Huge buildings affect Earth's rotation?

Question

Does constructing huge buildings affect the rotation of the Earth, similar to skater whose angular rotation increases when her arms are closed comparatively than open?

Answer 1

From conservation of angular momentum we have $(I+\Delta I)(\omega+\Delta \omega) = I\omega,$ or $$\frac{\Delta \omega}{\omega} = - \frac{\Delta I}{I+\Delta I} \simeq -\frac{\Delta I}{I}.$$ We make the following simplifying assumptions:

• The earth is a sphere of uniform density of mass $M$ and radius $R$.

• The building is constructed on the equator by digging out a sphere of earth of mass $m$ and radius $r$ and raising it a distance $2r$. We assume $r\ll R$.

With these assumptions we find $$\begin{eqnarray*} \frac{\Delta \omega}{\omega} &\simeq& - \frac{m(R+r)^2-m (R-r)^2}{\frac{2}{5}M R^2} \\ &\simeq& - 10\frac{r^4}{R^4}. \end{eqnarray*}$$

Assuming that $r$ is 200 m (the geometric mean of 100 m, 100 m, and 750 m) we find
$$\begin{eqnarray*} \frac{\Delta \omega}{\omega} &\simeq& -10^{-17}. \end{eqnarray*}$$ Atomic clocks are accurate to about one part in $10^{14}$, so there is no hope in measuring such an effect.

Answer 2

Well, if we make a quick estimate of the mass of a huge building.

Let's say the building has a base of $100\times100 \;\text{m}^2$ and a height of $1500 \;\text{m}$, this is already substantially bigger than the current biggest building. Then we have a volume of $1.5\times 10^7\text{m}^3$. If we make the assumption, again very rough and on the high side, that the tower is solid concrete with a density of $2400\; \text{kg}/\text{m}^3$ then the total mass is $3.6\times 10^{10}\;\text{kg}$.

If we compare this to the mass of the earth which is $\approx 6\times 10^{24}\;\text{kg}$ you can already see that it is unlikely to have much influence.

Now since we're dealing with rotation we should actually look at the center of mass of the tower. The CM will be located at half the height in our example so at only $750\;\text{m}$ height. Now this is significantly lower than the average mountain so I think it is safe to say that the effect of tall buildings on the rotation of the earth is negligible.