The moon is tidally locked to the Earth, and its once-a-month rotation would make for long days for colonizers. So let's use flywheels to get the moon to rotate in a twentyfour-hour day.
EXECUTIVE SUMMARY
On the moon, we'll set up a big bunch of flywheels with their axes parallel to the moon's
axis. While the moon is still tide-locked, we'll spin up most of the flywheels, turning them "eastward", like we want the moon to be turning when we are done, and we'll leave them coasting in the low gravity and vacuum. Then we'll break the moon out of its tide lock by starting and stopping other flywheels to make the moon oscillate back and forth, like pumping up a swing. We'll work the moon's heavy ends up "over the top", and start it slowly going around, unlocked. Then we'll go back to all the eastward-turning flywheels we started, and apply brakes to stop them from turning. The rotational momentum of those flywheels will transfer into the moon, getting it up to the eastward-turning speed we want.
There's a little more to it, and lots of options. There's science that I am going to over-simplify, and serious math that I am going to totally ignore.
=================
INTRO
The moon turns once a month, and we are going to speed it up to turning once an Earth day. If it were not in tide lock, we could simply spin up a bunch of flywheels to westward, and let the equal-and-opposite rotational reaction speed up the moon's eastward rotation. We'd have to store the flywheels forever (which is easy to do on the moon) or dispose of them creatively and possibly usefully, but that would be a straightforward job (and a short idea). Since the moon is tidally locked, we need to break it free, which is complicated. But we can also use the tide lock to our advantage.
FLYWHEELS ON THE MOON
A flywheel stores energy as rotational momentum. When a flywheel is sped up, there must be an equal-and-opposite rotational reaction somewhere, and when a flywheel is slowed down, its angular momentum has to go somewhere. The Earth is too heavy to react noticeably to the torque of our day-to-day flywheels, but the moon could be torqued, if we can build enough flywheels.
The moon is almost ideal for flywheels. The gravity is low, the rotation is slow, vacuum is plentiful, and the weather is non-existent. The someday-colonized moon will have a need for flywheels, to store solar power through the long lunar nights (and even the short nights this project will bring about).
We'll have the flywheels for this project set up so their axes are parallel to the moon's axis. (Let's go ahead and use the current once-a-month axis, and keep turning eastward, and make a twentyfour-hour day.) The flywheels near the poles will sit flat like a pancake, those at the equator will stand like a rolling wheel.
Some of the flywheels can have external, detachable motors, and can be built to be started up slow and easy. Some others will need to be more fast-starting, with powerful motors built in, motors that can act as regenerative brakes. The cost and materials of flywheels is a variable in planning this.
Electrical power for the flywheels can be of solar or atomic origin. The availability of power is another variable. The construction of the flywheels can be done by hordes of self-replicating robots, or by deported criminals (self-replicating or not).
TIDE LOCK
The moon is an ovoid, like an egg. The end that is closer to Earth is getting pulled a bit more by Earth's gravity, while the end that is further away is getting a bit more centrifugal force as the moon orbits Earth. The two ends are both hanging down, so to speak, in line with the tidal forces that formed them when the moon was soft. Rotating the moon will require moving the ends out of line, and that will mean lifting them, like swinging up a hanging pendulum. That lifting doesn't happen naturally, so the moon is "locked". For now ... .
The first part of this idea is to lift the two heavy ends of the moon up to tidal equilibrium --- to unlock the moon --- so we can start it rotating. We won't just torque the moon up from a dead stop --- doing that in one burst would take a lot of power and many, many flywheels. We'll gradually work it up through increasing oscillations, using a few flywheels and spreading out the need for power. It'll be like getting a rigid-armed swing to go over the top bar and do a "360". Actually, it's more like trying to get a gigantic hard-cooked egg rolling over and over long-ways, since that has two ends and only requires 180 degrees of rocking to succeed (except the gravity is 90 degrees over).
Part of this idea takes advantage of the fact that the moon is in tide lock. The flywheels that will later be used to accelerate the moon's rotation will be spun up while the moon is still locked, to bend the rules regarding angular momentum.
