Genetic drift, bottleneck effect and founder effect

WEBVTT
Kind: captions
Language: en

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- [Voiceover] We've
already made several videos

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over evolution,

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and just to remind ourselves

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what evolution is talking about,

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it's the change in heritable traits

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of a population over generations.

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And a lot of times, you'll
hear people say evolution

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and Natural Selection
really in the same breath,

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but what we wanna make a little
bit clear in this video is

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that Natural Selection is
one mechanism of evolution.

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It's the one most talked about

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because it is viewed as
the primary mechanism.

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Natural Selection.

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But what we're gonna talk about

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in this video is another
mechanism called Genetic Drift.

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So there's Natural Selection,

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and there is Genetic Drift.

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Now we've done many videos
on Natural Selection,

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but it's this idea that you
have variation in a population,

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you have different heritable traits,

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and I'm gonna depict those
with different colors here.

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We have a population of
living circles here, (laughs)

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and they could come in
blue or maybe magenta.

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Maybe they come in another variation too,

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maybe there is yellow circles,

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and Natural Selection is all about

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which of these traits are
most fit for the environment

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so that they can reproduce.

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So there might be something
about being, say, blue,

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that allows those circles
to reproduce faster,

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or to be less likely to
be caught by predators,

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or to be able to stalk prey better.

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Even if they're only slightly
more likely to reproduce,

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over time, over many generations,

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their numbers will increase and dominate,

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and the other numbers are less likely,

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or the other trait is
less likely to survive,

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and so we will have this Natural Selection

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for that blue trait.

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So this is all about traits
being the fittest traits.

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Now Genetic Drift is also
change in heritable traits

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of a population over generations,

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but it's not about the
traits that are most fit

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for an environment are the
ones that necessarily survive.

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Genetic Drift is really about random.

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Random changes.

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Random changes,

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and a good example of that
I have right over here

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that we got from,

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I'll give proper credit,

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this is from OpenStax College Biology,

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and this shows how Genetic
Drift could happen.

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So right over here,

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I'm showing a very small
population of 10 rabbits,

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and we have the gene for color,

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and we have two versions of that gene,

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or we could call them two alleles.

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You have the capital B version,

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and you have the lower case B,

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and capital B is dominant.

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This is kind of a very Mendelian example

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that we're showing here.

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And so if you have two
of lower case genes,

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two of the white alleles,

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you're going to be white.

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If you have two of the brown
alleles, the capital Bs,

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you're going to be brown,

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and if you're a heterozygote,

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you're still going to be brown.

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So as you can see here, there
are several heterozygotes

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in this fairly small population.

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But if you just count the capital Bs

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versus the lower case Bs,

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you see that we have an
equal amount of each.

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And so the frequency,

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if you were to pick a random
allele from this population,

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you're just as likely

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to pick a capital B than a lower case B.

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Even though the phenotype,

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you see a lot more brown,

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but these six brown here
have both the upper case B

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and the lower case B.

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Now let's say they're in a population

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where whether you are brown
or whether you are white,

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it confers no advantage.

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There's no more likelihood
of surviving and reproducing

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if you're brown than white,

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but just by chance,

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by pure random chance,

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the five bunnies on the top are the ones

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that are able to reproduce,

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and the five bunnies on
the bottom are not the ones

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that are able to reproduce.

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And you might be saying hey,
why did I pick those top five?

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I didn't pick them, I'm
just giving an example.

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It could've been the bottom five.

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It could've been only these two,

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or the only two white ones were the ones

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that were able to reproduce.

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It's by pure random chance,

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or it could be because of
traits that are unrelated

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to the alleles that we are talking about.

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But from the point of
view of these alleles,

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it looks like random chance.

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And so in the next generation,

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those five rabbits reproduce

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and you could have a situation like this,

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and just by random chance, as you can see,

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the capital B allele
frequency has increased

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from 50% of the alleles
in the population to 70%.

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And then it could be
another random chance,

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and I'm not saying this is
necessarily going to happen.

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It could happen the other way.

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It could happen even though

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that first randomness happened,

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maybe now all of a sudden
this white rabbit is able

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to reproduce a lot,

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but maybe not.

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Maybe these two brown rabbits

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that are homozygous for
the dominant trait are able

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to reproduce,

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and one again it has
nothing to do with fitness.

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And so they're able to reproduce,

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and then all of a sudden,

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the white allele is completely
gone from the environment.

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And the reason why this happened isn't

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because the white allele somehow
makes the bunnies less fit.

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In fact, it might have
even conferred a little bit

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of an advantage.

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It might have been,

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from the environment that the
bunnies are in point of view,

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it might have even been a better trait,

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but because of random chance,

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it disappears from the population.

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And the general idea
with the Genetic Drift,

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so once again, just to compare,

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Natural Selection,

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you are selecting,

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or the environment is selecting traits

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that are more favorable for reproduction,

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while Genetic Drift is random changes.

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Random changes in reproduction
of the population.

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Now, as you can imagine,

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I just gave an example with 10 bunnies,

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and what I just described
is much more likely

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to happen with small populations.

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So much more likely.

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More likely

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with small populations.

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And we have videos on
statistics on Khan Academy,

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but the likelihood of this happening

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with 10 bunnies versus the likelihood

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of what I just described happening

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with 10 million bunnies is very different.

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It's much more likely to
happen with a small population.

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So a lot of the contexts
of Genetic Drift are

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when people talk about small populations.

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In fact, many times Biologists are worried

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about small populations specifically

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because of Genetic Drift.

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For random reasons, you
could have less diversity,

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less variation in your population,

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and even favorable traits
could be selected for

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by random chance.

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There's two types of Genetic
Drift that are often called out

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that cause extreme
reductions in population,

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and significantly reduce the populations.

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One is called the Bottleneck Effect.

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Let me write this down.

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So the Bottle,

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Bottleneck,

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the Bottleneck Effect,

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and then the other is
called the Founder Effect.

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Do that over here.

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The Founder,

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Founder Effect.

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They are both ideas

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where you have significant
reduction in population

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for slightly different reasons.

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Bottleneck Effect is you have
some major disaster or event

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that kills off a lot of the population,

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so only a little bit of the
population is able to survive.

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And the reason why it's
called Bottleneck is imagine

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if you had a bottle here.

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If you had a bottle here and, I dunno,

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inside of that bottle,

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you had marbles of different colors.

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So you have some yellow marbles,

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you have some magenta marbles,

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you have some, I don't know, blue marbles.

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These are the colors
that I tend to be using.

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You have some blue marbles,

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so you have a lot of variation
in your original population.

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But if you think about
pouring them out of a bottle,

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maybe somehow there's some major disaster,

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and only two of these survive,

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or let's say only four of these survive,

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and so you could view that as,

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"Well, what are the marbles
that are getting poured

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"out of the bottle?"

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It's really just a metaphor.

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Obviously, we're not putting populations

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of things in bottles.

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But after that disaster,

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only a handful survive,

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and they might not have any traits

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that are in any way more
desirable or more fit

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for the environment than everything else,

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but they just by random chance,

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because of this disaster,

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they are the ones that survived.

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And so all of a sudden,

00:08:47.535 --> 00:08:49.758
you have a massive reduction
not only in the population,

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but also in the variation
in that population,

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and many alleles might
have even disappeared,

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and so you have an extreme form

00:08:57.595 --> 00:09:01.161
of Genetic Drift actually occurring.

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Another example is Founder Effect,

00:09:05.390 --> 00:09:09.411
which is the same idea of a
population becoming very small,

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but the Founder Effect isn't
because of a natural disaster.

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Let's say you had a population.

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Once again, you have a
lot of different alleles

00:09:17.777 --> 00:09:18.610
in that population.

00:09:18.610 --> 00:09:20.943
You have a lot of variation,

00:09:21.895 --> 00:09:26.231
you have a lot of variation
in that population.

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So let me just keep coloring it.

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You have a lot of variation
in this population,

00:09:31.210 --> 00:09:32.467
and let's say that,

00:09:32.467 --> 00:09:35.001
you know, they're all
hanging out in their region,

00:09:35.001 --> 00:09:36.668
and maybe, you know,

00:09:38.334 --> 00:09:40.365
they are surrounded by mountains.

00:09:40.365 --> 00:09:42.104
I'm just making this up as I go,

00:09:42.104 --> 00:09:44.864
but let's say a couple of
these blue characters were

00:09:44.864 --> 00:09:46.716
out walking one day,

00:09:46.716 --> 00:09:49.024
and they maybe get separated

00:09:49.024 --> 00:09:50.762
from the rest of their population.

00:09:50.762 --> 00:09:54.611
Maybe they discover a little
undiscovered mountain pass,

00:09:54.611 --> 00:09:57.925
and they go settle a new
population someplace.

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So that's why it's called
the Founder Effect.

00:09:59.410 --> 00:10:01.542
These are the founders
of a new population,

00:10:01.542 --> 00:10:03.363
and once again, by random chance,

00:10:03.363 --> 00:10:04.921
they just have a lot less variation.

00:10:04.921 --> 00:10:05.985
They're a smaller population

00:10:05.985 --> 00:10:08.106
and they happen to be disproportionately

00:10:08.106 --> 00:10:10.011
or all blue in this case,

00:10:10.011 --> 00:10:14.178
and so now this population
is going to (mumbles)

00:10:15.630 --> 00:10:17.800
Just the process of this was Genetic Drift

00:10:17.800 --> 00:10:21.341
where many alleles will have disappeared

00:10:21.341 --> 00:10:24.314
because you have such a small
population of blues here.

00:10:24.314 --> 00:10:26.140
And also because you have
such a small population,

00:10:26.140 --> 00:10:29.150
you're likely to have
even more Genetic Drift.

00:10:29.150 --> 00:10:31.110
So it's a really interesting
thing to think about.

00:10:31.110 --> 00:10:33.345
Evolution and Natural
Selection are often talked

00:10:33.345 --> 00:10:34.524
about hand in hand,

00:10:34.524 --> 00:10:38.490
but Natural Selection isn't the
only mechanism of Evolution.

