This post is inspired by this question asked by user Akash on Physics.SE in which he asks whether humans walking and driving around on the surface of the Earth have an effect on the Earth's rotation. In my answer, I estimated that the total moment of inertia of the entire population of the Earth, plus its vehicles, is around ten billion times less than the moment of inertia of the Earth itself. As a result, any change in the rotation of the Earth due to changes in its moment of inertia caused by motions of its inhabitants would be unmeasurably miniscule.
As a followup question, user Michiel T asked whether any one of our strongest nukes could have a significant effect on the Earth's rotation. In the post, I would like to offer a somewhat more detailed answer to this followup question.
I will show that even if all of the energy released by a large nuclear bomb were subtracted from the rotational energy of the Earth, its rotational speed would decrease by such a small amount that an Earth day would lengthen by a fraction of a microsecond.
To show this, we assume that the Earth is a sphere of mass distributed uniformly throughout its volume. It follows that the moment of inertia of the Earth is
where is the Earth's radius. Assuming that the Earth exhibits fixed axis rotation with angular velocity , its total rotational kinetic energy is given by
Now let denote the energy released in a strong nuclear blast. Imagine that we could somehow channel all of this energy into slowing down the Earth without wasting any of it, then the rotational energy of the Earth would decrease and would result in a reduction of its angular velocity. Its new angular velocity would satisfy
On the other hand, the period of the Earth's orbit and its modified period after our hypothetical nuclear blast are related to the angular velocities and by
Combining all of these facts, we obtain the following relationship between the periods and :
and with some algebra, and can solve for ;
So we see that the new period of the Earth is some (dimensionless) factor times the original period. Now for the fun part: we need to estimate the size of this dimensionless factor that tells us by what factor the period changes. According to a survey I performed of some online resources, the energy released in a large nuclear blast is on the order of (terajoules). Moreover, the mass of the Earth is , and its radius is about . Lastly, the Earth's rotational period is hours which is about . Combining these numbers, we find that
Notice that our estimate of the energy released in a nuclear blast was at best an order-of-magnitude estimate. At this level of approximation, the expression we wrote down can just as correctly be written
In other words, the period of the Earth's rotation, one day, would change by less than a trillionth of a day. Or put another way, noting that a day contains about seconds, we see that the day would lengthen by approximately ten nanoseconds. Wow! That's a really really small change. Pretty impressive huh? That just goes to show you what an enormous amount of energy the Earth possesses by virtue of its spin around its axis.
Somehow, I take comfort in how difficult it would be for the human race to alter the rotation of the Earth using a nuclear device.
Some comments on my assumptions
Note that in the course of this calculation, I assumed that the Earth exhibits fixed-axis rotation. This is not entirely accurate as the Earth exhibits a small "wobble" of its rotational axis called Chandler Wobble, but including Chandler wobble would not significantly affect the order-of-magnitude estimate ultimately made.
I also assumed that all of the energy of the nuclear blast was converted into reducing the rotational energy of the Earth. In reality, this would be impossible and the vast majority of the energy of a nuclear blast would be released as heat. This fact, however, only strengthens the conclusion that such a blast would have a negligible effect on the Earth's rotational speed.
An interesting fact
According to this wikipedia article on the Earth's rotation, the speed of the Earth's rotation increased by about 3 microseconds due to the 2004 Indian Ocean earthquake. In that case, however, the increase in rotational speed was due to a change in the moment of inertia of the Earth caused by the Earthquake. This got me thinking that perhaps a nuclear blast could be used to change the Earth's moment of inertia as opposed to putting its energy towards reducing the rotational energy of the Earth. Perhaps I will examine this possibility in another post...