If the polar ice caps melt suddenly

Answer

### Step-by-Step Solution:

1. **Understanding the Scenario**:
- When the polar ice caps melt suddenly, the water from the ice caps redistributes across the Earth's surface. This redistribution affects the Earth's moment of inertia.

2. **Moment of Inertia**:
- The moment of inertia (I) of an object is a measure of how difficult it is to change its rotational motion. When the ice caps melt, the water moves from the poles towards the equator, which increases the Earth's moment of inertia.

3. **Conservation of Angular Momentum**:
- According to the law of conservation of angular momentum, if no external torque acts on a system, the total angular momentum (L) of that system remains constant. Mathematically, this can be expressed as:
\[ L = I \omega = \text{constant} \]
- Here, \( \omega \) is the angular velocity of the Earth.

4. **Relating Moment of Inertia and Angular Velocity**:
- If the moment of inertia increases (let's denote the new moment of inertia as \( I' \)), then the angular velocity must decrease to keep the angular momentum constant:
\[ I \omega = I' \omega' \]
- Rearranging gives:
\[ \omega' = \frac{I}{I'} \omega \]

5. **Understanding the Implications**:
- Since \( I' > I \) (the moment of inertia has increased), it follows that \( \frac{I}{I'} < 1 \). Therefore, \( \omega' < \omega \), meaning the angular velocity decreases.

6. **Effect on the Length of a Day**:
- The angular velocity \( \omega \) is related to the time period (T) of rotation (length of a day) by the formula:
\[ \omega = \frac{2\pi}{T} \]
- If \( \omega \) decreases, then \( T \) must increase. This means the length of a day will become longer.

7. **Conclusion**:
- Therefore, if the polar ice caps melt suddenly, the moment of inertia of the Earth increases, leading to a decrease in angular velocity, which results in an increase in the length of a day. The new length of a day will be more than 24 hours.