00:10:38.490 --> 00:10:40.038
You also have Genetic Drift,

00:10:40.038 --> 00:10:40.904
which is really about,

00:10:40.904 --> 00:10:43.018
not selecting for favorable traits,

00:10:43.018 --> 00:10:44.935
it is about randomness.


Rotational kinetic energy

WEBVTT
Kind: captions
Language: en

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- [Voiceover] When a major
league baseball player throws

00:00:02.103 --> 00:00:05.637
a fast ball, that ball's
definitely got kinetic energy.

00:00:05.637 --> 00:00:07.581
We know that cause if you get in the way,

00:00:07.581 --> 00:00:09.429
it could do work on
you, that's gonna hurt.

00:00:09.429 --> 00:00:10.583
You gotta watch out.

00:00:10.583 --> 00:00:14.212
But here's my question: does
the fact that most pitches,

00:00:14.212 --> 00:00:15.855
unless you're throwing a knuckle ball,

00:00:15.855 --> 00:00:19.157
does the fact that most
pitches head toward home plate

00:00:19.157 --> 00:00:21.805
with the baseball spinning
mean that that ball

00:00:21.805 --> 00:00:24.197
has extra kinetic energy?

00:00:24.197 --> 00:00:26.571
Well it does, and how
do we figure that out,

00:00:26.571 --> 00:00:28.986
that's the goal for this video.

00:00:28.986 --> 00:00:31.267
How do we determine what the rotational

00:00:31.267 --> 00:00:33.647
kinetic energy is of an object?

00:00:33.647 --> 00:00:35.770
Well if I was coming at
this for the first time,

00:00:35.770 --> 00:00:37.803
my first guest I'd say okay,

00:00:37.803 --> 00:00:40.561
I'd say I know what regular
kinetic energy looks like.

00:00:40.561 --> 00:00:42.875
The formula for regular kinetic energy is

00:00:42.875 --> 00:00:45.954
just one half m v squared.

00:00:45.954 --> 00:00:48.567
So let's say alright, I want
rotational kinetic energy.

00:00:48.567 --> 00:00:50.963
Let me just call that k rotational

00:00:50.963 --> 00:00:52.497
and what is that gonna be?

00:00:52.497 --> 00:00:54.964
Well I know for objects that are rotating,

00:00:54.964 --> 00:00:58.764
the rotational equivalent of
mass is moment of inertia.

00:00:58.764 --> 00:01:01.461
So I might guess alright instead of mass,

00:01:01.461 --> 00:01:04.354
I'd have moment of inertia
cause in Newton's second law

00:01:04.354 --> 00:01:06.813
for rotation I know that
instead of mass there's

00:01:06.813 --> 00:01:09.091
moment of inertia so maybe I replace that.

00:01:09.091 --> 00:01:12.188
And instead of speed
squared, maybe since I have

00:01:12.188 --> 00:01:15.284
something rotating I'd
have angular speed squared.

00:01:15.284 --> 00:01:16.898
It turns out this works.

00:01:16.898 --> 00:01:20.014
You can often derive, it's
not really a derivation,

00:01:20.014 --> 00:01:22.516
you're just kind of guessing
educatedly but you could

00:01:22.516 --> 00:01:25.797
often get a formula for the
rotational analog of some

00:01:25.797 --> 00:01:29.780
linear formula by just
substituting the rotational analog

00:01:29.780 --> 00:01:32.415
for each of the variables,
so if I replaced mass with

00:01:32.415 --> 00:01:35.247
rotational mass, I get
the moment of inertia.

00:01:35.247 --> 00:01:37.721
If I replace speed with rotational speed,

00:01:37.721 --> 00:01:40.743
I get the angular speed and
this is the correct formula.

00:01:40.743 --> 00:01:43.303
So in this video we needed
to ride this cause that

00:01:43.303 --> 00:01:44.913
is not really a derivation,
we didn't really

00:01:44.913 --> 00:01:47.720
prove this, we just showed
that it's plausible.

00:01:47.720 --> 00:01:50.111
How do we prove that
this is the rotational

00:01:50.111 --> 00:01:52.991
kinetic energy of an
object that's rotating

00:01:52.991 --> 00:01:54.349
like a baseball.

00:01:54.349 --> 00:01:56.997
The first thing to recognize
is that this rotational

00:01:56.997 --> 00:01:59.684
kinetic energy isn't really a new kind of

00:01:59.684 --> 00:02:02.301
kinetic energy, it's
still just the same old

00:02:02.301 --> 00:02:05.721
regular kinetic energy for
something that's rotating.

00:02:05.721 --> 00:02:07.051
What I mean by that is this.

00:02:07.051 --> 00:02:09.820
Imagine this baseball
is rotating in a circle.

00:02:09.820 --> 00:02:13.320
Every point on the baseball
is moving with some speed,

00:02:13.320 --> 00:02:15.462
so what I mean by that is
this, so this point at the top

00:02:15.462 --> 00:02:18.613
here imagine the little
piece of leather right here,

00:02:18.613 --> 00:02:20.367
it's gonna have some speed forward.

00:02:20.367 --> 00:02:23.485
I'm gonna call this mass
M one, that little piece

00:02:23.485 --> 00:02:27.188
of mass right now and I'll
call the speed of it V one.

00:02:27.188 --> 00:02:29.744
Similarly, this point on
the leather right there,

00:02:29.744 --> 00:02:32.288
I'm gonna call that M two,
it's gonna be moving down

00:02:32.288 --> 00:02:35.713
cause it's a rotating circle,
so I'll call that V two

00:02:35.713 --> 00:02:38.370
and points closer to the
axis are gonna be moving

00:02:38.370 --> 00:02:41.027
with smaller speed so
this point right here,

00:02:41.027 --> 00:02:43.779
we'll call it M three, moving
down with a speed V three,

00:02:43.779 --> 00:02:46.771
that is not as big as V two or V one.

00:02:46.771 --> 00:02:48.082
You can't see that very well,

00:02:48.082 --> 00:02:51.587
I'll use a darker green
so this M three right here

00:02:51.587 --> 00:02:54.921
closer to the axis, axis
being right at this point

00:02:54.921 --> 00:02:58.781
in the center, closer to the
axis so it's speed is smaller

00:02:58.781 --> 00:03:01.411
than points that are
farther away from this axis,

00:03:01.411 --> 00:03:03.373
so you can see this is kinda complicated.

00:03:03.373 --> 00:03:05.539
All points on this baseball
are gonna be moving with

00:03:05.539 --> 00:03:08.126
different speeds so points
over here that are really

00:03:08.126 --> 00:03:10.796
close to the axis, barely moving at all.

00:03:10.796 --> 00:03:12.946
I'll call this M four and it would be

00:03:12.946 --> 00:03:15.093
moving at speed V four.

00:03:15.093 --> 00:03:17.676
What we mean by the
rotational kinetic energy is

00:03:17.676 --> 00:03:19.892
really just all the regular
kinetic energy these

00:03:19.892 --> 00:03:23.846
masses have about the center
of mass of the baseball.

00:03:23.846 --> 00:03:26.681
So in other words, what
we mean by K rotational,

00:03:26.681 --> 00:03:29.455
is you just add up all of these energies.

00:03:29.455 --> 00:03:32.021
You have one half, this
little piece of leather

00:03:32.021 --> 00:03:33.737
up here would have some kinetic energy

00:03:33.737 --> 00:03:37.487
so you do one half M
one, V one squared plus.

00:03:38.415 --> 00:03:41.050
And this M two has some kinetic energy,

00:03:41.050 --> 00:03:43.152
don't worry that it points downward,

00:03:43.152 --> 00:03:45.954
downward doesn't matter for
things that aren't vectors,

00:03:45.954 --> 00:03:49.258
this V gets squared so
kinetic energy's not a vector

00:03:49.258 --> 00:03:51.779
so it doesn't matter that
one velocity points down

00:03:51.779 --> 00:03:54.443
cause this is just the
speed and similarly,

00:03:54.443 --> 00:03:58.950
you'd add up one half M
three, V three squared,

00:03:58.950 --> 00:04:00.517
but you might be like this is impossible,

00:04:00.517 --> 00:04:02.926
there's infinitely many
points in this baseball,

00:04:02.926 --> 00:04:05.388
how am I ever going to do this.

00:04:05.388 --> 00:04:07.379
Well something magical is about to happen,

00:04:07.379 --> 00:04:09.526
this is one of my favorite
little derivations,

00:04:09.526 --> 00:04:12.133
short and sweet, watch what happens.

00:04:12.133 --> 00:04:15.067
K E rotational is really just the sum,

00:04:15.067 --> 00:04:17.661
if I add all these up
I can write is as a sum

00:04:17.661 --> 00:04:21.494
of all the one half M V
squares of every point

00:04:22.457 --> 00:04:25.416
on this baseball so imagine
breaking this baseball

00:04:25.416 --> 00:04:27.756
up into very, very small pieces.

00:04:27.756 --> 00:04:30.072
Don't do it physically but
just think about it mentally,

00:04:30.072 --> 00:04:33.039
just visualize considering
very small pieces,

00:04:33.039 --> 00:04:35.919
particles of this baseball
and how fast they're going.

00:04:35.919 --> 00:04:38.938
What I'm saying is that
if you add all of that up,

00:04:38.938 --> 00:04:41.359
you get the total
rotational kinetic energy,

00:04:41.359 --> 00:04:42.967
this looks impossible to do.

00:04:42.967 --> 00:04:44.552
But something magical is about to happen,

00:04:44.552 --> 00:04:45.766
here's what we can do.

00:04:45.766 --> 00:04:48.352
We can rewrite, see the problem here is V.

00:04:48.352 --> 00:04:50.755
All these points have a different speed V,

00:04:50.755 --> 00:04:52.711
but we can use a trick,
a trick that we love

00:04:52.711 --> 00:04:55.125
to use in physics, instead
of writing this as V,

00:04:55.125 --> 00:04:57.773
we're gonna write V as, so
remember that for things

00:04:57.773 --> 00:05:01.564
that are rotating, V
is just R times omega.

00:05:01.564 --> 00:05:04.133
The radius, how far from the axis you are,

00:05:04.133 --> 00:05:06.885
times the angular velocity,
or the angular speed

00:05:06.885 --> 00:05:09.358
gives you the regular speed.

