Deformation
A-Level Physics Topic 6 16:50 English narration · English + 中文 subtitles burned in
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A bungee jumper steps off a bridge and falls — faster and faster, straight toward the ground.
一个蹦极者从桥上一跃而下——越落越快，直冲地面。
Then, at the last moment, an elastic cord snaps tight and pulls them safely back up.
然后，在最后一刻， 一根有弹性的绳索猛地绷紧，把他们安全地拉了回来。
How does a simple stretchy cord stop a falling person?
一根简单的弹力绳， 怎么能拦住一个下落的人？
As the cord stretches, it stores their energy of motion, hidden away as elastic potential energy.
绳索被拉伸时，把他们的运动能量储存起来，藏成弹性势能。
The more it stretches, the more it stores, until the fall is gently halted.
拉得越长，储存得越多，直到下落被温柔地止住。
That is the physics of deformation.
这就是形变的物理学。
This is how solids stretch, squeeze, and store energy.
这就是固体如何拉伸、压缩，并储存能量。
Today: Hooke's law and the spring constant, stress, strain, and the Young modulus, and the energy hidden in a stretched material.
今天：胡克定律与劲度系数、应力、应变、 杨氏模量，以及藏在被拉伸的材料里的能量。
Let's begin.
让我们开始吧。
When a force acts along the length of a solid, the solid changes shape.
当力沿固体的长度方向作用时，固体改变形状。
We call that change deformation.
我们把这种改变叫做形变。
A tensile force stretches the object and makes an extension.
拉伸力把物体拉长，产生伸长量。
A compressive force squeezes the object and makes a compression — treated as a negative extension.
压缩力把物体压短，产生压缩——当作负的伸长量处理。
The applied force is the load.
施加的力就是负载。
The extension is the change from the natural length.
伸长量是相对于自然长度的变化。
Keep those four words straight: load, extension, tensile, compressive.
把这四个词分清：负载、伸长、拉伸、压缩。
Both are measured from the natural, unstretched length, never from the loaded length.
这两个量都是从自然的、未拉伸的长度量起，绝不是从加载后的长度量起。
And the spring's own weight is taken as negligible unless a question says otherwise — negligible being the exam's way of telling you to treat it as zero.
而且除非题目另有说明，弹簧自身的重量按可以忽略处理—— 「可以忽略」是考试告诉你把它当作零的说法。
Hang a load on a spring, and it stretches.
在弹簧上挂一个负载，它就伸长。
For small loads, something simple happens: the extension is exactly proportional to the force.
对于小负载，会发生一件很简单的事： 伸长量与力恰好成正比。
Double the load, and the spring stretches twice as far.
负载加倍，弹簧就伸长一倍。
This is Hooke's law.
这就是胡克定律。
But watch — there is a point, the limit of proportionality, beyond which the neat straight line bends, and the rule breaks down.
但注意——有一个点，叫比例极限，超过它，那条整齐的直线就弯曲，规律不再成立。
State it the way the scheme prints it: the extension is directly proportional to the applied force — the load — provided the limit of proportionality is not exceeded.
要照评分标准印出来的说法陈述： 伸长量与所施加的力——也就是负载——成正比， 前提是没有超过比例极限。
The proviso is not optional; without it the statement is wrong for every real material past that point.
这个前提不是可有可无的；去掉它，对任何真实材料在那一点之后都是错的。
A real coiled metal spring is the classic example.
一根真正的金属螺旋弹簧就是经典例子。
Within the small-extension region, the extension is proportional to the force applied.
在小伸长区域内，伸长量与所加的力成正比。
That single idea — force proportional to extension — is Hooke's law written in words.
这一个想法——力与伸长成正比——就是用文字写的胡克定律。
Most exam springs are assumed to obey it unless the graph or the wording says otherwise.
大多数考题里的弹簧， 除非图或文字另有说明，都默认遵守它。
Past the limit of proportionality, that assumption fails.
