in this video we're looking at a classic idea for how we might create artificial gravity on a
space station so the idea is that the astronauts live inside a giant rotating donut and in order
to move in this circular motion those astronauts must feel a normal force pointing to the center
of curvature in order to provide the centripetal acceleration now if we can spin this thing at just
the right rate so that that normal force is equal to mg where m is the mass of the astronaut and g
is the acceleration of gravity at the surface of the earth then the force upward on that person's
feet is exactly the same as it would be standing on the surface of earth so physiologically this
is going to be equivalent to standing on the surface of the earth and it solves a lot of the
problems that happen with the human body in zero g's including the loss of bone density for example
now one of the constraints of this problem is that large rotation rates can cause their own problems
there are some interesting issues with coordination that happen because the coriolis
and problems with vertigo and so on and from the reading i've done on this
it looks like the maximum rotation rate is about two rotations per minute
that would allow you to have ordinary people visiting this space station without experiencing a
lot of unpleasant side effects apparently you can safely go at a much higher rotation rate than this
if you select for the people who have the highest tolerance to high rotations and they
go through special training so you can think of this as a very conservative estimate if we
make a space station spin at two rotations per minute then basically everybody's going
to be comfortable there so then of course the whole physics problem of this is to figure out
what diameter is going to allow you to create an earth-like gravity with two rotations per minute
as your rotation rate and we're going to approach this by looking of course at a newton's second law
analysis of this astronaut so we write down f net equals ma well the only force they're experiencing
is that normal force and then the easiest formula to remember for their centripetal acceleration
is v squared over r but that's not the most useful formula in this problem
we can express the speed of this astronaut by looking at one full rotation and saying well the
distance they cover is one circumference or two pi r the time it takes is one period capital t
and so we get a four pi squared r squared over t squared in that numerator and we still have
an r hanging out in the denominator so one of those r's is going to cancel and this gives us a
useful alternative for the formula for centripetal acceleration it's 4 pi squared r over t squared
so if i go back to newton's second law the net force on the astronaut again is just the normal
force it's the only force acting that's going to be equal to the mass times the acceleration
which we can then write as 4 pi squared r over t squared now we can plug in the fact
that we want a normal force creating an artificial gravity similar to earth so we
want the normal force to be equal to the weight that you would feel on earth in other words mg
where g is about 9.8 meters per second squared and the m's cancel out of this which means it works
the same for every astronaut on board then we want to solve for the required radius of this doughnut
so i multiply both sides by t squared i get gt squared and then i divide by 4 pi
squared now it's just a matter of plugging in everything we know about the problem so i have
the g is 9.8 that's meters per second squared and then the period squared well i have to express the
period in seconds and the rotation rate was given to me as two rotations per minute well how many
seconds does it take for one rotation if you spin twice every minute that's 30 seconds per rotation
so we plug in our 30 seconds for the period that's going to be squared and then i have this
divided by 4 pi squared now if i look at the units here i have second squared from that period term
second square in the denominator from the acceleration term and i get meters popping
out of this so that's good and keeping a little extra precision here i get 223.4
meters for the radius now in the problem we were asked for the diameter of this donut shaped space
station so i have to double that result and i'm going to round the final answer to three
significant digits here and this gives me a final result of 447 meters so you might say well it's
never going to be practical to build rotating space stations that have a diameter of 447 meters
and certainly that's not a practical way to travel to mars right now again if you start
to do some reading on this you'll find that a lot of people can tolerate much higher rotation
rates and so you can make your diameter smaller in order to make this more practical but the
smaller you go and the faster you go the higher percentage of people start to suffer really bad
physiological disturbances so think of this number as the smallest rotating space station
on which any ordinary person would be comfortable if you enjoyed this video or at least found it
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