Collective learning

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Kind: captions
Language: en

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There are many things that
differentiate human beings

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from other species.

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But the one thing that probably
differentiates humans even

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from our closest relatives in
the animal kingdom in a really

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big way is the notion that
humans are collective learners.

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And to understand
that, let's think

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about how even our closest
relative in the animal

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kingdom, the chimpanzee,
might communicate.

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So you might have
one chimpanzee,

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and over the course of
his or her lifetime,

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they're able to learn a
bunch of cool experiences.

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And they're even able to learn
to use tools, manipulate tools,

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and-- who knows--
maybe even make tools.

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Maybe even get a twig some
place and take off the leaves

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and then use that to go get ants
out of a hole or whatever else.

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So they're able to learn all
this stuff over a lifetime.

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Now unfortunate
for chimpanzees--

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well, what is fortunate
for chimpanzees is they

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do teach some of these
things that they've

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learned to other members
of their group, often

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their offspring.

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But what's unfortunate
for chimpanzees is they

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don't have a great way to
communicate with each other.

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So for most chimpanzees, the
way that they're able to teach

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is essentially by showing--
not showing and telling, just

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showing.

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And so because this is such
a un-precise or not exact

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and such an inefficient
way of communication,

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really all of the nuances of
what this chimpanzee might

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be able accumulate over
his or her lifetime

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aren't able to be conveyed
to the next generation

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or to the other
chimpanzees around.

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So you have tremendous
energy loss.

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And in particular, all
that can be conveyed

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are maybe the to
specific movements

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or what you might be able
to observe in the present.

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All of the other things
that maybe the chimpanzee is

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learning about-- the times of
year where this is appropriate,

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and maybe they can
convey some of that

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by showing them at the
right times of year--

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but other nuanced aspects
of it, or particular ways

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to hold something, or twist
something, can only be shown.

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It can't be described
in a very precise way.

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So you have all of this
loss of experience,

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this loss of information.

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And then over the course
of these animals' life,

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they may be able to learn
the same amount again.

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But then when they
need to communicate it,

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they have the
exact same problem.

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It's hard to
communicate with what

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they have at their disposal,
which is really just showing

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the other chimpanzees
what they've done.

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And so once again, you
have a loss of information.

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And what you have in this
type of circumstance is,

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generation after generation,
even though there is learning

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over the course of an
individual chimpanzee's life,

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and even though they can
communicate some of that

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to each other, that
form of communication

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loses so much
information and so much

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nuance that you never have
an overall accumulation

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of knowledge and wisdom in
this chimpanzee population.

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Now humans, on the
other hand, have

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something called a
symbolic language.

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And I'll talk about
this in a second.

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But for now, it's safe to say
that human language is far more

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precise and far more
efficient than just being

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able to show someone something.

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Imagine if you had to
learn how to do something

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without being able to
communicate verbally,

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if you just had to look
at someone else's actions.

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And then you'd have a good
idea of how difficult it

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is for chimpanzees
to teach each other.

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But in the case of
human beings, we

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have this thing called
symbolic language.

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That's a very precise, very
efficient way of communicating.

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So from one human
being to another,

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you can actually
communicate a good deal.

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Maybe not every single nuance
and every single experience,

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but a good chunk of it.

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So right here, I'm drawing
about that much of it

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to some other human being.

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Maybe it is the offspring.

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Maybe it is another member
of the tribe or the group

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or whatever it is.

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And then this human
being might come up

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with some other innovations.

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They're able to build off
of all of this learning

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from that previous generation
or from that other human being

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that's around.

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And they're able to come
up with their own nuances

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and their own innovations.

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And this one right
over here might come up

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with his or her own the
nuances and innovations.

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And because they have a good
communication mechanism,

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this one could even communicate
to that one what he's learned

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or what she's learned
and communicate

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a good chunk of that.

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Maybe not all of it, but
maybe a reasonable bit.

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They can describe exactly
how they do something,

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the times of years,
when it's good to do it,

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and when it's not
good to do it, how

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to plan for the future,
what's the history

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of this new learning.

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And so what you
have going on here

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is because of this strong
communication mechanism,

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precise, efficient
communication,

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what you have is a human
group, or eventually

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a human civilization, is able
to have a collective memory.

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In the case of the chimpanzees,
they're every generation.

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Every chimpanzee is having
to relearn the things

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that the other chimpanzees
might have already done

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in previous generations.

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They're not able to really
move forward or build

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on those in significant ways.

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In humans, as information is
learned and experience gained,

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a good bit of that is able to
be passed on to other humans.

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And so this might be passed on.

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So all of this might be passed
on or good chunk of this

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could be passed on to
the next generation.

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And I'm not even talking
about written language yet.

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This could still just be oral
communication, which is still

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a very strong, precise,
efficient means

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of communication.

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Written communication
takes it to another level.

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But then this person
over here, maybe she

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comes up with other innovations.

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And at some point,
you might say, well,

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look if everyone keeps
having innovations

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and they keep learning
what everyone learned

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in previous
generations, maybe this

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will tap out the
total amount of memory

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that a human being even has.

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And there's actually
a case that maybe this

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is one of the reasons
why humans even

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have larger memories,
because there

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is all of this
collective knowledge

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to gain from one generation to
another, from one human being

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to another.

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But there are some
limits to this.

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And this is the other element
where this collective aspect

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of collective memory
and collective learning

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becomes really powerful.

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A human being, because of the
strong communication mechanism,

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is not just limited to the
knowledge and the experience

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in their memory.

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They are able to tap
into-- So this human

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being right over here does
not have this skill set.

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And that skill set
maybe gets passed

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on to another human being.

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So let me copy and paste that.

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This other human being that's
maybe living at the same time.

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And when that becomes
relevant, they

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could actually tap into it.

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And maybe they could learn
it from that human being.

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Or maybe it's in a
different part of society

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and this human being
can build certain tools

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or build certain things
using this information,

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using that knowledge
right over there.

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And then this
human being doesn't

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need to know that information.

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They can just leverage the
output of that information

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to then build on top of it.

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So what it allows
human beings to do

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is not only convey information,
and build on information,

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from generation to
generation, human to human,

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it allows all of
the human brains

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collectively at any
given point of time

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to be one collective
memory bank that

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can be used to
develop or innovate

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and for specific
domains and adapt

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to specific parts
of the ecosystem

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or to teach each other.

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So all of a sudden,
this is really unique,

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as far as we can tell,
in the animal kingdom.

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All of a sudden, it's
not all about the brain

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or the memory of one
individual member of a species.

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It now becomes about
the brain or the memory

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of the entire civilization
or the entire group

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of the species.

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And just an example of
that, as far as I know,

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there's no human being who
knows how to do everything

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that all human beings
know how to do.

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I could imagine that
there is a chimpanzee that

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knows how to do everything
that any other chimpanzee knows

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how to do.

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There are no humans that can
be a fighter pilot, a doctor,

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a gymnast, a lawyer, understands
philosophy, and speaks

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20 different languages.

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As far as I know, that
human being does not exist.

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And that's OK,
because they can tap

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into the experiences, the
abilities of other human beings

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to build up their civilization.

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None of us, as far
as we know, knows

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how to do everything
that we need

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to actually build
our civilization.

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But the information is
in our collective memory

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to actually do it.

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Now the next thing
you might say is, OK,

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I started with this premise
that we have a strong, precise,

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efficient means
of communication,

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and that other animals don't.

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But don't other animals actually
have some form of language?

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So for example, even monkeys.

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Wouldn't they screech
when they are in danger?

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That's a form of communication,
maybe a form of language.

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Maybe certain animals--
birds, monkeys-- maybe

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they have a song that they sing
that can convey certain things.

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Maybe it's when they
are looking for a mate.

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Isn't that a form
of communication?

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And these are.

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These are a form
of communication

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and a form of language.

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But these don't
really come in play

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in terms of the
teaching/learning.

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You don't see one chimpanzee
making screeching sounds

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or learning sounds.

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They might do a little bit, just
to warn, maybe as a warning.

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But there's no deep
nuance or deep precision

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that's being able to convey by
these one-off sounds or even

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one-off gestures.

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And what's particularly
powerful about human language

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is that it is a
symbolic language.

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And when I say it's
a symbolic language,

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I'm even saying it
in a broader sense

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that even just written symbols.

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I'm talking about even
though the sounds themselves.

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So let's go to a time where
we did not even have writings.

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And when we talk about
symbolic languages,

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let's think about a
non-symbolic language.

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So in a non-symbolic language,
you might have some sound.

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Let's call it Sound 1.

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And it has some meaning.

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Let's call it Meaning 1.

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So this might be a
certain type of scream.

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It means that a
predator is approaching.

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Then you might have something
like a Sound 2 or Gesture 2.

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And then it has
some other meaning.

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It has Meaning 2.

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It might be a
certain type of song,

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which means that I'm in the mood
to reproduce or whatever else.

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You might have Gesture 3
that has some direct meaning.

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It might mean that I have found
food or something like that.

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So Meaning 3.

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What humans have,
they can do this,

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where particular sounds
have particular meanings.

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So for example, in humans,
you could have Sound 1.

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It refers to Meaning 1.

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I'll just refer to
it as Meaning 1.

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You could have Sound 2
that refers to Meaning 2.

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You could have Sound 3
that refers to Meaning 3.

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So these are just
direct representations.

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But what is really powerful
about symbolic languages,

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is that these oral symbols
can be combined according

00:11:06.200 --> 00:11:11.040
to set rules or grammars to have
an infinite number of meanings.

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So this is what really makes
human language transcend

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other languages
and really makes it

00:11:16.160 --> 00:11:18.819
this robust, precise
communication mechanism,

00:11:18.819 --> 00:11:20.110
is you could have combinations.

00:11:20.110 --> 00:11:22.900
Sound 1, Sound 2,
Sound 3 will now

00:11:22.900 --> 00:11:24.595
have another meaning, Meaning 4.

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Then you could maybe
have a combination

00:11:26.220 --> 00:11:31.980
we have Sound 3, Sound 1, and
Sound 2 might have Meaning 5.

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And if you have tens
of thousands of sounds,

00:11:34.780 --> 00:11:37.332
and really our oral words
are those sounds in a given

00:11:37.332 --> 00:11:38.790
language, then all
of a sudden, you

00:11:38.790 --> 00:11:40.540
can have infinite
meanings by putting them

00:11:40.540 --> 00:11:42.030
in different combinations.

00:11:42.030 --> 00:11:43.946
And if you think this
is a little bit abstract

00:11:43.946 --> 00:11:45.460
imagine that Sound
1 is the sound

00:11:45.460 --> 00:11:46.825
of me saying the word "dog."

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And I'm not even
going to write it down

00:11:48.450 --> 00:11:50.033
because I want to
imagine a world even

00:11:50.033 --> 00:11:51.450
before written communication.

00:11:51.450 --> 00:11:54.250
So Sound 1 is the sound "dog."

00:11:54.250 --> 00:11:57.020
Sound 2 is a sound, "eats."

00:11:57.020 --> 00:12:00.330
And Sound 3 is the sound "man."

00:12:00.330 --> 00:12:03.320
So literally, Sound 1,
if you heard, "dog,"

00:12:03.320 --> 00:12:06.070
you'd think, OK, I'd
visualize a dog of some type.

00:12:06.070 --> 00:12:09.710
And even there, you'd have
some visualization of a dog.

00:12:09.710 --> 00:12:11.140
And we all have one maybe.

00:12:11.140 --> 00:12:13.270
Sound 2, if you heard
"eats," you'd say,

00:12:13.270 --> 00:12:15.360
OK, I'd imagine
eating in some way.

00:12:15.360 --> 00:12:18.137
And Sound 3, "man," you have
some visualization of it.

00:12:18.137 --> 00:12:19.720
And if it was a
non-symbolic language,

00:12:19.720 --> 00:12:22.020
that's all you could get
out of those three sounds.

00:12:22.020 --> 00:12:24.730
But now, in a symbolic
language, we can combine those.