一过比例极限，这个假设就失效。
We write Hooke's law as force equals the spring constant, times the extension.
我们把胡克定律写成：力等于劲度系数乘以伸长量。
The spring constant tells you how stiff the spring is — a big constant means a stiff spring.
劲度系数告诉你弹簧有多硬—— 系数大意味着弹簧硬。
Rearrange, and the constant is simply the force divided by the extension.
变形一下，系数就等于力除以伸长量。
Hang two newtons, get four centimetres, and the spring constant is fifty newtons per metre.
挂两牛顿，伸长四厘米， 劲度系数就是每米五十牛顿。
On a force-extension graph, the constant is just the gradient of the straight line.
在力—伸长图上，这个系数就是那条直线的斜率。
Work the example carefully.
仔细算这道例题。
A spring stretches by four point zero centimetres under a load of two point zero newtons.
一根弹簧在两牛顿负载下伸长四点零厘米。
First convert the extension to metres: four centimetres is zero point zero four zero metres.
先把伸长量换成米： 四厘米是零点零四零米。
Then k equals force over extension — two point zero divided by zero point zero four zero — which is fifty newtons per metre.
然后劲度系数等于力除以伸长——两除以零点零四零—— 就是每米五十牛顿。
Always convert to S-I units before you divide, or the power of ten will be wrong.
除之前一定先换成国际单位，否则十的幂会错。
Here is a load–extension graph.
这是一张负载—伸长图。
From the origin it rises as a straight line up to point P — the limit of proportionality.
从原点开始，直线上升到点比——比例极限。
Past P the line curves and flattens.
过了比，线弯曲并变平。
In the straight region, the gradient is the spring constant k.
在直线区域，斜率就是劲度系数。
Past P, Hooke's law no longer holds, even if the material is still elastic for a little further.
过了比，胡克定律不再成立，即使材料还能弹性再走一段。
Read P carefully: that is where the neat proportionality ends.
仔细读比：整齐的比例关系就在那里结束。
Watch the axes.
注意坐标轴。
A force against extension graph has gradient k.
力对伸长的图斜率是劲度系数。
Swap the axes — extension against force — and the gradient is one over k.
轴对调——伸长对力——斜率就是劲度系数的倒数。
A common trap plots length against force, not extension.
常见陷阱是画长度对力，而不是伸长。
Length equals original length plus force over k, so the gradient is still one over k, but the line no longer starts at the origin.
长度等于原长加力除以劲度系数，所以斜率仍是倒数， 但直线不再过原点。
Read it off carefully, and do not treat the intercept as zero extension by mistake.
仔细读图，别把截距错当成零伸长。
For a stretched wire, two better quantities describe what is happening inside.
对于一根被拉伸的金属丝，有两个更好的量能描述内部发生了什么。
Stress is the force spread over the cross-section — the force divided by area, measured in pascals.
应力是分摊在横截面上的力—— 力除以面积，用帕斯卡量度。
Strain is how much it stretches compared to its first length — the extension divided by the original length.
应变是它相对于原长伸长了多少——伸长量除以原来的长度。
Strain is just a ratio, so it has no unit at all.
应变只是一个比值，所以它完全没有单位。
Picture the wire.
想象这根金属丝。
Original length L-nought from clamp to free end, cross-sectional area A, and a hanging load F.
从夹具到自由端的原长为零标，横截面积为艾，挂着负载艾夫。
Stress is F over A — how hard the material is pulled per square metre.
应力是力除以面积——材料每平方米被拉得多紧。
Strain is extension x over L-nought — how much each unit of length has grown.
应变是伸长量除以原长——每一单位长度长了多少。
Using stress and strain lets you compare wires of different size on the same scale.
用应力和应变，就能在同一尺度上比较不同尺寸的金属丝。
Divide the stress by the strain, and you get the Young modulus — a single number for how stiff a material is.
用应力除以应变，你就得到杨氏模量——一个衡量材料有多硬的单一数值。
It equals the load, times the original length, divided by the area and the extension.