00:05:09.358 --> 00:05:12.145
This formula is really
handy, so we're gonna replace

00:05:12.145 --> 00:05:16.185
V with R omega, and this
is gonna give us R omega

00:05:16.185 --> 00:05:18.352
and you still have to
square it and at this point

00:05:18.352 --> 00:05:19.993
you're probably thinking
like this is even worse,

00:05:19.993 --> 00:05:21.079
what do we do this for.

00:05:21.079 --> 00:05:24.023
Well watch, if we add this
is up I'll have one half M.

00:05:24.023 --> 00:05:26.848
I'm gonna get an R squared
and an omega squared,

00:05:26.848 --> 00:05:28.958
and the reason this is
better is that even though

00:05:28.958 --> 00:05:32.626
every point on this baseball
has a different speed V,

00:05:32.626 --> 00:05:35.491
they all have the same
angular speed omega,

00:05:35.491 --> 00:05:38.315
that was what was good about
these angular quantities

00:05:38.315 --> 00:05:41.618
is that they're the same for
every point on the baseball

00:05:41.618 --> 00:05:43.870
no matter how far away
you are from the axis,

00:05:43.870 --> 00:05:46.042
and since they're the
same for every point I can

00:05:46.042 --> 00:05:48.634
bring that out of the
summation so I can rewrite

00:05:48.634 --> 00:05:51.609
this summation and bring
everything that's constant

00:05:51.609 --> 00:05:54.818
for all of the masses out
of the summation so I can

00:05:54.818 --> 00:05:58.220
write this as one half times the summation

00:05:58.220 --> 00:06:01.803
of M times R squared
and end that quantity,

00:06:02.782 --> 00:06:06.597
end that summation and just
pull the omega squared out

00:06:06.597 --> 00:06:08.565
because it's the same for each term.

00:06:08.565 --> 00:06:11.444
I'm basically factoring
this out of all of these

00:06:11.444 --> 00:06:13.857
terms in the summation, it's like up here,

00:06:13.857 --> 00:06:15.548
all of these have a one half.

00:06:15.548 --> 00:06:17.487
You could imagine factoring out a one half

00:06:17.487 --> 00:06:18.985
and just writing this whole quantity as

00:06:18.985 --> 00:06:22.135
one half times M one V one squared plus

00:06:22.135 --> 00:06:24.167
M two V two squared and so on.

00:06:24.167 --> 00:06:26.055
That's what I'm doing
down here for the one half

00:06:26.055 --> 00:06:28.615
and for the omega squared,
so that's what was good

00:06:28.615 --> 00:06:31.077
about replacing V with R omega.

00:06:31.077 --> 00:06:32.540
The omega's the same for all of them,

00:06:32.540 --> 00:06:33.816
you can bring that out.

00:06:33.816 --> 00:06:35.514
You might still be
concerned, you might be like,

00:06:35.514 --> 00:06:37.993
we're still stuck with
the M in here cause you've

00:06:37.993 --> 00:06:39.990
got different Ms at different points.

00:06:39.990 --> 00:06:42.160
We're stuck with all
these R squareds in here,

00:06:42.160 --> 00:06:44.628
all these points at the
baseball are different Rs,

00:06:44.628 --> 00:06:46.328
they're all different
points from the axis,

00:06:46.328 --> 00:06:48.558
different distances from
the axis, we can't bring

00:06:48.558 --> 00:06:51.449
those out so now what do we
do, well if you're clever

00:06:51.449 --> 00:06:53.792
you recognize this term.

00:06:53.792 --> 00:06:56.615
This summation term is
nothing but the total moment

00:06:56.615 --> 00:06:59.296
of inertia of the object.

00:06:59.296 --> 00:07:01.628
Remember that the moment
of inertia of an object,

00:07:01.628 --> 00:07:04.394
we learned previously,
is just M R squared,

00:07:04.394 --> 00:07:06.410
so the moment of inertia of a point mass

00:07:06.410 --> 00:07:09.122
is M R squared and the moment of inertia

00:07:09.122 --> 00:07:12.483
of a bunch of point
masses is the sum of all

00:07:12.483 --> 00:07:15.402
the M R squareds and that's
what we've got right here,

00:07:15.402 --> 00:07:19.514
this is just the moment of
inertia of this baseball

00:07:19.514 --> 00:07:22.115
or whatever the object is,
it doesn't even have to be

00:07:22.115 --> 00:07:24.287
of a particular shape, we're gonna add all

00:07:24.287 --> 00:07:26.998
the M R squareds, that's
always going to be

00:07:26.998 --> 00:07:28.637
the total moment of inertia.

00:07:28.637 --> 00:07:30.925
So what we've found is
that the K rotational

00:07:30.925 --> 00:07:34.066
is equal to one half times this quantity,

00:07:34.066 --> 00:07:35.947
which is I, the moment of inertia,

00:07:35.947 --> 00:07:38.284
times omega squared and that's the formula

00:07:38.284 --> 00:07:40.004
we got up here just by guessing.

00:07:40.004 --> 00:07:41.850
But it actually works
and this is why it works,

00:07:41.850 --> 00:07:43.859
because you always get
this quantity down here,

00:07:43.859 --> 00:07:46.204
which is one half I omega
squared, no matter what

00:07:46.204 --> 00:07:47.676
the shape of the object is.

00:07:47.676 --> 00:07:49.420
So what this is telling
you, what this quantity

00:07:49.420 --> 00:07:52.346
gives us is the total
rotational kinetic energy

00:07:52.346 --> 00:07:55.666
of all the points on that
mass about the center

00:07:55.666 --> 00:07:58.591
of the mass but here's
what it doesn't give you.

00:07:58.591 --> 00:08:01.036
This term right here does not include

00:08:01.036 --> 00:08:03.451
the translational kinetic
energy so the fact that

00:08:03.451 --> 00:08:06.292
this baseball was flying
through the air does not

00:08:06.292 --> 00:08:08.142
get incorporated by this formula.

00:08:08.142 --> 00:08:10.264
We didn't take into account the fact that

00:08:10.264 --> 00:08:12.391
the baseball was moving through the air,

00:08:12.391 --> 00:08:13.976
in other words, we
didn't take into account

00:08:13.976 --> 00:08:16.791
that the actual center
of mass in this baseball

00:08:16.791 --> 00:08:19.200
was translating through the air.

00:08:19.200 --> 00:08:21.365
But we can do that easily
with this formula here.

00:08:21.365 --> 00:08:24.279
This is the translational kinetic energy.

00:08:24.279 --> 00:08:26.930
Sometimes instead of writing
regular kinetic energy,

00:08:26.930 --> 00:08:29.841
now that we've got two we
should specify this is really

00:08:29.841 --> 00:08:31.791
translational kinetic energy.

00:08:31.791 --> 00:08:34.361
We've got a formula for
translational kinetic energy,

00:08:34.361 --> 00:08:37.701
the energy something has due
to the fact that the center

00:08:37.701 --> 00:08:40.522
of mass of that object is
moving and we have a formula

00:08:40.522 --> 00:08:42.972
that takes into account the
fact that something can have

00:08:42.972 --> 00:08:45.494
kinetic energy due to its rotation.

00:08:45.494 --> 00:08:48.316
That's this K rotational,
so if an object's rotating,

00:08:48.316 --> 00:08:50.483
it has rotational kinetic energy.

00:08:50.483 --> 00:08:52.718
If an object is translating it has

00:08:52.718 --> 00:08:54.500
translational kinetic energy,

00:08:54.500 --> 00:08:56.515
i.e. if the center of mass is moving,

00:08:56.515 --> 00:08:59.986
and if the object is
translating and it's rotating

00:08:59.986 --> 00:09:02.435
then it would have those both
of these kinetic energies,

00:09:02.435 --> 00:09:04.948
both at the same time and
this is the beautiful thing.

00:09:04.948 --> 00:09:08.430
If an object is translating
and rotating and you want

00:09:08.430 --> 00:09:11.390
to find the total kinetic
energy of the entire thing,

00:09:11.390 --> 00:09:14.004
you can just add these two terms up.

00:09:14.004 --> 00:09:17.147
If I just take the translational
one half M V squared,

00:09:17.147 --> 00:09:20.573
and this would then be the
velocity of the center of mass.

00:09:20.573 --> 00:09:22.157
So you have to be careful.

00:09:22.157 --> 00:09:23.749
Let me make some room
here, so let me get rid

00:09:23.749 --> 00:09:25.130
of all this stuff here.

00:09:25.130 --> 00:09:28.741
If you take one half M,
times the speed of the center

00:09:28.741 --> 00:09:31.655
of mass squared, you'll
get the total translational

00:09:31.655 --> 00:09:33.239
kinetic energy of the baseball.

00:09:33.239 --> 00:09:36.386
And if we add to that the
one half I omega squared,

00:09:36.386 --> 00:09:39.184
so the omega about the
center of mass you'll get

00:09:39.184 --> 00:09:43.688
the total kinetic energy, both
translational and rotational,

00:09:43.688 --> 00:09:46.624
so this is great, we can
determine the total kinetic energy

00:09:46.624 --> 00:09:49.889
altogether, rotational
motion, translational motion,

00:09:49.889 --> 00:09:52.580
from just taking these two terms added up.

00:09:52.580 --> 00:09:54.049
So what would an example of this be,

00:09:54.049 --> 00:09:55.796
let's just get rid of all this.

00:09:55.796 --> 00:09:59.180
Let's say this baseball,
someone pitched this thing,

00:09:59.180 --> 00:10:02.582
and the radar gun shows that
this baseball was hurled

00:10:02.582 --> 00:10:04.799
through the air at 40 meters per second.

00:10:04.799 --> 00:10:07.452
So it's heading toward home
plate at 40 meters per second.

00:10:07.452 --> 00:10:09.858
The center of mass of
this baseball is going

00:10:09.858 --> 00:10:12.551
40 meters per second toward home plate.

00:10:12.551 --> 00:10:15.094
Let's say it's also, someone
really threw the fastball.

00:10:15.094 --> 00:10:18.107
This thing's rotating
with an angular velocity

00:10:18.107 --> 00:10:20.190
of 50 radians per second.