00:12:24.730 --> 00:12:27.160
We could say, "dog eats man."

00:12:27.160 --> 00:12:30.290
So once again, we just reused
the three sounds, the three

00:12:30.290 --> 00:12:31.150
symbols.

00:12:31.150 --> 00:12:34.580
But now, they're referring to
a whole new, much more complex,

00:12:34.580 --> 00:12:37.170
meaning than just referring
to certain objects

00:12:37.170 --> 00:12:38.230
or certain actions.

00:12:38.230 --> 00:12:40.645
Or you could have,
"man eats dog."

00:12:40.645 --> 00:12:43.180
It's not pleasant, but I guess
in a desperate situation.

00:12:43.180 --> 00:12:46.070
But once again, it
is another meaning

00:12:46.070 --> 00:12:49.090
that we can get out
of the same sounds.

00:12:49.090 --> 00:12:50.800
And what these
symbolic languages do,

00:12:50.800 --> 00:12:52.900
besides giving you an
infinite number of meanings,

00:12:52.900 --> 00:12:54.566
they're allowed to
give you more nuance,

00:12:54.566 --> 00:12:58.070
and really refer to things that
are abstract, and including--

00:12:58.070 --> 00:13:00.080
and maybe most
importantly-- things

00:13:00.080 --> 00:13:03.130
like the present,
the future, the past,

00:13:03.130 --> 00:13:07.000
kind of hypothetical things that
really are necessary in order

00:13:07.000 --> 00:13:09.790
to really communicate
or optimally communicate

00:13:09.790 --> 00:13:14.910
all of the experiences or
the learnings from one entity

00:13:14.910 --> 00:13:16.680
to another.


Loop de loop answer part 2

WEBVTT
Kind: captions
Language: en

00:00:00.410 --> 00:00:03.400
In the last video, we figured out the absolute minimum

00:00:03.400 --> 00:00:07.340
speed in order to stay on the circular path right over here especially near the top

00:00:07.560 --> 00:00:10.980
was 27.6 km/h

00:00:11.130 --> 00:00:13.420
What I want to do in this video is

00:00:13.420 --> 00:00:17.090
I just clipped out the parts where he's actually on the loop de loop

00:00:17.220 --> 00:00:19.440
and I want to figure out his average [speed]

00:00:19.440 --> 00:00:21.740
So I'm going to use the video editor right here

00:00:21.740 --> 00:00:25.060
to time how long it takes to complete the loop de loop

00:00:25.230 --> 00:00:29.220
And then we can use that and we know about the circumference of this loop de loop

00:00:29.220 --> 00:00:32.460
and we're going to assume that it is perfectly circular for our purpose

00:00:32.480 --> 00:00:36.130
although it looks like it's a little bit egg-shaped in reality or elliptical

00:00:36.330 --> 00:00:40.170
For our calculation, we're going to assume that it's perfectly circular

00:00:40.170 --> 00:00:42.710
I'll leave it to you to think about how it would change

00:00:42.710 --> 00:00:45.210
if you had an elliptical shape like this

00:00:45.350 --> 00:00:47.370
So anyway, let's watch the video again

00:00:47.370 --> 00:00:52.080
Remember this is from fifth gear which shows on Channel 5 in United Kingdom

00:00:52.420 --> 00:00:54.850
So there you go

00:00:54.850 --> 00:00:57.770
Let's watch it again. It's fun to watch. There you go

00:00:57.990 --> 00:01:01.630
And right over here, we have a little timer for my video editor

00:01:01.870 --> 00:01:06.160
And this right over here is in seconds and I was corrected on a earlier video

00:01:06.230 --> 00:01:09.540
This right over here is not in hundredth of seconds. This is in frames

00:01:09.540 --> 00:01:11.630
And there's 30 frames per second

00:01:11.670 --> 00:01:15.240
So it starts at 0 seconds 0 frames

00:01:15.240 --> 00:01:20.250
and then when we play it, it goes to 2 seconds and 14 frames

00:01:20.510 --> 00:01:22.410
There's 30 frames per second

00:01:22.560 --> 00:01:28.350
So it's 2 and 14/30 of a second. It's how long it takes the car to do the loop

00:01:28.900 --> 00:01:34.510
So 1 second, and then 2 seconds, 2 and 14/30. So almost 2.5 second

00:01:34.960 --> 00:01:36.850
So let's write that down

00:01:36.860 --> 00:01:50.440
So the time required to do the loop de loop is roughly 2 and 14/30 seconds

00:01:50.460 --> 00:01:52.530
And what is the distance that it travels?

00:01:52.530 --> 00:01:56.320
If we assume that this thing is circular although it looks like it's a little bit egg-shaped

00:01:56.320 --> 00:01:58.250
if we assume that it is circular

00:01:58.270 --> 00:02:02.950
then the distance traveled is the circumference of the circular loop de loop

00:02:03.190 --> 00:02:08.970
The circumference is 2 pi times the radius which is equal to 2 pi

00:02:09.070 --> 00:02:13.030
and in the previous video, we figured out the radius was 6 m

00:02:13.390 --> 00:02:20.710
So it's 2 pi times 6 m which is equal to 12 pi meters

00:02:20.840 --> 00:02:23.510
If you wanted to figure out its average speed--

00:02:23.510 --> 00:02:26.130
the velocity is constantly changing because the direction is changing

00:02:26.150 --> 00:02:28.140
but the magnitude of the velocity--

00:02:28.140 --> 00:02:37.250
if we wanted to figure out the average magnitude of the velocity or the average speed

00:02:38.000 --> 00:02:44.150
the total distance traveled is 12 pi meters

00:02:44.170 --> 00:02:48.010
divided by the time required to travel the 12 pi meters

00:02:48.030 --> 00:02:53.720
so that is 2 and 14/30 seconds

00:02:53.720 --> 00:02:57.190
Now let's get our calculator out to actual calculate that value

00:02:57.670 --> 00:03:06.930
So we're going to have the distance 12 pi m divided by

00:03:06.930 --> 00:03:15.480
2 + 14/30 just to get the exact value

00:03:15.660 --> 00:03:20.010
And then this gives us in meters per second

00:03:20.030 --> 00:03:23.710
15.3 m/s

00:03:24.140 --> 00:03:31.070
So the average speed is approximately 15.3 m/s

00:03:31.170 --> 00:03:37.900
which is almost twice as fast as the minimum speed we figured out

00:03:37.900 --> 00:03:40.100
because you want that margin of safety

00:03:40.210 --> 00:03:42.870
and you want to be able to have some traction with the road

00:03:42.900 --> 00:03:46.260
Although you don't want to go too fast, because then the G force is going to be too big

00:03:46.260 --> 00:03:49.070
then this--maybe we'll talk about that in the future video

00:03:49.260 --> 00:03:54.590
I'll just relate this into kilometers per hour. Let's figure out what that is

00:03:55.580 --> 00:03:58.730
I want to use the calculator here

00:03:58.970 --> 00:04:00.890
So that's in the meters per second

00:04:01.100 --> 00:04:06.270
Let's figure out how many meters per hour by multiplying it by 3600 seconds per hour

00:04:06.610 --> 00:04:11.170
So that's how many meters per hour, and divide it by 1000 which you can see right over there

00:04:11.550 --> 00:04:16.030
that is 55 km/h

00:04:16.200 --> 00:04:20.270
If you want to do it in miles, it's rough approximation, divide it by 1.6

00:04:20.520 --> 00:04:26.470
It's about 35 mph give or take or 55 km/h

00:04:26.830 --> 00:04:32.560
So this is approximately 55 km/h

00:04:32.730 --> 00:04:36.640
So the driver here luckily they did the physics ahead of time

00:04:36.770 --> 00:04:38.900
and he had the margin of safety

00:04:38.900 --> 00:04:43.340
He was well in excess of the minimum velocity just to maintain the circular motion

00:04:43.340 --> 00:04:46.650
So he probably has some nice traction with the track up here


Loop de loop answer part 1

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:02.420
What I want to do
now is figure out,

00:00:02.420 --> 00:00:05.320
what's the minimum
speed that the car has

00:00:05.320 --> 00:00:08.200
to be at the top of this
loop de loop in order

00:00:08.200 --> 00:00:09.540
to stay on the track?

00:00:09.540 --> 00:00:11.160
In order to stay in
a circular motion.

00:00:11.160 --> 00:00:13.625
In order to not
fall down like this.

00:00:13.625 --> 00:00:15.000
And I think we
can all appreciate

00:00:15.000 --> 00:00:18.320
that is the most difficult part
of the loop de loop, at least

00:00:18.320 --> 00:00:20.240
in the bottom half
right over here.

00:00:20.240 --> 00:00:22.970
The track itself
is actually what's

00:00:22.970 --> 00:00:27.960
providing the centripetal force
to keep it going in a circle.

00:00:27.960 --> 00:00:30.910
But when you get
to the top, you now

00:00:30.910 --> 00:00:33.880
have gravity that is
pulling down on the car,

00:00:33.880 --> 00:00:35.240
almost completely.

00:00:35.240 --> 00:00:37.710
And the car will have to
maintain some minimum speed

00:00:37.710 --> 00:00:42.970
in order to stay in
this circular path.

00:00:42.970 --> 00:00:45.300
So let's figure out what
that minimum speed is.

00:00:45.300 --> 00:00:46.830
And to help figure
that out, we have

00:00:46.830 --> 00:00:50.354
to figure out what the radius of
this loop de loop actually is.

00:00:50.354 --> 00:00:52.520
And it actually does not
look like a perfect circle,

00:00:52.520 --> 00:00:54.850
based on this little screen
shot that I got here.

00:00:54.850 --> 00:00:57.090
It looks a little
bit elliptical.

00:00:57.090 --> 00:01:00.790
But it looks like the radius
of curvature right over here

00:01:00.790 --> 00:01:02.520
is actually smaller
than the radius

00:01:02.520 --> 00:01:05.760
of the curvature of the
entire loop de loop.

00:01:05.760 --> 00:01:07.300
That if you made
this into a circle,

00:01:07.300 --> 00:01:10.250
it would actually be maybe
even a slightly smaller circle.

00:01:10.250 --> 00:01:12.770
But let's just assume, for
the sake of our arguments

00:01:12.770 --> 00:01:18.330
right over here, that this
thing is a perfect circle.

00:01:18.330 --> 00:01:21.510
And it was a perfect
circle, let's think

00:01:21.510 --> 00:01:23.460
about what that
minimum velocity would

00:01:23.460 --> 00:01:27.680
have to be up here at the
top of the loop de loop.

00:01:27.680 --> 00:01:32.250
So we know that the magnitude
of your centripetal acceleration

00:01:32.250 --> 00:01:36.220
is going to be equal
to your speed squared

00:01:36.220 --> 00:01:39.910
divided by the
radius of the circle

00:01:39.910 --> 00:01:42.080
that you are going around.

00:01:42.080 --> 00:01:44.160
Now at this point
right over here,

00:01:44.160 --> 00:01:46.160
at the top, which is
going to be the hardest

00:01:46.160 --> 00:01:48.940
point, the magnitude
of our acceleration,

00:01:48.940 --> 00:01:54.550
this is going to be 9.81
meters per second squared.

00:01:54.550 --> 00:01:56.590
And the radius,
we can estimate--

00:01:56.590 --> 00:01:58.450
I copied and pasted
the car, and it

00:01:58.450 --> 00:02:01.770
looks like I can get it to stack
on itself four times to get

00:02:01.770 --> 00:02:05.740
the radius of this
circle right over here.

00:02:05.740 --> 00:02:08.639
And I looked it up on the
web, and a car about this size

00:02:08.639 --> 00:02:11.710
is going to be about
1.5 meters high

00:02:11.710 --> 00:02:14.660
from the bottom of the
tires to the top of the car.

00:02:14.660 --> 00:02:17.110
And so it looks like--
just eyeballing it

00:02:17.110 --> 00:02:19.930
based on these copying
and pasting of the cars,

00:02:19.930 --> 00:02:23.580
that the radius of this
loop de loop right over here

00:02:23.580 --> 00:02:25.220
is 6 meters.

