Chapter 1 | The Murder of Emmett Till | American Experience | PBS

So Music, this is the muddy backwoods tallahatchie river, where a weighted body was found alleged to be that of young emmett dill. I saw a hole which i presumed was a bullet hole and i could look through that hole and see daylight on the other side and i wonder, was it necessary to shoot it here? Is money mississippi the home of roy bryant? It was here that the chicago negro boy, emmett till, is alleged to have paid unwelcome attention to rory bryant’s most attractive wife.

When white women’s was on the streets, you had to get off the street.

That was a way of life, and all a white woman would have to say was that kind of looked at me me, so we’re talking about a way of life that, in this part of the country that was enforced by law.

This was the home of moe’s wright.

It was from this shack.

The state alleges emmett till was taken by roy bryant and j w milom.

The house was as dark.

There’s a thousand midnight she couldn’t see it was like a nightmare.

I mean it’s, i mean someone come and stand over you with a pills in one hand, a flashlight, you’re, 16 years old.

This is a terrifying experience. Music.

The teal case held the whole system up for inspection by the rest of the country and by the rest of the world.

It was the beginning of the focusing on the problems between the races in the deep south that culminated in the ultimate civil rights battles of the of of the rest of the 50s and and into the 60s.

I think black people -‘s reaction – was so visceral and i think it was probably more than anything else in terms of the mass civil rights movement, the spark that launched it.

Everybody knew we were under attack and that attack was symbolized by the attack on a 14 year old boy Music when one drives through the lowest hills and looks out at the sweep of those fields below flat as a pancake.

As far as the eye can see it’s breathtaking those who have not been to the delta find themselves gasping at the site as they come over the lowest hills and see that expanse of flat agricultural land.

It was the summer of 1955 when emmett till arrived in mississippi from chicago.

His family had worked cotton for generations, but this trip would be emmett’s, introduction to the delta known as the most southern place on earth.

This is mississippi.

Today, a situation exists in mississippi. That is, unlike the situation in most states in the nation.

In some sections of the state, there is a preponderance of colored citizens.

This situation has brought problems.

It has created challenges, but most important of all.

It has inspired a social system to meet the challenge in every community.

In mississippi there is segregation of the races drinking fountains are segregated, restrooms are segregated, the local theater is segregated in any way said anything that they didn’t.

Like you, didn’t disagree with them on a whole.

You just didn’t do that.

If a white person did something to you, you had no recourse at all.

People disappeared. We don’t know what happened to them.

They just disappeared Music.

In the 75 years before emmett till set foot in mississippi, more than 500 black people had been lynched in the state.

Most were men who had been accused of associating with white women.

Music part of that culture was that the women were put on pedestals and they were some sort of uh idealization of of whatever it means to be a woman or to be female.

There was an almost uh, irrational fear of black men, uh, as if every black man was ready to attack or rape, a white woman.

If you gave him a chance, i can remember when my father died sammy the black man he worked for him.

Was there and i threw my arms around his neck and he pulled away from me.

He could not have that.

You know physical show of affection of sharing grief or whatever black men did not touch white women, Music, many white southerners, perhaps most deep south southerners, had convinced themselves that black people were relatively happy in their in their segregated relationships with white people, most white people. I think had had convinced themselves that this was a defensible social system in which they lived.

I had a cousin that was living in mississippi and was walking down.

The side walked down near downtown and tunica and didn’t get off the side, walking the man slapped him and knocked him off the sidewalk, and he got up instead of killing the white man like he wanted.

He just started walking and never stopped until he got to memphis and never stopped until he got up to chicago Music.

Hundreds of thousands of black people fled mississippi for chicago in the years between the world wars.

One way: train fare of 11 10 took them to a different world, neighborhoods and schools were segregated, but the city offered a kind of freedom.

Black mississippians could only dream about chicago was a land of promise and they thought that milk and honey was everywhere, and so it was a lot of excitement leaving the south leaving the cotton fields.

You could hold your head up in chicago Music.

Mamie carthan arrived in chicago at the age of two.

An only child. Young mamie was the hope of her family of former sharecroppers.

