02 Hooke's Law

Summary

本视频脚本介绍了胡克定律，即弹簧的弹性行为。通过实验观察不同负载下弹簧的伸长情况，发现弹簧的伸长与施加的力成正比，这个比例常数被称为弹簧常数K。然而，当比较不同尺寸的弹簧时，会发现即使材料相同，它们的弹簧常数也不同，这引发了对材料力学性能更深入探讨的问题。视频以弹簧实验为引子，引导观众思考如何通过应力和应变来统一比较不同材料的力学行为。

Takeaways

• 📚 胡克定律是描述弹簧在受力时行为的基本法则。
• 🔍 当弹簧受到拉伸或压缩时，其长度会发生变化，这种变化称为伸长或压缩。
• 📏 弹簧的原始长度用L₀表示，伸长或压缩后的长度变化用ΔL或X表示。
• 📈 力与伸长之间的关系通常是线性的，这种线性关系的斜率称为弹簧常数（K）。
• 🧮 胡克定律的数学表达式为F = KX，其中F是作用力，X是伸长量，K是弹簧常数。
• 🔧 弹簧常数K是一个特定的值，它反映了弹簧材料的弹性特性。
• 📊 当比较不同弹簧时，如果它们由相同材料制成，但尺寸不同，它们的弹簧常数K会有所不同。
• 🔬 为了解决不同尺寸弹簧的比较问题，引入了应力和应变的概念。
• 🌐 应力是单位面积上的力，而应变是材料尺寸变化与原始尺寸的比率。
• 🔄 通过引入应力和应变，可以更公平地比较不同尺寸和形状的弹簧材料的机械行为。

Q & A

胡克定律是什么？

胡克定律是描述弹性物体在受到外力作用时形变与作用力成正比关系的定律，通常表达为 F = kX，其中 F 是作用力，k 是弹簧常数，X 是物体的形变或伸长量。

什么是弹簧的原长？

弹簧的原长，也称为休息长度，是指弹簧在没有受到外力作用时的自然长度，通常用 L₀ 表示。

弹簧的伸长量通常用什么字母表示？

弹簧的伸长量通常用 ΔL 或 X 表示，它描述了弹簧在受到外力作用后相对于原长增加的长度。

如何得到弹簧的力与伸长量的关系图？

通过对不同重量的负载施加在弹簧上并测量相应的伸长量，然后将这些数据点绘制在力与伸长量的坐标图上，可以得到一条直线，这条直线的斜率即为弹簧常数。

弹簧常数 K 代表什么？

弹簧常数 K 代表弹簧的刚度，它是一个衡量弹簧抵抗形变能力的量，数值越大表示弹簧越硬，即需要更大的力才能产生相同的形变。

为什么不同尺寸的弹簧即使由相同材料制成，它们的弹簧常数也会不同？

不同尺寸的弹簧即使由相同材料制成，它们的弹簧常数也会不同，因为弹簧的刚度不仅取决于材料，还受到弹簧的尺寸、形状和横截面积等因素的影响。

在比较两个不同尺寸的弹簧时，我们如何确定它们的材料属性是否相同？

在比较两个不同尺寸的弹簧时，可以通过确保它们由相同材料制成，并且施加相同的力来观察它们的形变量，从而比较它们的材料属性。

为什么在胡克定律中，斜率 B 通常为零？

在胡克定律中，斜率 B 通常为零是因为在理想情况下，弹簧在没有负载时不会自然伸长或压缩，因此当力为零时，伸长量也应为零，使得直线关系通过原点。

如果两个弹簧由不同材料制成，它们的力与伸长量关系会有什么不同？

如果两个弹簧由不同材料制成，即使尺寸相同，它们的力与伸长量关系也会不同，因为不同材料的弹性模量和屈服强度不同，导致弹簧常数 K 有显著差异。

胡克定律在实际应用中有哪些局限性？

胡克定律在实际应用中的局限性包括：只适用于弹性范围内的形变，对于超出弹性极限的塑性形变或断裂不适用；对于非线性材料，如橡胶等，胡克定律不适用；对于复杂形状或结构的物体，胡克定律可能需要进行修正或使用更复杂的模型。

