Two solid spheres are made of the same material. The ratio of their diameters is 2: 1. The ratio of their moments of inertia about their respective diameters is

Options

A. `1:4`
B. `8:1`
C. `32:1`
D. `16:1`

Explanation

To solve the problem, we need to find the ratio of the moments of inertia of two solid spheres made of the same material, given that the ratio of their diameters is 2:1.

### Step-by-Step Solution:

1. **Understanding the Moment of Inertia Formula**:
The moment of inertia \( I \) of a solid sphere about its diameter 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. **Identifying the Ratios**:
Let the diameter of the first sphere be \( d_1 \) and the second sphere be \( d_2 \). Given the ratio of their diameters:
\[ \frac{d_1}{d_2} = \frac{2}{1} \]
Since the radius \( r \) is half of the diameter, we can express the radii as:
\[ \frac{r_1}{r_2} = \frac{d_1/2}{d_2/2} = \frac{2/2}{1/2} = \frac{2}{1} \]

3. **Finding the Mass Ratio**:
The mass of the spheres can be found using the density \( \rho \) and the volume \( V \). The volume of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
Therefore, the masses of the spheres can be expressed as:
\[ m_1 = \rho V_1 = \rho \left( \frac{4}{3} \pi r_1^3 \right) \]
\[ m_2 = \rho V_2 = \rho \left( \frac{4}{3} \pi r_2^3 \right) \]
The ratio of the masses is:
\[ \frac{m_1}{m_2} = \frac{V_1}{V_2} = \frac{\frac{4}{3} \pi r_1^3}{\frac{4}{3} \pi r_2^3} = \frac{r_1^3}{r_2^3} \]

4. **Substituting the Radius Ratio**:
Since we have \( \frac{r_1}{r_2} = \frac{2}{1} \):
\[ \frac{m_1}{m_2} = \left( \frac{r_1}{r_2} \right)^3 = \left( \frac{2}{1} \right)^3 = \frac{8}{1} \]

5. **Calculating the Moment of Inertia Ratio**:
Now we can find the ratio of the moments of inertia:
\[ \frac{I_1}{I_2} = \frac{m_1 r_1^2}{m_2 r_2^2} \]
Substituting the ratios we found:
\[ \frac{I_1}{I_2} = \frac{8}{1} \cdot \left( \frac{r_1}{r_2} \right)^2 = \frac{8}{1} \cdot \left( \frac{2}{1} \right)^2 = \frac{8}{1} \cdot \frac{4}{1} = \frac{32}{1} \]

6. **Final Result**:
Thus, the ratio of the moments of inertia of the two spheres is:
\[ I_1 : I_2 = 32 : 1 \]