Let `I_1 and I_2` be the moments of inertia of two bodies of identical geometrical shape, the first made of aluminum and the second of iron.

Options

A. `I_1ltI_2`
B. `I_1=I_2`
C. `I_1gtI_2`
D. Relation between `I_1` and `I_2` depends on the actual shapes of the bodies

Answer

To solve the problem, we need to analyze the moments of inertia of two bodies made of different materials but having identical geometrical shapes. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understand the Moment of Inertia Formula**:
The moment of inertia \( I \) of a body can be expressed as:
\[ I = m \cdot k^2 \]
where \( m \) is the mass of the body and \( k \) is the radius of gyration.

2. **Relate Mass to Density and Volume**:
The mass \( m \) of an object can be expressed in terms of its density \( \rho \) and volume \( V \):
\[ m = \rho \cdot V \]
Therefore, we can rewrite the moment of inertia as:
\[ I = \rho \cdot V \cdot k^2 \]

3. **Identify Common Parameters**:
Since both bodies have identical geometrical shapes, they will have the same volume \( V \) and the same radius of gyration \( k \). Thus, we can denote:
\[ I_1 = \rho_{Al} \cdot V \cdot k^2 \]
for aluminum and
\[ I_2 = \rho_{Fe} \cdot V \cdot k^2 \]
for iron.

4. **Compare Densities**:
It is known that the density of iron \( \rho_{Fe} \) is greater than the density of aluminum \( \rho_{Al} \):
\[ \rho_{Fe} > \rho_{Al} \]

5. **Establish the Relationship Between Moments of Inertia**:
Since both bodies have the same volume and radius of gyration, we can conclude:
\[ I_2 = \rho_{Fe} \cdot V \cdot k^2 > \rho_{Al} \cdot V \cdot k^2 = I_1 \]
Therefore, we have:
\[ I_2 > I_1 \]

6. **Final Conclusion**:
From the above analysis, we conclude that the moment of inertia of the iron body \( I_2 \) is greater than that of the aluminum body \( I_1 \). Thus, the correct answer is:
\[ I_2 > I_1 \]