She graduated from high school at the top of her class and became one of the first black women in town to hold a civil service job in 1940 mamie married soldier louis till and one year later, their son emmett was born in 1945.

Mamie got word that private till had died in europe.

All she received of his possessions was a signet ring inscribed with his initials lt emmett.

Her only child was four years old.

Childhood case of polio left him with a stutter, but by the time he was a teenager, emmett till had grown into a cocky self assured boy who loved to be the center of attention.

When we first met, we went gym uh in mr long’s gym period.

I remember emmett raising his shirt up to about his navel and start making his belly roll just waves of fat rolling and it just broke us up.

I mean the whole gym went crazy.

He was that kind of kid anything going on. He’s in the middle of all of it and he just loved loved play ball.

He just loved jokes.

He would pay people to tell him jokes.

If there was a group there emmy was in front and he was the lively one.

He was the one that everybody kind of looked to, not your one leader, you .

As found on YouTube

Email Dyno Backdoor Pass

Under Stencil Wipe: Understanding and Optimization – Tech 2 Tech

Music, hello and welcome to tech tech presented by kaisen, where we will explore common cleaning questions and answers. Let’s get started.

I would like to introduce kaizen’s own Jeff, Dearing, welcome and thank you for joining us today for another techto te with Team Kaizen.

The topic of conversation today is under stencil, wiping both basically understanding the process and working with a handful of parameters to make sure we can optimize that process and before we get started.

I’d like to mention that there’s a four part series on our techto te website: titled stencil cleaning series that if you need some more in depth data based on the information I share with you today, please feel free to reference those shorts on The internet that are on our techto Tech website, again titled stencil cleaning series.

Let’s get started.

Our outline today is going to be a four simple step approach.

First, we need to select a chemistry that dissolves the flux.

Next, we’ll need to optimize.

The settings for each chemistry tested based on the fluid delivery system as as well as the paper solvent interaction.

Third, we always want to consider ending the wipe sequence with a vacuum and last we want to make sure we test the chemistry in the printer. Instead of off the printer using SPI testing, as well as an under stencil inspection, what do we need to know about printer chemistry? Well, here we have a simple breakdown of the features of our three options.

Our first option is a solvent based chemistries.

You need to keep in mind that some solvents evaporate quicker than others.

They usually provide a wider range of flux.

Solubility.

You also need to remember the excuse me.

The paper can impact the solvent usage and the biggest issue with solvent based chemistries is not adjusting the default settings, which is something we’ll touch on as we go through this presentation now, moving into water based chemistries, some materials split, the evaporation is not everything And can vary lot again.

The paper absorption rate affects the usage of the water based chemistries.

There’s also a risk of leaving the chemistry on the stencil which can mix with the paste and could cause cross contamination.

This also holds true, when you add water to IPA again, the biggest issue with water based chemistries is not adjusting the default settings and, lastly, we have Ipa. Ipa does not not dissolve many pastes and can cause a gooey mess.

It typically affects the first print after a wipe, sometimes it evaporates too quickly, which also can lead to it consuming more volume than an engineered chemistry.

When you take into account the amount needed to wet the paper and clean plus the evaporation.

Next, I’d like to talk about the common problems.

An engineered chemistry can fix first and foremost, some common solder, paste defects, poor gasketing from flux, buildup and insufficient.

Pads engineered chemistry can help increase your prints between wipes from 1 to two to.

However many we can determine when we work to optimize your process.

It also increases the number of prints between off the printer cleaning and, most importantly, engineered chemistry provides a more consistent transfer, efficiency between prints and wipes.

All of this, even without Nano coatings on your stencils, now let’s take a closer look at some of the limitations of IPA.

We’re all familiar with the no clean fluxes that have changed compet compositions. Lately they can leave flux streaking on the stencil.

They can damage the printer rails and tooling and in some cases they cause a gooey mess.

Like I mentioned before, IPA dries fast again having a higher consumption rate.

It requires more volume to wet the stencil, like I had mentioned before, it can be a poor cleaner.