它等于负载， 乘以原长，再除以面积和伸长量。
For a steel wire, two metres long, stretched one millimetre by fifteen newtons, it comes to two times ten to the eleven pascals.
对于一根两米长的钢丝，被十五牛顿拉伸一毫米， 结果是二乘以十的十一次方帕斯卡。
And here is the key: the Young modulus belongs to the material itself.
关键在这里：杨氏模量属于材料本身。
Steel is steel, whatever its shape.
钢就是钢，不管它是什么形状。
On a stress–strain graph the straight part from the origin has gradient equal to the Young modulus.
在应力—应变图上，从原点出发的直线部分斜率就等于杨氏模量。
Past the limit of proportionality the line curves over.
过了比例极限，线会弯下来。
Metals sit near ten to the eleven pascals — steel about two times ten to the eleven.
金属大约在十的十一次方帕斯卡——钢约二乘以十的十一次方。
The modulus is a material property: it does not depend on the wire's length or thickness.
模量是材料性质： 与金属丝的长度或粗细无关。
The spring constant does depend on shape: k equals E A over L-nought.
劲度系数却与形状有关：系数等于模量乘面积再除以原长。
Reading two materials on the same axes: the gradient of the straight part is the Young modulus, so the steeper line is the stiffer material, and a wire with double the Young modulus gives half the strain for the same stress.
在同一坐标轴上读两种材料： 直线段的斜率就是杨氏模量，所以更陡的那条线对应更硬的材料， 而杨氏模量大一倍的金属丝，在同样的应力下应变只有一半。
To measure it, use a long, thin wire.
要测量它，用一根又长又细的金属丝。
Clamp one end, run it over a pulley, and hang weights from the free end.
夹住一端，让它绕过一个滑轮，从自由端挂上砝码。
First, measure the original length with a metre rule, and the diameter with a micrometer, at several places.
首先，用米尺量出原长，用螺旋测微器在几个地方量出直径。
Then add loads one at a time and record the extension.
然后一个一个地加负载， 记录伸长量。
Plot force against extension: its gradient gives you the Young modulus.
画出力对伸长量的图：它的斜率就给出杨氏模量。
Why long and thin?
为什么要又长又细？
So the extension is large enough to measure well.
这样伸长量才够大，便于精确测量。
Look at the apparatus carefully.
仔细看这套装置。
A paper flag on the wire is read against a fixed scale as masses hang past the pulley.
金属丝上的纸旗对着固定标尺读数，滑轮外挂着砝码。
Area A is pi d squared over four — use the average diameter from several micrometer readings.
面积艾是圆周率乘直径平方除以四——用几次螺旋测微器读数的平均直径。
On the force–extension graph, the straight gradient is E A over L-nought, so E equals gradient times L-nought over A.
在力—伸长图上，直线斜率是模量乘面积除以原长，所以模量等于斜率乘原长再除以面积。
Repeat readings to cut random error, and check the wire is uniform.
重复读数以减小随机误差，并检查金属丝是否均匀。
A "describe the experiment" answer earns its marks for the quantities measured and how: the original length with a metre rule, the diameter with a micrometer at several points along the wire, the load from the masses or a newton meter, and the extension from a marker read against a fixed scale.
「描述这个实验」的答案，得分点在于测了哪些量、怎么测的： 原长用米尺，直径用千分尺在金属丝上的好几处测量， 负载用砝码或者弹簧秤，伸长量用标记对着固定的刻度尺读。
Then the precautions: measure the diameter in several places and take a mean, use a long thin wire so the extension is large enough to measure, read the marker at eye level to avoid parallax, and wear eye protection because a wire under tension can snap.
然后是注意事项：直径要在多处测量并取平均， 用又长又细的金属丝，让伸长量大到足以测得准， 读标记时视线要与刻度平齐以避免视差， 而且要戴护目镜，因为受拉的金属丝可能会断。
Stretch a material more and more, and it passes through stages.
