施加的恒力，做功为W = Fs cos θ。若θ = 0°，cos 0° = 1，所以W = Fs；若θ = 90°，cos 90° = 0，不做功。例如，水平搬运书包时，竖直方向的力不做功，因为没有竖直位移。同理，法向力和向心力通常不做功，因为它们始终垂直于运动方向。
When multiple forces act on an object, the total work done is the algebraic sum of the work done by each force. Alternatively, one can first find the net force and then calculate the work done by the net force; both methods give the same result.
当物体受到多个力作用时，总功等于各力所做功的代数和。也可以先求合力，再计算合力所做的功；两种方法所得结果相同。
3. Work Done by a Varying Force – Force–Displacement Graphs | 变力做功 – 力-位移图像
If the force is not constant, the work done can be found from the area under a force–displacement (F–s) graph. The area between the graph and the s-axis, taken with appropriate sign (positive for area above the axis, negative for area below), gives the total work. For a linear spring obeying Hooke’s law, the force varies from 0 to kx, so the work done in stretching the spring is the area of a triangle:
W = ½ Fmax × x = ½ k x²
This result is the elastic potential energy stored in the spring.
如果力不是恒定的，可以通过力-位移（F-s）图像下的面积来求功。图线与s轴之间的面积，根据正负号规则（轴上方面积为正，轴下方面积为负）给出总功。对于遵守胡克定律的线性弹簧，力从0变为kx，因此拉伸弹簧所做的功为一个三角形的面积：
W = ½ Fmax × x = ½ k x²
该结果即为储存在弹簧中的弹性势能。
4. 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, the kinetic energy is given by:
Eₖ = ½ m v²
Kinetic energy is a scalar, always non-negative, and has the unit joule. The net work done on an object is equal to its change in kinetic energy – this is the work–energy theorem.
动能（Eₖ）是物体因运动而具有的能量。对于质量为m、速度为v的物体，动能的表达式为：
Eₖ = ½ m v²
动能是标量，始终非负，单位是焦耳。物体所受合外力做的功等于其动能的变化量——这就是功-能定理。
5. Gravitational Potential Energy | 重力势能
Gravitational potential energy (Ep or Ug) is the energy stored in an object due to its position in a gravitational field. Near the Earth’s surface, for a change in height Δh, the change in gravitational potential energy is:
ΔEp = m g Δh
al energy can be chosen arbitrarily; often the ground or lowest point is taken as reference. Lifting an object increases its Ep, while lowering decreases it.
重力势能（Ep 或 Ug）是物体因在引力场中的位置而具有的能量。在地球表面附近，对于高度变化Δh，重力势能的变化量为：
ΔEp = m g Δh
其中g为重力加速度（约 9.81 m s⁻²）。势能零点可任意选择；通常选取地面或最低点为参考位置。提升物体使其重力势能增加，下降则减小。
6. Elastic Potential Energy | 弹性势能
essed spring. For a spring obeying Hooke’s law (F = k x), the elastic potential energy is:
Ee = ½ k x²
where k is the spring constant (N m⁻¹) and x is the extension or compression from the natural length. This energy is always non-negative
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