00:02:25.220 --> 00:02:27.960
So this right over
here is 6 meters.

00:02:27.960 --> 00:02:29.864
So you multiply both
sides by 6 meters.

00:02:29.864 --> 00:02:32.030
Or actually, we could keep
it just in the variables.

00:02:32.030 --> 00:02:34.071
So let me just rewrite
it-- just to manipulate it

00:02:34.071 --> 00:02:39.310
so we can solve for v. We have
v squared over r is equal to a.

00:02:39.310 --> 00:02:41.300
And then you multiply
both sides by r.

00:02:41.300 --> 00:02:44.894
You get v squared is
equal to a times r.

00:02:44.894 --> 00:02:47.310
And then you take the principal
square root of both sides.

00:02:47.310 --> 00:02:49.750
You get v is equal to
the principal square root

00:02:49.750 --> 00:02:52.690
of a times r.

00:02:52.690 --> 00:02:54.770
And then if we plug
in these numbers,

00:02:54.770 --> 00:02:56.890
this velocity that we
have to have in order

00:02:56.890 --> 00:03:02.390
to stay in the circle is going
to be the square root of 9.81

00:03:02.390 --> 00:03:09.266
meters per second
squared, times 6 meters.

00:03:09.266 --> 00:03:11.140
And you can verify that
these units work out.

00:03:11.140 --> 00:03:14.276
Meters times meters is meter
squared, per second squared.

00:03:14.276 --> 00:03:15.650
You take the square
root of that,

00:03:15.650 --> 00:03:17.233
you're going to get
meters per second.

00:03:17.233 --> 00:03:20.160
But let's get our calculator
out to actually calculate this.

00:03:20.160 --> 00:03:24.560
So we are going to get
the principal square root

00:03:24.560 --> 00:03:30.200
of 9.81 times 6 meters.

00:03:30.200 --> 00:03:35.260
It gives us-- now here's
our drum roll-- 7.67.

00:03:35.260 --> 00:03:37.950
I'll just round to three
significant digits,

00:03:37.950 --> 00:03:41.682
7.67 meters per second squared.

00:03:41.682 --> 00:03:43.640
And significant digits
is a whole conversation,

00:03:43.640 --> 00:03:46.540
because this is just a very,
very rough approximation.

00:03:46.540 --> 00:03:49.540
I'm not able to measure
this that accurately at all.

00:03:49.540 --> 00:03:52.410
But I get roughly 7
point-- I'll just round,

00:03:52.410 --> 00:03:54.560
7.7 meters per second.

00:03:54.560 --> 00:03:59.582
So this is approximately
7.7 meters per second.

00:03:59.582 --> 00:04:01.540
And just to give a sense
of how that translates

00:04:01.540 --> 00:04:04.850
into units that we're used
to when we're driving cars,

00:04:04.850 --> 00:04:08.635
we can convert 7.7
meters per second.

00:04:11.310 --> 00:04:13.610
If we want to say how many
meters we go to an hour,

00:04:13.610 --> 00:04:17.769
well, there's 3,600
seconds in an hour.

00:04:17.769 --> 00:04:20.490
And then if you want to
convert that into kilometers--

00:04:20.490 --> 00:04:22.390
this will be in meters--
you divide by 1,000.

00:04:22.390 --> 00:04:27.860
One kilometer is
equal to 1,000 meters.

00:04:27.860 --> 00:04:29.940
And you see here,
the units cancel out.

00:04:29.940 --> 00:04:33.270
You have meters, meters,
seconds, seconds.

00:04:33.270 --> 00:04:35.780
You're left with
kilometers per hour.

00:04:35.780 --> 00:04:37.730
So let's actually
calculate this.

00:04:37.730 --> 00:04:39.320
And so we get our
previous answer.

00:04:39.320 --> 00:04:41.480
We want to multiply
it times 3,600

00:04:41.480 --> 00:04:45.090
to figure out how many
meters in an hour.

00:04:45.090 --> 00:04:47.145
And then you divide
by 1,000 to convert it

00:04:47.145 --> 00:04:49.000
to kilometers per hour.

00:04:49.000 --> 00:04:51.240
So you divide by 1,000.

00:04:51.240 --> 00:04:56.380
And we get 27.6
kilometers per hour.

00:04:56.380 --> 00:05:01.270
So this is equal
to 27.6 kilometers

00:05:01.270 --> 00:05:03.264
per hour, which is
surprisingly slow.

00:05:03.264 --> 00:05:04.680
I would have thought
it would have

00:05:04.680 --> 00:05:06.220
to be much, much, much faster.

00:05:06.220 --> 00:05:09.120
But it turns out, it does not
have to be much, much faster.

00:05:09.120 --> 00:05:11.970
Only 27.6 kilometers per hour.

00:05:11.970 --> 00:05:13.640
Now the important
thing to keep in mind

00:05:13.640 --> 00:05:16.550
is this is just fast
enough, at this point,

00:05:16.550 --> 00:05:18.700
to maintain the circular motion.

00:05:18.700 --> 00:05:21.290
But if this were a perfect
circle right over here,

00:05:21.290 --> 00:05:24.730
and you were going at exactly
27.6 kilometers per hour,

00:05:24.730 --> 00:05:27.297
you would not have much
traction with the road.

00:05:27.297 --> 00:05:29.380
And if you don't have much
traction with the road,

00:05:29.380 --> 00:05:31.390
the car might slip
and might not be

00:05:31.390 --> 00:05:33.320
able to actually
maintain its speed.

00:05:33.320 --> 00:05:37.060
So you definitely
want your speed

00:05:37.060 --> 00:05:39.200
to be a good bit larger
than this in order

00:05:39.200 --> 00:05:41.190
to keep a nice
margin of safety--

00:05:41.190 --> 00:05:43.110
in order to especially
have traction

00:05:43.110 --> 00:05:45.430
with the actual loop
de loop, and to be

00:05:45.430 --> 00:05:47.466
able to maintain your speed.

00:05:47.466 --> 00:05:49.090
Now what I want to
do in the next video

00:05:49.090 --> 00:05:52.480
is actually time the car to
figure out how long does it

00:05:52.480 --> 00:05:54.525
take it to do this loop de loop.

00:05:54.525 --> 00:05:56.400
And we're going to assume
that it's a circle.

00:05:56.400 --> 00:05:57.640
And we're going
to figure it out.

00:05:57.640 --> 00:05:59.223
And we're going to
figure out how fast

00:05:59.223 --> 00:06:03.390
it's actual average velocity
was over the course of this loop

00:06:03.390 --> 00:06:04.950
de loop.


Loop de loop question

WEBVTT
Kind: captions
Language: en

00:00:01.267 --> 00:00:06.000
What I want to do in this video is show you a very cool video, then pose a question.

00:00:06.000 --> 00:00:08.667
But I want to make sure that everything is up and up,

00:00:08.667 --> 00:00:10.867
in terms of the copyright laws.

00:00:10.867 --> 00:00:14.000
So the video I want to show you is from the show called the fifth gear,

00:00:14.000 --> 00:00:18.000
that appears on channel 5, I believe in the United Kingdom.

00:00:18.000 --> 00:00:21.667
I think I'm okay on good copyright grounds to claim

00:00:21.667 --> 00:00:23.159
fair use here, because

00:00:23.159 --> 00:00:26.164
obviously the point of this is I'm going to ask a math question

00:00:26.164 --> 00:00:28.185
about the vide were going to see

00:00:28.185 --> 00:00:28.685
and so fair use

00:00:28.685 --> 00:00:29.185
laws

00:00:29.185 --> 00:00:31.108
the purpose and character of use

00:00:31.108 --> 00:00:34.708
including whether such a commercial nature or non profit educational

00:00:34.708 --> 00:00:36.933
purposes we're definitely the later

00:00:36.933 --> 00:00:39.067
non profit educational purposes

00:00:39.067 --> 00:00:40.533
we are actually

00:00:40.533 --> 00:00:43.667
we aren't trying to in any way replace the work being used

00:00:43.667 --> 00:00:47.133
I'm just going to show a little snippet of it

00:00:47.133 --> 00:00:49.836
so we can use it as the basis for a problem

00:00:49.836 --> 00:00:52.533
this is kinda the meat of fair use law

00:00:52.533 --> 00:00:56.200
the viewer may fairly cite from the original work if it is

00:00:56.200 --> 00:01:00.200
design if his design be really and truly to use the passages

00:01:00.200 --> 00:01:02.600
for the purposes for fair and reasonable criticism

00:01:02.600 --> 00:01:03.708
in this case

00:01:03.708 --> 00:01:06.867
it's fair and reasonable math problem or physics problem

00:01:06.867 --> 00:01:12.267
on the other hand it is as clear he thus cites the most important parts of the work. with the view not to criticize

00:01:12.267 --> 00:01:14.933
but to supercede the

00:01:14.933 --> 00:01:16.733
the use of the original work and substitute

00:01:16.733 --> 00:01:18.333
the review for it

00:01:18.333 --> 00:01:21.667
such a use will be deemed in law a piracy

00:01:21.667 --> 00:01:26.333
so in no way do I want to supersede it. You should go look it up, go watch the show.

00:01:26.333 --> 00:01:27.933
this is not good enough

00:01:27.933 --> 00:01:29.667
I don't even have the sound on

00:01:29.667 --> 00:01:31.600
it's going to be a small window over here

00:01:31.600 --> 00:01:32.400
where I'm just going to show this

00:01:32.400 --> 00:01:33.533
really really

00:01:33.533 --> 00:01:36.267
the part where they actually do a loop

00:01:36.267 --> 00:01:37.067
de loop.

00:01:37.067 --> 00:01:39.200
now with that out of the way

00:01:39.200 --> 00:01:40.733
this is where you can find more about fifth gear

00:01:40.733 --> 00:01:43.533
which is on channel 5

00:01:43.533 --> 00:01:46.067
so now let me show you the video

00:01:46.067 --> 00:01:49.600
I'm going to have the sound down and so this is the video over here

00:01:49.600 --> 00:01:51.800
this is a pretty neat video

00:01:51.800 --> 00:01:55.067
this guy is going to try to do a loop de loop and he's actually going to succeed

00:01:55.067 --> 00:01:56.533
at doing the loop de loop

00:01:56.533 --> 00:01:59.200
and what I want you to do, the question I want to pose

00:01:59.200 --> 00:02:00.933
is look at this video

00:02:00.933 --> 00:02:03.467
and get a stop watch out

00:02:03.467 --> 00:02:05.733
or whatever else you need out a ruler whatever else you might need to do

00:02:05.733 --> 00:02:07.200
and try to figure out what

00:02:07.200 --> 00:02:10.067
we're going to try to figure out two things

00:02:10.067 --> 00:02:12.200
we're going to find out the minimum velocity the car needs to do the

00:02:12.200 --> 00:02:13.467
loop de loop

00:02:13.467 --> 00:02:15.667
and we're also going to try to figure out what his actual velocity was

00:02:15.667 --> 00:02:19.133
over the course of the loop de loop

00:02:19.133 --> 00:02:21.333
so let's actually just watch this video

00:02:21.333 --> 00:02:22.267
so there he goes

00:02:22.267 --> 00:02:23.333
he's starting

00:02:23.333 --> 00:02:24.267
he's accelerating

00:02:24.267 --> 00:02:25.000
it looks like he's

00:02:25.000 --> 00:02:25.600
just crossed

00:02:25.600 --> 00:02:28.867
he's pushing almost forty miles an hour

00:02:28.867 --> 00:02:31.200
at least I'm assuming that's miles per hour

00:02:31.200 --> 00:02:32.533
this is a British show

00:02:32.533 --> 00:02:34.133
and then he

00:02:34.133 --> 00:02:35.667
hits

00:02:35.667 --> 00:02:38.867
the loop de loop

00:02:38.867 --> 00:02:40.400
and then he finishes it

00:02:40.400 --> 00:02:41.800
and he's done

00:02:41.800 --> 00:02:44.000
so let's think about for the next video

00:02:44.000 --> 00:02:46.533
let's think about if we can figure out what the minimum

00:02:46.533 --> 00:02:51.733
what the minimum velocity or i should say the magnitude of the velocity

00:02:51.733 --> 00:02:53.267
or the speed had to be

00:02:53.267 --> 00:02:53.767
to actually

00:02:53.767 --> 00:02:57.533
complete the loop right over here and then we can actually get a stopwatch later on to

00:02:57.533 --> 99:59:59.999
see what his actall velocity was


Space station speed in orbit

WEBVTT
Kind: captions
Language: en

00:00:00.540 --> 00:00:03.020
Now that we know the
magnitude of the acceleration

00:00:03.020 --> 00:00:07.480
due to gravity at 400 kilometers
above the surface of the earth,

00:00:07.480 --> 00:00:09.440
where the space
station might hang out,

00:00:09.440 --> 00:00:12.780
what I want to do in this video
is think about how fast does

00:00:12.780 --> 00:00:16.051
the space station need to be
moving in order to keep missing

00:00:16.051 --> 00:00:18.550
the earth as it's falling, or
another way to think about it,

00:00:18.550 --> 00:00:20.390
in order to stay
in orbit, in order

00:00:20.390 --> 00:00:24.840
to maintain its circular
motion around the earth.