把一种材料越拉越长，它会经历几个阶段。
At first it is elastic: let go, and it springs back to its first length.
起初它是弹性的：一松手，它就弹回原来的长度。
Push past the elastic limit, and it turns plastic: now it stays stretched, keeping a permanent change of shape.
越过弹性极限，它就变成塑性的：现在它保持被拉伸的状态，留下永久的形状改变。
On the graph, the loading and unloading lines no longer match — and the gap between them is energy lost as heat inside the material.
在图上，加载线和卸载线不再重合——它们之间的间隙，就是在材料内部损失为热的能量。
Mark three regions on the force–extension graph.
在力—伸长图上标出三个区域。
First, elastic and straight — Hooke is obeyed up to the limit of proportionality, labelled P.
第一，弹性且直线——胡克定律成立，直到比例极限比。
Second, elastic but curved — between P and the elastic limit E.
第二，弹性但弯曲——在比和弹性极限弹之间。
Unload here and the object still returns to its first length.
这里卸载，物体仍回到原长。
Third, plastic — past E.
第三，塑性——过了弹。
Unload now and a permanent extension B stays on the axis.
现在卸载，轴上会留下永久伸长永。
Hooke's law only holds in the straight elastic region.
胡克定律只在直线弹性区成立。
In the lab, a universal tensile testing machine does this cleanly.
在实验室里，万能拉伸试验机把这事做得干净利落。
The sample is held between two grips on a tall rigid frame.
试样夹在高大刚性机架的两个夹具之间。
The machine pulls the grips apart while it records force and extension continuously.
机器把夹具拉开，同时连续记录力和伸长。
From that trace you can read the limit of proportionality, the elastic limit, and any permanent set after unloading.
从那条曲线上，你可以读出比例极限、 弹性极限，以及卸载后的任何永久变形。
When a material is taken into the plastic region and then unloaded, the unloading line is parallel to the original Hooke line but shifted right.
当材料进入塑性区再卸载时，卸载线与原来的胡克直线平行，但向右平移。
That shift is the permanent extension left when the load returns to zero.
这个平移就是负载回到零时留下的永久伸长。
The area between the loading and unloading curves is the energy turned into thermal energy inside the material — energy you cannot get back as useful elastic potential energy.
加载曲线与卸载曲线之间的面积， 就是在材料内部变成热能的能量——你无法再把它收回为有用的弹性势能。
A rubber band does the same thing with one difference that matters: its loading and unloading curves are also different, but it does return to its original length, so its deformation is elastic.
橡皮筋也是同一回事，但有一个关键的不同： 它的加载和卸载曲线同样不重合，但它确实会回到原来的长度，所以它的形变是弹性的。
The area between the two curves is again energy dissipated as thermal energy, and that effect has a name — elastic hysteresis.
两条曲线之间的面积同样是以热能形式耗散掉的能量， 而这个效应有个名字——弹性滞后。
It is why a squash ball warms up as it is played with.
壁球在打的过程中会变热，就是这个原因。
How much energy does a stretched spring store?
一根被拉伸的弹簧储存了多少能量？
It is the work you did stretching it — the area under the force-extension graph.
就是你拉伸它所做的功——力—伸长图下方的面积。
For a Hooke's-law spring, that area is a triangle: one half, the force, times the extension.
对于遵守胡克定律的弹簧，这块面积是一个三角形：二分之一，力，乘以伸长量。
Which is also one half, the spring constant, times the extension squared.
也就是二分之一，劲度系数，乘以伸长量的平方。
Stretch a fifty-newton-per-metre spring by twenty centimetres, and it stores one joule of elastic potential energy.
把一根每米五十牛顿的弹簧拉长二十厘米， 它就储存一焦耳的弹性势能。
See the shaded triangle under the straight force–extension line.
看直线力—伸长线下的阴影三角形。
Its area is one half F times x — the work done stretching the material from zero to that extension.
