Environment

So the Stanford post assumes that we continue to consume roughly 2% more energy per year. At that rate, in 1000 years we would go from consuming 1.753×10^13 W to consuming 6.98×10^21 W. This would be 40,000 times the energy the sun puts on the Earth. Because most energy quickly turns into heat, this would heat up the entire surface of the Earth to the point where it is uninhabitable. I feel that tidal locking would be the least of our concerns at that point.

Professor Liu seems to have made a simple mistake: What his model showed was unsustainable was not tidal energy, but actually his assumed exponential growth rate of energy consumption to ludicrous levels, levels that would spell disaster for the Earth.

That said, the website's math checks out. The linear approach is a very basic year 1 physics problem that can be quickly confirmed.

The values we need for this calculation:

The mass of the earth (M) is: 5.97×10^24 kg

The radius of the earth (R) is 6.37×10^6 m

The angular velocity of the earth (w) is 7.29×10^-5 rad/sec

The current total worldwide primary energy consumption is 1.753 × 10^13 W. This is pretty close to the article's assumption

The equations necessary:

The moment of inertia of a solid sphere of uniform density is: 2/5 MR^2

Rotational kinetic energy is calculated by: 1/2 I w^2

After some very basic plug-and-chug:

This provides a moment of inertia of the earth (I) of: 9.69×10^37 kilogram meters squared

And a total rotational kinetic energy of: 2.575×10^29 kg m^2 /s^2 This is pretty close to what the Stanford website calculated.

So if we used the suggested 1% here, it would take around 5.0 x 10^10 years to tidally lock the earth to the moon with our current energy consumption. But that's not what was assumed in the article. It was also assumed that we would continue to expand our energy consumption by a constant 2% per year. This requires basic calculus.

We have energy consumption that starts at the previously mentioned: 1.753 × 10^13 W

Below, n is equal to the number of years.

This leads us to a consumption growth formula of: 1.753×10^13 * 1.02^n

To indefinitely integrate that formula, we simply divide it by ln(1.02), which gives us: 8.85236×10^14 1.02^n (we will drop the +c because it's not necessary here)

And now we just need to solve the following equation for n: 2.575×10^29 = 8.85236×10^14 1.02^n

Solving gives us a real solution of: around 1681 years. This is close enough for me to say that the math checks out, considering that I didn't start with exactly the same base formulas. But ultimately this is besides the point. The math is right, but the premise of a constant 2% growth is ultimately unsustainable. Short of building planet-scale radiators to shed heat, the earth would become uninhabitable by virtue of the sheer energy consumption alone.