00:00:24.840 --> 00:00:28.300
So we know from our
studies of circular motion

00:00:28.300 --> 00:00:31.110
so far, what's keeping it going
in circular motion, assuming

00:00:31.110 --> 00:00:33.780
that it has a constant
speed, is some type

00:00:33.780 --> 00:00:35.490
of centripetal acceleration.

00:00:35.490 --> 00:00:36.865
And that centripetal
acceleration

00:00:36.865 --> 00:00:39.860
is the acceleration
due to gravity.

00:00:39.860 --> 00:00:42.680
And we figured out what
it was at 400 kilometers.

00:00:42.680 --> 00:00:45.410
And so we know that that
centripetal acceleration-- let

00:00:45.410 --> 00:00:46.890
me write it here
in pink-- we know

00:00:46.890 --> 00:00:49.920
that the magnitude of that
centripetal acceleration

00:00:49.920 --> 00:00:53.590
has to be equal to the speed or
the magnitude of the velocity

00:00:53.590 --> 00:00:58.710
squared divided by
divided r, where

00:00:58.710 --> 00:01:01.770
r is the radius of
the circular path.

00:01:01.770 --> 00:01:03.380
So in this case,
it'd be the radius

00:01:03.380 --> 00:01:05.069
of the orbit, which
would be the radius

00:01:05.069 --> 00:01:07.197
of the Earth, plus the altitude.

00:01:07.197 --> 00:01:09.280
So that, we already figured
out in the last video,

00:01:09.280 --> 00:01:14.180
is 6,771 kilometers.

00:01:14.180 --> 00:01:16.280
Let's just solve this
for v, and then we

00:01:16.280 --> 00:01:18.110
can put in the numbers
in our calculators.

00:01:18.110 --> 00:01:21.590
So you multiply both sides by
r, and you flip the two sides.

00:01:21.590 --> 00:01:24.210
You get v squared is
equal to the magnitude

00:01:24.210 --> 00:01:26.830
of our acceleration
times the radius.

00:01:26.830 --> 00:01:29.110
The magnitude of our
velocity or speed

00:01:29.110 --> 00:01:33.680
is equal to the square root
of our acceleration times--

00:01:33.680 --> 00:01:35.350
or the magnitude of
our acceleration--

00:01:35.350 --> 00:01:37.221
times the radius.

00:01:37.221 --> 00:01:38.720
And so let's get
our calculator out,

00:01:38.720 --> 00:01:40.511
and you can verify that
the units work out.

00:01:40.511 --> 00:01:43.340
This is meters per second
squared times meters, which

00:01:43.340 --> 00:01:45.132
gives us meters squared
per second squared.

00:01:45.132 --> 00:01:46.673
Take the square root
of that, and you

00:01:46.673 --> 00:01:49.060
get meters per second, which
is the appropriate units.

00:01:49.060 --> 00:01:52.140
But let's get our calculator
out and actually calculate this.

00:01:52.140 --> 00:01:52.640
Let me see.

00:01:52.640 --> 00:01:55.080
My calculator's sitting
on my other screen.

00:01:55.080 --> 00:01:56.920
There you go.

00:01:56.920 --> 00:02:01.570
And then we want to calculate
the principal square root

00:02:01.570 --> 00:02:06.115
of the acceleration due to
gravity at this altitude,

00:02:06.115 --> 00:02:07.490
the magnitude of
the acceleration

00:02:07.490 --> 00:02:12.740
due to gravity at that
altitude is 8.69 meters

00:02:12.740 --> 00:02:17.160
per second squared times the
radius of our circular path.

00:02:17.160 --> 00:02:19.170
That's going to be the
radius of Earth, which

00:02:19.170 --> 00:02:24.980
is 6,371 kilometers plus the
400 kilometers of altitude

00:02:24.980 --> 00:02:27.130
that we have in this scenario.

00:02:27.130 --> 00:02:29.930
So that gives us 6.-- and we
did this in the last video--

00:02:29.930 --> 00:02:35.830
6.771 times 10 to
the sixth meters.

00:02:35.830 --> 00:02:38.360
And it's important that
everything here is in meters.

00:02:38.360 --> 00:02:41.040
Our acceleration is in
meters per second squared.

00:02:41.040 --> 00:02:42.750
This right over
here is in meters.

00:02:42.750 --> 00:02:45.130
So the units don't
do anything strange.

00:02:45.130 --> 00:02:47.315
And then we get a drum
roll for how fast--

00:02:47.315 --> 00:02:49.190
and this is going to be
in meters per second.

00:02:49.190 --> 00:02:51.940
We already thought about
how the units will work out.

00:02:51.940 --> 00:02:56.340
We get 7,670-- I could say 71.

00:02:56.340 --> 00:02:57.840
But actually I'm
just going to stick

00:02:57.840 --> 00:02:59.006
to three significant digits.

00:02:59.006 --> 00:03:04.320
7,670 meters per second.

00:03:04.320 --> 00:03:05.980
So let me write that down.

00:03:05.980 --> 00:03:08.220
The necessary velocity
to stay in orbit

00:03:08.220 --> 00:03:13.780
is 7,670 meters per second.

00:03:13.780 --> 00:03:15.930
So let's just conceptualize
that for a second.

00:03:15.930 --> 00:03:19.790
Every second, it's
going over 7,000 meters.

00:03:19.790 --> 00:03:23.550
Or every second, it's
going over 7 kilometers.

00:03:23.550 --> 00:03:24.430
Every second.

00:03:24.430 --> 00:03:27.020
It's going at this super-- if
we assume that's the direction

00:03:27.020 --> 00:03:30.610
it's traveling, it's going at
this super-, superfast speed.

00:03:30.610 --> 00:03:35.360
And if we want to translate
that into kilometers per hour,

00:03:35.360 --> 00:03:39.144
you just take 7,670
meters per second.

00:03:39.144 --> 00:03:40.810
If you want to know
how many meters it's

00:03:40.810 --> 00:03:42.351
going to do in an
hour, you just say,

00:03:42.351 --> 00:03:47.090
well, there's 3,600
seconds per hour.

00:03:47.090 --> 00:03:48.880
And so if you
multiply that, that's

00:03:48.880 --> 00:03:50.972
how many meters it
will travel in an hour.

00:03:50.972 --> 00:03:52.430
But if you want
that in kilometers,

00:03:52.430 --> 00:03:53.790
you just divide by 1,000.

00:03:53.790 --> 00:03:58.440
You have 1 kilometer
for every 1,000 meters.

00:03:58.440 --> 00:04:00.110
Meters will cancel out.

00:04:00.110 --> 00:04:01.470
Seconds will cancel out.

00:04:01.470 --> 00:04:04.500
And you are left with
kilometers per hour.

00:04:04.500 --> 00:04:06.847
So let's do that.

00:04:06.847 --> 00:04:08.180
So that was our previous answer.

00:04:08.180 --> 00:04:15.370
We multiply by 3,600 and
then divide by 1,000.

00:04:15.370 --> 00:04:17.880
So we really could have
just multiplied by 3.6.

00:04:17.880 --> 00:04:26.440
And then we get 27,000, roughly
27,600 kilometers per hour.

00:04:26.440 --> 00:04:29.440
So this is really an
unfathomable speed.

00:04:29.440 --> 00:04:32.110
And you might be wondering, how
does such a big thing maintain

00:04:32.110 --> 00:04:36.370
that type of speed, because even
a jet plane, which is nowhere

00:04:36.370 --> 00:04:39.710
near this fast, has to have
these huge engines to maintain

00:04:39.710 --> 00:04:41.780
its speed.

00:04:41.780 --> 00:04:43.410
How does this thing maintain it?

00:04:43.410 --> 00:04:45.570
And the difference between
this and a jet plane

00:04:45.570 --> 00:04:50.770
is that a jet or a car or if
I throw a ball or anything

00:04:50.770 --> 00:04:52.697
like that-- but a jet plane.

00:04:52.697 --> 00:04:54.280
Let's focus on a jet
plane so we don't

00:04:54.280 --> 00:04:55.654
have to worry
about other things.

00:04:55.654 --> 00:04:58.865
A jet plane has to
travel through the air.

00:04:58.865 --> 00:05:00.240
It has to travel
through the air.

00:05:00.240 --> 00:05:03.070
And actually, it uses the air as
kind of its form of propulsion.

00:05:03.070 --> 00:05:06.560
It sucks in the air, and then it
spits out the air really fast.

00:05:06.560 --> 00:05:08.770
But it has all of
this air resistance.

00:05:08.770 --> 00:05:10.910
So if the engines were
to just shut down,

00:05:10.910 --> 00:05:15.600
all the air would
bump into the plane

00:05:15.600 --> 00:05:19.110
and provide essentially
friction to slow down the plane.

00:05:19.110 --> 00:05:21.850
What the space station or the
space shuttle or something

00:05:21.850 --> 00:05:24.230
in space has going for
itself is that it's

00:05:24.230 --> 00:05:27.230
traveling in an almost
complete vacuum.

00:05:27.230 --> 00:05:31.250
Not 100% complete vacuum,
but almost complete vacuum.

00:05:31.250 --> 00:05:34.950
So it has pretty much
no air resistance,

00:05:34.950 --> 00:05:37.840
negligible air resistance
to have to deal with.

00:05:37.840 --> 00:05:41.020
So we know from Newton's
laws, an object in motion

00:05:41.020 --> 00:05:42.220
tends to stay in motion.

00:05:42.220 --> 00:05:44.500
So once this thing
gets going, it

00:05:44.500 --> 00:05:46.130
doesn't have air
to slow it down.

00:05:46.130 --> 00:05:49.270
It'll keep staying that speed.

00:05:49.270 --> 00:05:52.650
In fact, if it did
not have gravity,

00:05:52.650 --> 00:05:56.530
which is causing this
centripetal acceleration

00:05:56.530 --> 00:05:58.970
right over here, it would
just go in a straight path.

00:05:58.970 --> 00:06:01.494
It would go in a straight
path forever and ever.

00:06:01.494 --> 00:06:03.160
And that brings up
an interesting point.

00:06:03.160 --> 00:06:05.290
Because if you are
in orbit like this,

00:06:05.290 --> 00:06:08.580
traveling at this very,
very, very fast speed,

00:06:08.580 --> 00:06:11.050
you have to make sure that
you don't vary from this speed

00:06:11.050 --> 00:06:11.620
too much.

00:06:11.620 --> 00:06:17.240
If you slow down, you will
slowly spiral into the Earth.

00:06:17.240 --> 00:06:19.390
And if you speed up a
lot beyond that speed,

00:06:19.390 --> 00:06:24.720
you will slowly spiral
away from the Earth.

