How to make the Moon spiral into Earth? I recently watched a video of what would happen if the Moon spiraled into Earth. But the video is pretty sketchy on the physics of just what would have to happen for that to occur. At first I thought I understood (just slow the Moon down enough), but my rudimentary orbital mechanics isn't enough to convince me that's sufficient (e.g., wouldn't the Moon just settle into a lower orbit?).
What forces would have to be applied to the Moon to get it to spiral into the Earth, at what times? What basic physics are involved? (And why should I have already known this if I could simply remember my freshman Physics?)
 A: The moon's mean orbital speed is just over 1 km/s. You need to get rid of most of that to have the moon collide with the Earth. Not all of it, but most of it.
With technology that we have today, and with anything we could reasonably expect to get, changing the orbit of the moon would require some sort of reaction mass. There is, of course, very probably lots of interesting physics we are unaware of. But we are unaware of it. So we need to use reaction mass.
This question claims that electric propulsion could achieve 20 km/s for the nozzle velocity of a rocket. So, if you could construct an electric propulsion rocket that achieved this, you would need to throw away about one 20th of the mass of the moon to slow the rest by 1 km/s.  The moon's escape velocity is about 2.68 km/s, and the solar system after that is about 16.6 km/s, so the stuff being ejected goes out of the solar system.
So you would need to construct a huge-honking electric propulsion rocket and fire one 20th of the mass of the moon "forward" so as to slow it's orbit. The exhaust would escape the solar system, so not be a concern. (Unless it happened to hit another planet.) You might need to adjust things to keep the jet aimed correctly and prevent the moon rotating out from under you. The time this takes depends on how big you make the rocket. If it was exceptionally large such that it produced one millionth of one g, then it would take 3.17 years. This would require $3.67 \times 10^{13}$ kg/s in the exhaust. And would require an energy input of $7.34 \times 10^{19}$ Watts, assuming 100% efficiency. That is about 2.5 million times the electrical power of the entire Earth.
Now it is probably possible to do some "jim-jam" with orbits and get a collision with less change of velocity. But your question specified a "spiral." So the analysis here is in the ballpark.
A: Using the parallel axis theorem:
$$L(moon)=mvr$$
The moon must either loose mass $m$ or tangential speed $v$ for some reasons, therefore loose angular momentum $L(moon)$ which is the only phenomenon it is holding it in orbit and not pulled down by Earths larger gravity.
The orbital angular momentum of the Moon is $2.9 x 10^{34}$ $kg^{2}/s$.
However moon loosing orbital angular momentum would mean that the Earth would gain spin angular momentum (i.e. shorter day) and because the two angular momenta are locked together via mutual gravity it would not cause the moon to spiral down to Earth in a deadly collision spiral. The moon would just orbit the Earth in a lower orbit.
What you ask would require the moon to be made stand still.
Actually the opposite is happening, Earth is loosing spin angular momentum therefore the moon is gaining orbital angular momentum every year and increases its orbital distance by 3-4 cm every year.
Nevertheless, forcing the moon to orbit in a much lower orbit will have catastrophic results for the Earth and its inhabitants.
The referenced video is misleading by presenting the moon gradually orbiting in a lower orbit without telling you that an extra amount of force must be applied each time to make the moon orbit distance from the Earth smaller. Too much science popularization giving false conclusions.
A: 
wouldn't the Moon just settle into a lower orbit?

Yes.  To make the moon spiral into the earth requires continuous application of some drag or retarding force, not just some singular event.

What forces would have to be applied to the Moon to get it to spiral into the Earth, at what times?

