This is just for fun :)
Warning: This post will feature Numbers, including Big Numbers.
I got all my data here from wikipedia. I’m not going to cite anything because it’s late and I’m doing this for fun. Don’t take any of this as fact. At most any of these numbers will be off by a percent, we’re working with orders of magnitude here so precision is less important.
Given a gyroscope of diameter 4093m (radius of 2041.5m), made of this imaginary substance I made up. The density of steel is 7840kg/m3, so let’s use that, bc why not.
If the disk is 1m tall (we’ll change this later), it has a volume of 1m⋅(2041.5m)2⋅π, which gives 13,093m3, which brings the mass of such a disk up to 1.026⋅1011Kg, about 275 Empire State Buildings!
The moment of inertia of a disk is calculated as I = m⋅r2/2, so for the disk I = 2.138⋅1017 Kg⋅m^2. For thoso who aren't familliar with the term, moment of inertia is a measure of how hard it is to get an object spinning, the same way regular inertia is a measure of how hard it is to start moving an object.
Rotational momentum can be calculated from L = I⋅w, where w is the angular velocity in rads/s. Let’s assume the outermost edge of the disk is going at a reasonable 0.25c, or 7.5⋅107 m/s. Any more and we need to get relativistic, which I don’t have the mindspace to do tonight. This means the disk is traveling at only 7.5⋅107m/s/2041.5m = 3.673⋅104 rads/s! This yields a total rotational momentum of 7.8545⋅1021 kg⋅m2/s. A completely incomprehensible number.
Remember though! This is only for a disk one meter in height. I’m sure a being capable of making a disk of such width would be able to make it tall, too! If we call it 200m tall, short enough to still be called a disk, we can bring the grand total to 1.571⋅1024 kg⋅m2/s.
Is that impressive? Well, the earth’s angular momentum is of the magnitude 1033 kg⋅m2/s. So a disk of this size would only alter the earth’s rotation by one part in a billion. That’s equivalent to reducing the length of the day by 86ns. Over a century this would add up to a 3 second discrepancy. 27 leap seconds have been added over the last 60 years for various reasons, mostly the year getting longer, so this would barely even be noticeable.
If you’re not familiar with conservation of momentum, how this would work is by “leeching off” some momentum from the spinning earth. Momentum must be conserved, so if we made the disk rotate in the same direction as the earth but faster (much faster), then the earth must slow down to compensate. We could also speed up the earth by making this disk rotate in the opposite direction of the earth; it must speed up to compensate. This is an interesting sort of motion, a spinning opject isn't "going anywhere", in the way we think about it, but it still moves very fast. Some quirk of geometry, maybe? There's a reason why pi shows up everywhere.
This whole thing assumes that our gyroscope is built on the axis of the earth’s rotation. If it’s not, then we could really fuck with the movement of the earth, in ways I don’t understand. Let’s keep it simple.
Even if this rotation wouldn’t mess with the day/night cycle, it would store energy as E = I⋅w2, which in this case is 2.884⋅1026 Joules! That’s 1.37 Million times the Tsar Bomba, the largest nuclear bomb ever detonated. So yeah, that would fuck shit up if it got loose! Cool!
The moral of this story? The earth is really fucking big. Like holy shit it’s a big fucking rock my god.
The other moral? Actually not really a moral, these numbers are too big to really understand, but wecan enjoy them non the less. People tend to ignore different types of motion in favour of translational motion since that’s what animals do most often. There are three classical kinds of motion: translational (like a bullet), rotational (like a spinning top) and vibrational (like a spring). All matter is always moving in all of these ways. People joke that physicists model everything as harmonic oscillators but it’s true! Everything vibrates, just as much as the molecules in the air move forwards like bowling balls, constantly colliding. And, in the same way, everything spins, from planets down the the electrons in your atoms (electrons spin in a really fucked up way though).
Also worth asking: could a disk like this ever be actually made? No. But hey maybe we just haven’t discovered the right carbon-neodymium yet? Still no. You could not hold a solid disk going 0.25c together. It would be plasma long before it reached that point, and also a billion other reasons why this wouldn’t work. It would glow so brightly it would melt a hole in the earth’s crust (I imagined this underground) before I got that far.
