### Melting polar ice caps will lead to rise in sea levels .... BULLSHIT!!!!

A lot has been said and written by the global hand-wringers about the catastrophic rise in ocean levels should the polar icecaps melt. The ice caps consist mostly of floating ice, and the hand wringers would have us believe that should they melt, the result would be a rise in ocean levels. However, as it turns it (usually turns out, should I say) the hysteria generated is null because, as I will show in this report, the ocean levels will not rise one iota.

To prove the above, imagine a cube of material of density *d _{i}* floating in a body of water of density

*d*. The cube has side length

_{w}*L*and a portion of it,

*l*, is floating above the surface of the water. We’d like to determine by how much the surface of the water will rise,

_{2}*r*, assuming a rectangular body of water (e.g. in a tub) with surface area

*A*.

Describing the two forces acting on the block, we have

_{}

_{}

where

*F*= downward force

_{d}*F*= upward force

_{u}*L*= length of the side of the floating cube

*g*= acceleration of gravity, aka 9.8 m/s

^{2}

*d*= density of water

_{w}*d*= density of floating cube

_{i}The object will float if there is no net force on it, i.e.

_{}

_{}

and if a portion of the object is above the water or right on the surface of the water, i.e.

_{}

Simplifying we get

_{} and _{}

_{}

This is a restatement of the Archimedes principle, which states that the force acting on a floating object is equal to the weight of the water displaced.

Now, for this problem we need to restate the amount of displaced water into a measure of the rise in water level. Assume the object is floating in a rectangular tub of water with surface area *A*. We represent the rise of level of the water in the tub by *r*.

_{}

_{}

Substituting for *l _{1}* from the equation above and solving for

*r*, we have

_{}

_{}

Now, we can further simply this equation by representing *d _{i}* in terms of

*L*and the mass of the floating object. By definition, the density of a cube is its mass,

*m*, divided by the cube of the length of a side,

*L*.

_{}

_{}

Substituting into the equation for *r*, we get

_{}

_{}

This is an important result because it states that the rise in water level, *r*, is independent of the density, shape or size of the floating object; it is only dependent on the mass, *m*, of the object. Another way to state this is that **if I place two different objects into the water, the rise in water level will be the same if the mass of the two objects is the same**.

Now, how does this apply to floating ice which melts over time? The above results apply to a cube of ice floating in the ocean. Suppose now I melted that cube of ice and managed to enclose it in a cubic container of zero mass. Since the density of ice increases as it melts into water, the side of this “water-cube” would be less than L. But, the mass would not change. If I now “floated” this “water-cube” on the surface of the ocean, it would float right on the surface, i.e. *l _{2}* = 0.

But, most importantly, because of the result above, the rise in ocean level due to the ice cube is **IDENTICAL** to the rise in ocean level due to the “water cube”.

So, if the floating ice caps melted, **the rise in ocean levels would be .... ZERO**.

QED.