It dries out apertures, which results in poor print release; it’s not recommended by printer companies and, most importantly, it’s, flammable back to our outline now that we’ve discussed selecting the chemistry that helps dissolve the flux I’d like to discuss In a little more detail, optimizing the settings for each of those chemistries tested based on the fluid delivery system and the paper solvent interaction.

It’s important to understand that 90 of users never change the factory default settings.

What you see here is a simple diagram, as well as a chart of some of the parameters that come default from the factory, a few of them that we like to address there’s, always a Min and a Max, but we need to make sure we Address the wipe speeds for the wet the dry and the vacuum, we also want to address vacuum pressure, and one of our goals is to decrease that wipe frequency.

Without doing all of that, your process can lead to inconsistent printing.

Now, if there’s too much solvent, you’re going to leave excess solvent behind on the bot bottom side of the stencil, and it can mixx with the paste causing defects too little solvent.

Will not effectively remove the flux from the bottom of the stencil or apertures again causing defects. Please keep in mind.

Solvent should be consistent along the entire length of the paper and remember that those settings can be adjusted based on the solvent paper interaction.

The tech tip here is good coverage on your paper is typically somewhere between 15 and 20 millim in width.

Now I’d like to take a closer look at some of our printer solvent settings.

When discussing water based solvents, the default solvent, speed stroke, we found to be effective as well as the default index speed, but when it comes to the solvent speed bar it’s much more effective at 5 in per second, which saturates that paper at approximately 16 Mm in width, this indicates that water based materials may need to have more rotations than solvents to be effective, but less than standard default settings.

When we look at solvent based chemistries, the default solvent, speed, stroke again is effective, as well as the index speed, but we also need to pay attention to the solvent speed bar with solvent based chemistries.

It’s much more effective at 3 in per second, which again saturates the paper at a 16 mm width, but this indicates that solvent based materials may need to have less rotations than the default settings or water based solvents to be effective, and an important note Here is the paper should be damp and not wet.

Now, when discussing the wipe sequence in paper, there’s a few facets that we need to look at one most printers allow the operator to program a variety of wipe.

Sequ is using a wet stroke, a dry stroke and a vacuum step.

The paper that we’re using you need to keep in mind that not all papers are the same. You have your synthetic fiber paper.

You have your natural fiber paper and it’s extremely important to remember that the type of wipe paper can critically impact the speed and the wipe sequence of the process.

You’re handling again.

Why so much talk about the paper? It’s where the paper meets the chemistry or the rubber meets the road.

Each paper type has different absorption rates and properties, and each chemistry has different surface tensions and densities.

Using the right paper can cut your paper and chemistry consumption in half.

Now, let’s look at some simple printer setup tips.

First, we want to check the blades for proper condition and ensure that they’re tight.

Next, let’s confirm the board.

Support is efficient by pressing down on the stencil in solid areas above the board always remember that the wipe recipe for before and after a pause should be wet, wet dry, dry vacuum vacuum. We also need to keep in mind and try to keep the board snug against its rails and, as you,’ll hear again, ending in a vac step is highly recommended to make sure we optimize the process.

So, back to our four simple steps: we’ve selected, the chemistry we’ve discussed, optimizing the settings for each chemistry.

Third, we always want to consider ending the wipe sequence with a vacuum and, lastly, we want to make sure to test the chemistry in the printer rather than off the printer, using SPI testing and under stencil inspection.

When it comes comes to SBI testing, we need to keep in mind two things: qfps show the best.

Bleedout Tendencies and bgas show the transfer issues.

Bgas are considered the best test for improvements when changing variables and if further optimizations needed it can be achieved by exploring alternative wi sequences.

In conclusion, for today, it’s important that we understand the key elements in the solder past printing process and how they interact to make sure we select the ideal chemistry after we do that it’s, pertinent that we maint maintain control of the process.

It’s where most of the rework comes from and it’s where the money is in smt, keep in mind if specific defects continue to occur, follow logical, troubleshooting guidelines and when push comes to shove, make sure you consult your local Kaizen team.

Member for technical support before we finish today, I’d like to make sure we review some frequently asked questions on this topic.

First, how do I know what the best chemistry is for my printer? Well, alcohol is for sure, not the best one. The first step to finding the best chemistry is to get in touch with your Cleaning Supplier, and they can send you in the right direction.

