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Hello!
I'm Diana, and you're watching "Physics, Girl!"
So the idea for this video came from a conversation
with a friend where we were talking about tall buildings,
because Earth is spinning, and tall buildings
are spinning around with it especially
fast at the equator from the perspective of someone
looking down at Earth.
So you could imagine a building so tall where the end of it
is going around it like the speed of light.
Obviously not.
You can't go at the speed of light.
But it's interesting to think about how
tall that building might be, if only Newtonian physics applied.
And at some point, a different question
popped up, which is that if you dropped
a penny from the top of the building, how tall would
the building need to be in order for the penny to go into orbit?
So this could actually work, hypothetically.
Like I said, the Earth is rotating,
and everything's sticking out of it.
Like me and the buildings are all rotating
at the same angular speed-- that is,
they do a full rotation of 360 degrees every 24 hours.
Because that's what a day is!
Well, 23 hours, 56 minutes, and 4 seconds.
But even though everything is spinning at the same angular
rate, different parts of the building and different parts
of the inside of the Earth are moving
at different linear speeds, depending
on how far away they are from Earth's axis of rotation.
That's this.
This is just like when you go on a merry-go-round,
and you move faster on the edges than sitting in the middle.
So people are spinning around the equator at a pretty fast
1,040 miles per hour.
But if you dug down deep into the Earth
and settled down in the nice molten core,
you could find a place where you're going a comfortable 100
miles an hour.
So all this means that if you are standing on the equator,
your head would be further away from Earth's axis of rotation
than your feet, so it would be moving faster linearly--
at a rate of a tenth of a millimeter per second,
so it's not noticeable.
But also, if you were standing on one of Earth's poles,
it wouldn't be noticeable either, because you would only
be spinning in this direction.
So now instead of digging down into the Earth,
imagine you go the other way, and you build up
in a super tall building-- the one we talked
about at the beginning of the video--
where the top of your building is going the speed of light!
You can't do that.
[ALARM SOUNDING]
But how tall would this building have
to be if Newton were doing the calculation,
before Einstein discovered the theory of relativity?
Well, he'd find that your building would
have to be-- OK, well, it would do one revolution
per 86,164 seconds.
And our revolution is 2 pi r per 86,164 seconds,
which we'll set to this.
So my radius is this.
Subtract the radius of the Earth to get
the height of the building-- though our number doesn't
change much-- to get 4,111,175,472,000 meters,
or 27.5 astronomical units.
That means the top of your building
would stick out past the sun, past Jupiter's orbit,
and somewhere between Uranus and Neptune's orbit.
That is an awesome calculation!
But unfortunately, not realistic,
because Newtonian physics is so 1800s.
And there would be so many problems
before even getting to that point,
like structural integrity of the building,
or not having an atmosphere because you're out in space
at that point.
So the problems are many.
For example, we established before with the merry-go-round
that as you build a building up higher,
the top of your building is going faster
than the bottom of your building.
And when you're going around in a circle,
you feel like you're being pushed toward the outside
of the circle.
Just like when you're turning a car to the left,
you feel like you're being pushed to the right.
Do you see where I'm going with this?
That feeling is just your inertia
wanting to keep you going in a straight line
as you get accelerated around in the circle.
But Newton's laws don't really work in an accelerating
reference frame.
So we can define a fake force that you feel outward
and call it the centrifugal force.
We add this fake force into Newton's laws
and pretend that it's actually a force in every other way
to do our calculations.
Centrifugal force-- the force that physics teachers hate.
Such is the weirdness of rotating frames,
that you need to make up forces in order to use Newton's laws.
But think about it.
The idea of creating artificial gravity
is often a circular craft that's spinning,
and that's kind of what we're doing here.
The top of the building is spinning around,
and that fake centrifugal force feels
like it's pulling you away from Earth
like a fake or artificial force of gravity.
And it's pulling in the opposite direction
of the real force of gravity.
We don't notice this counteraction even standing
at the equator of the Earth where
Earth is spinning, because the effect is only
about 0.03 meters per second squared, which would change
the precious 9.81 meters per second squared
to 9.78 meters per second squared.
But, like, the bulges of mass around the Earth effect
gravity on the same order, so it's not noticeable.
OK, so now as you climb up the building, from an outsider's
perspective, your linear speed increases,
so your acceleration increases.
And from your perspective, your centrifugal acceleration
counteracts the acceleration of gravity more and more.
As you go up in your spinning building,
it feels like gravity gets weaker and weaker
until the point when you feel weightless.
Gravity is also getting weaker as you move away from Earth.
But way before it goes to 0, you get to a point
where the centrifugal force balances
the gravitational force.
This is the point when if you drop the penny out
of the top of the building, it will go into orbit.
So from your perspective, as you drop the penny out the window,
it'll look like it's just floating there,
because it would be moving sideways with the tower.
And it would have just enough sideways motion
so that it doesn't fall down to Earth,
but not so much that it would fly off into space.
That's orbiting.
And that magic altitude is 35,786 kilometers
above Earth's surface.
At that height, you could put any object of any mass
into orbit.
Maybe you want to attempt to orbit yourself!
Anything you want, out the window!
Orbit!
Another fun thing to think about--
if you keep going up in your tower past this point,
you'd start to feel an acceleration
in the other direction.
At some point, you'd get out so far
that you'd feel 9.8 meters per second squared
pushing you against the ceiling.
But now the quadrillion dollar question is,
could you ever build a building of heights like this?
Well, short of using futuristic carbon nanotubes,
we probably won't be making anything this tall
anytime soon.
Thanks for watching, and happy physicsing!
While researching this video, I found a video
by [INAUDIBLE] that had an eerily similar title
to the question I was wondering about.
But as Michael does, he talked about a ton
of different things, including the really tall buildings
that we have here on Earth.
It's a great video.
It's called "How High Can We Build?
I'll link it in the description.
You should check it out.
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