What is the moment of inertia of a solid sphere of density `rho` and radius R about its diameter ?

Options

A. `(105)/(176) R^(5) rho`
B. `(176)/(105) R^(5) rho`
C. `(105)/(176) R^(2) rho`
D. `(176)/(105) R^(2) rho`

Explanation

To find the moment of inertia of a solid sphere of density \( \rho \) and radius \( R \) about its diameter, we can follow these steps:

### Step 1: Understand the Moment of Inertia
The moment of inertia \( I \) about an axis is a measure of how difficult it is to change the rotational motion of an object about that axis. For a solid sphere, we need to find the moment of inertia about its diameter.

### Step 2: Use the Formula for Moment of Inertia
The moment of inertia about the center of mass (CM) for a solid sphere is given by the formula:
\[ I_{CM} = \frac{2}{5} M R^2 \]
where \( M \) is the mass of the sphere and \( R \) is its radius.

### Step 3: Calculate the Mass of the Sphere
The mass \( M \) of the sphere can be calculated using its volume and density:
\[ M = \text{Volume} \times \text{Density} = V \times \rho \]
The volume \( V \) of a solid sphere is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Thus, the mass becomes:
\[ M = \frac{4}{3} \pi R^3 \rho \]

### Step 4: Substitute Mass into the Moment of Inertia Formula
Now, substitute the expression for mass \( M \) into the moment of inertia formula:
\[ I_{AB} = I_{CM} = \frac{2}{5} \left(\frac{4}{3} \pi R^3 \rho\right) R^2 \]
This simplifies to:
\[ I_{AB} = \frac{2}{5} \times \frac{4}{3} \pi R^5 \rho \]

### Step 5: Simplify the Expression
Now, simplify the expression:
\[ I_{AB} = \frac{8}{15} \pi R^5 \rho \]

### Step 6: Substitute \(\pi\) for Approximation
For approximation, we can use \( \pi \approx \frac{22}{7} \):
\[ I_{AB} = \frac{8}{15} \times \frac{22}{7} R^5 \rho \]

### Step 7: Final Calculation
Now, calculate the final expression:
\[ I_{AB} = \frac{176}{105} R^5 \rho \]

### Conclusion
Thus, the moment of inertia of the solid sphere about its diameter is:
\[ I_{AB} = \frac{176}{105} R^5 \rho \]

### Answer
The correct option is:
\[ \text{Option 2: } \frac{176}{105} R^5 \rho \]

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