Second, our default settings process parameter recommendations from the manufacturer.

They’re made to change.

You will save a lot of money if you spend time making sure you optimize those settings, as we discussed in this slideshow third, do engineered solvents clean better than IPA.

Absolutely an engineered solvent will help your printer process compareed to the IPA every time, and why is it unsafe to use IPA well, first and foremost, it’s very flammable, but it also does not solub solubilize Pac, as well as in engineered chemistry with current pce Technology thanks again for your time and please feel free to reach out to your team Kaizen representative with any further technical questions or your cleaning needs thanks Jeff, and thank you guys for tuning in this techto Tech session.

If you would like to discuss this topic further or have any questions not answered in the session, please contact your local kaisen regional manager or send an email to Tech, the number2 teen com, and we will have someone fall up with you as soon as possible and If you like, this video, be sure to follow us on our social media platforms for more expert planing content, Music .

As found on YouTube

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Is Dating Different When You’re Blind? | Ep 1 – Blind Love

People say It’d be easier to date a blind, person 39, cause they just care about your inner beauty, Well we do Man I basically just want someone to spend. Time with And we don’t, Woman Like breakups really suck and they hurt and whatever – and I am jaded Blind people problems laughing Becoming independent – was an act of rebellion.

I am looking for love.

Is that kind of a weird thing? Looking for love, xylophone, music? 42 inaudible My name is Mario I’m 30, years old single and totally blind Whoop that was a.

Good one.

Mario.

I have four foster.

Children that I’m in the process of adopting So, it becomes a quadruple challenge to actually date.

We trying to change that.

Thank you, Jesus laughing! What’s going on Open the door jerk, What’s up man, What’s goin’on Nothin?’much! Well, I have another date.

What happened to the last one Too much going on The blindness thing was a stretch for her Some women shy away from me because of my blindness Phone I’ve turned off the voiceover. Now the phone is officially no longer in blind man set Yeah.

Take a look at her picture.

Oh this is her Mario Yeah Mm.

Hmm, I love mm.

Hmm.

I like her.

She’s a good choice.

I do not list in my online profiles that I am totally blind sighing, So you just never talked about it at all.

No, Oh I mean it’s a throw of the dice man with anybody.’cause.

I mean me personally. You know you can’t just be showing up blind and whatever You know what I’m saying You can’t show up blind without notification, is what you’re saying Hell? No, You can’t just show up blind man.’Cause the part about it.

Is I go back to it? They are so into me.

They’re so into the picture.

We put up, compliments, galore and the minute I say I’m blind it’s like it’s, rejection Like everybody.

Nobody wants to be rejected, Nobody wants to be rejected, but to be rejected the moment you say I’m blind is frustrating, But it’s a whole new world for them.

So you think I should tell her ahead of time.

Uh traffic noises Jenna, We’re setting up Soloman’s tinder right now.

My name is Solomon.

I’m 32 years old.

Looking for a woman that wakes up and goes, You know what I really wan na be with a blind person for the rest of my life Jenna. I am a 32 year old man.

Looking for a woman, I can bring home to mom Solomon Ugh Woman want the typical guy that takes the kids to T ball practice and I’m not able to do all that on my own.

How many dates you been going on Zero? I’ve been on one in like 2 years.

They just think.

Oh, this is a blind guy.

You know Screw this, But if they, you know see me in action, see me in person, personality and all that They realize like.

Where has this guy been all my life? You know Man, Maybe you should put something in there.

You know lost my sight.

Two years ago, hasn’t slowed me down much Yeah that’s good.

I’m, still the great guy. I was before somber music, Except now I’m fresh out.

I was 20 years old.

I had perfect vision before that Every few years it would get a little worse.

The doc told me that it would probably go in my 40s.

I was like Oh by then I’ll be married.

I’ll have kids and all that Two years ago, when I was 30, it went all the way.

Totally blind Can not see anything And I’m, like Oh crap.

How am I gon na find somebody now No hits No hits on Tinder It’s been days already.

No hits No hits jazzy music Joni.

