homework and exercises - Effect on length of day as the polar ice caps melt

If the polar ice caps of the Earth melt, how will the length of the day be affected?

Answer

The short answer is about less than a second.

You can do a quick-and-dirty calculation by assuming that the mass of the polar ice caps (roughly $10^{19}$ kg or about $\frac{1}{200,000}$ of earth's mass). We expect then that this would slow earth's rotation by that same amount (because the mass flows from the poles to the equator): $$ \frac{1}{200,000}\cdot86000\,{\rm s}\approx0.43\,{\rm s} $$

You can do a more complicated solution involving rotational kinetic energy, $K=\frac12I\omega^2$, but this will not change the answer significantly (increases it to something like 2/3 a second) than the quick and dirty solution.