Elastic Potential Energy | 弹性势能

📚 Elastic Potential Energy | 弹性势能

When a spring or any deformable object is stretched or compressed, work is done by the applied force. If the object remains within its elastic limit, this work can be stored as recoverable elastic potential energy. Understanding this idea brings together force, extension, work and energy conservation, which are central themes in CIE A-Level Physics.

当弹簧或任何可变形物体被拉伸或压缩时，外力对其做功。如果物体仍在弹性极限内，这些功可以以可恢复的弹性势能形式储存起来。理解这一概念将力、伸长量、功和能量守恒联系起来，它们是 CIE A-Level 物理的核心内容。

1. Deformation and Elastic Behaviour | 形变与弹性行为

Deformation refers to a change in the shape or size of an object caused by applied forces. For springs and wires, we usually consider tensile deformation when a stretching force is applied, or compressive deformation when a squeezing force is applied.

形变是指物体在力的作用下形状或尺寸发生改变。对于弹簧和金属丝，我们通常考虑施加拉力时发生的拉伸形变，或施加压力时发生的压缩形变。

An object behaves elastically if it returns to its original shape and size after the deforming forces are removed. If it does not return to its original shape, it has been loaded beyond its elastic limit and shows plastic behaviour.

如果物体在撤去外力后能恢复原来的形状和尺寸，我们就说它表现出弹性行为。如果物体不能恢复原状，说明它已被加载到超过弹性极限，表现出塑性行为。

2. Hooke’s Law and Spring Constant | 胡克定律与劲度系数

Hooke’s law states that the extension x of a spring is directly proportional to the applied force F, provided that the force does not exceed the limit of proportionality.

胡克定律指出，只要外力不超过比例极限，弹簧的伸长量 x 与所施加的力 F 成正比。

F = kx

In this equation, k is the spring constant, also called stiffness. Its SI unit is newton per metre, N m⁻¹.

在这个公式中，k 是劲度系数，也叫弹簧常数。它的 SI 单位是牛顿每米，N m⁻¹。

A large k means the spring is stiff and needs a large force to produce a given extension. A small k means the spring is soft and extends easily under the same force.

k 值大表示弹簧较硬，要产生一定的伸长量需要较大的力。k 值小表示弹簧较软，在相同力作用下容易伸长。

3. Extension, Compression and Natural Length | 伸长量、压缩量与自然长度

The symbol x in Hooke’s law always means the change in length from the natural length, not the total length of the spring. If a spring has natural length L₀ and is stretched to length L, then the extension is x = L – L₀.

胡克定律中的符号 x 总是指从自然长度算起的长度变化量，而不是弹簧的总长度。如果弹簧的自然长度为 L₀，拉伸后的长度为 L，则伸长量为 x = L – L₀。

For compression, x is positive but represents the decrease in length from the natural length. The same magnitude of extension or compression stores the same amount of elastic potential energy for an ideal spring.

对于压缩，x 也为正值，但它表示从自然长度算起的缩短量。对于理想弹簧，相同大小的伸长量和压缩量储存相同大小的弹性势能。

x = change in length from natural length

In exam questions, this distinction is important. A common error is to substitute the total stretched length of the spring instead of the extension.

在考试题目中，这一区别很重要。一个常见错误是把弹簧拉伸后的总长度代入公式，而不是代入伸长量。

4. Work Done and Average Force | 做功与平均力

When a spring is extended from zero extension to x, the applied force increases linearly from 0 to F. The work done by this varying force cannot be found by simply multiplying the final force by the extension.

当弹簧从零伸长量拉伸到 x 时，施加的力从 0 线性增加到 F。这个变力所做的功不能简单地用最终力乘以伸长量来计算。

Because the force increases uniformly, the average force during the extension is half the final force. Therefore the work done is W = ½ F x.

由于力均匀增加，伸长过程中的平均力等于最终力的一半。因此做功为 W = ½ F x。

W = ½ F x

This work is stored as elastic potential energy provided that no energy is lost as heat or permanent deformation.

只要没有能量以热或永久形变的形式损失，这些功就储存为弹性势能。

5. Elastic Potential Energy Formula | 弹性势能公式

Substituting F = kx into W = ½ F x gives the elastic potential energy stored in a stretched or compressed spring obeying Hooke’s law.

将 F = kx 代入 W = ½ F x，可得到遵守胡克定律的拉伸或压缩弹簧中储存的弹性势能。

E = ½ kx²

Here E is the elastic potential energy in joules, k is the spring constant in N m⁻¹, and x is the extension or compression in metres. The energy is a scalar quantity and is always positive.

其中 E 是弹性势能，单位为焦耳；k 是劲度系数，单位为 N m⁻¹；x 是伸长量或压缩量，单位为米。能量是标量，且始终为正。

For example, if a spring with k = 200 N m⁻¹ is compressed by 0.050 m, the stored energy is E = ½ × 200 × 0.050² = 0.25 J.

例如，若一个 k = 200 N m⁻¹ 的弹簧被压缩 0.050 m，储存的能量为 E = ½ × 200 × 0.050² = 0.25 J。

If the extension changes from x₁ to x₂, the energy change is given by ΔE = ½ kx₂² – ½ kx₁². This is more useful when the spring does not start from zero extension.

如果伸长量从 x₁ 变化到 x₂，能量变化为 ΔE = ½ kx₂² – ½ kx₁²。当弹簧不从未伸长状态开始时，这个公式更有用。

6. Force-Extension Graphs and Area | 力-伸长图像与面积

The work done on a spring is equal to the area under the force-extension graph. For a spring obeying Hooke’s law, the graph is a straight line through the origin, so this area is a triangle with base x and height F.