I still have an idea of my dream. Guy in my head, Dance Instructor Off to the left a little bit, Joni Good style, darker, hair, thin, not too tall’cause, I’m short And dancing is like it:’s, non, negotiable, laughing Dance, Instructor Inside turn and down.

My name is Joni and I’m 32 years old.

Do you have anything going on this weekend? Uh yeah, I’m gon na bring my friend James to a dance social Tomorrow, Yeah Yeah, yeah, yeah, fun, fun fun.

He’s kind of been asking me out.

You know The person I date definitely has to be able to dance, or at least be willing to try it So like he decided that he was gon na learn how to dance to, like you know, impress me And so Showing some interest by doing a little Class here and there – Yes exactly So he’s trying, I feel, like I’m better at dancing than I am at relationships.

I’ve got ta figure out.

If I like him or not,’cause, I don’t really know laughing.

I’m, not sure, Like you know, if I say yes, then that’s gon na mean more than I’m ready to like Mm commitment.

Yeah At this point, settling down and making a commitment is kinda, scary, Dance, Instructor All right, bring it in laughing Good job.

Awesome thanks. Jonas Have a good day.

Thank you.

You’re cool Jenna.

Did you call my girlfriend? Actually, I’d love to call her now I’m kinda nervous Wan na call her now Yeah Jenna was on a mission to set me up with her friend A girl named Angelina.

It’s good, because I met her the other night and she wants to hang out again.

Well, it won’t be awkward, No yeah Angelina On Phone, Hello, Angelina, it’s: Solomon Jenna,’s friend from the other night, How you doin’Angelina Hi.

How are you Doin,’good, how you doin’Angelina Good Good, hey? I was callin 39.

I’m, not sure if you have plans tonight or anything, but a few of us were gon na go out and I want to see if you want to come Angelina.

That sounds like fun All right.

I will see you there Angelina For sure I’ll see you too. All right have a good day.

Angelina, You too, All right, bye, Yeah, laughing Yeah Got a date.

I’m a blind person, but I got beauty secrets laughing.

I’m obsessed with what I look like You have such pretty hair.

It’s jet black and curly, Say mh hm.

I got lucky with the genes.

My teeth have to feel a certain way.

My face has to feel a certain way.

An outfit I put on has to feel a certain way Being attractive, makes it easier for people to look past.

My blindness – Hey move over some’cause. Your glare from the bathroom is messing up the color Okay, I’d be glad to move over once.

I know which direction: Oh, I’m sorry move to the left laughing There.

You go.

Stop Okay, button that bottom button; Yeah that looks nice.

What the pants look like.

Mario, You, don’t think the pants are too fitted Nora.

I can’t even see the pants’cause.

You got the phone down on the floor.

Mario All right, you on your way, Nora, Yeah, upbeat, music Beauty is something people develop an understanding of based on what they see in popular culture.

I take pride in my appearance, but I’m not as worried about looking like movie stars, as maybe somebody who sees them every day, because I can’t see them car honking, Ooh hell You honking that horn, like a lunatic This, a nice neighborhood, Get In the car, Mario Turn the music down Nora, Okay, you had me waitin’over here for a long time again, Mario Mario, That’s, all right, you got gas. My brother will pretty much just call me and say I’m going on a date, and I want my sister to take me.

A lot.

Are those glasses, clean, Mario, I don’t know you tell me you’re the one with the vision laughing.

Oh, my God, Mario, Let me get your opinion on her.

I did not tell her that I was blind.

Why didn’t you tell her.

You was blind How you know she wan na deal with a blind man.

Mario, If you were in her position, I would get mad And I would leave.

Are you serious? Yes, I really would, But you can’t catfish people, You have a blind brother and you about to get cussed out.

Well, I’m about to get cussed out you tryin’to catfish the girl. She don’t know what’s goin’on She gon na leave you at the table, upbeat music.

I brought James to our dance, social, hoping he won.’t feel too self conscious about dancing in front of people.

Okay, I’ve changed my mind laughing It’s, gon na be okay.

James has been taking dance classes.

To impress me Thank you, sir, And it’s working laughing muffled conversation.

She’s here.