00:10:22.264 --> 00:10:24.376
We know the mass of a
baseball, I've looked it up.

00:10:24.376 --> 00:10:28.781
The mass of a baseball
is about 0.145 kilograms

00:10:28.781 --> 00:10:31.795
and the radius of the baseball,
so a radius of a baseball

00:10:31.795 --> 00:10:35.388
is around seven centimeters,
so in terms of meters that

00:10:35.388 --> 00:10:38.865
would be 0.07 meters, so
we can figure out what's

00:10:38.865 --> 00:10:41.240
the total kinetic energy,
well there's gonna be

00:10:41.240 --> 00:10:43.202
a rotational kinetic
energy and there's gonna be

00:10:43.202 --> 00:10:45.048
a translational kinetic energy.

00:10:45.048 --> 00:10:47.875
The translational kinetic
energy, gonna be one half

00:10:47.875 --> 00:10:50.835
the mass of the baseball
times the center of mass speed

00:10:50.835 --> 00:10:53.993
of the baseball squared which
is gonna give us one half.

00:10:53.993 --> 00:10:57.626
The mass of the baseball was
0.145 and the center of mass

00:10:57.626 --> 00:11:00.650
speed of the baseball is 40,
that's how fast the center

00:11:00.650 --> 00:11:02.712
of mass of this baseball is traveling.

00:11:02.712 --> 00:11:06.712
If we add all that up we
get 116 Jules of regular

00:11:06.712 --> 00:11:08.894
translational kinetic energy.

00:11:08.894 --> 00:11:11.246
How much rotational
kinetic energy is there,

00:11:11.246 --> 00:11:13.281
so we're gonna have
rotational kinetic energy

00:11:13.281 --> 00:11:16.088
due to the fact that the
baseball is also rotating.

00:11:16.088 --> 00:11:19.587
How much, well we're gonna
use one half I omega squared.

00:11:19.587 --> 00:11:22.484
I'm gonna have one half, what's
the I, well the baseball is

00:11:22.484 --> 00:11:26.328
a sphere, if you look up the
moment of inertia of a sphere

00:11:26.328 --> 00:11:29.665
cause I don't wanna have
to do summation of all

00:11:29.665 --> 00:11:32.999
the M R squareds, if you
do that using calculus,

00:11:32.999 --> 00:11:34.873
you get this formula.

00:11:34.873 --> 00:11:36.995
That means in an algebra
based physics class

00:11:36.995 --> 00:11:38.900
you just have to look this
up, it's either in your book

00:11:38.900 --> 00:11:41.635
in a chart or a table or you
could always look it up online.

00:11:41.635 --> 00:11:45.763
For a sphere the moment of
inertia is two fifths M R squared

00:11:45.763 --> 00:11:48.619
in other words two fifths
the mass of a baseball

00:11:48.619 --> 00:11:50.459
times the raise of the baseball squared.

00:11:50.459 --> 00:11:53.627
That's just I, that's the
moment of inertia of a sphere.

00:11:53.627 --> 00:11:56.235
So we're assuming this
baseball is a perfect sphere.

00:11:56.235 --> 00:11:59.358
It's got uniform density,
that's not completely true.

00:11:59.358 --> 00:12:00.954
But it's a pretty good approximation.

00:12:00.954 --> 00:12:03.019
Then we multiply by this omega squared,

00:12:03.019 --> 00:12:04.754
the angular speed squared.

00:12:04.754 --> 00:12:07.137
So what do we get, we're
gonna get one half times

00:12:07.137 --> 00:12:11.222
two fifths, the mass of
a baseball was 0.145.

00:12:11.222 --> 00:12:13.284
The radius of the baseball
was about, what did we say,

00:12:13.284 --> 00:12:18.027
.07 meters so that's .07
meters squared and then finally

00:12:18.027 --> 00:12:20.494
we multiply by omega squared
and this would make it

00:12:20.494 --> 00:12:23.238
50 radians per second and we square

00:12:23.238 --> 00:12:25.821
it which adds up to 0.355 Jules

00:12:28.705 --> 00:12:31.416
so hardly any of the
energy of this baseball

00:12:31.416 --> 00:12:33.034
is in its rotation.

00:12:33.034 --> 00:12:36.449
Almost all of the energy is
in the form of translational

00:12:36.449 --> 00:12:38.521
energy, that kinda makes sense.

00:12:38.521 --> 00:12:40.895
It's the fact that this
baseball is hurling toward

00:12:40.895 --> 00:12:43.901
home plate that's gonna
make it hurt if it hits you

00:12:43.901 --> 00:12:46.049
as opposed to the fact
that it was spinning when

00:12:46.049 --> 00:12:48.545
it hits you, that doesn't
actually cause as much damage

00:12:48.545 --> 00:12:50.705
as the fact that this
baseball's kinetic energy

00:12:50.705 --> 00:12:54.466
is mostly in the form of
translational kinetic energy.

00:12:54.466 --> 00:12:57.154
But if you wanted the total
kinetic energy of the baseball,

00:12:57.154 --> 00:12:59.135
you would add both of these terms up.

00:12:59.135 --> 00:13:02.641
K total would be the
translational kinetic energy

00:13:02.641 --> 00:13:04.937
plus the rotational kinetic energy.

00:13:04.937 --> 00:13:09.104
That means the total kinetic
energy which is the 116 Jules

00:13:10.046 --> 00:13:12.546
plus 0.355 Jules which give us

00:13:14.425 --> 00:13:15.592
116.355 Jules.

00:13:18.343 --> 00:13:20.590
So recapping if an object is both rotating

00:13:20.590 --> 00:13:23.156
and translating you can
find the translational

00:13:23.156 --> 00:13:26.787
kinetic energy using one
half M the speed of the

00:13:26.787 --> 00:13:29.564
center of mass of that
object squared and you can

00:13:29.564 --> 00:13:32.071
find the rotational
kinetic energy by using

00:13:32.071 --> 00:13:34.552
one half I, the moment of inertia.

00:13:34.552 --> 00:13:36.161
We'll infer whatever shape it is,

00:13:36.161 --> 00:13:38.640
if it's a point mass
going in a huge circle

00:13:38.640 --> 00:13:41.035
you could use M R
squared, if it's a sphere

00:13:41.035 --> 00:13:43.635
rotating about its center
you could use two fifths

00:13:43.635 --> 00:13:46.209
M R squared, cylinders
are one half M R squared,

00:13:46.209 --> 00:13:49.007
you can look these up
in tables to figure out

00:13:49.007 --> 00:13:52.032
whatever the I is that
you need times the angular

00:13:52.032 --> 00:13:56.319
speed squared of the object
about that center of mass.

00:13:56.319 --> 00:13:58.423
And if you add these
two terms up you get the

00:13:58.423 --> 00:14:01.423
total kinetic energy of that object.


More on Normal force (shoe on floor)

WEBVTT
Kind: captions
Language: en

00:00:00.196 --> 00:00:02.215
- [Voiceover] Check out
this fine looking sneaker

00:00:02.215 --> 00:00:04.826
right here, we're gonna
use this shoe to illustrate

00:00:04.826 --> 00:00:07.220
some more challenging
normal force problems

00:00:07.220 --> 00:00:09.829
and we're gonna take this
as an opportunity to discuss

00:00:09.829 --> 00:00:11.935
a lot of the misconceptions
that people have about

00:00:11.935 --> 00:00:13.129
the normal force.

00:00:13.129 --> 00:00:15.312
So one misconception is that people forget

00:00:15.312 --> 00:00:17.332
normal force is a contact force.

00:00:17.332 --> 00:00:20.317
You only have a normal
force when two surfaces

00:00:20.317 --> 00:00:22.534
are in contact, so when
the shoe's in contact with

00:00:22.534 --> 00:00:24.844
the floor, there will be
a normal force on the shoe

00:00:24.844 --> 00:00:26.529
and a normal force on the floor.

00:00:26.529 --> 00:00:28.680
Or if the shoe were in
contact with the wall,

00:00:28.680 --> 00:00:30.575
there would be a normal force on the wall

00:00:30.575 --> 00:00:32.056
and a normal force on the shoe.

00:00:32.056 --> 00:00:34.150
But if the shoe were just
falling through the air,

00:00:34.150 --> 00:00:36.000
here's what happens for a lot of people.

00:00:36.000 --> 00:00:37.650
Let's say the shoe's just falling,

00:00:37.650 --> 00:00:39.088
and you got a question
and the question said,

00:00:39.088 --> 00:00:41.230
draw the forces that
are exerted on the shoe

00:00:41.230 --> 00:00:42.757
while it's falling through the air.

00:00:42.757 --> 00:00:44.785
People get so used to having normal forces

00:00:44.785 --> 00:00:48.661
that they make a mistake, they
do this, they say alright,

00:00:48.661 --> 00:00:50.172
let me draw it over here.

00:00:50.172 --> 00:00:52.196
They say there's a gravitational
force and that's just fine.

00:00:52.196 --> 00:00:54.417
There will be a gravitational
force, there's always gravity,

00:00:54.417 --> 00:00:56.148
the Earth's always pulling down,

00:00:56.148 --> 00:00:58.337
and it pulls down with an amount mg.

00:00:58.337 --> 00:01:00.734
But they're so used to
having normal forces,

00:01:00.734 --> 00:01:03.780
I mean normal forces pop up
in so many different questions

00:01:03.780 --> 00:01:05.769
it's almost just like a reflex,

00:01:05.769 --> 00:01:08.435
people just automatically put,
whoop, there's a normal force

00:01:08.435 --> 00:01:10.068
there's gotta be a normal force, right?

00:01:10.068 --> 00:01:11.509
There's always a normal force?

