02 Hooke's Law
Summary
TLDR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
🔍 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。
🔬 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。
Mindmap
Keywords
💡胡克定律
💡弹簧常数
💡位移
💡直线方程
💡材料性质
💡应力
💡应变
💡线性关系
💡力
💡原始长度
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
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