Look at Solomon,’s face he’s getting nervous, Mario In the pursuit of love, Nora Get it together.

We’re going in here blind date.

Blind date, I’ve only dated white girls.

I’m. Sorry, I’m.

Being honest, am I allowed to just say that Ever dated a blind person before No, I have not.

No, I have not.

No Would you.

I’ve never held a girl.’s hand before Never kissed a girl before.

Well then, how did you know that you wanted to marry him? I have no idea Like breakups really suck and they hurt and whatever – and I am jaded Sorry, I lost ya.

I feel like I just didn’t land that very well.

She hates the word relationship, It does definitely freak me out Should probably come visit me.

I should Solomon Once they know that I’m blind.

When I met him, no idea I’m instantly in the friend zone. She might be on the prowl for a husband, which you know I’m, on the prowl for a wife, laughing .

As found on YouTube

Instagram Elite Mentoring™ by Niklas Pedde

Understanding the Area Moment of Inertia

Let's say we have a plank of wood which we
would like to use to cross a canal. It has a rectangular cross-section and so
we could either use it like we have done here, or we could rotate it onto its side, like
this. Intuitively we can tell that the plank will
be stiffer if the load is applied to the shorter side of the cross-section. Some cross-sections are much more efficient
at resisting bending than others.

The further the material is spread from the
bending axis, the stiffer a cross-section tends to be. The cross-section on the right has more material
located far from the bending axis and so is better at resisting bending, even though both
cross-sections have the same area. This concept of resistance to bending can
be quantified by calculating the area moment of inertia, which is also sometimes called
the second moment of area. The area moment of inertia reflects how the
area of a cross-section is distributed relative to a particular axis, and so is a measure
of how much resistance the cross-section has to bending. The I-beam locates the majority of the material
as far as possible from the bending axis, and so is a very efficient cross-section. This is why it is so commonly used in construction. In this video we're going to take a detailed
look at the area moment of inertia. Let's start by seeing how it can be calculated
for an arbitrary cross-section like the one shown here.

The first thing to note is that the area moment
of inertia is not a unique property of a cross-section. It quantifies the resistance to bending about
a particular axis, and so its value changes depending on where we place this reference
axis. We can approximate the area moment of inertia
of a cross-section by splitting it into small elements. Each element contributes to the total area
moment of inertia by a quantity equal to its area dA multiplied by Y^2, where Y
is the distance to the reference axis, which is the X axis in this case.

We can sum up the values for all of the small
elements to obtain the area moment of inertia for the entire cross-section. It is denoted by the letter I, and because
the X axis is our reference axis, we will give it the subscript X. We can define the area moment of inertia more
precisely using integration, like this. And if the Y axis is our reference axis, we
can calculate I-Y in the same way. The area moment of inertia has the unit of
length to the fourth power, and because of the squared term it is always a positive quantity. Let's work through an example where we calculate
the I-X and I-Y values for a rectangular cross-section, using those equations. We can consider that the rectangle is made
up of multiple thin strips which each have a height equal to dy. Each strip has an area equal to b * dy,
which gives us the following integral. The limits of the integral are from the bottom
to the top of the rectangle, so from negative h over 2, to h over 2.

If we solve this definite integral we end
up with I-X being equal to b multiplied by h to the power 3, divided by 12. We can obtain the equation for I-Y by switching
the height and the width terms. So there you go, we just calculated the area
moment of inertia for a rectangular cross-section using integration. But calculating integrals can be difficult,
so to help us out area moment of inertia equations are often provided in reference texts for
a range of common shapes. A few examples are shown here. Usually the equations are provided for centroidal
axes, which are axes that pass through the centroid of the cross section. Remember that the centroid is the geometric
centre of a cross-section. X-C and Y-C are called the centroidal axes. So what do we do if we need to obtain an area
moment of inertia equation for an axis that is not a centroidal axis, but it is too difficult
to calculate it using integration? Fortunately there is a method we can use to
calculate an adjusted area moment of inertia for any axis that is parallel to a centroidal
axis, like this one.