00:01:11.509 --> 00:01:13.571
But there isn't always a
normal force, if the shoe

00:01:13.571 --> 00:01:16.783
is not in contact with the
surface, you don't have a normal

00:01:16.783 --> 00:01:20.160
force, it's not until this
shoe makes it to the ground

00:01:20.160 --> 00:01:22.968
or touches another surface
that you'll have that

00:01:22.968 --> 00:01:25.117
normal force, so if we
stick this shoe right here

00:01:25.117 --> 00:01:27.429
and we let it rest on the ground,

00:01:27.429 --> 00:01:29.465
now you'll have a normal
force and that normal force

00:01:29.465 --> 00:01:32.676
will point up, and this
is what people wanna say

00:01:32.676 --> 00:01:35.145
and it's true when the
surfaces are in contact

00:01:35.145 --> 00:01:36.586
but if they're not in contact,

00:01:36.586 --> 00:01:38.149
you don't have a normal force.

00:01:38.149 --> 00:01:40.452
And then here's another misconception,

00:01:40.452 --> 00:01:43.295
people think the normal
force is always equal to mg,

00:01:43.295 --> 00:01:45.269
because again, it's equal to mg

00:01:45.269 --> 00:01:48.035
in so many different
scenarios that people just

00:01:48.035 --> 00:01:50.523
wanna say well it's always
equal to mg, and again,

00:01:50.523 --> 00:01:51.889
it's just like a reaction.

00:01:51.889 --> 00:01:54.602
People see normal force, they
just automatically replace it

00:01:54.602 --> 00:01:57.116
with mg, and that'll be
true in the simple case,

00:01:57.116 --> 00:01:59.546
but I'll show you coming up
how that's not gonna be true

00:01:59.546 --> 00:02:01.438
and what you do if it's not true.

00:02:01.438 --> 00:02:03.791
So for instance if we wanted
to find what's the normal force

00:02:03.791 --> 00:02:06.182
if this shoe has a mass m,

00:02:06.182 --> 00:02:08.407
so let's assume that
the shoe has a mass m,

00:02:08.407 --> 00:02:10.225
what would the normal force be?

00:02:10.225 --> 00:02:11.751
We can use Newton's second law,

00:02:11.751 --> 00:02:13.854
we can always use Newton's second law,

00:02:13.854 --> 00:02:17.901
so we'll say that acceleration
equals the net force

00:02:17.901 --> 00:02:21.294
divided by the mass, and in this case,

00:02:21.294 --> 00:02:23.223
since these are vertical forces,

00:02:23.223 --> 00:02:24.867
I'm gonna consider the acceleration

00:02:24.867 --> 00:02:26.523
in the vertical direction
and the net force

00:02:26.523 --> 00:02:28.169
in the vertical direction.

00:02:28.169 --> 00:02:30.233
And so what is the acceleration
for the shoe vertically

00:02:30.233 --> 00:02:33.118
if it's just sitting here in
a room, sitting on the ground,

00:02:33.118 --> 00:02:35.717
at rest and not changing it's motion,

00:02:35.717 --> 00:02:37.807
not changing it's velocity,
the acceleration's just gonna

00:02:37.807 --> 00:02:40.038
be zero, so the vertical acceleration,

00:02:40.038 --> 00:02:42.637
acceleration, excuse me, should be zero.

00:02:42.637 --> 00:02:46.219
For the net force, I've
got an upward normal force,

00:02:46.219 --> 00:02:48.040
so I'm gonna make that positive,

00:02:48.040 --> 00:02:50.675
if fn represents the
magnitude of the normal force,

00:02:50.675 --> 00:02:54.224
this would be positive Fn, I'm
just gonna put positive here

00:02:54.224 --> 00:02:56.031
even though I don't really
need it but to show you

00:02:56.031 --> 00:02:58.204
that it's upward, we're gonna
consider upward to be positive

00:02:58.204 --> 00:03:00.630
and then I've got this
downward gravitational force,

00:03:00.630 --> 00:03:04.053
and if mg represents the size
of the gravitational force,

00:03:04.053 --> 00:03:06.737
I'm gonna put a negative
here to represent that that

00:03:06.737 --> 00:03:09.746
gravitational force is
down, and then I divide by

00:03:09.746 --> 00:03:13.966
the mass of the shoe and
if I do this I get that

00:03:13.966 --> 00:03:16.426
these two forces, this net
force divided by the mass

00:03:16.426 --> 00:03:19.194
has to be zero, according
to Newton's second law.

00:03:19.194 --> 00:03:22.085
But I can multiply both sides
by the mass and if I do that,

00:03:22.085 --> 00:03:24.685
the left hand side is still zero,

00:03:24.685 --> 00:03:27.609
and I'll get that this is
equal to the normal force

00:03:27.609 --> 00:03:31.359
minus mg, so I'll have
normal force minus mg,

00:03:32.978 --> 00:03:35.087
and if I finally solve
for the normal force,

00:03:35.087 --> 00:03:38.806
I'll get that the normal
force is gonna equal mg,

00:03:38.806 --> 00:03:42.095
and a lot of people are like
yeah I already knew that, duh.

00:03:42.095 --> 00:03:45.067
Normal force is always equal
to mg, but it's only equal

00:03:45.067 --> 00:03:47.298
to mg in this case because
those were the only two forces,

00:03:47.298 --> 00:03:49.435
look at the assumptions we made.

00:03:49.435 --> 00:03:51.829
Only two forces were
the normal force and the

00:03:51.829 --> 00:03:54.427
gravitational force and we
assumed that the acceleration

00:03:54.427 --> 00:03:57.270
was zero, if you relax
any of those requirements,

00:03:57.270 --> 00:04:00.614
normal force is no longer
going to be equal to mg.

00:04:00.614 --> 00:04:02.679
And it was on a horizontal surface,

00:04:02.679 --> 00:04:05.321
if you relax that requirement,
again, there's no reason

00:04:05.321 --> 00:04:07.584
to think this has to
be in the Y direction,

00:04:07.584 --> 00:04:09.487
you could have normal
forces in the X direction.

00:04:09.487 --> 00:04:12.407
So let's slowly, one point at a time,

00:04:12.407 --> 00:04:14.687
try to relax some of these
requirements and see what

00:04:14.687 --> 00:04:16.999
that does to the normal force.

00:04:16.999 --> 00:04:19.935
In other words, what if we
just added another force?

00:04:19.935 --> 00:04:22.205
What if we let the shoe
sit here on the ground

00:04:22.205 --> 00:04:23.888
and I push down on it?

00:04:23.888 --> 00:04:26.899
So I'm pushing down on this
shoe, I'm gonna say I'm pushing

00:04:26.899 --> 00:04:31.656
down with a force, I'll
just call it F1, so a force

00:04:31.656 --> 00:04:34.586
of magnitude, F1, and
it's pointing downward,

00:04:34.586 --> 00:04:36.574
how would that change this now?

00:04:36.574 --> 00:04:38.297
So this is, we're stepping it
up, this is gonna be a little

00:04:38.297 --> 00:04:40.649
harder, what do we do?

00:04:40.649 --> 00:04:42.863
Well the acceleration is still
zero, let's say it's still

00:04:42.863 --> 00:04:45.292
just sitting there, so we
don't have to do anything

00:04:45.292 --> 00:04:47.019
with the left hand side that's still zero,

00:04:47.019 --> 00:04:49.081
multiplying by m still makes that zero,

00:04:49.081 --> 00:04:50.817
but now up here in this force up here,

00:04:50.817 --> 00:04:53.119
I'm gonna have another
force, I'm gonna have F1,

00:04:53.119 --> 00:04:55.639
that points down so in my force diagram

00:04:55.639 --> 00:04:58.862
I'd have another force
that points down, F1,

00:04:58.862 --> 00:05:01.465
that means I'd have to
subtract it when I find

00:05:01.465 --> 00:05:03.574
the net vertical force, I'd have F1,

00:05:03.574 --> 00:05:06.473
this would be a negative F1 right here,

00:05:06.473 --> 00:05:09.482
and when I solve for Fn
I'd add mg to both sides

00:05:09.482 --> 00:05:12.331
to cancel it and then
I'd add F1 to both sides

00:05:12.331 --> 00:05:14.769
to cancel this F1, this negative F1,

00:05:14.769 --> 00:05:16.686
and I'd get mg plus F1.

00:05:17.653 --> 00:05:19.600
So I get the normal
force is gonna be bigger,

00:05:19.600 --> 00:05:22.319
bigger by an amount F2
and that makes sense,

00:05:22.319 --> 00:05:25.043
if you push down on a, oh no F2, wow,

00:05:25.043 --> 00:05:28.873
F1, sorry about that, it's
gonna be bigger by an amount F1,

00:05:28.873 --> 00:05:32.309
so if I push down with an
extra 10 Newtons of force,

00:05:32.309 --> 00:05:34.161
there's more pressure, right?

00:05:34.161 --> 00:05:36.306
That makes sense, the
pressure between the ground

00:05:36.306 --> 00:05:37.877
and the shoe is gonna be
greater, you're squashing

00:05:37.877 --> 00:05:40.556
these two surfaces together
with greater force,

00:05:40.556 --> 00:05:42.535
so the ground's gotta
push up to keep the shoe

00:05:42.535 --> 00:05:45.789
out of the surface, that's
what this normal force does,

00:05:45.789 --> 00:05:48.058
it exerts a force to keep the
object out of the surface,

00:05:48.058 --> 00:05:51.390
to keep the object from
penetrating that surface,

00:05:51.390 --> 00:05:54.975
so if I push down on an
object, into a surface,

00:05:54.975 --> 00:05:57.904
that normal force increases
and it increases by the amount

00:05:57.904 --> 00:06:00.005
you're pushing down, so that makes sense.

00:06:00.005 --> 00:06:01.984
If you had an upward
force, let's say you had an

00:06:01.984 --> 00:06:04.289
upward force, someone's
pulling up on the shoe

00:06:04.289 --> 00:06:06.892
while you push down, you're
fighting over the shoe,

00:06:06.892 --> 00:06:09.331
you're wrestling over it with
somebody because they just,

00:06:09.331 --> 00:06:12.343
they love the shoe, they
recognize the beauty of the shoe,

00:06:12.343 --> 00:06:15.475
well crafted shoe, so if
there's an F2 pointing up,

00:06:15.475 --> 00:06:17.491
we now have another force in our diagram,

00:06:17.491 --> 00:06:20.303
that force would point
up, we would call this F2,

00:06:20.303 --> 00:06:22.323
over here how would this change?