00:06:24.720 --> 00:06:27.620
Because then the centripetal
acceleration due to gravity

00:06:27.620 --> 00:06:30.660
won't be enough to keep you
in a perfect circular path.

00:06:30.660 --> 00:06:34.250
So you really have to stay
pretty close to that speed

00:06:34.250 --> 00:06:36.070
right there.


Acceleration due to gravity at the space station

WEBVTT
Kind: captions
Language: en

00:00:00.550 --> 00:00:03.400
Most physics books will tell
you that the acceleration

00:00:03.400 --> 00:00:06.470
due to gravity near the
surface of the Earth

00:00:06.470 --> 00:00:10.240
is 9.81 meters per
second squared.

00:00:10.240 --> 00:00:12.000
And this is an approximation.

00:00:12.000 --> 00:00:14.030
And what I want to
do in this video

00:00:14.030 --> 00:00:17.710
is figure out if this is the
value we get when we actually

00:00:17.710 --> 00:00:20.940
use Newton's law of
universal gravitation.

00:00:20.940 --> 00:00:24.380
And that tells us that the
force of gravity between two

00:00:24.380 --> 00:00:26.595
objects-- and let's just
talk about the magnitude

00:00:26.595 --> 00:00:30.210
of the force of gravity
between two objects--

00:00:30.210 --> 00:00:35.910
is equal to the universal
gravitational constant times

00:00:35.910 --> 00:00:39.680
the mass of one of the
bodies, M1, times the mass

00:00:39.680 --> 00:00:44.340
of the second body divided by
the distance between the center

00:00:44.340 --> 00:00:46.150
of masses of the bodies squared.

00:00:52.480 --> 00:00:55.840
So let's use this, the
universal law of gravitation

00:00:55.840 --> 00:00:59.340
to figure out what the
acceleration due to gravity

00:00:59.340 --> 00:01:01.430
should be at the
surface of the Earth.

00:01:01.430 --> 00:01:03.430
And I have a g right over here.

00:01:03.430 --> 00:01:06.870
I have the mass of the Earth,
which I've looked up over here.

00:01:06.870 --> 00:01:08.649
And we also have the
radius of the Earth.

00:01:08.649 --> 00:01:10.190
And for the sake of
this, we're going

00:01:10.190 --> 00:01:13.650
to assume that the distance
between the body, if we're

00:01:13.650 --> 00:01:17.300
at the the surface of the
Earth, the distance between that

00:01:17.300 --> 00:01:20.020
and the center of
the Earth is just

00:01:20.020 --> 00:01:25.130
going to be the
radius of the Earth.

00:01:25.130 --> 00:01:28.272
And so this will give us
the magnitude of the force.

00:01:28.272 --> 00:01:30.730
If we want to figure out the
magnitude of the acceleration,

00:01:30.730 --> 00:01:32.855
which this really is-- I
actually didn't write this

00:01:32.855 --> 00:01:33.560
is a vector.

00:01:33.560 --> 00:01:35.810
So this is just the magnitude
of the acceleration.

00:01:35.810 --> 00:01:37.893
If you wanted the acceleration,
which is a vector,

00:01:37.893 --> 00:01:42.590
you'd have to say downwards or
towards the center of the Earth

00:01:42.590 --> 00:01:43.830
in this case.

00:01:43.830 --> 00:01:47.090
But if you want
the acceleration,

00:01:47.090 --> 00:01:49.920
we just have to
remember that force

00:01:49.920 --> 00:01:53.630
is equal to mass
times acceleration.

00:01:53.630 --> 00:01:55.560
And if you wanted to
solve for acceleration

00:01:55.560 --> 00:01:58.280
you just divide both
sides times mass.

00:01:58.280 --> 00:02:05.050
So force divided by mass
is equal to acceleration.

00:02:05.050 --> 00:02:08.330
Or if you take the
magnitude of your force

00:02:08.330 --> 00:02:10.110
and you divide by
mass, you're going

00:02:10.110 --> 00:02:12.130
to get the magnitude
of your acceleration.

00:02:12.130 --> 00:02:13.310
This is a scalar quantity.

00:02:13.310 --> 00:02:15.560
This is a scalar
quantity right over here.

00:02:15.560 --> 00:02:20.740
So if you want the acceleration
due to gravity, you divide.

00:02:20.740 --> 00:02:25.100
Let's write this in terms of
the force of gravity on Earth.

00:02:25.100 --> 00:02:26.940
So the magnitude of
the force of gravity

00:02:26.940 --> 00:02:29.310
on Earth, this one
right over here.

00:02:29.310 --> 00:02:30.980
So this will be in
the case of Earth.

00:02:30.980 --> 00:02:33.060
I just wrote Earth
really, really small.

00:02:33.060 --> 00:02:35.500
So one of these masses
is going to be Earth.

00:02:35.500 --> 00:02:38.140
It's going to be this
mass right over here.

00:02:38.140 --> 00:02:41.520
And so if you wanted
the acceleration

00:02:41.520 --> 00:02:43.920
due to gravity at the
surface of the Earth,

00:02:43.920 --> 00:02:48.170
you would just have to
divide by the mass that

00:02:48.170 --> 00:02:50.890
is being accelerated
due to that force.

00:02:50.890 --> 00:02:52.940
And in this case, it
is the other mass.

00:02:52.940 --> 00:02:55.920
It is the mass that's
sitting on the surface.

00:02:55.920 --> 00:02:59.150
So let's divide both
sides by that mass.

00:02:59.150 --> 00:03:02.050
Let's divide both
sides by that mass.

00:03:02.050 --> 00:03:05.400
And this will give
us the magnitude

00:03:05.400 --> 00:03:09.270
of the acceleration on
that mass due to gravity.

00:03:09.270 --> 00:03:15.550
So this is equal to the
magnitude of acceleration,

00:03:15.550 --> 00:03:17.189
due to gravity.

00:03:17.189 --> 00:03:19.730
And the whole reason why this
is actually a simplifying thing

00:03:19.730 --> 00:03:22.620
is that these two, this M2
right over here and this M2

00:03:22.620 --> 00:03:23.750
cancels out.

00:03:23.750 --> 00:03:27.680
And so the magnitude
of our acceleration

00:03:27.680 --> 00:03:31.120
due to gravity using Newton's
universal law of gravitation

00:03:31.120 --> 00:03:34.110
is just going to be this
expression right over here.

00:03:34.110 --> 00:03:38.570
It's going to be the
gravitational constant times

00:03:38.570 --> 00:03:42.250
the mass of the Earth
divided by the distance

00:03:42.250 --> 00:03:45.900
between the object's
center of mass

00:03:45.900 --> 00:03:47.710
and the center of the
mass of the Earth.

00:03:47.710 --> 00:03:48.780
And we're going to
assume that the object is

00:03:48.780 --> 00:03:50.654
right at the surface,
that its center of mass

00:03:50.654 --> 00:03:51.930
is right at the surface.

00:03:51.930 --> 00:03:54.900
So this is actually going to
be the radius of the Earth

00:03:54.900 --> 00:03:58.430
squared, so divided
by radius squared.

00:03:58.430 --> 00:04:03.320
Sometimes this is also viewed
as the gravitational field

00:04:03.320 --> 00:04:04.650
at the surface of the Earth.

00:04:04.650 --> 00:04:06.280
Because if you
multiply it by a mass,

00:04:06.280 --> 00:04:08.537
it tells you how much force
is pulling on that mass.

00:04:08.537 --> 00:04:10.370
But with that out of
the way, let's actually

00:04:10.370 --> 00:04:13.360
use a calculator to
calculate what this value is.

00:04:13.360 --> 00:04:16.089
And then what I want to do
is figure out, well, one,

00:04:16.089 --> 00:04:17.630
I want to compare
it to the value

00:04:17.630 --> 00:04:19.470
that the textbooks
give us and see, maybe,

00:04:19.470 --> 00:04:21.950
why it may or may
not be different.

00:04:21.950 --> 00:04:23.680
And then think
about how it changes

00:04:23.680 --> 00:04:25.191
as we get further
and further away

00:04:25.191 --> 00:04:26.440
from the surface of the Earth.

00:04:26.440 --> 00:04:28.880
And in particular, if
we get to an altitude

00:04:28.880 --> 00:04:31.320
that the space shuttle or the
International Space Station

00:04:31.320 --> 00:04:35.600
might be at, and this is at
an altitude of 400 kilometers

00:04:35.600 --> 00:04:37.740
is where it tends to
hang out, give or take

00:04:37.740 --> 00:04:39.910
a little bit, depending
on what it is up to.

00:04:39.910 --> 00:04:42.250
So first, let's just
figure out what this value

00:04:42.250 --> 00:04:45.590
is when we use a universal
law of gravitation.

00:04:45.590 --> 00:04:49.730
So let's get my calculator out.

00:04:49.730 --> 00:04:51.900
So we know what g is.

00:04:51.900 --> 00:04:58.740
It is 6.6738 times 10
to the negative 11.

00:04:58.740 --> 00:05:04.340
This EE button means, literally,
times 10 to the negative 11.

00:05:04.340 --> 00:05:07.517
So this is 6.6738 times
10 to the negative 11.

00:05:07.517 --> 00:05:09.100
And then I want to
multiply that times

00:05:09.100 --> 00:05:13.410
the mass of Earth, which
is right over here.

00:05:13.410 --> 00:05:20.280
That is 5.9722 times
10 to the 24th.

00:05:20.280 --> 00:05:23.460
So times 10 to the 24th power.

00:05:23.460 --> 00:05:27.060
And we want to divide that by
the radius of Earth squared.

00:05:27.060 --> 00:05:29.900
So divided by the
radius of Earth

00:05:29.900 --> 00:05:31.377
is-- so this is in kilometers.

00:05:31.377 --> 00:05:33.210
And I just want to make
sure that everything

00:05:33.210 --> 00:05:35.710
is the same units.

00:05:35.710 --> 00:05:38.980
So 6,371 kilometers--
actually, let me scroll over.

00:05:38.980 --> 00:05:40.855
Well, you can't see the
kilometers right now.

00:05:40.855 --> 00:05:42.200
But this is kilometers.

00:05:42.200 --> 00:05:45.900
It is the same thing
as 6,371,000 meters.

00:05:45.900 --> 00:05:47.750
If you just multiply
this by 1,000.

00:05:47.750 --> 00:05:54.990
Or you could even
write this as 6.371.

00:05:54.990 --> 00:06:02.690
6.371 times 10 to
the sixth meters.

00:06:02.690 --> 00:06:04.010
And we're going to square this.

00:06:04.010 --> 00:06:05.301
That's the radius of the Earth.

00:06:05.301 --> 00:06:08.220
The distance between the center
of mass of Earth and the center

00:06:08.220 --> 00:06:10.060
of mass of this object,
which is sitting

00:06:10.060 --> 00:06:11.960
at the surface of the Earth.

00:06:11.960 --> 00:06:14.100
And so let's get our drum roll.

00:06:14.100 --> 00:06:16.959
And we get 9.8.

00:06:16.959 --> 00:06:19.250
And if we round, we actually
get something a little bit

00:06:19.250 --> 00:06:21.420
higher than what the
textbooks give us.

00:06:21.420 --> 00:06:23.250
We get 9.82.

00:06:23.250 --> 00:06:24.330
Let's just round.

00:06:24.330 --> 00:06:31.019
So we get 9.82-- 9.82
meters per second squared.

00:06:31.019 --> 00:06:33.060
And so you might say,
well, what's going on here?

00:06:33.060 --> 00:06:35.260
Why do we have this
discrepancy between what

00:06:35.260 --> 00:06:37.470
the universal law of
gravitation gives us

00:06:37.470 --> 00:06:40.460
and what the average
measured acceleration

00:06:40.460 --> 00:06:43.680
due to the force of gravity
at the surface of the Earth.

