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 rotational kinetic energies of both the solid sphere and the solid cylinder, and then find the ratio of these energies.

### Step 1: Identify the moment of inertia for both objects

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

2. **Moment of Inertia of the Solid Cylinder (I_cylinder)**:
The moment of inertia of a solid cylinder rotating about its geometrical axis is given by:
\[ I_{\text{cylinder}} = \frac{1}{2} m R^2 \]

### Step 2: Define the angular velocities

Let the angular speed of the sphere be \( \omega \). According to the problem, the angular speed of the cylinder is twice that of the sphere:
\[ \omega_{\text{cylinder}} = 2\omega \]

### Step 3: Calculate the kinetic energy of rotation for both objects

1. **Kinetic Energy of the Sphere (E_sphere)**:
The rotational kinetic energy is given by the formula:
\[ E = \frac{1}{2} I \omega^2 \]
Therefore, for the sphere:
\[ E_{\text{sphere}} = \frac{1}{2} I_{\text{sphere}} \omega^2 = \frac{1}{2} \left(\frac{2}{5} m R^2\right) \omega^2 = \frac{1}{5} m R^2 \omega^2 \]

2. **Kinetic Energy of the Cylinder (E_cylinder)**:
For the cylinder:
\[ E_{\text{cylinder}} = \frac{1}{2} I_{\text{cylinder}} \omega_{\text{cylinder}}^2 = \frac{1}{2} \left(\frac{1}{2} m R^2\right) (2\omega)^2 \]
Simplifying this:
\[ E_{\text{cylinder}} = \frac{1}{2} \left(\frac{1}{2} m R^2\right) (4\omega^2) = \frac{1}{2} m R^2 \omega^2 \]

### Step 4: Find the ratio of the kinetic energies

Now we can find the ratio of the kinetic energies:
\[ \frac{E_{\text{sphere}}}{E_{\text{cylinder}}} = \frac{\frac{1}{5} m R^2 \omega^2}{\frac{1}{2} m R^2 \omega^2} \]
The \( m R^2 \omega^2 \) terms cancel out:
\[ \frac{E_{\text{sphere}}}{E_{\text{cylinder}}} = \frac{\frac{1}{5}}{\frac{1}{2}} = \frac{1}{5} \cdot \frac{2}{1} = \frac{2}{5} \]

### Final Answer
The ratio of their kinetic energies of rotation is:
\[ \frac{E_{\text{sphere}}}{E_{\text{cylinder}}} = \frac{2}{5} \]