00:06:22.323 --> 00:06:24.009
Again, still acceleration to zero,

00:06:24.009 --> 00:06:26.078
but I'd have an upward
force now so I'd have to add

00:06:26.078 --> 00:06:29.505
F2 vertically because
that's another force,

00:06:29.505 --> 00:06:31.531
I'd have a plus F2 right here,

00:06:31.531 --> 00:06:34.007
and then over here when I solve for this,

00:06:34.007 --> 00:06:37.229
I add mg to both sides,
I add F1 to both sides,

00:06:37.229 --> 00:06:40.291
and I have to subtract F2 from both sides

00:06:40.291 --> 00:06:43.791
so now I'd have Fn is mg plus F1 minus F2,

00:06:44.719 --> 00:06:47.973
this also makes sense,
if you pull up on a shoe,

00:06:47.973 --> 00:06:50.980
you're relieving some of the
pressure between the shoe

00:06:50.980 --> 00:06:53.245
and the other surface,
the shoe and the floor.

00:06:53.245 --> 00:06:56.627
So if I pull up with 20
Newtons I'm gonna reduce

00:06:56.627 --> 00:06:58.899
the normal force by 20
Newtons because I'm relieving

00:06:58.899 --> 00:07:01.781
some of that pressure between
the shoe and the floor.

00:07:01.781 --> 00:07:03.510
Let's make it even harder.

00:07:03.510 --> 00:07:06.064
Let's make this thing scary,
sometimes you get really

00:07:06.064 --> 00:07:08.871
crazy problems and you don't
know what to do, let's say,

00:07:08.871 --> 00:07:12.051
we have another force, let's
say this force is gonna be

00:07:12.051 --> 00:07:15.968
a diagonal force, so
we're gonna pull this way.

00:07:17.624 --> 00:07:21.541
That was not a well drawn
force, let me draw it like this.

00:07:21.541 --> 00:07:24.016
So we got a force this way at an angle.

00:07:24.016 --> 00:07:26.516
Now we're talking, this is F3.

00:07:28.182 --> 00:07:32.677
F3, at an angle of, we'll
call it Fi, so the angle from

00:07:32.677 --> 00:07:35.344
this horizontal line here is Fi.

00:07:37.131 --> 00:07:38.830
Now what do we do?

00:07:38.830 --> 00:07:41.384
So I've got this crooked
angle in here, now this F3 is

00:07:41.384 --> 00:07:44.845
gonna be pointing this way,
so I'll add another force

00:07:44.845 --> 00:07:48.183
to my force diagram, and I
can figure out how to include

00:07:48.183 --> 00:07:53.018
this into my vertical force
version of Newton's second law,

00:07:53.018 --> 00:07:56.106
I can't include the entire
F3 force, here's a mistake

00:07:56.106 --> 00:07:58.914
people make, they wanna
just add F3 or subtract F3,

00:07:58.914 --> 00:08:02.126
but I can't do that, this
is the vertical form of

00:08:02.126 --> 00:08:04.608
Newton's second law, this is
only applying to the vertical

00:08:04.608 --> 00:08:07.791
direction, the Y direction,
but F3 is pointing both

00:08:07.791 --> 00:08:09.776
vertically and horizontally.

00:08:09.776 --> 00:08:12.656
So I have to only include
the vertical part of F3

00:08:12.656 --> 00:08:15.146
in this formula so what I
have to do is say that alright

00:08:15.146 --> 00:08:17.197
F3 is gonna have a vertical component,

00:08:17.197 --> 00:08:20.620
that vertical component,
I'll call it F3 Y,

00:08:20.620 --> 00:08:22.779
for F3 in the vertical direction.

00:08:22.779 --> 00:08:24.965
And it's also gonna have
a horizontal component.

00:08:24.965 --> 00:08:29.338
I'll just call that F3 X for
F3 in the horizontal direction.

00:08:29.338 --> 00:08:32.671
So if I wanna solve for F3 Y,
I'll just use the definition

00:08:32.671 --> 00:08:35.589
of sine and I know to use
sine because this side is

00:08:35.589 --> 00:08:38.144
the opposite to this angle, I know sine

00:08:38.144 --> 00:08:40.819
relates opposite side, so
I'm gonna write this as

00:08:40.819 --> 00:08:45.232
sine of phi is gonna equal
the opposite side is F3 Y

00:08:45.232 --> 00:08:48.982
so F3 in the Y direction
divided by F3 total,

00:08:50.416 --> 00:08:53.916
the total magnitude of F3,
and if I solve this for F3 Y,

00:08:53.916 --> 00:08:58.083
I get F3 in the Y direction
is gonna equal F3 times sine

00:08:59.808 --> 00:09:03.561
of Fi, now I can include this
in my force, my net force,

00:09:03.561 --> 00:09:07.572
because this points upward,
so since it points up,

00:09:07.572 --> 00:09:10.095
this vertical component's
gonna add a plus,

00:09:10.095 --> 00:09:14.341
F3 sin theta over here,
or sorry not theta,

00:09:14.341 --> 00:09:18.508
it's gonna be F3 sin Fi, and
I'll have a plus F3 sin fi

00:09:20.447 --> 00:09:24.883
right here, and when we
subtract this F3 sin phi from the

00:09:24.883 --> 00:09:27.476
other side to get it over to here,

00:09:27.476 --> 00:09:31.141
we're gonna get minus F3
sine phi and that makes sense

00:09:31.141 --> 00:09:34.860
because if we pull up, we
know we're reducing some

00:09:34.860 --> 00:09:37.510
of the pressure, we'll
reduce the normal force,

00:09:37.510 --> 00:09:39.812
so this component, look at
this component points up,

00:09:39.812 --> 00:09:41.709
it's reducing some of the pressure

00:09:41.709 --> 00:09:44.266
on the bottom of the shoe,
the normal force goes down,

00:09:44.266 --> 00:09:47.523
and so we subtract the F3 sin fi.

00:09:47.523 --> 00:09:50.454
And one more way to step this
problem up to the next level

00:09:50.454 --> 00:09:53.258
would be to say that this
room isn't really just a room,

00:09:53.258 --> 00:09:54.905
maybe it's an elevator,

00:09:54.905 --> 00:09:56.926
and this elevator is accelerating upward

00:09:56.926 --> 00:10:00.132
with some acceleration
A zero, in that case,

00:10:00.132 --> 00:10:02.564
nothing would change on
this right hand side.

00:10:02.564 --> 00:10:04.914
Sometimes people think
if there's acceleration,

00:10:04.914 --> 00:10:07.807
there's gonna be some new force,
but if these are the forces

00:10:07.807 --> 00:10:10.741
those are the forces, the only
thing that changes over here

00:10:10.741 --> 00:10:13.334
if this was in an elevator
that's accelerating up,

00:10:13.334 --> 00:10:16.543
is that instead of zero,
you'd replace this with a zero

00:10:16.543 --> 00:10:19.391
or whatever the
acceleration is, that's it,

00:10:19.391 --> 00:10:21.529
that's the only change, you
could still solve for fn

00:10:21.529 --> 00:10:23.958
the same way, when you
multiply by m it wouldn't be

00:10:23.958 --> 00:10:26.767
zero on the left anymore,
you'd have a m a zero,

00:10:26.767 --> 00:10:30.394
and then a plus m a zero when you solve

00:10:30.394 --> 00:10:32.255
for the normal force.

00:10:32.255 --> 00:10:34.569
Alright, I think we've
pretty much exhausted

00:10:34.569 --> 00:10:36.875
this example of a shoe on the floor,

00:10:36.875 --> 00:10:39.222
it's probably harder than
any example you'll see

00:10:39.222 --> 00:10:41.367
but now you know how to
handle any type of force

00:10:41.367 --> 00:10:43.908
you might meet or acceleration.


Heat capacity at constant volume and pressure

WEBVTT
Kind: captions
Language: en

00:00:00.664 --> 00:00:03.743
- [Voiceover] Imagine you
had a monatomic ideal gas

00:00:03.743 --> 00:00:06.862
in this cylinder here, an
there was this tightly fitted

00:00:06.862 --> 00:00:10.604
piston above it that prevented
any gas from getting out.

00:00:10.604 --> 00:00:13.233
Well we know that the
total internal energy

00:00:13.233 --> 00:00:16.372
for a monatomic ideal gas is just three

00:00:16.372 --> 00:00:19.455
halves P times V or three halves NKT,

00:00:21.030 --> 00:00:23.280
or three halves little nRT,

00:00:25.485 --> 00:00:27.798
and we know that saying U internal

00:00:27.798 --> 00:00:31.804
the internal energy is really
just code for the total

00:00:31.804 --> 00:00:35.024
kinetic energy of the monatomic ideal gas,

00:00:35.024 --> 00:00:36.603
that these are the same thing.

00:00:36.603 --> 00:00:37.940
When we talk about internal energy

00:00:37.940 --> 00:00:40.775
we're talking about how fast
are these particles moving,

00:00:40.775 --> 00:00:43.023
in other words, what's the
total kinetic energy of

00:00:43.023 --> 00:00:45.979
all of these particles added up.

00:00:45.979 --> 00:00:48.491
My question is how do we go about

00:00:48.491 --> 00:00:50.313
changing the internal energy?

00:00:50.313 --> 00:00:52.865
Let's say we wanted to increase
the total kinetic energy,

00:00:52.865 --> 00:00:54.323
what could we do?

00:00:54.323 --> 00:00:56.591
Well, we could say increase the pressure,

00:00:56.591 --> 00:00:58.434
or the volume, or the
temperature, yeah but I mean

00:00:58.434 --> 00:01:02.504
physically, actually, in
the lab, what do we do?

00:01:02.504 --> 00:01:04.530
And there's basically two ways to change

00:01:04.530 --> 00:01:05.866
the internal energy.

00:01:05.866 --> 00:01:07.608
If you want to add internal energy,

00:01:07.608 --> 00:01:09.977
i.e. get these particles moving faster,

00:01:09.977 --> 00:01:13.177
we can heat it up so
put this above a flame

00:01:13.177 --> 00:01:14.696
or on a hot plate,

00:01:14.696 --> 00:01:18.280
and heat will flow into the gas which will

00:01:18.280 --> 00:01:21.318
cause these particles to
move faster and faster.