00:06:43.680 --> 00:06:46.870
And the discrepancy here, the
discrepancy between these two

00:06:46.870 --> 00:06:49.550
numbers, is really
because Earth is not

00:06:49.550 --> 00:06:53.419
a uniform sphere
of uniform density.

00:06:53.419 --> 00:06:55.210
And that's what we have
to assume over here

00:06:55.210 --> 00:06:57.360
when we use the universal
law of gravitation.

00:06:57.360 --> 00:07:03.050
It's actually a little bit
flatter than a perfect sphere.

00:07:03.050 --> 00:07:05.050
And it definitely does
not have uniform density.

00:07:05.050 --> 00:07:07.508
The different layers of the
Earth have different densities.

00:07:07.508 --> 00:07:09.460
You have all sorts of
different interactions.

00:07:09.460 --> 00:07:11.950
And then you also, if you
measure effective gravity,

00:07:11.950 --> 00:07:15.810
there's also a little bit of a
buoyancy effect from the air.

00:07:15.810 --> 00:07:17.750
Very, very, very,
very negligible, I

00:07:17.750 --> 00:07:20.320
don't know if it would have
been enough to change this.

00:07:20.320 --> 00:07:23.450
But there's other minor,
minor effects, irregularities.

00:07:23.450 --> 00:07:24.700
Earth is not a perfect sphere.

00:07:24.700 --> 00:07:26.300
It is not of uniform density.

00:07:26.300 --> 00:07:28.990
And that's what accounts
for the bulk of this.

00:07:28.990 --> 00:07:31.440
Now, with that out of the
way, what I'm curious about

00:07:31.440 --> 00:07:35.280
is what is the
acceleration due to gravity

00:07:35.280 --> 00:07:37.870
if we go up 400 kilometers?

00:07:37.870 --> 00:07:41.290
So now, the main difference
here, g will stay the same.

00:07:41.290 --> 00:07:42.940
The mass of Earth
will stay the same,

00:07:42.940 --> 00:07:45.010
but the radius is now
going to be different.

00:07:45.010 --> 00:07:47.510
Because now we're placing the
center of mass of our object--

00:07:47.510 --> 00:07:50.010
whether it's a space station
or someone sitting in the space

00:07:50.010 --> 00:07:52.540
station, they're going to
be 400 kilometers higher.

00:07:52.540 --> 00:07:55.490
And I'm going to exaggerate
what 400 kilometers looks like.

00:07:55.490 --> 00:07:57.106
This is not drawn to scale.

00:07:57.106 --> 00:07:59.480
But now the radius is going
to be the radius of the Earth

00:07:59.480 --> 00:08:02.020
plus 400 kilometers.

00:08:02.020 --> 00:08:05.290
So now, for the case
of the space station,

00:08:05.290 --> 00:08:09.130
r is going to be not
6,371 kilometers.

00:08:09.130 --> 00:08:11.680
It's going to be 6,000--
we're going to add 400

00:08:11.680 --> 00:08:16.320
to this-- 6,771
kilometers, which

00:08:16.320 --> 00:08:22.200
is the same thing as
6,771,000 meters, which

00:08:22.200 --> 00:08:27.550
is the same thing as 6.771
times 10 to the sixth meters.

00:08:27.550 --> 00:08:32.080
This is-- 1, 2, 3, 4, 5,
6-- 10 to the sixth meters.

00:08:32.080 --> 00:08:35.659
So let's go back
to our calculator.

00:08:35.659 --> 00:08:39.090
So second entry, that's
the last entry we had.

00:08:39.090 --> 00:08:42.330
And instead of 6.371
times 10 to the sixth,

00:08:42.330 --> 00:08:45.030
let's add 400
kilometers to that.

00:08:45.030 --> 00:08:47.842
So then we get 6.7.

00:08:47.842 --> 00:08:50.280
So we're adding 400 kilometers.

00:08:50.280 --> 00:08:51.336
So it was 371.

00:08:51.336 --> 00:08:53.970
Now it's 771 times
10 to the sixth.

00:08:53.970 --> 00:08:56.290
And what do we get?

00:08:56.290 --> 00:09:00.710
We get 8.69 meters
per second squared.

00:09:00.710 --> 00:09:09.140
So now the acceleration here is
8.69 meters per second squared.

00:09:09.140 --> 00:09:11.580
And you can verify that
the units work out.

00:09:11.580 --> 00:09:13.320
Because over here,
gravity is in meters

00:09:13.320 --> 00:09:15.624
cubed per kilogram
second squared.

00:09:15.624 --> 00:09:17.540
You multiply that times
the mass of the Earth,

00:09:17.540 --> 00:09:18.620
which is in kilograms.

00:09:18.620 --> 00:09:21.290
The kilograms cancel out
with these kilograms.

00:09:21.290 --> 00:09:24.300
And then you're dividing
by meters squared.

00:09:24.300 --> 00:09:26.690
So you divide this
by meters squared.

00:09:26.690 --> 00:09:28.940
You're left with meters
per second squared.

00:09:28.940 --> 00:09:30.472
So the units work out as well.

00:09:30.472 --> 00:09:32.180
So there's an important
thing to realize.

00:09:32.180 --> 00:09:33.346
And this is a misconception.

00:09:33.346 --> 00:09:35.080
We do a whole video
on it earlier,

00:09:35.080 --> 00:09:37.640
when we talk about the
universal law of gravitation,

00:09:37.640 --> 00:09:42.580
is that there is gravity when
you are in orbit up here.

00:09:42.580 --> 00:09:44.920
The only reason why it feels
like there's not gravity

00:09:44.920 --> 00:09:46.420
or it looks like
there's not gravity

00:09:46.420 --> 00:09:48.700
is that this space
station is moving

00:09:48.700 --> 00:09:51.870
so fast that it's
essentially in free fall.

00:09:51.870 --> 00:09:55.555
But it's moving so fast that
it keeps missing the Earth.

00:09:55.555 --> 00:09:57.180
And in the next video,
we'll figure out

00:09:57.180 --> 00:10:01.140
how fast does it have to
travel in order for it to stay

00:10:01.140 --> 00:10:04.620
in orbit, in order for it to not
plummet to Earth due to this,

00:10:04.620 --> 00:10:07.150
due to the force of gravity,
due to the acceleration that

00:10:07.150 --> 00:10:10.200
is occurring, this centripetal,
this center-seeking

00:10:10.200 --> 00:10:11.600
acceleration?


Calculus proof of centripetal acceleration formula

WEBVTT
Kind: captions
Language: en

00:00:00.538 --> 00:00:04.054
What I want to do in this video is a calculus proof of the famous

00:00:04.054 --> 00:00:07.862
centripetal acceleration formula that tells us the magnitude

00:00:07.862 --> 00:00:10.824
of centripetal acceleration, the actual direction will change

00:00:10.824 --> 00:00:14.901
it's always going to be pointing inwards, but the magnitude of centripetal acceleration is equal to the

00:00:14.901 --> 00:00:19.235
magnitude of the velocity-squared divided by the radius

00:00:19.358 --> 00:00:23.082
I want to be very clear, this is a scalar formula right over here; we're talking about the magnitude

00:00:23.082 --> 00:00:26.410
of the acceleration and the magnitude of the velocity. If these were vectors,

00:00:26.410 --> 00:00:30.045
we would have arrows drawn over it. So this really, I don't want people to get confused

00:00:30.045 --> 00:00:34.313
because this is a v, this is really referring to the speed-squared.

00:00:34.313 --> 00:00:36.859
and this is the magnitude, and these are all

00:00:36.859 --> 00:00:41.094
scalar quantities. So to do that, let's imagine some

00:00:41.094 --> 00:00:45.366
object, maybe it's some object in orbit around a planet or something,

00:00:45.366 --> 00:00:48.830
so let's say that's the planet, and you have some object

00:00:48.830 --> 00:00:52.087
that is in orbit around the planet, and it is going in a counter-clockwise

00:00:52.087 --> 00:00:56.273
direction, and so let's specify its position vector as

00:00:56.273 --> 00:00:58.657
a function as time.

00:01:01.519 --> 00:01:04.199
So that is its position vector and it is going to

00:01:04.199 --> 00:01:07.999
change as a function of time as this thing spins around

00:01:07.999 --> 00:01:11.569
We're going to assume, for the purposes of this

00:01:11.569 --> 00:01:13.430
proof.

00:01:18.630 --> 00:01:19.893
So that is our

00:01:19.893 --> 00:01:23.307
y-axis, and this is our x-axis.

00:01:23.645 --> 00:01:27.532
We're going to define theta as the angle between the positive

00:01:27.532 --> 00:01:30.051
x-axis and our vector.

00:01:31.189 --> 00:01:35.199
And we're going to assume this thing is in orbit with the radius of r. So the

00:01:35.199 --> 00:01:39.284
magnitude of our position vector, even though the direction is going to change

00:01:39.284 --> 00:01:43.400
the magnitude of our position vector is not going to change. It's always going to have

00:01:43.400 --> 00:01:47.068
length r. So this is going in a circle with radius r.

00:01:47.345 --> 00:01:50.345
The magnitude of our position vector,

00:01:50.345 --> 00:01:54.635
which is changing as a function of time, is going to be

00:01:54.635 --> 00:01:58.628
r. So how can we write

00:01:58.628 --> 00:02:03.348
the position vector in terms of its components at any given time?

00:02:03.348 --> 00:02:06.881
We can write the position vector, and I'll do it in engineering notation,

00:02:06.881 --> 00:02:09.562
and so you might want to review those videos if some of this looks foreign and I'll

00:02:09.562 --> 00:02:14.026
do a bit of basic trigonometry in breaking down the vector into its components

00:02:14.026 --> 00:02:17.557
and I encourage you to review some of those videos if some of that looks a bit

00:02:17.557 --> 00:02:23.006
daunting. If you take the position vector at any time,

00:02:23.006 --> 00:02:25.263
the magnitude is r,

00:02:25.263 --> 00:02:28.475
this angle is theta,

00:02:28.475 --> 00:02:33.274
its x-component, in blue,

00:02:33.274 --> 00:02:37.905
this vector right over here, the magnitude of the vector, I should say,

00:02:37.905 --> 00:02:41.380
is going to be r cosine of theta

00:02:41.380 --> 00:02:44.512
We learned that this came from basic trigonometry when we started

00:02:44.512 --> 00:02:48.443
two-dimensional projectile motion, we saw how to break these

00:02:48.443 --> 00:02:55.263
vectors down into its components, and the y-component of this vector

00:02:55.294 --> 00:02:58.457
is going to be r sine of theta

00:03:00.704 --> 00:03:03.989
So this is going to be r sine of

00:03:03.989 --> 00:03:09.113
theta. So the position vector at any time

00:03:09.113 --> 00:03:11.972
can be written as a sum of its x- and y-components.

00:03:11.972 --> 00:03:15.083
So it's the magnitude of its x-component, it's going to be

00:03:15.083 --> 00:03:20.174
r cosine of theta, and I could write theta as a function of

00:03:20.174 --> 00:03:23.886
time if I'd like, but I'm just going to write r cosine of theta

00:03:23.886 --> 00:03:27.035
actually, let me write it that way, so it shows theta is a function of time

00:03:27.035 --> 00:03:30.503
This thing is moving, and there's going to be that times the

00:03:30.503 --> 00:03:34.870
i-unit vector, we're in engineering notation over here.

00:03:34.870 --> 00:03:38.353
So that's the i-unit vector, it tells us that the x-component

00:03:38.353 --> 00:03:42.507
is going in the positive x direction. Plus

00:03:42.507 --> 00:03:45.479
the magnitude of the y-component, which is

00:03:45.479 --> 00:03:50.443
r sine of theta, which is going to be a function of time.

00:03:50.443 --> 00:03:54.715
So to be clear, the function of time applies to the theta.

00:03:57.484 --> 00:03:58.371
And that is going

00:03:58.371 --> 00:04:00.603
in the j-direction.

00:04:02.818 --> 00:04:05.284
So that is our j-unit vector.