对弹簧做的功等于力-伸长图像下的面积。对于遵守胡克定律的弹簧，图像是过原点的直线，因此该面积是以 x 为底、F 为高的三角形。

Area = ½ × base × height = ½ Fx = ½ kx²

If a material does not obey Hooke’s law, the graph is curved. The stored energy is still the area under the curve, but the formula ½ Fx may no longer be valid.

如果材料不遵守胡克定律，图像会弯曲。此时储存的能量仍然是曲线下的面积，但公式 ½ Fx 可能不再适用。

When the force is loaded and unloaded, the area between the loading and unloading curves represents energy lost, often as heat due to internal friction or plastic deformation.

当加载和卸载时，加载曲线与卸载曲线之间的面积代表损失的能量，通常是由于内摩擦或塑性形变而以热的形式散失。

7. Strain Energy in Materials | 材料中的应变能

For a stretched wire, elastic potential energy can also be expressed in terms of stress and strain. The energy stored per unit volume is called strain energy density.

对于被拉伸的金属丝，弹性势能也可以用应力和应变来表示。单位体积储存的能量称为应变能密度。

u = ½ stress × strain = ½ σε

Here σ is stress, ε is strain, and u is strain energy per unit volume. For a material obeying Hooke’s law, stress is proportional to strain, so the triangular area argument still applies.

其中 σ 是应力，ε 是应变，u 是单位体积的应变能。对于遵守胡克定律的材料，应力与应变成正比，因此三角形面积法仍然适用。

If the Young modulus Y is known, the strain energy density can be written as u = ½ Y ε² or u = σ² / 2Y. These forms are useful when comparing energy stored in different materials under the same stress or strain.

若已知杨氏模量 Y，应变能密度可写成 u = ½ Y ε² 或 u = σ² / 2Y。当比较不同材料在相同应力或应变下储存的能量时，这些形式很有用。

8. Energy Conservation with Springs | 弹簧系统的能量守恒

In an isolated spring-mass system, the total mechanical energy is conserved if friction and air resistance are negligible. For a horizontal spring, the total energy alternates between kinetic energy and elastic potential energy.

在孤立的弹簧-质量系统中，如果摩擦和空气阻力可以忽略，总机械能守恒。对于水平弹簧，总能量在动能和弹性势能之间相互转化。

E_total = ½ mv² + ½ kx²

At maximum displacement, the mass is momentarily at rest, so all the energy is elastic potential energy. At the equilibrium position, x = 0, so all the energy is kinetic energy.

在最大位移处，质量块瞬时静止，因此所有能量都是弹性势能。在平衡位置处 x = 0，因此所有能量都是动能。

For a vertical spring, gravitational potential energy must also be included. A common exam problem is a mass falling onto a spring, where initial gravitational potential energy becomes elastic potential energy and possibly kinetic energy at the lowest point.

对于竖直弹簧，还必须包括重力势能。一个常见的考试题目是质量块落到弹簧上，初始重力势能转化为弹性势能，到达最低点时可能还有动能。

9. Experimental Determination | 实验测定

A simple experiment to determine the spring constant k involves hanging different known masses from the spring and measuring the extension with a ruler. The force applied is the weight mg of each mass.

测定弹簧劲度系数 k 的一个简单实验是：在弹簧下悬挂不同已知质量，用刻度尺测量伸长量。施加的力是每个质量块的重力 mg。

The force F is plotted on the vertical axis against extension x on the horizontal axis. The gradient of the straight-line section gives the spring constant k.

将力 F 画在纵轴上，伸长量 x 画在横轴上。直线部分的斜率给出弹簧劲度系数 k。

gradient = ΔF / Δx = k

The area under the best-fit straight line up to a chosen extension can be used to estimate the elastic potential energy stored at that extension. The measurements should be repeated and extensions should be converted to metres before calculating energy in joules.

最佳拟合直线下方到某一选定伸长量的面积可以用来估算该伸长量下储存的弹性势能。测量应重复进行，计算能量前应将伸长量换算为米，以得到焦耳单位。

10. Common Misconceptions and Exam Tips | 常见误区与考试提示

A very common mistake is confusing spring length with extension. The formula E = ½ kx² requires x to be the change in length, not the total length of the spring.

一个十分常见的错误是把弹簧长度与伸长量混淆。公式 E = ½ kx² 中的 x 必须是长度变化量，而不是弹簧总长度。

Another frequent error is using E = Fx instead of E = ½ Fx. This forgets that the force increases from zero to its final value, so the average force is half the final force.

另一个常见错误是使用 E = Fx 而不是 E = ½ Fx。这忽略了力从零增加到最终值的过程，因此平均力应为最终力的一半。

Students should also remember to convert all lengths to metres, square the extension when using E = ½ kx², and check whether a graph is linear before applying the triangle-area formula.

学生还应注意将所有长度换算为米，使用 E = ½ kx² 时对伸长量进行平方，并在应用三角形面积公式前检查图像是否为线性。

Finally, if the force-extension graph is nonlinear, the stored energy is still the area under the curve, but it cannot be found from ½ Fx unless the graph is a straight line through the origin.

最后，如果力-伸长图像是非线性的，储存的能量仍然是曲线下的面积，但除非图像是过原点的直线，否则不能使用 ½ Fx 来求该能量。

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Elastic Potential Energy | 弹性势能