00:01:21.318 --> 00:01:23.402
That's one way to do it, to add heat.

00:01:23.402 --> 00:01:27.036
The other way to do it
is to do work on the gas.

00:01:27.036 --> 00:01:28.904
I could take this piston and push it down,

00:01:28.904 --> 00:01:30.704
and if you push this down hard enough

00:01:30.704 --> 00:01:32.706
it will squash this gas together,

00:01:32.706 --> 00:01:35.744
and those impacts with this
piston while it's moving down

00:01:35.744 --> 00:01:38.602
will cause them to start
moving faster and faster,

00:01:38.602 --> 00:01:41.617
that will also add
internal energy to the gas.

00:01:41.617 --> 00:01:44.542
So if we wanted to write
down a formula that told us

00:01:44.542 --> 00:01:47.220
how you could get a change
in the internal energy,

00:01:47.220 --> 00:01:49.762
if I want to change the internal energy,

00:01:49.762 --> 00:01:52.057
Delta U, which is really just saying

00:01:52.057 --> 00:01:53.905
changing the kinetic energy,

00:01:53.905 --> 00:01:55.073
well there's two ways to do it.

00:01:55.073 --> 00:01:56.558
I can add heat.

00:01:56.558 --> 00:01:58.380
If I added ten joules of heat

00:01:58.380 --> 00:02:00.856
I'd add ten joules to the internal energy,

00:02:00.856 --> 00:02:03.691
but I've also got to take into
account this work being done,

00:02:03.691 --> 00:02:07.156
and so I can do plus the
work done on the gas,

00:02:07.156 --> 00:02:09.316
and that's it, this is
actually the first law

00:02:09.316 --> 00:02:11.769
of thermodynamics, it's the
law of conservation of energy

00:02:11.769 --> 00:02:14.424
it says there's only
two ways to add energy,

00:02:14.424 --> 00:02:16.179
internal energy to a gas.

00:02:16.179 --> 00:02:18.294
Let me talk a little bit more
about this work done though

00:02:18.294 --> 00:02:20.611
cause getting the sign right is important.

00:02:20.611 --> 00:02:22.727
If you're doing work on
the gas compressing it

00:02:22.727 --> 00:02:24.684
you're adding energy to the gas,

00:02:24.684 --> 00:02:28.241
but if you let the gas
push up on the piston

00:02:28.241 --> 00:02:31.255
and this gas expands
pushing the piston up,

00:02:31.255 --> 00:02:33.010
then the gas is doing the work,

00:02:33.010 --> 00:02:35.463
that's energy leaving the system.

00:02:35.463 --> 00:02:37.780
So if the gas does work
you have to subtract

00:02:37.780 --> 00:02:39.445
work done by the gas.

00:02:39.445 --> 00:02:42.371
If the outside force does work on the gas

00:02:42.371 --> 00:02:44.801
you add that to the internal energy.

00:02:44.801 --> 00:02:45.927
So you got to pay attention to

00:02:45.927 --> 00:02:47.456
which way the energy is flowing.

00:02:47.456 --> 00:02:49.188
Work done on gas, energy goes in.

00:02:49.188 --> 00:02:51.889
Work done by the gas, energy goes out,

00:02:51.889 --> 00:02:53.891
and you'd have to subtract that over here.

00:02:53.891 --> 00:02:55.781
Let's say the gas did expand.

00:02:55.781 --> 00:02:58.843
Let's say the gas in here
was under so much pressure

00:02:58.843 --> 00:03:01.587
that the force it exerted
on this piston was enough

00:03:01.587 --> 00:03:06.110
to push that piston upward
by a certain amount.

00:03:06.110 --> 00:03:09.125
So let's say it started right
here and it went up to here

00:03:09.125 --> 00:03:10.700
so that piston went from here to there

00:03:10.700 --> 00:03:13.872
no gasses escaping cause
this is tightly fitting,

00:03:13.872 --> 00:03:16.596
but the gas was able to push
it up a certain distance d,

00:03:16.596 --> 00:03:18.622
how much work was done?

00:03:18.622 --> 00:03:20.400
You know the definition of work.

00:03:20.400 --> 00:03:22.986
Work is defined to be the
force times the distance

00:03:22.986 --> 00:03:24.898
through which that force was applied.

00:03:24.898 --> 00:03:27.801
So the work that the
gas did was F times d,

00:03:27.801 --> 00:03:30.191
but we want this to be in
terms of thermal quantities

00:03:30.191 --> 00:03:32.661
like pressure, and
volume, and temperature.

00:03:32.661 --> 00:03:33.899
So what could we do?

00:03:33.899 --> 00:03:36.104
We can say that this volume,

00:03:36.104 --> 00:03:37.881
not only did the piston raise up,

00:03:37.881 --> 00:03:40.807
but there was an extra
volume generated within here

00:03:40.807 --> 00:03:42.899
that I'm going to call Delta V,

00:03:42.899 --> 00:03:45.464
and I know that this
Delta V has got to equal

00:03:45.464 --> 00:03:49.087
the area of the piston times the distance

00:03:49.087 --> 00:03:51.045
through which that piston moved

00:03:51.045 --> 00:03:54.397
because this height
times that area gives me

00:03:54.397 --> 00:03:56.310
this volume right in here.

00:03:56.310 --> 00:03:57.390
Why am I doing this?

00:03:57.390 --> 00:04:01.418
Cause look I can write d
as equal to Delta V over A,

00:04:01.418 --> 00:04:05.693
and I can take this, I can
substitute this formula for d

00:04:05.693 --> 00:04:09.293
into here, and something magical happens,

00:04:09.293 --> 00:04:12.960
I'll get work equals F
times Delta V over A,

00:04:14.739 --> 00:04:18.316
but look F over A, we
know what F over A is,

00:04:18.316 --> 00:04:20.926
that's pressure so I get
that the work done by the gas

00:04:20.926 --> 00:04:22.929
is the pressure times Delta V.

00:04:22.929 --> 00:04:24.999
This is an equation that I
like because it's in terms

00:04:24.999 --> 00:04:27.699
of thermodynamic quantities
that we're already dealing with.

00:04:27.699 --> 00:04:30.760
So work done you can figure
out by taking P times Delta V

00:04:30.760 --> 00:04:33.887
but strictly speaking
this is only true if this

00:04:33.887 --> 00:04:36.025
pressure remained constant, right?

00:04:36.025 --> 00:04:37.375
If the pressure was changing,

00:04:37.375 --> 00:04:39.289
then what am I supposed to plug in here,

00:04:39.289 --> 00:04:41.560
the initial pressure, the final pressure?

00:04:41.560 --> 00:04:42.471
If the pressure's staying constant

00:04:42.471 --> 00:04:44.764
this gives you an exact
way to find the work done.

00:04:44.764 --> 00:04:46.789
You might object and say wait,

00:04:46.789 --> 00:04:49.359
how is it possible for a
gas to expand and remain

00:04:49.359 --> 00:04:50.506
at the same pressure?

00:04:50.506 --> 00:04:51.901
Well, you basically have to heat it up

00:04:51.901 --> 00:04:53.791
while the gas expands,

00:04:53.791 --> 00:04:56.394
that allows the pressure
to remain constant

00:04:56.394 --> 00:04:57.811
as the gas expands.

00:04:57.811 --> 00:05:01.141
And now we're finally ready
to talk about heat capacities.

00:05:01.141 --> 00:05:02.921
So let's get rid of this,

00:05:02.921 --> 00:05:05.844
and heat capacity is defined to be,

00:05:05.844 --> 00:05:08.207
imagine you had a certain
amount of heat being added.

00:05:08.207 --> 00:05:12.640
So a certain amount of heat
gets added to your gas.

00:05:12.640 --> 00:05:14.485
How much does the temperature increase?

00:05:14.485 --> 00:05:16.307
That's what the heat capacity tells you.

00:05:16.307 --> 00:05:18.782
So capital C is heat capacity

00:05:18.782 --> 00:05:22.000
and it's defined to be the amount of heat

00:05:22.000 --> 00:05:23.350
that you've added to the gas,

00:05:23.350 --> 00:05:26.311
divided by the amount of change

00:05:26.311 --> 00:05:28.772
in the temperature of that gas.

00:05:28.772 --> 00:05:31.247
And actually, something
you'll hear about often

00:05:31.247 --> 00:05:32.797
is the molar heat capacity,

00:05:32.797 --> 00:05:36.218
which is actually divided
by an extra n here.

00:05:36.218 --> 00:05:37.907
So instead of Q over Delta T

00:05:37.907 --> 00:05:41.663
it's Q over n the number
of moles times Delta T.

00:05:41.663 --> 00:05:43.904
Pretty simple but think about it.

00:05:43.904 --> 00:05:45.142
If we had a piston in here,

00:05:45.142 --> 00:05:47.167
are we going to allow that piston to move

00:05:47.167 --> 00:05:48.540
while we add the heat,

00:05:48.540 --> 00:05:51.105
or are we not going to
allow the piston to move?

00:05:51.105 --> 00:05:52.927
There's different ways
that this can happen,

00:05:52.927 --> 00:05:55.313
and because of that there's
different heat capacities.

00:05:55.313 --> 00:05:57.878
If we don't allow this piston to move,

00:05:57.878 --> 00:06:00.983
if we weld this thing
shut so it can't move

00:06:00.983 --> 00:06:03.863
we've got heat capacity
at constant volume,

00:06:03.863 --> 00:06:07.441
and if we do allow this
piston to move freely while

00:06:07.441 --> 00:06:09.849
we add the heat so that the pressure

00:06:09.849 --> 00:06:11.446
inside of here remains constant,

00:06:11.446 --> 00:06:14.708
we'd have the heat capacity
at constant pressure.

00:06:14.708 --> 00:06:17.004
And these are similar but different,

00:06:17.004 --> 00:06:19.231
and they're related, and
we can figure them out.

00:06:19.231 --> 00:06:22.179
So let's clear this
away, let's get a nice,

00:06:22.179 --> 00:06:25.127
here we go, two pistons
inside of cylinders.