The adjusted area moment of inertia I-X can
be calculated by summing the moment of inertia of the centroidal axis, and the product
of the cross-sectional area A and the square of distance d between both axes. This is called the parallel axis theorem. It's useful because we can take the equations
found in reference texts for centroidal axes and adjust them to obtain the moments of inertia
for any parallel axes.

Let's look at an example. Earlier we determined an equation for the
area moment of inertia of a rectangular cross-section, about an axis passing through the centroid
of the rectangle. We can use the parallel axis theorem to calculate
I for an axis shifted to the bottom of the cross-section. All we have to do is take the equation for
I we derived earlier, and add the area of the rectangle b*h multiplied by the distance
h over 2, squared. One useful property of area moments of inertia
is that they can be added to and subtracted from each other. This means we can calculate the value of I
for a shape like this by taking I for section A and subtracting I for section B. If we are calculating I based on equations,
we need to make sure that the equations we are using correspond to the correct reference
axis.

In the case of this T section for example,
we can't just add the I values for two rectangles given in a textbook because these equations
are for centroidal axes only, and the reference axis doesn't pass through the centroid of
the top rectangle. Fortunately we can use the parallel axis theorem
to overcome this. If h and b are the height and the width of
the rectangles, the distance between the reference axis and the centroid of rectangle 2 is (h1+h2)/2. We can calculate I for the composite cross-section
using the parallel axis theorem, like this. This is another reason the parallel axis theorem
is really important.

It gives us an easy way to calculate area
moments of inertia for all sorts of different composite shapes. It's worth clarifying at this point that the
area moment of inertia should not be confused with the mass moment of inertia, which is
a parameter used to describe the resistance of a body to changes in rotational velocity. There are similarities in the way the two
parameters are calculated, but they have different units and completely different uses. So, when might you actually need to use the
area moment of inertia? Well it is a particularly important parameter
for the analysis of beams and columns.

This equation, for example, defines the deflection
of a beam for an applied bending moment M. You will notice that the term E-I appears
in many equations. It is called "flexural rigidity". It quantifies the resistance of a beam to
bending. As we have just seen, I is the resistance
due to the geometry of the beam cross-section. But the stiffness of the beam material also
contributes to its total resistance to bending, and is captured by Young's modulus E.
Flexural rigidity also appears in the analysis of columns, where it can be used to calculate
the critical buckling load. Another important parameter related to the
area moment of inertia that appears in the analysis of columns is the radius of gyration. It represents the theoretical distance at
which we could condense the entire area of a cross-section into a narrow strip, to get
the same moment of inertia as the original cross-section.

It can be calculated using this equation. So far we have only calculated the area moments
of inertia relative to the X and Y axes. We can calculate a third area moment of inertia
for a reference axis that is perpendicular to the plane of the cross-section. This quantity is called the polar moment of
inertia, and it is usually denoted using the letter J. It represents the resistance of
the cross-section to twisting about the reference axis.

It is calculated in the same way as I-X and
I-Y, but using the distance Rho to the axis, rather than the perpendicular distance to
the X or Y axes. We can calculate J using integration, like
this. Rho squared is equal to X squared plus Y squared,
so we can expand the equation. And by doing a little more work we can figure
out that J is equal to I-X plus I-Y. This is known as the perpendicular axis theorem. The polar moment of inertia is mainly relevant
for situations which involve torsion.

Check out my video on torsion if you want
to learn more about this. To fully master the area moment of inertia
there is one last thing we need to cover, which is the rotation of the reference axes. We can use the transformation equations shown
here to calculate moments of inertia for rotated axes. The I-X-Y term in these equations is the product
of inertia, and it is calculated using this equation. Rotating the reference axes works in a very
similar way to transformation of stresses, which I covered in a video on stress transformation. This is because like stress, the area moment
of inertia is a tensor quantity. In fact you can even use Mohr's circle to
determine moments of inertia for rotated axes, like you can with stresses. It has I on the horizontal axis, and the product
of inertia I-XY on the vertical axis.

You can use Mohr's circle to find the principal
moments of inertia, which are the maximum and minimum values for any angle of rotation. That's it for what ended up being quite a
detailed review of the area moment of inertia. Thanks for watching, and don't forget to subscribe
if you haven't already!.

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