The ratio of the radii of gyration of a spherical shell and solid sphere of the same mass and radius about a tangential axis is

Options

A. `sqrt(6) : sqrt(5)`
B. `1 : sqrt(2)`
C. `sqrt(5) : sqrt(6)`
D. None of these

Answer

To find the ratio of the radii of gyration of a spherical shell and a solid sphere of the same mass and radius about a tangential axis, we can follow these steps:

### Step 1: Identify the Moment of Inertia for Both Objects
- For a solid sphere, the moment of inertia \( I_{cm} \) about its center of mass is given by:
\[ I_{cm} = \frac{2}{5} m r^2 \]
- For a spherical shell, the moment of inertia \( I_{cm} \) about its center of mass is given by:
\[ I_{cm} = \frac{2}{3} m r^2 \]

### Step 2: Apply the Parallel Axis Theorem
- The moment of inertia about a tangential axis can be calculated using the parallel axis theorem:
\[ I_t = I_{cm} + m d^2 \]
where \( d \) is the distance from the center of mass to the tangential axis. For both the solid sphere and the spherical shell, \( d = r \).

#### For the Solid Sphere:
\[ I_t = I_{cm} + m r^2 = \frac{2}{5} m r^2 + m r^2 = \frac{2}{5} m r^2 + \frac{5}{5} m r^2 = \frac{7}{5} m r^2 \]

#### For the Spherical Shell:
\[ I_t' = I_{cm} + m r^2 = \frac{2}{3} m r^2 + m r^2 = \frac{2}{3} m r^2 + \frac{3}{3} m r^2 = \frac{5}{3} m r^2 \]

### Step 3: Calculate the Radius of Gyration
- The radius of gyration \( k \) is defined as:
\[ I = m k^2 \]
Therefore, we can express \( k \) as:
\[ k = \sqrt{\frac{I}{m}} \]

#### For the Solid Sphere:
\[ k = \sqrt{\frac{I_t}{m}} = \sqrt{\frac{\frac{7}{5} m r^2}{m}} = \sqrt{\frac{7}{5}} r \]

#### For the Spherical Shell:
\[ k' = \sqrt{\frac{I_t'}{m}} = \sqrt{\frac{\frac{5}{3} m r^2}{m}} = \sqrt{\frac{5}{3}} r \]

### Step 4: Find the Ratio of the Radii of Gyration
- The ratio of the radius of gyration of the spherical shell to that of the solid sphere is:
\[ \frac{k'}{k} = \frac{\sqrt{\frac{5}{3}} r}{\sqrt{\frac{7}{5}} r} = \frac{\sqrt{\frac{5}{3}}}{\sqrt{\frac{7}{5}}} \]
- Simplifying this gives:
\[ \frac{k'}{k} = \sqrt{\frac{5 \cdot 5}{3 \cdot 7}} = \sqrt{\frac{25}{21}} \]

### Final Answer
The ratio of the radii of gyration of a spherical shell and a solid sphere about a tangential axis is:
\[ \frac{k'}{k} = \sqrt{\frac{25}{21}} \]