ANGULAR MOMENTUM
Angular momentum is what is conserved when a spinning ice-skater pull in her arms. We won't be doing that, but we also won't be generating or destroying any angular momentum (I think). Any time a flywheel starts up, the moon will turn in an equal-and-opposite reaction. Any time a flywheel is stopped by braking against something attached to the moon, the moon will take that momentum and turn with it. Or stop turning, since we got the flywheel going by twisting against the moon.
If we were to do this on Earth, we could spin up all the flywheels going westward, which would speed up the Earth's rotation eastward. Hilarity would ensue for as long as we could keep the flywheels going, but when we'd stop them, we'd go back to turning once in twenty-four hours. In other words, the angular momentum will be conserved. If we have to conserve it in a cave full of flywheels for the rest of forever, we must.
By the way, the math that I am not doing balances the masses of the moon and of the flywheels, and the effective diameters where those masses could be said to be concentrated---which bits can be figured as moment arms or torque. The speed of rotation, often said as RPM, is the other multiplier. The momentum of a lot of flywheels turning fast one way will equal the momentum of the moon turning slow the other way.
EASTWARDS FLYWHEELS
The fact that the moon is in tide lock can be used to seemingly refute everything that I just said about angular momentum, though. In brief, the eastward-turning flywheels can be spun up against tide lock, with no equal-and-opposite reaction accumulating, and then later braked to provide rotational impetus to the unlocked moon. We are going to end up with a turning moon, and no flywheels left turning the other way. (I seriously don't know where the reciprocal angular momentum goes, but I am sure this will work.)
When we start up the eastward-turning flywheels, the moon will rotate --- just a bit --- westward around its axis (its center of mass) in an equal-and-opposite reaction. If not tide-locked, it would accelerate its turn as long as we kept winding up the speed of the flywheel, and keep turning once we detached the flywheel's motor, as long as the flywheel kept turning --- all right and tight with the laws of physics.
Except, remember, the moon is tide locked. To keep turning would mean moving the heavy ends "upward", pulling the end nearest us away from the Earth's gravity. The moon is going to twist in response to the flywheels starting up, and then it will stop, stopped by the tide lock. The flywheels can build up angular momentum without the moon building up any angular momentum in the other direction.
(The angular momentum of the moon MUST be going somewhere. I suspect that the twisting and lifting affects the gravitational interplay of the moon with the Earth, and affects the moon's orbit a little, but I can't sketch it yet.)
When all the eastward-turning flywheels are up to speed, and we shut off the starting motors, the moon will rock back from the westward reaction twist, and swing eastward through dead center toward another gradual stop against tide lock. (Again, it's like a giant egg trying to roll longwise.)
The eastward-turning flywheels are the cheap ones, without built-in motors and brakes. They are started at leisure, can be turned up slowly, and are stored for the final days of the project. (They can be mounted on carts to be taken to the spin-up and braking stations, and stored in caves for safety.) Once the moon is out of tide lock, and is still barely rotating, they will be braked to a halt to transfer their eastward momentum to the moon.
ROCK IT
The second set of flywheels will break the moon out of tide lock through increasing oscillation. They will be aggressively starting and stopping, using their built-in motors as regenerative brakes, so these are the expensive flywheels. When they stop and start, there will be an equal-and-opposite reaction that will twist the moon. (These will need solid foundations under them.)
These flywheels will be set up in pairs, one for turning eastward and one for turning westward. They will interconnect with electrical wiring so that when one is stopped by its regenerative brake, the electricity generated is sent to the other to spin it up, turning in the other direction. The paired setup lets a steady electrical input be used to feed into one flywheel or the other, while bouncing the accumulated energy back and forth, and while reversing the torque reaction. We can get the inertial reaction to the stop/start of two flywheels, in the direction desired, at practically no cost in electricity, once one of the pair is up to speed.
(I have no math to calculate the period for the oscillations, but I think it can be calculated easily, and could be determined on-site through experimentation. It may also be known already, through the moon's natural motions, I just couldn't find it. It should be longer and longer with increasing amplitude, even though pendulums are supposed to have a fixed period. I think. I am guessing about a day at first, and two days when approaching equilibrium, and assuming that the robots can adjust.)