00:06:25.127 --> 00:06:26.185
We'll put a piston in here,

00:06:26.185 --> 00:06:27.602
but I'm going to weld this one shut.

00:06:27.602 --> 00:06:29.357
This one can't move.

00:06:29.357 --> 00:06:31.746
We'll have another one over
here, it can move freely.

00:06:31.746 --> 00:06:33.205
So over on this side,

00:06:33.205 --> 00:06:35.028
we'll have the definition of

00:06:35.028 --> 00:06:37.390
heat capacity, regular heat capacity,

00:06:37.390 --> 00:06:39.620
is the amount of heat you add divided by

00:06:39.620 --> 00:06:41.733
the change in temperature that you get.

00:06:41.733 --> 00:06:43.466
So on this side we're
adding heat, let's say

00:06:43.466 --> 00:06:46.908
heat goes in, but the piston does not move

00:06:46.908 --> 00:06:48.933
and so the gas in here is stuck,

00:06:48.933 --> 00:06:50.896
it can't move, no work can be done.

00:06:50.896 --> 00:06:52.669
Since this piston can't move,

00:06:52.669 --> 00:06:55.279
external forces can't do work on the gas,

00:06:55.279 --> 00:06:57.799
and the gas can't do work
and allow energy to leave.

00:06:57.799 --> 00:07:01.399
Q is the only thing adding
energy into this system,

00:07:01.399 --> 00:07:02.405
or in other words,

00:07:02.405 --> 00:07:04.348
we've got heat capacity at constant

00:07:04.348 --> 00:07:06.507
volume is going to equal,

00:07:06.507 --> 00:07:08.735
well, remember the first
law of thermodynamics said

00:07:08.735 --> 00:07:12.650
that Delta U, the only way
to add internal energy,

00:07:12.650 --> 00:07:16.205
or take it away is that you
can add or subtract heat,

00:07:16.205 --> 00:07:18.500
and you can do work on the gas.

00:07:18.500 --> 00:07:22.190
In this case, Q, if I
subtract W from both sides

00:07:22.190 --> 00:07:25.107
I get Delta U minus W over Delta T,

00:07:25.973 --> 00:07:29.751
but since we're not
allowing this piston to move

00:07:29.751 --> 00:07:33.021
the work done has got to be zero.

00:07:33.021 --> 00:07:35.736
So there's no work done at
all so the heat capacity

00:07:35.736 --> 00:07:38.369
at constant volume is going to be Delta U

00:07:38.369 --> 00:07:41.564
over Delta T, what's Delta U?

00:07:41.564 --> 00:07:43.904
Let's just assume this
is a monatomic ideal gas,

00:07:43.904 --> 00:07:46.897
if it's monatomic we've
got a formula for this.

00:07:46.897 --> 00:07:51.064
Delta U is just three halves,
P times V over Delta T.

00:07:52.230 --> 00:07:53.827
That's not the only way I can write it.

00:07:53.827 --> 00:07:57.994
Remember I can also write it
as three halves NK Delta T

00:08:01.345 --> 00:08:04.493
over Delta T, and
something magical happens,

00:08:04.493 --> 00:08:06.338
check it out the Delta T's go away

00:08:06.338 --> 00:08:08.164
and you get that this is a constant.

00:08:08.164 --> 00:08:11.108
That the heat capacity for
any monatomic ideal gas

00:08:11.108 --> 00:08:13.561
is just going to be three halves,

00:08:13.561 --> 00:08:16.509
Capital NK, Boltzmann's constant,

00:08:16.509 --> 00:08:18.669
N is the total number of molecules.

00:08:18.669 --> 00:08:22.836
Or you could have rewrote
this as little n R Delta T.

00:08:24.137 --> 00:08:25.914
The T's would still have cancelled

00:08:25.914 --> 00:08:29.222
and you would have got
three halves, little n,

00:08:29.222 --> 00:08:32.507
the number of moles,
times R, the gas constant.

00:08:32.507 --> 00:08:35.860
So the heat capacity at
constant volume for any

00:08:35.860 --> 00:08:39.325
monatomic ideal gas is
just three halves nR,

00:08:39.325 --> 00:08:41.782
and if you wanted the molar heat capacity

00:08:41.782 --> 00:08:44.635
remember that's just divide
by an extra mole here

00:08:44.635 --> 00:08:48.775
so everything gets divided
by moles everywhere

00:08:48.775 --> 00:08:52.128
divided by moles, that
just cancels this out,

00:08:52.128 --> 00:08:55.144
and the molar heat
capacity at constant volume

00:08:55.144 --> 00:08:57.618
is just three halves R.

00:08:57.618 --> 00:08:59.284
So that's heat capacity
at constant volume,

00:08:59.284 --> 00:09:02.096
what about heat capacity
at constant pressure?

00:09:02.096 --> 00:09:03.604
Now we're going to look at this side.

00:09:03.604 --> 00:09:07.564
Again, we're going to
allow this gas to have heat

00:09:07.564 --> 00:09:10.647
enter the cylinder, but
we're going to allow this

00:09:10.647 --> 00:09:13.191
piston to move up while
it does that so that

00:09:13.191 --> 00:09:15.800
the pressure inside of
here remains constant

00:09:15.800 --> 00:09:18.275
and this is going to be the heat capacity

00:09:18.275 --> 00:09:19.944
at constant pressure.

00:09:19.944 --> 00:09:22.573
Well, again, we're going to get that it's

00:09:22.573 --> 00:09:26.656
Q over Delta T, and just
like the first law said,

00:09:27.553 --> 00:09:30.136
Q has to equal Delta U minus W.

00:09:31.611 --> 00:09:34.964
So we get Delta U minus W, the work done,

00:09:34.964 --> 00:09:38.317
over Delta T, this time W is not zero.

00:09:38.317 --> 00:09:39.857
What's W going to be?

00:09:39.857 --> 00:09:43.064
Remember W is P times Delta V.

00:09:43.064 --> 00:09:46.192
So this is a way we can find
the work done by the gas,

00:09:46.192 --> 00:09:48.826
P times Delta V, so this is going to equal

00:09:48.826 --> 00:09:51.143
Delta U, we know what that is.

00:09:51.143 --> 00:09:53.325
If this is again, a monatomic ideal gas,

00:09:53.325 --> 00:09:57.158
this is going to equal
three halves nR Delta T

00:10:00.075 --> 00:10:03.068
plus this is P times Delta V,

00:10:03.068 --> 00:10:04.070
but we have to be careful,

00:10:04.070 --> 00:10:06.533
in this formula this work is referring to

00:10:06.533 --> 00:10:08.153
work done on the gas,

00:10:08.153 --> 00:10:11.124
but in this case, work
is being done by the gas,

00:10:11.124 --> 00:10:12.991
so I need another negative.

00:10:12.991 --> 00:10:16.434
Technically the work done on
the gas would be a negative

00:10:16.434 --> 00:10:19.967
amount of this since energy
is leaving the system.

00:10:19.967 --> 00:10:21.924
So that negative cancels this negative

00:10:21.924 --> 00:10:25.012
and I get plus P times Delta V,

00:10:25.012 --> 00:10:27.595
all of that over Delta T, phew.

00:10:28.765 --> 00:10:29.937
So what do we get?

00:10:29.937 --> 00:10:31.854
Three halves nR Delta T

00:10:34.031 --> 00:10:37.180
plus I want to rewrite P times Delta V,

00:10:37.180 --> 00:10:38.643
but I know how to do that.

00:10:38.643 --> 00:10:42.468
The Ideal Gas Law says PV equals nRT,

00:10:42.468 --> 00:10:45.326
well if that's true, then P times Delta V

00:10:45.326 --> 00:10:48.364
is going to equal nR Delta T.

00:10:48.364 --> 00:10:53.066
So can rewrite this as nR
Delta T divided by Delta T,

00:10:53.066 --> 00:10:56.284
almost there, all of
the Delta T's go away.

00:10:56.284 --> 00:10:58.129
Look what I'm left with.

00:10:58.129 --> 00:10:59.705
I'm left with C.

00:10:59.705 --> 00:11:01.707
Heat capacity at constant
pressure is going to be

00:11:01.707 --> 00:11:05.172
equal to three halves nR plus nR,

00:11:05.172 --> 00:11:07.422
that's just five halves nR,

00:11:08.300 --> 00:11:11.585
and if I wanted the
molar heat capacity again

00:11:11.585 --> 00:11:14.221
I could divide everything,
everything around here

00:11:14.221 --> 00:11:17.818
by little n, and that would just give me

00:11:17.818 --> 00:11:19.775
the molar heat capacity constant pressure

00:11:19.775 --> 00:11:22.071
would be five halves R.

00:11:22.071 --> 00:11:23.578
And notice they're almost the same.

00:11:23.578 --> 00:11:28.303
The heat capacity at constant
volume is three halves nR,

00:11:28.303 --> 00:11:30.261
and the heat capacity at constant pressure

00:11:30.261 --> 00:11:31.993
is five halves nR.

00:11:31.993 --> 00:11:34.199
They just differ by nR.

00:11:34.199 --> 00:11:36.743
So the difference
between the heat capacity

00:11:36.743 --> 00:11:40.410
at constant volume which
is three halves nR,

00:11:41.489 --> 00:11:45.070
and the heat capacity at constant pressure

00:11:45.070 --> 00:11:47.070
which is five halves nR,

00:11:48.217 --> 00:11:50.884
is just Cp minus Cv which is nR,

00:11:54.419 --> 00:11:57.266
just nR, and if you wanted
to take the difference

00:11:57.266 --> 00:12:00.536
between the molar heat
capacities at constant volume

00:12:00.536 --> 00:12:03.276
and pressure, it would just be R.

00:12:03.276 --> 00:12:04.828
The difference would just be R

00:12:04.828 --> 00:12:06.156
because everything
would get divided by the

00:12:06.156 --> 00:12:07.641
number of moles.

00:12:07.641 --> 00:12:09.914
So there's a relationship,
an important relationship.

00:12:09.914 --> 00:12:11.736
It tells you the difference between

00:12:11.736 --> 00:12:13.379
the heat capacity at constant pressure

00:12:13.379 --> 00:12:15.604
and the heat capacity at constant volume.