它的面积是二分之一力乘伸长——把材料从零拉到该伸长所做的功。
That stored energy is elastic potential energy, provided you stayed within the limit of proportionality.
只要你还在比例极限内，储存的能量就是弹性势能。
Outside that region the area is still work done, but the graph is no longer a triangle.
出了那个区域，面积仍是做的功， 但图不再是三角形。
Three equal forms of the same energy.
同一能量的三种等价形式。
One half force times extension.
二分之一力乘伸长。
One half k x squared.
二分之一劲度系数乘伸长的平方。
And F squared over two k.
以及力的平方除以二倍劲度系数。
Use whichever the data gives you.
数据给什么就用哪种。
If you know F and k but not x, the third form is fastest: energy equals force squared divided by two k.
若知道力和劲度系数却不知道伸长， 第三种最快：能量等于力的平方除以二倍劲度系数。
All three assume Hooke's law held throughout the stretch.
三种都假定全程遵守胡克定律。
Worked numbers: spring constant fifty newtons per metre, stretched by zero point two zero metres, within the limit of proportionality.
算一遍数字：劲度系数每米五十牛顿，伸长零点二零米，在比例极限内。
Elastic potential energy equals one half times fifty times zero point two zero squared.
弹性势能等于二分之一乘五十乘零点二零的平方。
That is one half times fifty times zero point zero four, which is one point zero joules.
也就是二分之一乘五十乘零点零四， 等于一点零焦耳。
Check the units: newtons per metre times metres squared gives joules.
检查单位：牛顿每米乘米的平方就是焦耳。
For a graph that is not a straight line — a stretched rubber band, or a spring past its limit of proportionality — the area is still the work done.
对于不是直线的图——拉长的橡皮筋，或过了比例极限的弹簧——面积仍是做的功。
Find it by counting grid squares, or by using trapezia under the curve.
用数方格的方法求，或用曲线下的梯形。
The same idea always holds: the area under the force–extension graph is the work done on the material.
同一个想法始终成立： 力—伸长图下方的面积就是对材料做的功。
A common multiple-choice trap compares two springs.
常见选择题会比较两根弹簧。
If they are stretched by the same force, the less stiff spring extends further, so it stores more energy — because energy is one half F times x.
若被同样的力拉伸，较软的弹簧伸得更长， 所以储存更多能量——因为能量是二分之一力乘伸长。
If they reach the same extension, the stiffer spring needs a larger force, so the stiffer spring stores more.
若达到相同伸长， 较硬的弹簧需要更大的力，所以较硬的弹簧储存更多。
Always ask: same force, or same extension?
总要问：同力，还是同伸长？
When a stretched spring is released onto a mass, the elastic potential energy becomes kinetic energy — and gravitational potential energy if the mass rises.
当被拉伸的弹簧松开放到一个质量上时，弹性势能变成动能——如果质量上升，还有重力势能。
To find the speed, set one half k x squared equal to one half m v squared, plus any m g h if height changes.
要求速度，令二分之一劲度系数乘伸长平方等于二分之一质量乘速度平方， 若高度变化再加上质量乘重力加速度乘高度。
Energy is conserved; it just changes form.
能量守恒；它只是改变形式。
Sometimes springs are combined.
有时弹簧会被组合起来。
Two springs in series, one hanging below the other, share the same load, but their extensions add — so together they are softer, and the inverse of the total constant is the sum of the inverses.
两根弹簧串联，一根挂在另一根下面，承受同样的负载， 但它们的伸长量相加——所以合起来更软，总系数的倒数等于各自倒数之和。
Two springs in parallel, side by side, share the load — so together they are stiffer, and their constants simply add.
两根弹簧并联，并排放置，分担负载——所以合起来更硬，它们的系数直接相加。
Three identical springs are worth working through, because the answer surprises people.
三根相同的弹簧值得算一遍，因为结果出乎很多人意料。
In series each spring carries the full load, so the extensions add and the combination is the most stretchy.
