When I was a kid in Brooklyn, I and a couple of other nerds would wonder about crazy things. Here's one: if every person in China climbed up on a six-foot ladder and then jumped off at the same time, would it nudge the Earth into a different orbit?
Well, it wouldn't, but not just for the obvious reason, that all the sore feet in China couldn't generate enough energy to make any difference. It would only create a windfall for Chinese podiatrists.
There is, however, a more fundamental reason why it wouldn't work.
First, let's see how strong their impact would be on the Earth's surface. It's a simple matter of calculating the energy gained by bodies falling a certain distance under the influence of gravity. (One of the kids insisted that they can't fall down; they must fall up, because they're on the bottom side of the globe. We beat him up.)
China is the most populous country on Earth, with a population of 1.35 billion souls (that's 2.7 billion soles) and an average body weight of 135 pounds, according to Alvanon, the global apparel-sizing experts (who incidentally peg the weight of today's average American at 178 pounds!). According to my calculation, the choreographed Chinese pounce would hit the ground with an energy of 1.6 trillion joules (a joule is an international unit of energy equal to about 1 thousandth of a Btu).
That may sound like a lot of energy, but it's only 2 trillionths of the total amount of energy released in what seismologists call the seismic moment by the 9.0 earthquake in Fukushima, Japan on March 11, 2011. Earthquakes have been occurring for billions of years, yet there is no evidence that even they have been able to nudge the Earth into a different orbit. So, obviously, those leaping Chinese certainly couldn't do it.
But here's the kicker: this whole question is a red herring, because no amount of earthquake, footquake, atomic bomb, or any other kind of energy unleashed on or below the Earth's surface could possibly change its orbit. It's physically impossible. And here's why.
Our planet has contentedly circled the sun in the same old orbit, more or less, for billions of years, because it is a faithful devotee of Isaac Newton's First Law of Motion: "An object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it." Note that Sir Ike specified an external force, a force from outside the object. That's critical.
So would trillions of Chinese feet pounding on the surface of the Earth constitute an external force on the planet? No, because those feet, the bodies attached to them, and even the tectonic plates that crash into one another during earthquakes are all part of the orbiting body we call Earth. How can any of them exert an external force on something they are a part of? It would be like trying to make your car move faster by pushing on the dashboard. You can't change the motion of something you're a part of by exerting force -- an internal force -- on a different part of it. That would constitute the old lift-yourself-up-by-the-bootstraps caper.
But wait! Although catastrophic events taking place on or beneath the Earth's surface cannot change its orbit, they can change its rotation. So believe it or not, Chinese jumpers and Japanese earthquakes could in principle change the rotational speed of the Earth.
The rotational speed of a spinning object, related to what physicists call its angular momentum, is affected by how it is shaped, that is, how its mass is distributed within its volume. For an object of any given mass, the more compact it is, the faster it can spin. That's why a figure skater spins faster when she gathers her mass into a tighter configuration by pulling her arms in closer to her center.
Thus, if some of the Earth's mass were shifted farther toward or away from its center, it would spin faster or slower. That's exactly what happened one year ago when the Fukushima earthquake thrust a plate of the Earth's crust deeper underground by jamming it beneath an adjacent plate. That event speeded up the planet's rotation by 1.8 microseconds per revolution. And because we define our 24-hour day as the time of one revolution of the Earth, our day is now 1.8 microseconds shorter than it was last year.
Now go explain to your boss why you weren't able to get that report finished on time: "The days were just too short this year!"
Just something to keep in mind.
That would be
delta I = m_Chinese * (r_Earth + 1.8m)^2 - m_Chinese * r_Earth^2 = m_Chinese * (r_Earth ^2 + 2 * r_Earth * 1.8m + 1.8m^2 - r_Earth ^2) = m_Chinese * (2 * r_Earth * 1.8m + 1.8m^2) = m_Chinese * (2 * r_Earth + 1.8m) *1.8m
So that's on the order of
delta I = 1.3 *10^9 * 65kg * 2 * 6.4e^6 m * 1.8m = 1.9 * 10^18 kg*m^2
So all the Chinese on six foot ladders change the moments of inertia of Earth by about 1,8*10^18 kg*m^2, which is a fraction of
1.9*10^18/8*10^37= 2.4 * 10^-20
Yep... that's not going to make a dent. Not even if you have an atomic clock.
Somebody correct me if I got some of this wrong.
I_Earth = 8 x 10^37 kg*m^2
Let's check that with an approximation... of all the mass being out at one Earth radius:
m_Earth = 6 * 10^24kg
r_Earth = 6400km = 6.4 * 10^6m
This would make
I = 6 * 10^24kg * (6.4 * 10^6m)^2 = 6 * 6.4 * 6.4 * 10^(24+2*6) kg * m^2 = 246 * 10^36 kg * m^2 = 24.6 * 10^37kg * m^2.
That's an over-estimate by a factor of about three... which is explained by the difference between a point mass and the actual radial mass distribution of the planet! Close enough.
For a single mass it turns out that
I = m * r^2
where r is the distance of the mass m from the axis of rotation.
For several masses m_i it's the sum
I = sum_over_i m_i * r_i^2
and for a mass distribution it's an integral
I = Integral rho(r) * d(r)^2 dV(r)
In all these cases the moments of inertia have the units mass times length^2, i.e. in SI units kg*m^2.
Angular momentum L of a rotating body is given by
L = I * omega
I is a constant called the "moments of inertia", which described the mass distribution of the body. omega is the angular frequency of the rotation, which is simply 2*pi*f or 2*pi/T, where f is the rotation frequency (in cycles per second and T is the rotation period in seconds).
This is no different than ordinary momentum
p = m * v
where m is the mass and v is the velocity of the body, except that we have replaced m and v with their rotational equivalents.
If L is a constant, and it has to be unless a second body is involved, like the moon or the sun or another planet etc., then omega is simply inverse proportional to I... i.e. when I increases by a small fraction, then omega has to decrease and when I decreases, omega has to increase.
I read recently that the earth is continually slowing down because of the drag of the oceans, so maybe it'll even out again in a million years?