When the ice melts at the poles it spreads the water all around the world. This moves mass from the poles to the equator and everywhere else.
Here is my maths to show what this would do to the day length;
Take the moment of initeria of a hollow sphere of negligable thickness 2m[SUB]1mm[/SUB]r[SUP]2[/SUP]/3 and divide by the total moment of inertia for the entire earth m[SUB]e[/SUB]r[SUP]2[/SUP]/2.
The r[SUP]2[/SUP] cancells as it's the same. The m[SUB]1mm[/SUB] is the mass of a 1mm layer of water over the whole earth. The m[SUB]e[/SUB] is the mass of the whole earth.
So, m[SUB]1mm[/SUB] x4
________________ = (360 x 10[SUP]12[/SUP]kg / 6 x 10[SUP]24[/SUP]) x 4/3 = 6 x 10[SUP]-11[/SUP]
m[SUB]e[/SUB] x3
This is the fraction that the world's spin is slowed by.
Multiply this by the number of seconds in a year 31.5 x 10[SUP]6[/SUP]
So that is 1.9 x 10[SUP]-3[/SUP] or 1.9 thousanths of a second per mm of sea level rise.
This is not noticable in human terms but it is easily measurable with atomic clocks. It has not happened. We are supposed to have had at least 180mm of this ea level rise since 1900. It has not happened. There is no explaination for why this has not happened other than the measurement of the ice melt from Greenland etc is wrong.
My maths is very rusty, like 30 years since I did any of this, so please correct me if I've droped one.
Not even close to being correct.
1. Compute Earth's moment of inertia.
For a sphere of uniform density, I = .4 MR²
But since Earth's density is non-uniform, the correct moment of inertia factor is .3307, hence for Earth, I = .3307 MR²
Mass = 5.97e24 kg
Radius = 6371 km
therefore I = 8.02e31 kg km²
2. Compute moment of inertia for a thin shell of water 1 mm thick
Volume of 1 km x 1 km x 1 mm of water = 1000 x 1000 x .001 = 1000 m³.
Mass of 1000 m³ water = 1e6 kg
Area of Earth's oceans: 361,060,000 km²
Mass of 1mm water added to earth's oceans: 1e6 x 3.6106e8 = 3.6106e14 kg
For a thin spherical shell, I = 2/3 MR²
Here we assume that the oceans are evenly distributed across the surface, which isn't true, but it's not far wrong.
Therefore moment of inertia of 1mm water shell = 9.77e21 kg km²
Notice already: TEN orders of magnitude difference!
3. Angular momentum is
conserved. Compute angular momentum L = Iω where ω is angular velocity in radians per second. For earth, ω = 2 π / 86400.
Therefore angular momentum L = 8.02e31 2 π / 86400 = 5.83e27 kg km² rad/sec.
Assuming for convenience that polar ice is perfectly axial, its angular momentum is zero. Melting the ice adds the moment of inertia of the shell to the moment of inertia to the Earth as a whole. So post-melt moment of inertia is: 8.02e31 + 9.77e21 = 8.02e31 (plus a lot of decimals) kg km². Since angular momentum is conserved, the new tiny-bit larger moment of inertia must be balanced by a new tiny-bit smaller angular velocity.
So:
Initial angular velocity: 7.242205216643040e-5 rad/sec
Final angular velocity: 7.272205215756699e-5 rad/sec
with a difference of 8.8634e-15 rad/sec
which amounts to a longer day by a whopping 10.53 microseconds (that's millionths of a second).
Is this detectable
? It's just barely within measureability limits, but it is swamped by daily effects of weather (~1 order of magnitude larger), seasonal effects (~2 orders of magnitude larger) and long-term lunar effects (~3 orders of magnitude larger). So it's lost in the noise.