A solid sphere of mass `m` and radius `R` is rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic emergies of rotation `(E_("sphere")//E_("cylinder"))` will be.

Options

A. `1 : 4`
B. `3 : 1`
C. `2 : 3`
D. `1 : 5`

Explanation

To solve the problem, we need to calculate the kinetic energies of rotation for both the solid sphere and the solid cylinder, and then find the ratio of these energies.

### Step 1: Determine the Moment of Inertia for the Sphere
The moment of inertia \( I \) for a solid sphere rotating about its diameter is given by the formula:
\[ I_{\text{sphere}} = \frac{2}{5} m R^2 \]

### Step 2: Determine the Moment of Inertia for the Cylinder
The moment of inertia \( I \) for a solid cylinder rotating about its geometrical axis is given by the formula:
\[ I_{\text{cylinder}} = \frac{1}{2} m R^2 \]

### Step 3: Calculate the Kinetic Energy of the Sphere
The kinetic energy \( E \) of rotation is given by the formula:
\[ E = \frac{1}{2} I \omega^2 \]
For the sphere, substituting the moment of inertia:
\[ E_{\text{sphere}} = \frac{1}{2} \left(\frac{2}{5} m R^2\right) \omega^2 = \frac{1}{5} m R^2 \omega^2 \]

### Step 4: Calculate the Kinetic Energy of the Cylinder
Since the angular speed of the cylinder is twice that of the sphere, we have \( \omega_{\text{cylinder}} = 2\omega \). Now we can calculate the kinetic energy for the cylinder:
\[ E_{\text{cylinder}} = \frac{1}{2} \left(\frac{1}{2} m R^2\right) (2\omega)^2 \]
Calculating this gives:
\[ E_{\text{cylinder}} = \frac{1}{2} \left(\frac{1}{2} m R^2\right) (4\omega^2) = \frac{1}{2} m R^2 \omega^2 \]

### Step 5: Find the Ratio of Kinetic Energies
Now we can find the ratio of the kinetic energies of the sphere to that of the cylinder:
\[ \frac{E_{\text{sphere}}}{E_{\text{cylinder}}} = \frac{\frac{1}{5} m R^2 \omega^2}{\frac{1}{2} m R^2 \omega^2} \]
The \( m R^2 \omega^2 \) terms cancel out:
\[ \frac{E_{\text{sphere}}}{E_{\text{cylinder}}} = \frac{1/5}{1/2} = \frac{2}{5} \]

Thus, the ratio of their kinetic energies of rotation is:
\[ \frac{E_{\text{sphere}}}{E_{\text{cylinder}}} = \frac{1}{5} \]

### Final Answer
The ratio of the kinetic energies of rotation \( \frac{E_{\text{sphere}}}{E_{\text{cylinder}}} \) is \( 1:5 \).