I don’t know, you go two months without a single question and then three come along at once. Anyway, Glory Be To Smurf, For He Is Awesome asks
So I was watching a program that said we’d dammed so many rivers that it had altered the spin of the earth by a fraction of a second. That got me thinking, firstly on how the hell that made a difference and secondly if there was anything humanity could do to really alter the spin of the earth?
I’d be interested in knowing if this program went into detail at all rather than just blurting that out as a known fact, but: yes, damming rivers will alter the speed of rotation of the Earth. This is down to a curious quantity known as moment of inertia, which I think has been mentioned on here once or twice but which I’ve never gone into in detail.
Wikipedia’s summary of moment of inertia is that it is “a property of a distribution of mass in space that measures its resistance to rotational acceleration about an axis”. What that means, shorn of science-speak, is that while the acceleration you derive from applying a given amount of force to an object will be dependent on that object’s mass (as from F = ma), if you’re trying to rotate it then the shape of the object will also become a determining factor. Some shapes are easier to set spinning than others because they have lower moments of inertia. Amongst other things, this is why acrobats or divers tuck their legs in close to their body when they perform a somersault; that compact shape is easier to spin – and so will spin faster – than if their limbs were splayed out every which way.
As a demonstration we’ll use the very simple example of the moment of inertia of a pendulum, largely because it’s the only one where I can still follow the maths. If you have a simple weight on the end of a massless line, then the moment of inertia of the pendulum will be
where m is the mass of the weight and r is the distance separating that mass from the pendulum pivot point. That dependence on r is key; it is not just mass that affects the moment of inertia of the pendulum, but also where that mass is in relation to the pivot point.
Now, if you’re applying a turning force to an object, that turning force will be known as torque. Torque for the pendulum is calculated by
where α is the pendulum’s angular acceleration. Note here that this equation is equivalent to F = ma for a rotating body, with the moment of inertia I standing in for the mass m. Finally, the angular momentum of the pendulum can be found by
where ω is the angular velocity of the pendulum. Once again, this is equivalent to the equation p = mv for a non-rotating body, with moment of inertia once again standing in for mass.
This moment of inertia/mass equivalence makes it somewhat easier to understand why the MoI makes rotating bodies harder/easier to spin faster/slower, but it’s not a perfect comparison. For one thing a closed system tends not to be able to alter its mass on the fly, whereas it can change its moment of inertia just by changing its shape. And this is where we come back to the Earth and all those dammed rivers; the Earth is a closed system, and by damming the rivers we have changed the distribution of the Earth’s mass to a very small – yet measurable – degree. There are now billions of tons of water flowing at a level a couple of hundred metres above where they would be if the river had remained undammed. This means the Earth’s moment of inertia has increased. But since the Earth is a closed system1 , it’s still rotating with the same angular momentum that it did before we dammed the rivers. Looking back at that angular momentum equation above: L stays the same. I has gone up. This means that the angular velocity ω – the speed of rotation — must decrease, and that the Earth is spinning more slowly as a result of the dams.
How large a shift in rotation are we talking about? Not that large; your typical dam would produce a change of just a few hundredths of a microsecond in the time it takes for the Earth to complete a full rotation. And remember, damming a river is just about the easiest way to shift that much mass around the Earth’s surface that humans can manage since water flows rather than having to be dug up and transported. Even something ridiculous like draining the Mediterraneanwouldn’t do much to the rotation since all that water would get spread out over the rest of the seas and oceans, resulting in only a small shift in the overall distribution of mass around the Earth’s surface. So no, barring magic future science there’s nothing humans can do to seriously shift the Earth’s rotation. There’s just too much mass involved.
- Discounting the shifting gravitational interactions it has with everything else in the Solar System, anyway, but those change over million-year timescales whereas most of the major dams were built over the last couple of hundred years. ↩