If the only (significant) force on the moon is the earth's gravity, then it will move in an ellipse.  To change that orbit requires a force.  For a spiral, you need a constant drag.  Real sources of such drag on moons might be an atmosphere or tidal losses.
A: Another way: maintain a large electric charge on the moon so that it radiates away low-frequency radio waves and looses energy that way.
This method is impractical; but so are all methods. This method is even more impractical than most, but I add it for the interest of the physics. The idea is that any accelerating charge radiates electromagnetic radiation, and the energy for the radiation has to come from somewhere; in this case it comes from kinetic energy of the moving charge. For a charge on an otherwise circular orbit around an attractive centre the net result is the inward spiraling motion mentioned in the question. The frequency of the radiation is equal to the orbital frequency, so very low here. The power is given by Larmor's formula:
$$
P = \frac{q^2 a^2}{6 \pi \epsilon_0 c^3}
$$
where $a$ is the acceleration, i.e. the centripetal acceleration of the moon in this case, which is currently about $2.725 \times 10^{-3}\,$m/s$^2$. Unfortunately, for any reasonable value of the electric charge $q$ this produces a power which is hopelessly too low. For example with $q$ of order $10^{10}$ Coulombs you could get a power of order $0.2$ watts and this would suffice to remove the moon's kinetic energy ($3.65 \times 10^{28}\,$joules) in about $10^{21}$ years. But it is expected that the Sun would engulf the Earth long before that.
Any attempt to make the charge larger would involve electric fields strong enough to rip electrons off the rocks of the moon, so would presumably not work.
A: The Earth-moon binary system is emitting gravitational waves, and so is steadily losing energy and the separation between them is decreasing. Eventually, (ignoring the death of the Sun, heat death of the Universe, the possibility of proton decay or the decay of our vacuum state, etc etc), the two bodies will collide.
We can estimate how long it will take the Earth and Moon to collide via this mechanism (under the wildly wrong assumption that the Earth and Moon will still exist by the end of this process), using Eq. 16 of http://www.bourbaphy.fr/damourgrav.pdf
\begin{equation}
t_c = \frac{5 c^5 D^4_{\rm moon}}{256 G^3 \mu M^2} = 2.8 \times 10^{15} \ {\rm years}
\end{equation}
where $D_{\rm moon}$ is the distance from the Earth to the Moon today, $\mu=m_1 m_2/(m_1 + m_2)$ is the reduced mass, and $M=m_1+m_2$ is the total mass (and $m_1$ and $m_2$ are the mass of the Earth and Moon, respectively). To put it in perspective, this is more than $200,000$ times longer than the age of the Universe.
I have neglected the finite size of the Earth and Moon, so really they will coalesce earlier, but not by enough to change the qualitative "wow, that's a long time" feeling you should have gotten :). Still, this at least gives an upper bound on how long the Earth-Moon system can remain stable.
A: The easiest way is to slow the moon down progressively until atmosphere does the job. But it would be catastrophic, of course.
Supposing that the moon is spiraling down on earth:
1 - Higher tides, huge waves and probable coast devastation.
2 - Once it get's close to Roche (18,470 km) limit it would star to crumble and fall into earth.
3 - We would get a beautiful set of rings at least for a few years.
4 - Most of life on earth would vanish.
5 - Earth would end up like venus, moonless inhospitable place (at least for a few hundred million years).
A: The key answer has to do with your presumption that "the moon would just settle into a lower orbit". This will only happen if, in addition to reducing the earth-moon distance (e.g., via a centripetal push), its radial velocity (and hence angular momentum) is correspondingly increased. If you just push it towards the earth below the equilibrium orbit (given its current angular velocity), then it would indeed spiral towards the earth.
The basic concept here is Kepler's law.
A: Making the moon spiral into the earth is actually a pretty easy task. You just have to embed earth and moon into about at least 10 quadrillion cubic kilometers of a newtonian fluid that doesn't freeze close to the absolute zero point of the thermodynamic temperature (-273.15°C), for example the engine oil SAE -233W (which is sometimes hard to find suppliers for, especially in those quantities).
As you have already correctly stated, any dissipative force that only acts during a limited period will just move the moon to a lower (generally elliptical) orbit. At least unless that new orbit is low enough to cause enough airstream on earth to whirl around hats, people or improperly secured garbage cans, in which case a sudden contact between moon and a molehill or Mount Everest is likely to end that fatal sequence of events.
