Can a Hydroelectric Dam Really Make the Days Longer?

How galore times person you said, “I privation the days were longer”? Well, NASA scientists accidental the operation of China’s Three Gorges Dam really did marque the days longer, by slowing the rotation of the Earth. Hallelujah! Extra clip you tin usage to larn a caller language, drawback up astatine work, oregon adjacent better, slumber in.

This raises a batch of questions. First of all, is it true? Did the dam really marque our days longer? And if so, why? And conscionable however overmuch other clip are we talking astir here? I mean, you request to program your schedule, right? Should we physique much elephantine dams to get adjacent much time?

Angular Velocity

To reply these questions, we request 3 basal physics ideas: (1) angular velocity, (2) angular momentum, and (3) the infinitesimal of inertia. I’ll explicate each of these.

First, here’s a small experimentation that you tin bash yourself. If you person a rotating table chair, get it spinning with your feet disconnected the crushed and your arms tucked in. Now, earlier you propulsion up, widen your arms. It slows you down, right? Here's a mentation with a quality connected a rotating platform:

The velocity of rotation is what we telephone angular velocity, represented by the Greek missive omega (ω). Why “angular”? Here’s a two-dimensional illustration below. Think of this ellipse arsenic centered connected an xy coordinate plane. I’ve drawn a radius (R) to a constituent connected the ellipse wherever it cuts the horizontal axis:

Now ideate this constituent moving counterclockwise astir the circle. As it does, the radius enactment sweeps astir similar the 2nd manus connected a (backward) clock—and astatine immoderate constituent successful time, it forms an space (θ) with the x axis. The faster the constituent moves astir the circle, the faster that space changes. Hence, “angular velocity.”

To beryllium specific, ω is defined by however overmuch the space changes (Δθ) successful a definite increment of clip (Δt). So if the spinning miss supra completes 1 rotation successful 1 second, we accidental she has an angular velocity of 360 degrees per second—or utilizing radians, 2π radians per second.

Actually, you cognize this conception from your car. Engine velocity is measured successful revolutions per minute. What it’s counting is however galore times the crankshaft rotates successful a minute, usually anyplace from 600 to 3,000 rpm’s for liable citizens.

The Moment of Inertia

Inertia is fundamentally the aforesaid happening arsenic mass. What does wide bash successful physics? Well, it's a spot of an entity that makes it much hard to alteration its motion. It’s overmuch harder to rotation a dense bowling shot than a airy baseball. On the different end, it would beryllium harder to stop the bowling shot too. More wide means much absorption to change—more inertia.

The “moment of inertia” is the aforesaid thing, but for rotational motion. Let’s accidental you person a car up connected blocks. If you drawback a tyre and use immoderate torque to rotation it, you summation its angular velocity. Now bash the aforesaid happening with a bicycle wheel. Way easier, right? And it’ll beryllium going faster than the dense car wheel. This is due to the fact that the motorcycle instrumentality has a little infinitesimal of inertia.

We cognize the inertia of an entity depends connected its mass, but the infinitesimal of inertia (I) besides depends connected wherever that wide is located, comparative to the axis of rotation. Say you person 2 wheels of adjacent radius and mass, but 1 has each the wide connected the outer portion of the instrumentality and the different is conscionable a coagulated disk. The disk has much of its wide person to the axis of rotation, truthful it volition person a smaller infinitesimal of inertia. See wherever this is going?

Angular Momentum

There’s 1 much happening to deliberation about: angular momentum (L). This is conscionable the merchandise of the angular velocity (ω) and the infinitesimal of inertia (I):

Now, if you person a strategy with nary outer interactions—like a idiosyncratic spinning successful a chair, with nary 1 pushing—the angular momentum (L) is constant. We accidental angular momentum is conserved. But look astatine the close broadside of the equation: For a fixed worth of L, if you summation the infinitesimal of inertia (I), the angular velocity (ω) must decrease.

Dams and Days

So backmost to our archetypal questions! The Earth is an isolated rotating strategy with a changeless angular momentum. This means that if you change the organisation of wide connected Earth, the infinitesimal of inertia volition change, and that volition alteration the angular velocity.

This is precisely what a dam does. The h2o wants to determination downhill to the sea, but a dam prevents that, truthful it holds the water’s wide farther distant from Earth’s axis of rotation—just similar spreading your arms successful a spinning chair. So yes, a large dam truly tin alteration the velocity of Earth’s rotation!

Now for the amusive part. Let’s estimation however overmuch longer an Earth time is due to the fact that of the Three Gorges Dam. First, here’s an highly exaggerated diagram:

The dam makes a large artifact of h2o (which we telephone a lake). Notice that the latitude (θ) of the dam tract determines however acold it is from the planet’s axis of rotation. As you determination from the northbound rod toward the equator, that region increases. So if you privation to marque the time arsenic agelong arsenic possible, successful summation to stacking up h2o astatine a precocious altitude, you should physique your dam adjacent the equator.

OK, let’s cipher the alteration successful the infinitesimal of inertia, ΔI. We tin find this with the pursuing equation. Don’t interest astir the math! I conscionable privation you to spot however it depends connected the latitude (θ), the wide (m) and extent (h) of the water, and the radius of the Earth (R):

We request to capable successful these variables with existent numbers. Luckily, we person Wikipedia. So, fixed an archetypal worth of Earth’s infinitesimal of inertia (I₁), I tin usage ΔI from supra to find the alteration successful the angular velocity (Δω) caused by the dam:

(If you privation to spot my numbers and calculations, they’re each successful this Python code.)

After crunching the numbers, I get an Earth time that is 5.3 x 10^–6 seconds longer. Yes, that’s 5.3 other microseconds! Due to our simplifying assumptions that’s apt an overestimate, but the summation is real. I assistance you, there’s not a full batch you tin execute successful 5 millionths of a second, but it shows that quality enactment truly tin change a planet’s rotation.

It besides means there’s a full clump of different things we could bash to get inactive much clip successful the day. We could physique skyscrapers each crossed the Great Plains. Or heck, we could conscionable person everyone connected world basal up astatine the aforesaid time. That would besides summation the infinitesimal of inertia and dilatory the Earth’s roll. I’m getting up from my table close now—care to articulation me?