00:04:07.853 --> 00:04:10.774
So now we have position as a function of theta

00:04:10.774 --> 00:04:14.122
which is actually a function of time. So let's take the derivative of

00:04:14.122 --> 00:04:19.627
this thing right here. So what is

00:04:19.627 --> 00:04:26.818
the derivative of our position vector with respect to time

00:04:26.818 --> 00:04:30.754
Well that's just going to be our velocity vector,

00:04:30.754 --> 00:04:34.625
as a function of time, and it's going to be equal to

00:04:34.625 --> 00:04:38.375
we just have to take the derivative of each of these parts with respect to

00:04:38.375 --> 00:04:42.222
time. And you just do the chain rule. So you're going to have

00:04:42.222 --> 00:04:45.646
the r sit outside cause that's just a constant. So you're going to have r

00:04:45.646 --> 00:04:50.277
the derivative of cosine of theta t with respect to theta t

00:04:50.277 --> 00:04:53.523
So I'm just doing the chain rule right over here. That's going to be

00:04:53.523 --> 00:04:58.361
negative sine of theta t

00:04:58.361 --> 00:05:01.746
and then as the chain rule, we also have to multiply that

00:05:01.746 --> 00:05:04.705
times the derivative of the theta of t

00:05:04.705 --> 00:05:10.122
with respect to t. So times d-theta, dt

00:05:10.122 --> 00:05:12.929
so this is just the chain rule right over here.

00:05:12.929 --> 00:05:16.882
So that's going to be how it's changing in the x-direction, and in the y-direction

00:05:16.882 --> 00:05:19.999
we do something very similar. In the y-direction

00:05:19.999 --> 00:05:23.305
we take the same derivative. We have the r scalar out front

00:05:23.305 --> 00:05:26.964
r, and then the derivative of sine of theta

00:05:26.964 --> 00:05:31.030
with respect to theta is going to be cosine of theta

00:05:31.030 --> 00:05:34.947
and I'll write it as a function of time, and then do the chain rule

00:05:34.947 --> 00:05:39.513
you'll also have to multiply that by the rate at which theta is changing with respect to t,

00:05:39.513 --> 00:05:42.885
times d-theta, dt,

00:05:42.885 --> 00:05:50.287
and this is all going to be times our j-unit vector. Now,

00:05:50.287 --> 00:05:54.418
there's something you might already realize, and you should rewatch the video

00:05:54.418 --> 00:05:57.691
on angular velocity if this is foreign to you, but

00:05:57.691 --> 00:06:01.759
d-theta, dt, this is our angular velocity.

00:06:01.759 --> 00:06:05.741
That's why I said to rewatch that video. This right over here, the rate at which

00:06:05.741 --> 00:06:09.333
the angle changes with respect to time, that is angular velocity.

00:06:09.333 --> 00:06:13.314
So this right over here is

00:06:13.314 --> 00:06:17.311
angular velocity. And for the sake of this

00:06:17.311 --> 00:06:21.362
video, this is an assumption we'll have to make for this formula right over here

00:06:21.362 --> 00:06:25.432
we're going to assume, that omega, which is the rate of change

00:06:25.432 --> 00:06:28.315
of our angle with respect to time, we're going to assume

00:06:28.315 --> 00:06:31.931
that this is constant.

00:06:32.754 --> 00:06:36.911
So this is an assumption we're making for this proof. This is we are going

00:06:36.911 --> 00:06:40.357
to assume that omega is constant. And if omega is

00:06:40.357 --> 00:06:44.227
constant, then we can treat it as a constant and we can factor

00:06:44.227 --> 00:06:47.413
it out of this expression. So let's factor out

00:06:47.413 --> 00:06:52.948
a negative omega-r from this expression over here. So we can rewrite

00:06:53.533 --> 00:06:56.420
our velocity as a function of time is equal to

00:06:56.420 --> 00:07:00.285
I'm going to factor out a negative omega

00:07:00.285 --> 00:07:04.420
times r, and if you factor out a negative omega-r,

00:07:04.420 --> 00:07:08.681
what you're left with is

00:07:08.681 --> 00:07:10.896
this first term,

00:07:14.215 --> 00:07:20.069
sine of theta-t

00:07:21.438 --> 00:07:25.932
And I didn't have to make it explicit that theta is a function of t, but this makes it explicit

00:07:25.932 --> 00:07:30.920
that theta is a function of t, and then times

00:07:30.920 --> 00:07:35.772
our i-unit vector,

00:07:35.772 --> 00:07:39.531
plus, so if we're factoring out

00:07:39.531 --> 00:07:44.312
a negative omega-r, this becomes negative

00:07:44.312 --> 00:07:46.817
cosine of theta,

00:07:46.817 --> 00:07:51.164
which is a function of t, and that

00:07:51.164 --> 00:07:55.544
is times our j-unit vector.

00:07:58.177 --> 00:07:59.389
So we factored out

00:07:59.420 --> 00:08:02.114
a negative omega-r. Now let's take the derivative of

00:08:02.114 --> 00:08:05.784
this with respect to time. So if we take

00:08:07.184 --> 00:08:09.999
the derivative of velocity

00:08:09.999 --> 00:08:13.964
with respect to time, this is clearly just what the acceleration is

00:08:13.964 --> 00:08:17.515
as a function of time, and we're going to assume that the magnitude of this thing

00:08:17.515 --> 00:08:21.048
is constant, but the actual direction is changing, so this is the acceleration

00:08:21.048 --> 00:08:26.564
as a function of time, is going to be equal to

00:08:28.795 --> 00:08:29.620
this negative

00:08:29.620 --> 00:08:32.656
omega-r, so what is the derivative of

00:08:32.656 --> 00:08:36.935
this thing right over here? So the derivative of sine with respect to theta,

00:08:36.985 --> 00:08:38.581
we're just doing the chain rule here,

00:08:38.581 --> 00:08:40.635
the derivative of sine with respect to theta

00:08:40.635 --> 00:08:46.828
is going to be cosine of theta

00:08:46.878 --> 00:08:49.049
as a function of t.

00:08:49.049 --> 00:08:52.464
And then chain rule, we also have to take that and multiply it with the derivative of theta

00:08:52.514 --> 00:08:54.701
with respect to t.

00:08:56.498 --> 00:08:59.788
I could write d-theta, dt here

00:08:59.788 --> 00:09:03.759
But that once again, is just omega.

00:09:03.759 --> 00:09:08.043
So that is just omega, and that of course

00:09:08.043 --> 00:09:12.774
is in the i-direction. And from that,

00:09:12.774 --> 00:09:14.808
and next to that, we take the derivative of

00:09:14.808 --> 00:09:19.515
cosine of theta of t with respect to theta, so that's going to be

00:09:19.515 --> 00:09:23.115
that would be negative sine of theta, so we would have a negative out front

00:09:23.115 --> 00:09:27.543
so it becomes positive sine of

00:09:27.543 --> 00:09:30.942
theta as a function of t.

00:09:30.942 --> 00:09:35.396
And then we have to do the chain rule, the derivative of theta with respect to t.

00:09:35.396 --> 00:09:38.125
We have to multiply by this, and for that we could write

00:09:38.125 --> 00:09:42.806
d-theta, dt right there, but that again is the same thing

00:09:42.806 --> 00:09:45.824
as omega. And all that is being multiplied by the

00:09:45.824 --> 00:09:48.728
j-unit vector.

00:09:51.120 --> 00:09:54.998
So now let's factor out this other omega, and we get something interesting,

00:09:54.998 --> 00:09:58.795
we get the acceleration vector as a function of

00:09:58.795 --> 00:10:02.513
time is equal to, and if we factor out another omega, we get

00:10:02.513 --> 00:10:06.455
negative omega-squared r,

00:10:06.455 --> 00:10:11.224
I'm just factoring out another negative omega,

00:10:11.224 --> 00:10:16.550
times, and I'll write it in parentheses here,

00:10:16.550 --> 00:10:21.040
cosine of theta

00:10:21.040 --> 00:10:23.025
as a function of t

00:10:23.025 --> 00:10:26.668
times our i-unit vector

00:10:26.668 --> 00:10:29.707
plus

00:10:29.707 --> 00:10:34.025
sine of theta, which is a function of t

00:10:34.025 --> 00:10:37.659
times our j-unit vector.

00:10:37.659 --> 00:10:45.467
Now what is all of this business right over here?

00:10:45.467 --> 00:10:49.401
Just look at this part right over here, well r times this,

00:10:49.401 --> 00:10:53.813
especially if you distributed the r, that is exactly this thing right over here

00:10:53.813 --> 00:10:57.462
If you distribute the r, you get exactly r cosine theta as a function

00:10:57.462 --> 00:11:01.241
of t times our i-unit vector plus our sine theta as a

00:11:01.241 --> 00:11:06.582
function of t times the j-unit vector. So everything that I squared-off in orange right over here,

00:11:06.582 --> 00:11:13.135
this is our position vector as a function of time.

00:11:13.135 --> 00:11:14.975
So all that work we did, we just got a

00:11:14.975 --> 00:11:19.135
very interesting result. We got that our acceleration vector

00:11:19.135 --> 00:11:23.384
as a function of time is equal to the negative

00:11:23.384 --> 00:11:32.767
of our constant angular velocity-squared

00:11:32.767 --> 00:11:34.829
times our position vector

00:11:34.829 --> 00:11:37.928
And just to be clear, angular velocity is kind of the pseudo vector,

00:11:37.928 --> 00:11:42.248
it tends to be treated like a scalar, especially when you're dealing with two-dimensionals

00:11:42.248 --> 00:11:47.311
like this, it's really a pseudo scalar, but let's just go with this.

00:11:47.311 --> 00:11:51.294
We're assuming this right over here is a constant scalar quantity.

00:11:51.909 --> 00:11:55.721
Now, we're very very very very close here.

00:11:55.721 --> 00:11:58.978
Now what we want to do is to relate this

00:11:58.978 --> 00:12:02.591
this is essentially the scalar version of it, so if we wanted

00:12:02.591 --> 00:12:05.774
to take the magnitudes of both sides,

00:12:05.774 --> 00:12:10.021
so we're saying the acceleration vector is equal to this constant times

00:12:10.021 --> 00:12:19.116
the position vector, so let's take the magnitude of both sides of this thing

00:12:19.116 --> 00:12:22.615
So then we get the magnitude of the acceleration vector,

00:12:22.615 --> 00:12:24.462
which I'm just going to call a sub c,

00:12:24.462 --> 00:12:26.880
is going to be equal to

00:12:26.880 --> 00:12:29.790
you could say the magnitude of this negative omega-squared

00:12:29.790 --> 00:12:32.645
but when you take the magnitude, it's like taking the absolute value

00:12:32.645 --> 00:12:35.464
in fact, absolute value is just the one-dimensional version of magnitude,

00:12:35.464 --> 00:12:38.107
that's just going to be positive omega-squared

00:12:38.107 --> 00:12:41.488
we don't care about the direction, sign gives us the direction, we just care about

00:12:41.488 --> 00:12:48.076
the actual size. So this is going to be

00:12:48.076 --> 00:12:50.255
the magnitude of negative omega-squared

00:12:50.255 --> 00:12:54.642
times the magnitude of our position vector

00:12:54.642 --> 00:12:58.517
the magnitude of omega-squared is just going to be omega-squared

00:12:58.517 --> 00:13:01.797
you can get rid of the sign, and the magnitude of our position vector

00:13:01.797 --> 00:13:05.167
we saw at the beginning of this video, is just r,

00:13:05.167 --> 00:13:08.563
our radius

00:13:09.317 --> 00:13:13.277
so this right over here is just going to be equal to

00:13:13.277 --> 00:13:16.051
the radius of the circle that we're going around.

00:13:16.189 --> 00:13:19.884
Now, we also know the angular velocity, or

00:13:19.884 --> 00:13:24.367
the magnitude of the angular velocity, is equal to the

00:13:24.367 --> 00:13:28.433
magnitude of our velocity, or the speed of our

00:13:28.433 --> 00:13:32.101
object, divided by the radius of the circle

00:13:32.101 --> 00:13:35.913
that it is going around. So we could substitute that right over here.