串联时每根弹簧都承受全部负载，所以伸长量相加，整体是最「软」的。
In parallel the load divides between them, so each extends by a third and the combination stores only a quarter of the energy a single spring would store for the same applied force.
并联时负载在它们之间分配，所以每根只伸长三分之一， 而在同样的施加力下，整体储存的能量只有单根弹簧的四分之一。
Memorise the two pictures.
记住这两幅图。
Series: same force through each spring, extensions add, so one over k-total equals one over k-one plus one over k-two — softer overall.
串联：每根弹簧受力相同，伸长相加，所以总系数的倒数等于各自倒数之和——整体更软。
Parallel: same extension for each spring, forces add, so k-total equals k-one plus k-two — stiffer overall.
并联：每根弹簧伸长相同，力相加，所以总系数等于各自系数之和——整体更硬。
Identical springs in parallel double k; identical springs in series halve k.
相同弹簧并联时劲度加倍；相同弹簧串联时劲度减半。
Three marks to secure.
三个要拿稳的分。
First, Hooke's law only holds up to the limit of proportionality — not beyond.
第一，胡克定律只在比例极限之内成立——超出就不行。
Second, keep stress, strain, and the Young modulus straight: stress is force over area, strain is extension over length, and their ratio is the modulus.
第二，把应力、应变和杨氏模量分清楚：应力是力除以面积，应变是伸长量除以长度， 它们的比值就是模量。
Third, the energy stored is the area under the force-extension graph — one half force times extension.
第三，储存的能量是力—伸长图下方的面积——二分之一力乘以伸长量。
Get these, and this topic is yours.
掌握这些，这个专题就是你的了。
These definitions are marked against fixed wording, so give one answer only and give it exactly.
这些定义是按固定措辞给分的，所以只给一个答案，而且要一字不差。
Hooke's law: the extension is directly proportional to the applied force, provided the limit of proportionality is not exceeded.
胡克定律：伸长量与所施加的力成正比，前提是没有超过比例极限。
Spring constant: the force per unit extension.
弹簧常数：单位伸长量所需的力。
Limit of proportionality: the point beyond which the extension is no longer proportional to the force.
比例极限：超过这一点之后，伸长量就不再与力成正比。
Elastic limit: the maximum force for which the material returns to its original length when the force is removed — a little beyond the limit of proportionality, and swapping the two is a standard lost mark.
弹性极限：撤去力之后材料仍能恢复原长的最大作用力—— 它比比例极限稍微靠后一点，把这两者弄混是标准的失分。
Elastic deformation: the material returns to its original length.
弹性形变：材料恢复到原来的长度。
Plastic deformation: it does not, and a permanent extension remains.
塑性形变：不能恢复，留下永久的伸长。
Stress: force per unit cross-sectional area.
应力：单位横截面积上的力。
Strain: extension per unit original length.
应变：单位原长上的伸长量。
Young modulus: the ratio of stress to strain within the limit of proportionality.
杨氏模量：在比例极限之内，应力与应变之比。
The traps.
陷阱。
Halve the diameter before using pi r squared, and convert square millimetres — one is ten to the minus six square metres.
用 πr² 之前先把直径减半，而且要换算平方毫米——一平方毫米是十的负六次方平方米。
On a length-force graph only the triangle above the original length is work, not the whole area down to the axis.
在长度—力的图上，只有原长以上的那个三角形才是功，不是一直到坐标轴的整块面积。
Stored energy goes as x squared, so double the extension is four times the energy.
储存的能量随 x 的平方变化，所以伸长量加倍，能量是四倍。
Never state Hooke's law without its condition.
陈述胡克定律时绝不能省掉它的条件。
And a thicker wire does not have a larger Young modulus — the modulus belongs to the material; the thicker wire has a larger spring constant.
另外，更粗的金属丝并不具有更大的杨氏模量—— 模量属于材料；更粗的丝具有更大的弹簧常数。