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
gonna 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 Newtonian Li
that means clocks run at the same rate
everywhere space and time are two
separate things and gravity is an actual
force that NASA's exert on each other
you know space time no relative alright
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 nano
meter 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 which brings me to submission
you guys know the drill
email your answers to PBS space time at
gmail.com before 5:00 p.m. New York City
time on the date that you see on the
screen use the subject line to 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 - 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
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 gonna speak technically
for a minute to viewers who actually
know 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 okay 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 einsteinium gravity
challenger 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 if 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 Einstein yin
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 from my
final episode of space time
you