00:13:35.913 --> 00:13:42.648
So if we square it, this is going to be (v over r)-squared, now we saw that in the video on angular velocity,

00:13:42.648 --> 00:13:44.687
times r

00:13:44.687 --> 00:13:47.424
and this is all going to be the magnitude of our acceleration,

00:13:47.424 --> 00:13:51.490
which is really our centripetal acceleration, our inward directed acceleration.

00:13:51.798 --> 00:13:56.021
So this is going to be equal to, and I think you see where this is going,

00:13:56.021 --> 00:13:59.869
This is equal to v-squared over r-squared

00:13:59.869 --> 00:14:02.970
times r, but this r cancels out

00:14:02.970 --> 00:14:06.688
with the r-squared, so you're just left with v-squared

00:14:06.688 --> 00:14:09.004
over r, and you're done!

00:14:09.004 --> 00:14:13.654
The magnitude of the centripetal acceleration is equal to your speed,

00:14:13.654 --> 00:14:17.161
the magnitude of your velocity, squared, divided by

00:14:17.161 --> 99:59:59.999
your radius. And, we are done.


Relationship between angular velocity and speed

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:03.630
Let's say we have some object that's moving in a circular path

00:00:03.630 --> 00:00:08.200
Let's say this is the center of the object path, the center of the circle

00:00:08.390 --> 00:00:12.880
So the object is moving in a circular path that looks something like that

00:00:12.880 --> 00:00:16.600
counterclockwise circular path--you could do that with clockwise as well

00:00:16.870 --> 00:00:23.010
I want to think about how fast it is spinning or orbiting around this center

00:00:23.070 --> 00:00:25.930
how that relates to its velocity?

00:00:25.950 --> 00:00:33.290
So let's say that this thing right over here is making five revolutions every second

00:00:33.310 --> 00:00:40.880
So in 1 second, 1 2 3 4 5. Every second it's making 5 revolutions

00:00:40.900 --> 00:00:45.700
So how could we relate that to how many radians it is doing per second?

00:00:45.740 --> 00:00:48.660
Remember radians is just one way to measure angles

00:00:48.660 --> 00:00:50.710
You could do with how degrees per second

00:00:54.550 --> 00:01:02.590
If we do it with radians, we know that each revolution is 2 pi radians

00:01:02.610 --> 00:01:10.340
If you go all the way around a circle, you have gone 2 pi radians

00:01:10.340 --> 00:01:16.800
which is really just you say you've gone 2 pi radii, whatever the radius of the circle is

00:01:16.820 --> 00:01:20.120
and that's where actually the definition of the radian comes from

00:01:20.140 --> 00:01:30.000
So if you go 5 revolutions per second and they're 2 pi per revolution

00:01:30.000 --> 00:01:33.350
then you can do a little bit of dimensional analysis. These cancel out

00:01:33.490 --> 00:01:41.450
and you get 5 times 2 pi which gets us to 5 times 2 pi gets us 10 pi

00:01:41.450 --> 00:01:47.760
radians per second

00:01:47.770 --> 00:01:52.200
And it works out the dimensional analysis and hopefully it also makes sense to you intuitively

00:01:52.350 --> 00:01:56.640
If you're doing five revolution a second, each of those revolutions is 2 pi radians

00:01:56.710 --> 00:02:00.030
so you're doing 10 pi radians per second. You're going

00:02:00.310 --> 00:02:04.150
1 2 3 4 5, so that gives us 10, or 2 pi 2 pi 2 pi 2 pi 2 pi radians

00:02:04.160 --> 00:02:08.550
every time, you're doing it five times a second. So you're doing it 10 pi radians per second

00:02:08.810 --> 00:02:12.730
So this right here, either five revs per second or 10 pi radians per second

00:02:12.780 --> 00:02:14.990
they're both essentially measuring the same thing

00:02:14.990 --> 00:02:18.990
how fast are you orbiting around this central point?

00:02:19.310 --> 00:02:23.020
And this measure of how fast you're orbiting around a central point

00:02:23.020 --> 00:02:29.830
is called angular velocity

00:02:29.970 --> 00:02:32.460
It's called angular velocity because if you think about it

00:02:32.540 --> 00:02:40.360
this is telling us how fast is our angle changing, or speed of angle changing

00:02:40.550 --> 00:02:43.280
When you're dealing with it in two dimensions and this is

00:02:43.430 --> 00:02:47.260
typically when in a recent early physics course how we do deal with it

00:02:47.260 --> 00:02:49.230
Even though it's called the angular velocity

00:02:49.310 --> 00:02:51.020
it tends to be treated as angular speed

00:02:51.020 --> 00:02:52.870
It actually is a vector quantity

00:02:53.020 --> 00:02:56.360
and it's a little unintuitive that the vector's actually popping out of the page

00:02:56.460 --> 00:02:59.860
for this. It's actually a pseudo-vector and we'll talk more about that in the future

00:02:59.920 --> 00:03:03.580
So it is a vector quantity and the direction of the vector

00:03:03.580 --> 00:03:06.580
is dependant on which way it's spinning. So for example

00:03:06.580 --> 00:03:09.200
when it's spinning in a counterclockwise direction

00:03:09.280 --> 00:03:13.530
there is a vector, the real angular vector does pop out of the page

00:03:13.530 --> 00:03:16.700
We start thinking about operating in three dimensions

00:03:16.730 --> 00:03:22.760
And if it's going clockwise, the angular velocity vector would pop into the page

00:03:22.800 --> 00:03:25.220
The way you think about that, right-hand rule

00:03:25.230 --> 00:03:28.260
Curl your fingers of your right hand in the direction that it's spinning

00:03:28.260 --> 00:03:33.380
and then your thumb is essentially pointing in the direction that

00:03:33.380 --> 00:03:36.610
the actual vector or the pseudo-vector's gonna going

00:03:36.610 --> 00:03:38.480
We'll not think too much about that

00:03:38.510 --> 00:03:42.500
For our purposes, when we're just thinking about two-dimensional plane right over here

00:03:42.950 --> 00:03:49.320
we can really think of an angular velocity as a--the official term is a pseudo-scaler

00:03:49.320 --> 00:03:55.820
but we can include that as a scaler quantity, as long as we do specify which way it is rotating

00:03:55.830 --> 00:03:59.040
So this right over here, this 10 pi radians per second

00:03:59.040 --> 00:04:01.810
we could call this its angular velocity

00:04:02.030 --> 00:04:06.300
And this tends to be denoted by an omega

00:04:06.330 --> 00:04:08.370
a lower case omega right there

00:04:08.380 --> 00:04:10.250
Upper case omega looks like this

00:04:14.390 --> 00:04:16.430
So there's a couple of ways you could think about it

00:04:16.450 --> 00:04:25.220
You could say angular velocity is equal to change in angle over a change in time

00:04:25.320 --> 00:04:32.290
So for example, this is telling us 10 pi radians per second

00:04:32.450 --> 00:04:41.820
Or if you want to do in the calculus sense and take instantaneous angular velocity

00:04:41.950 --> 00:04:47.510
it would be the derivative of your angle with respect to time

00:04:47.590 --> 00:04:50.950
How the angle is changing with respect to time

00:04:50.960 --> 00:04:56.100
With that out of the way, I want to see if we can see how this relates to speed

00:04:56.440 --> 00:05:00.460
How does this relate to the actual speed of the object?

00:05:00.660 --> 00:05:05.910
So to get the speed of the object, we just have to think about how far is this object traveling

00:05:05.910 --> 00:05:08.200
every revolution that it's doing

00:05:08.860 --> 00:05:15.000
And what we can do right over here--let's say that this radius is r

00:05:15.270 --> 00:05:19.760
So in every revolution, it is traveling 2 pi r

00:05:19.780 --> 00:05:24.080
Let's say this is r meters. Give ourselves some units right over there

00:05:24.420 --> 00:05:33.270
So the circumference over here is going to be 2 pi r meters

00:05:33.290 --> 00:05:55.530
Let's say that the angular velocity is equal to omega radians per second

00:05:56.070 --> 00:06:00.760
And so how many revolutions is that per second?

00:06:00.760 --> 00:06:03.450
We can go backwards from what we did over here

00:06:03.470 --> 00:06:10.710
We have one revolution is equal to 2 pi radians

00:06:10.860 --> 00:06:14.940
Just to be clear, sometimes angular velocity is actually measured in revolutions per second

00:06:14.940 --> 00:06:17.640
but the SI unit is in radians per second

00:06:17.660 --> 00:06:22.040
So here I want to convert omega from radians per second into revolutions per second

00:06:22.280 --> 00:06:32.670
Radians cancel out. We are left with--we get omega over 2 pi revolutions per second

00:06:32.920 --> 00:06:35.600
We know how many meters we get for a revolution

00:06:35.770 --> 00:06:40.520
We have 2 pi r meters per revolutions

00:06:40.830 --> 00:06:42.980
So we copy and paste this

00:06:43.360 --> 00:06:46.780
So our angular velocity, if we want revolutions per second

00:06:46.780 --> 00:06:50.970
it's going to be omega over 2 pi revolutions per second

00:06:51.540 --> 00:06:55.960
Omega is in radians per second if we put it into revolutions per second

00:06:55.960 --> 00:06:58.560
omega / 2 pi revolutions per second

00:06:58.900 --> 00:07:05.310
And then let's multiply that times--we want to convert this into meters per second

00:07:05.310 --> 00:07:09.040
So how many meters do we have per revolution?

00:07:09.320 --> 00:07:17.720
Well, we're gonna travel a whole circumference per revolution, so 2 pi r meters per revolution

00:07:17.740 --> 00:07:19.680
So these two cancel out

00:07:19.700 --> 00:07:21.990
2 pi cancels out with the 2 pi

00:07:21.990 --> 00:07:36.130
So you end up getting omega times r meters per second

00:07:36.500 --> 00:07:40.720
And just like that, we have the magnitude of the velocity

00:07:40.720 --> 00:07:44.750
or we could say the speed of the object as it goes around in a circle

00:07:45.010 --> 00:07:48.720
So what we can say is the magnitude of the velocity--

00:07:48.720 --> 00:07:51.460
I'll specify that by v--I want to be clear. This is not vector

00:07:51.460 --> 00:07:54.320
quantity. It's not the velocity. It's the magnitude of velocity

00:07:54.560 --> 00:07:59.910
Or we can say this is the speed. It's going to be equal to omega times r

00:08:00.460 --> 00:08:08.070
So the speed is equal to the angular velocity times r

00:08:08.070 --> 00:08:14.030
I guess we could say the magnitude of the angular velocity times the radius

00:08:14.220 --> 00:08:17.150
I don't want you to be confused. I am not saying that this is a vector quantity

00:08:17.150 --> 00:08:21.460
If this was a vector, I would put an arrow right over there

00:08:21.680 --> 00:08:24.240
And if this was a vector, I would put an arrow over there

00:08:24.240 --> 00:08:26.560
then I'll be referring to the thing that's popping out of the page

00:08:26.560 --> 00:08:29.450
but here I'm talking about the magnitude of the angular velocity

00:08:29.710 --> 00:08:42.670
and so writing in words, you get speed is equal to angular velocity--

00:08:42.670 --> 00:08:49.610
if you want to be particular, this is the magnitude of the angular velocity--

00:08:49.610 --> 00:08:54.870
times the radius of the circle that you are going around

00:08:54.870 --> 00:08:57.640
and if you want to solve for angular velocity

00:08:57.640 --> 00:09:01.540
you divide both sides by radius and you get angular velocity

00:09:01.760 --> 00:09:14.880
Omega is equal to speed which we're using v for, divided by the radius

00:09:14.900 --> 00:09:19.020
So we can actually use this information to do other interesting things later on

00:09:19.030 --> 00:09:22.860
But hopefully this gives you a sense of how all of this stuff is related