Outlines

00:00

🔍 胡克定律与弹簧常数

本段落介绍了胡克定律（Hooke's Law），这是描述弹性体（如弹簧）在受力时形变与恢复力之间关系的物理定律。首先，通过弹簧挂重物的例子，解释了弹簧在不同重量下的伸长情况。接着，定义了弹簧的原始长度（L0）和伸长量（ΔL 或 X），并指出在不同的负荷下，弹簧的伸长与施加的力之间存在线性关系。这种线性关系的斜率被称为弹簧常数（K）。胡克定律的数学表达式为 F = KX，其中 F 是力，X 是伸长量。然而，当比较不同弹簧时，即使它们由相同材料制成，也会发现它们的弹簧常数不同，这引出了对材料属性更深入理解的必要性，即下一话题——应力和应变。

05:01

🔬 材料的应力与应变

第二段落探讨了当两个由相同材料制成的不同尺寸的圆柱体受到相同力的作用时，它们伸长量的差异性。通过比较两个不同尺寸的圆柱体（样本A和样本B），说明了即使材料相同，由于尺寸不同，它们在受到相同力时的伸长量也会不同。这种差异性导致了不同的弹簧常数（Ka 和 Kb），这似乎表明它们具有不同的物理属性，尽管它们是由相同的材料制成。这种矛盾现象引出了对材料的应力和应变概念的讨论，这是理解材料在受力时如何响应的关键。

Keywords

💡胡克定律

胡克定律描述了弹簧在弹性限度内，所受的力与其伸长量成正比。视频中通过弹簧受力伸长的例子引入了胡克定律，即 F = KX，其中 F 是力，K 是弹簧常数，X 是伸长量。这个定律是讨论材料机械行为的基础。

💡弹簧常数

弹簧常数 K 是表征弹簧硬度的一个量，越大的 K 表示弹簧越难拉伸。在视频中，当讲到如何用不同负载测试弹簧的拉伸情况时，K 被定义为力与伸长量关系的斜率。不同的弹簧会有不同的弹簧常数，这反映了其不同的机械特性。

💡位移

位移是指物体由于外力作用而偏离其初始位置的距离。在视频中，位移（用 X 或 ΔL 表示）用于描述弹簧被拉长的长度。位移是理解胡克定律的关键参数，因为它直接影响受力与弹簧常数之间的关系。

💡直线方程

直线方程 y = mx + b 是描述线性关系的数学表达式，视频中将其用于解释胡克定律中的 F = KX，其中 K 是斜率，对应于弹簧常数。通过这种数学关系，视频展示了力和位移之间的线性比例。

💡材料性质

材料性质指的是不同材料在受力时表现出的机械行为。视频中提到虽然不同形状和大小的样品都是由相同的材料（如不锈钢或铝）制成，但其在受力后的位移表现不同，这引出了对比材料本质和形状影响的问题。

💡应力

应力是指单位面积上承受的力，通常用于描述材料在外力作用下的内在反应。在视频中，讨论了如何通过应力与应变来解决不同形状的材料在受力下表现出不同的位移的问题。

💡应变

应变是材料在受力时发生的形变量，相对于其原始长度的变化。视频中提到通过分析应力和应变的关系，可以更好地理解不同材料样品在相同负载下表现出的不同位移问题。

💡线性关系

线性关系是指两个变量之间呈正比例或反比例变化。在视频中，胡克定律表现为力与位移之间的线性关系，这意味着随着弹簧被拉长，所受的力成比例增加。

💡力

力是导致物体发生位移或形变的外界作用。在视频中，通过向弹簧施加不同的力，观察其拉伸程度，以此来讨论材料的机械行为及胡克定律的应用。

💡原始长度

原始长度（L₀）指的是在施加外力之前，弹簧或其他物体的自然长度。视频中提到，当弹簧在外力作用下发生拉伸时，会产生与原始长度相比的位移，这在讨论胡克定律时非常重要。

Highlights

Introduction to Hooke's Law using a spring example.

Demonstration of how a spring elongates when a weight is added.

Explanation of the original length and its symbol L₀.

Elongation is described using ΔL or X as a symbol for the change in length.

Force vs. elongation gives a linear relationship.

Introduction of the spring constant K as the slope of the linear relationship.

