`I_1, I_2` are moments of inertia of two solid spheres of same mass about axes passing through their centres. If first is made of wood and the second is made of steel, then

Options

A. `I_(1)=I_(2)`
B. `I_(1)tI_(2)`
C. `I_(1)gtI_(2)`
D. `I_(1)leI_(2)`

Answer

To solve the problem, we will analyze the moments of inertia of two solid spheres made of different materials (wood and steel) but having the same mass.

### Step-by-Step Solution:

1. **Understand the Formula for Moment of Inertia**:
The moment of inertia \( I \) for a solid sphere about an axis through its center is given by the formula:
\[ I = \frac{2}{5} M r^2 \]
where \( M \) is the mass of the sphere and \( r \) is its radius.

2. **Identify the Given Information**:
- Both spheres have the same mass \( M \).
- The first sphere is made of wood, and the second sphere is made of steel.

3. **Consider the Densities**:
- The density of wood (\( \rho_{wood} \)) is less than the density of steel (\( \rho_{steel} \)).
- Since both spheres have the same mass, the volume of each sphere can be expressed as:
\[ V = \frac{M}{\rho} \]
- For the wood sphere, the volume is:
\[ V_{wood} = \frac{M}{\rho_{wood}} \]
- For the steel sphere, the volume is:
\[ V_{steel} = \frac{M}{\rho_{steel}} \]

4. **Relate Volume to Radius**:
The volume \( V \) of a sphere is also given by:
\[ V = \frac{4}{3} \pi r^3 \]
- Thus, we can relate the radius to the volume:
\[ r = \left( \frac{3V}{4\pi} \right)^{1/3} \]
- Since the volume of the wood sphere is greater than that of the steel sphere (because \( \rho_{wood} < \rho_{steel} \)), it follows that:
\[ r_{wood} > r_{steel} \]

5. **Calculate the Moments of Inertia**:
- For the wooden sphere:
\[ I_1 = \frac{2}{5} M r_{wood}^2 \]
- For the steel sphere:
\[ I_2 = \frac{2}{5} M r_{steel}^2 \]

6. **Compare the Moments of Inertia**:
Since \( r_{wood} > r_{steel} \), it follows that:
\[ r_{wood}^2 > r_{steel}^2 \]
Therefore, we can conclude:
\[ I_1 > I_2 \]

### Conclusion:
The moment of inertia of the wooden sphere (\( I_1 \)) is greater than the moment of inertia of the steel sphere (\( I_2 \)):
\[ I_1 > I_2 \]