On meltng of ice on the pole of the earth, its moment of inertia will :-

Options

A. increase
B. decrease
C. remain unchanged
D. none of these

Answer

To solve the problem regarding the moment of inertia of the Earth when ice melts at the poles, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Moment of Inertia**:
The moment of inertia (I) of a body is a measure of how difficult it is to change its rotational motion about an axis. It depends on the mass distribution relative to the axis of rotation. The formula for moment of inertia for point masses is given by:
\[ I = \sum m_i r_i^2 \]
where \(m_i\) is the mass of the particle and \(r_i\) is the distance from the axis of rotation.

2. **Effect of Melting Ice**:
When ice at the poles melts, the water that results from the melting ice will flow towards the equator. This means that the mass of the Earth is redistributed from the poles (where the radius is smaller) to the equator (where the radius is larger).

3. **Analyzing the Change in Radius**:
The radius at the equator is greater than the radius at the poles. Since the moment of inertia is proportional to the square of the distance from the axis of rotation, moving mass from the poles to the equator increases the overall moment of inertia.

4. **Mathematical Representation**:
If we denote the mass of the ice that melts and moves to the equator as \(m\), the moment of inertia contribution from the ice at the poles (with a smaller radius \(r_p\)) will be:
\[ I_{poles} = m r_p^2 \]
After melting, this mass contributes to the moment of inertia at the equator (with a larger radius \(r_e\)):
\[ I_{equator} = m r_e^2 \]
Since \(r_e > r_p\), it follows that:
\[ I_{equator} > I_{poles} \]
Thus, the total moment of inertia of the Earth increases.

5. **Conclusion**:
Therefore, when ice melts at the poles and moves to the equator, the moment of inertia of the Earth will **increase**.

### Final Answer:
The moment of inertia of the Earth will **increase** when ice melts at the poles.

---