Hooke's Law is mathematically expressed as F = KX.

Comparison of two different springs reveals a problem: different elongations with the same force.

A smaller cross-sectional area elongates more than a larger one with the same applied force.

Even when two springs are made of the same material, they show different spring constants.

The issue arises because of varying sizes of the springs despite having the same material.

The problem with comparing different spring constants is emphasized.

The instructor suggests the next topic will involve stress and strain to resolve the discrepancy.

Stress and strain are introduced as the next steps to address material behavior more accurately.

The importance of material uniformity in mechanical behavior analysis is highlighted.

Transcripts

okay so let's uh I'd like to start off

talking about mechanical behaviour by

talking about Hookes law you may be

familiar with it maybe not by the way

we'll we'll get to it so I want to take

a look at

hooks

okay whether you know what that is just

yet or not and what you may have seen

say in high school something like this

you take a spring okay and you hang it

from maybe the edge of your desk oh god

hang it at your desk you put a weight on

it and then it gets a little bit longer

right and then you put say another

weight on there and it gets even longer

the weight being heavier so that's what

I'm trying to sketch here but another

way it on and that same spring elongates

while it's under load okay

so let's define a few things here that's

going to find the resting length or we

could call it in fact the original

length length and there's a usually we

use the letter L the script letter L

that I like to use and pull oh this is

zero has a subscript there L not you

might say and that's the original length

okay then the next thing you do is you

say okay well that's fine now what about

over here it's gotten longer what are

you gonna call that well that you could

call

yeah call that maybe the elongation

and you could use different letters for

that sometimes you use Delta L change in

length that makes a bit of sense or

often in this particular context X is

useiess book because it's a it's a

distance and it's elongated then what

you do is you take that all the data

that you've collected from different

loads you've put on the spring and you

want now force versus that elongation

and what you may know you've done this

is that it's quite nutrition you get a

straight line you get a linear

relationship and there's a slope to that

line we often call that slope K in fact

that's that K often gets a special name

and that is the spring constant

okay the reason for that being that of

course the equation for a straight line

is y equals MX plus B in a general form

B of 0 in this case and we're not

plotting y versus X R applauding after

the versus Dax

so it becomes F equals now instead of an

for the slope

we've got K X F equals KX and that is

actually a special equation and if you

live long enough isn't long enough ago

in the past I sound like to say you can

name things like straight lines after

yourself and so there you go that's

Hookes law but it's actually quite

important I joked but it is quite

important and again there is the spring

constant all right but there's a big

problem with this as a big problem with

this the problem is that sure it's fine

when you're looking at one particular

spring but what if I wanted to compare

two Springs and so I can see see that

the problem here is this if I took a

small sample this time this for

simplicity so we're not concerned with

the you know the mechanical of the shape

of the spring or anything I'm going to

just make it a cylinder it's a simple

cylinder that we're going to apply a

load to like this force

clear there we go okay and then we're

gonna get another another sample

cylinder as well but obviously it's a

lot bigger it's larger cross-sectional

area it's fatter if you will but we can

say apply the same force to that

the last thing I'll clarify here is that

we're gonna set this requirement that

both cylinders are made from the same

material from okay that's pretty mess

answer and everybody got messy there

same material and for example maybe it's

a 316l stainless steel or 6061-t6

aluminum whatever it's so it's exactly

the same material but what you can

appreciate is if I then take those two

samples and I plot them well I plotted

the relationship between force and

displacement sure it will be linear that

you know I mean for low amounts of load

it should be linear for most materials

especially metals but are they gonna be

the same that's let's say if I apply and

say that's the force I'm applying and

you know I want to see okay that's the

force I'm gonna apply well how much has

you know label these four you see

there's a and B how much is sample a

elongated and how much the sample B

along it which one has elongated more

for the same force for the same force

the narrow one the tiny little one you

would expect to elongate more and so we

did we'd be left with this type of thing

for sample a

this type of this relationship for

sample B and that's a problem because

then you end up with two different

spring constants K B and K a and so it

would seem as though it would seem as

though they have different properties

seems like different properties

that's a problem because we know they

were made from exactly the same material

so how are we gonna deal with this but

that'll be the next topic and it's gonna

be through stress and strain thank you