LOSSES
There will be losses, of course. There will be losses in the flywheel pairs, and there will be friction losses in the moon itself, due to crustal shifting and slipping in the core. The moon isn't responding exactly as it should to the pull of Earth's gravity, which is how we know it has a liquid core. (Take a hard-boiled egg and a non-boiled egg, and roll them around a while --- you can tell which is which.) That liquid center may kibosh this whole project, or it may require slower swings, more flywheels or more power. With enough spinning flywheels, all braked at once, the moon can be torqued through anything, but the budget may not be there.
The point of the oscillation process is that it could be done with a single pair of flywheels, if the moon were solid and there was no time constraint. With a non-solid core, there is some sloshing, and some amount of flywheels that can oscillate the moon despite the sloshing.
The oscillation flywheels are likely to be set up first, as they'd be useful as electrical storage before the project starts, or if it fails to start. The moon may not rock properly, after all. A few pairs could be run as a test of oscillation, to get a measurement of friction losses and an estimate of the pairs needed.
INTERMEDIATE EASTWARD FLYWHEELS
Another possible set of flywheels are intermediate in set up and operation. They are built to turn eastward, and are started aggressively during the back-and-forths of the unlock-oscillation process. They are started while the moon is rocking westward, to let the reaction to them assist in oscillating it up out of tide lock. Once the moon is out of tide lock, and is rotating slowly, these are braked to a halt to transfer their eastward momentum to the moon.
(The whole project could be done with just eastward flywheels spun up during westward swings, but I like the paired-flywheels part, and I think it would be cheaper.)
SAGGING AT THE TOP
When we get the moon oscillated up to where it is spending long times at the "top" of the swings, at right angles to the tide lock, the gravitational/centrifugal forces may stretch it into a new ovoid, which would stop the project for a while. I doubt that will happen, though, as the moon is pretty much solid, these days. If it got stretched back to a spherical shape, we'd be in good shape, but that is also unlikely. There may be moonquakes, some of which would represent friction losses and would require more flywheel power.
GOING OVER THE TOP
At some point, we'll have the moon balancing at the "top" of the swings. Again, the closest earthly analogy is trying to get a giant egg rolling end over end. We now have it going up almost on its ends, and we need to get it to go over in the direction we want.
To get the moon turning, we can simply stop a bunch of the eastward-turning flywheels we have in storage. We don't want to do so too soon and waste them against the remnants of tide lock. Careful measurements should be done to get the timing right.
SPEED IT UP
To speed up the moon in its eastward turning, we stop all the eastward-turning flywheels that we spun up against tide lock so long ago. The power stored in the flywheels can be used make electricity to power mass-drivers or electric thrusters to assist in the spin-up, or taken off as heat or electricity for other uses, or just wasted as heat. The torque will go into the moon in any case. (The simplest method is to throw a grenade into the flywheel-storage cave, and run like hell.)
To get double the torque reaction from our stored eastward-turning flywheels, we could use the regenerative brakes take off the power, use that to wind the previous one in line back up, turning to westward, then put the westward-turning flywheels back in storage (with a sign on the door that says, "No Grenades, Please").
FLYWHEEL DISPOSAL
The cheap flywheels that are completely stopped can be broken up for scrap, the more expensive ones with built-in motors can be used as power-storage devices. Flywheels that are turning can be stored a long time in the conditions found on the moon---the rotation can even be moved into newer flywheels as technology improves.
Still, storage is a problem, and there are ways to get the momentum of a westward-turning flywheel off the moon. My favorite is a long, smooth ramp on the equator---grease it up, lower a flywheel onto it, and watch the show --- even though that needs a catcher ship. But that's a whole 'nother Halfbakery idea.
BENEFITS OF THE PROJECT
A twentyfour-hour day will be good for future development of the moon, both for people and growing plants in a familiar day/night cycle. It will also help to moderate temperature extremes.
The increased rotational speed will make the construction of a space elevator for the moon much easier, as it can be much shorter. (Again, I haven't the math (or the incentive).)
The robot horde will be useful in further development, or as parts. The flywheels can be used for energy storage, broken up for materials, or even used to transfer momentum out to rotating space colonies.
And maybe someone can tell me where the rotational momentum came from.