The equations used would be: (assuming non-relativistic speeds)
L = π/2⋅d⋅v⋅h⋅r3
E = π/2⋅d⋅v^2⋅h⋅r2
Where d = density, r, h are radius, height, and v is the maximum speed the outer rim spins at. This could be a little javascript thingy! Pi is such a beautiful character, give it a moment of appreciation before we move on.
Oh we’re moving on? I thought I was going to bed. I feel a need to keep going, thought. Let’s try and actually solve the question.
So, clearly, an increasing radius greatly increases the momentum. Wait, if we wanted to reduce the length of the day/night cycle meaningfully, say by reducing it from 24h to 23h45m, what would we need to do?
Well, there are 1440 minutes in day, so we need a reduction of 15/1440 or 1.042%. This is going to require a rotational momentum of 0.01042⋅7.2⋅1033 Kg⋅m2/s, or 7.5⋅1031 Kg⋅m2/s.
Assuming we keep the same rule of “a disk has a height <10% its radius”, which I completely made up, we can simplify the angular momentum equation to:
7.5⋅1031 Kg⋅m2/s = 0.05π⋅d⋅r4⋅v
Using the same density and max velocity as before, we can solve for r.
8.120⋅1020 m4 = r4
r = 168.8 km. With a height of 16.8km. Damn! Take a moment to imagine how big this “““disk””” would have to be to reduce the length of a day. That’s a diameter of 337.6km. Vaguely the distance from Montreal to Boston as the crow flies, and twice as tall as mount everest. At the ends, it moves at a Quarter of the Speed Of Light, just to reduce the length of the day by 15 minutes.
This disk would need to have an energy of 3.332⋅1033J! Absolutely no way the earth’s mantle could produce that much energy. For reference the sun outputs about 3.3⋅1031J PER DAY! It would take 100 days for the ENTIRE SUN to generate enough energy to make this disk slow the earth’s rotation by 15 minutes per day.
What the fuck?
That’s ridiculous, you might say. There’s absolutely no way this could exist, that a mass of molten metal that large could just be spinning at such a high angular velocity somewhere, let alone one even larger. HA! Now this a geology lesson too! No, a lesson in humility! I’ve tricked you into (at least attempting) comprehending the true scale of this planet we live on!! When I told you to imagine this impossibly massive disk, you thought about something 17km tall and 336km wide, thought about what that means in actual terms, in distance you’ve actually traveled, how bafflingly large that would need to be. And yet THAT IS NOTHING. We humans with our automobiles and locomotives feels so content with our pitiful magnitudes of translational momentum, we don’t stop to think about the others. The earth’s core is IMMENSE BEYOND COMPREHENTION! TThe Power of the Sun??? THE POWER OF THE EARTH!!! THE IMMENSE ENERGY STORED IN THE MOVEMENT OF THIS LONE ROCK, GYRATING THROUGH THE VASTNESS OF SPACE. ALL THE WORLD'S MIGHT, ALL THE FORCES OF MAN, A SPECK BESIDE THE UNENDING UNCEASING UNIMAGINABLE SPINNING INHERENT TO OUR WORLD. BORN OF COSMIC DUST WHICH BEQUEATHED ITS MOMENTA ONTO YOU. OUR ROTATIONAL MOMENTUM IS THE CONSEQUENCE OF THE BIRTH OF TIME AND SPACE ITSELF. GOD WAS BORN REVOLVING COUNTER-CLOCKWISE. FEEL THE IRON CORE BENEATH YOUR FEET. TREMBLE AT ITS MIGHT. MAY YOU NEVER SLEEP SOUNDLY AGAIN.
I’m really sorry. This is just what happens when you try and do physics after 2am. I should probably wrap this up and post it now before I go insane. I hope you’ve developed... some sort of feelings? Towards rotational momentum.
(remember: the milky way is also rotating, and at speeds much faster and with radii much greater than anything I’ve talked about here. Then forget about it, quickly, lest you go mad)