What Happens If Polar Ice Melts?

[Aki]

Question #1:
If all of Earth's polar ice were suddenly to melt into the oceans, in the short term the length of the day would
a) increase b) remain the same c) decrease d)first decrease, then increase


The answer is a) , but why?


Question #2:
An ice cube of pure fresh water floats on pure fresh water in a glass. A huge ice shelf, also of pure fresh water, floats on the ocean away from Antarctica. Neglecting the contribution due to the density of air, as the ice cube and the iceberg melt,
a) the water level rises both in the glass and in the ocean
b) the water level does not change in either case
c) the water level stays the same in the glass but rises in the ocean
d) the water level decreases in the glass but stays the same in the ocean

The answer is c), but why?

 

[vincentchan]

question 1:
think about conservation of angular momentum
question 2:
think about density

 

[xanthym]

Question #1:
If all of Earth's polar ice were suddenly to melt into the oceans, in the short term the length of the day would
a) increase b) remain the same c) decrease d)first decrease, then increase

The answer is a) , but why?
Polar ice melting into the oceans will transfer mass from close to Earth's rotational axis (i.e., polar caps) to farther away from the axis (i.e., into the oceans). This will increase the Earth's Moment of Inertia "I" about its rotational axis. Because no external torques are involved, Angular Momentum is conserved:
{Angular Momentum After Melt} = {Angular Momentum Before Melt}
::: ⇒ I_AfterMelt*ω_AfterMelt = I_BeforeMelt*ω_BeforeMelt
::: ⇒ ω_AfterMelt = I_BeforeMelt*ω_BeforeMelt/I_AfterMelt
Since {I_AfterMelt > I_BeforeMelt}, the above equation indicates that {ω_AfterMelt < ω_BeforeMelt}. Thus, melting will cause the Earth's Angular Velocity to DEcrease and the day length to INcrease.


Question #2:
An ice cube of pure fresh water floats on pure fresh water in a glass. A huge ice shelf, also of pure fresh water, floats on the ocean away from Antarctica. Neglecting the contribution due to the density of air, as the ice cube and the iceberg melt,
a) the water level rises both in the glass and in the ocean
b) the water level does not change in either case
c) the water level stays the same in the glass but rises in the ocean
d) the water level decreases in the glass but stays the same in the ocean

The answer is c), but why?
Ice Cube: The floating ice cube displaces a volume "D" of pure water given by {g*D*ρ_GlassWater = M*g} or {D = M/ρ_GlassWater}, where "M" is mass of ice cube. When "C" grams of ice melt, displacement volume decreases by ΔD = C/ρ_GlassWater. Furthermore, additional liquid water volume (from melting ice) is added given by ΔW = C/ρ_CubeWater. Since {ρ_GlassWater = ρ_CubeWater} because both are pure fresh water, we have {|ΔD| = |ΔW|}, and the glass water level remains constant.
Ice Shelf: The process is similar to melting ice cube, except that {ρ_OceanWater > ρ_ShelfWater} since the former is Salt water and the latter is Pure water. Thus, {|ΔD| < |ΔW|}, and the water level will rise.

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[Aki]

Thanks xanthym for the good explanation,
however, I still have a question about the first question.
If the polar ices are melting, then wouldn't that cause the Earth to have a smaller "radius" (from the center of the Earth to the top of the polar ices)? Just like how a figure skater spins faster when her arms are pulled in, wouldn't the Earth spin faster when the ice is gone?

 

[whozum]

Well, if the ice caps melted, you've got a ton more water on the Earth than you had initially.

To relate to your belly dancer analogy, she would pull her arms in, but simultaneously gain a ton of weight.

 

[Aki]

When a figure skater pulls her arms in while she's spinning, it causes her to spin even faster, right?

 

[whozum]

This is correct

 

[Doc Al]

If the polar ices are melting, then wouldn't that cause the Earth to have a smaller "radius" (from the center of the Earth to the top of the polar ices)? Just like how a figure skater spins faster when her arms are pulled in, wouldn't the Earth spin faster when the ice is gone?

When the polar caps melt the water moves away from the axis of rotation. The analogy with the ice skater would be: She starts with her arms overhead, then spreads them wide apart. So her rotation slows down.