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Hey Space Timers.
It's time for another T-shirt challenge question.
And as you'll see, this one is a bit harder than the last one
we did.
This time around, you're going to need some math,
and you'll need to be familiar with high school level physics.
Let me give you some general background first,
and then I'll set up the specifics for you.
For the purposes of this challenge,
I want you to treat gravity and all physics Newtonianly.
That means clocks run at the same rate everywhere,
space and time are two separate things,
and gravity is an actual force that masses
exert on each other.
No space-time, no relativity.
All right.
Pretend you have a sphere with the same mass density
throughout.
That sphere is not rotating, and it's not orbiting
any other larger bodies.
For simplicity, I'm going to refer
to this sphere as a planet, but it could
be any other massive body.
A star, whatever.
Suppose that a particle is orbiting the planet
right at the surface.
And I know, technically it's not orbiting
if it's on the surface.
So fine, if it makes you feel better,
say that it's orbiting a billionth of a nanometer
above the surface.
I think you know what I mean.
Anyway.
In Newtonian gravity, you can work out
an expression for the orbital speed
of this particle in terms of the mass and radius of the planet,
or in terms of the density and radius of the planet.
You can also work out how much time it would take
to go halfway around the globe.
Keep that in mind, and now imagine a second particle
that we release from rest at the planet's surface
and that we allow to fall through the center
of the planet to the other side.
You can imagine doing this with a super thin evacuated tunnel
along a diameter of the planet.
But I think it's easier to pretend
that the planet is a uniformly dense fluid,
and that this particle can pass through that fluid
without friction.
Again, I think you know what I mean here.
This is not supposed to be a trick question.
Here's the challenge.
At the same time that the orbiting particle passes
this point, let's release the second particle from rest
from exactly the same height.
Remember, they're both on the planet's surface.
Now each of them will eventually arrive at the antipodal point
on the planet.
The question is, which one reaches the other side first.
Now, there's a small roadblock.
When the second particle is inside the planet,
how do you calculate the gravitational force on it?
After all, as it falls, some of the mass is above it.
Well, you can use calculus to figure that out.
It's something called Gauss's Law.
But let me tell you how that part of the problem
works so that you can solve the rest of the problem
without calculus, using only algebra.
Here's the deal.
At any given location inside the planet,
the particle will feel only the gravitational force
from whatever mass is closer to the center of the planet
than the particle is.
With that fact, plus the fact that the density is uniform,
plus some basic geometry about spheres,
you should be able to get a formula
for the gravitational force on the particle
when it's a distance little r from the center of the planet.
Here's a big hint.
The expression for the gravitational force
on the second particle when it's inside the planet
should algebraically resemble a familiar non-gravitational
force that you also study in high school physics.
In fact, drawing an algebraic analogy
between the gravitational and non-gravitational situations
is actually the key to figuring out
the travel time of the second particle
without using calculus.
You'll notice I haven't given you any numbers.
And that's because you don't need them.
The answer to which particle wins the race
comes out the same regardless of the mass
and radius of the planet, or of the masses
of the two particles.
The point is to figure out the general answer
through a combination of physical reasoning and algebra.
Now as far as I know, you can't get the answer
without doing algebra.
But if you think you have an airtight argument that
doesn't require algebra, you're welcome to submit it.
Which brings me to submission.
You guys know the drill.
Email your answers to pbsspacetime@gmail.com
before 5:00 PM New York City time on the date
that you see on the screen.
Use the subject line "Two-particle Newtonian gravity
challenge".
It's not case-sensitive, and you should not
include the quotation marks.
But other than that, use this exact subject line,
including the hyphen, because we filter these things
automatically.
Now from among the correct answers,
we will randomly select five people
to receive a PBS Digital Studios T-shirt.
As usual, answers must be accompanied
by correct explanations, or they don't count.
Also as usual, do not discuss the question
or post your answers in the comments page
here, or on Reddit, or on any public internet forum,
until after we announce the winners.
Be cool.
Finally, I want to discuss the Einsteinian version
of this question.
Namely, if the particles depart simultaneously as
measured by the clock at one end of the planet, which
one arrives first according to the clock
at the other end of the planet.
Now this question can also be answered,
but now you need some knowledge of general relativity,
and you need calculus.
So I'm going to speak technically
for a minute to viewers who actually
knows something about this.
Ready?
You have to solve the Einstein equations
in the presence of a spherically symmetric perfect fluid whose
energy density is the same when measured locally
by an observer that's instantaneously at rest
at any location in that fluid.
OK.
Once you find the metric, you can then
find the circular geodesics and the radial geodesics,
and work out who arrives first according
to the clock on the other side of the planet.
Now I've never actually worked this out,
but you guys can do it.
Why don't we have a second challenge.
You can submit your answers to that challenge by email with
the subject line "Two-particle Einsteinian gravity challenge".
Five random people with correct answers and explanations
will receive a T-shirt.
You can only enter one challenge, though, Newton
or Einstein.
Pick one.
Now I know you could always make up a second email address
and enter twice, but don't do that.
Because you can enter the Einsteinian challenge at all,
that means you can solve the Newtonian challenge
in about 60 seconds, which isn't much of a challenge.
So don't enter both.
Honor system.
Anyway, that's it.
As I told you, it's a harder challenge this time.
But you also have more time, and both the Newtonian
and Einsteinian versions of the question
are pretty fun exercises.
I encourage you to talk to your friends about it,
because physics is a social activity.
Just don't do it in a public internet forum.
That way, everyone has a chance.
Anyway, good luck to all of you, and have fun with it.
We'll announce the solution in two weeks' time.
But I will see you guys next week for my final episode
of Space Time.