If the polar ice caps of the Earth were to melt and the water returned to the oceans, the oceans would be made deeper by about 30 m. What effect would this have on the Earth's rotation? Make an estimate of the resulting change in the length of the day. (Concern has been expressed that warming of the atmosphere resulting from industrial pollution could cause the ice caps to melt.)

We will solve this problem step by step, using the principle of conservation of angular momentum.

Step 1: Understanding the Concept
The Earth rotates about its axis, and its angular momentum is given by:

L = Iω

where
L = Angular momentum
I = Moment of inertia
ω = Angular velocity

Since no external torque acts on the Earth, the angular momentum remains conserved. If the distribution of mass changes (due to melting ice caps), the moment of inertia (I) changes, which affects the angular velocity (ω) and thus the length of the day.

Step 2: Moment of Inertia of the Earth
For a uniform sphere, the moment of inertia is:

I = (2/5) MR²

where
M = Mass of the Earth
R = Radius of the Earth

When the ice melts and redistributes as water, it spreads over the oceans, effectively increasing the Earth’s moment of inertia.

Step 3: Change in Moment of Inertia
If the polar ice caps melt, the water will move from the poles (where the distance from the axis of rotation is small) to the oceans (closer to the equator). The increase in the depth of the ocean is given as 30 m. This increases the Earth's radius slightly.

The new moment of inertia, considering a thin shell of water of thickness h = 30 m, is:

dI = (2/3) M' R²

where M' is the mass of the melted ice.

The total mass of the melted ice can be estimated as:

M' = ρ × volume of water
= ρ × (surface area of oceans × depth)
= (1000 kg/m³) × (3.6 × 10^14 m² × 30 m)
= 1.08 × 10^19 kg

The change in the moment of inertia is:

dI = (2/3) × (1.08 × 10^19 kg) × (6.4 × 10^6 m)²
= 9.22 × 10^31 kg m²

The original moment of inertia of the Earth is:

I = (2/5) × (6 × 10^24 kg) × (6.4 × 10^6 m)²
= 9.74 × 10^37 kg m²

The new moment of inertia is:

I' = I + dI
= 9.74 × 10^37 + 9.22 × 10^31
≈ 9.74 × 10^37 kg m²

Step 4: Change in Angular Velocity
Since angular momentum is conserved:

L = Iω = constant

So,

Iω = I'ω'

Rearranging,

ω' = ω × (I / I')

Since I' > I, it follows that ω' < ω, meaning the Earth will rotate more slowly.

Step 5: Change in Length of the Day
The length of a day (T) is related to angular velocity:

T = 2π / ω

Since ω' < ω, the new day length T' is given by:

T' = T × (I' / I)

Approximating the small change:

ΔT = T' - T
≈ T × (ΔI / I)
≈ (86400 s) × (9.22 × 10^31 / 9.74 × 10^37)
≈ 8.18 × 10^(-4) s
≈ 0.82 milliseconds

Conclusion
If the polar ice caps melt, the Earth's rotation will slow down, and the length of the day will increase by approximately 0.82 milliseconds.