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

Answer

To solve the problem, we need to find the ratio of the rotational kinetic energies of a solid sphere and a solid cylinder, both having the same mass \( m \) and radius \( R \). The cylinder is rotating with an angular speed that is twice that of the sphere.

### Step-by-Step Solution:

1. **Identify the Kinetic Energy Formula for Rotation**:
The kinetic energy of a rotating object is given by the formula:
\[ E = \frac{1}{2} I \omega^2 \]
where \( I \) is the moment of inertia and \( \omega \) is the angular speed.

2. **Calculate the Moment of Inertia for the Sphere**:
The moment of inertia \( I \) for a solid sphere rotating about its diameter is:
\[ I_s = \frac{2}{5} m R^2 \]

3. **Calculate the Kinetic Energy of the Sphere**:
Let \( \omega_s \) be the angular speed of the sphere. The kinetic energy of the sphere \( E_s \) is:
\[ E_s = \frac{1}{2} I_s \omega_s^2 = \frac{1}{2} \left(\frac{2}{5} m R^2\right) \omega_s^2 = \frac{1}{5} m R^2 \omega_s^2 \]

4. **Calculate the Moment of Inertia for the Cylinder**:
The moment of inertia \( I \) for a solid cylinder rotating about its geometrical axis is:
\[ I_c = \frac{1}{2} m R^2 \]

5. **Calculate the Kinetic Energy of the Cylinder**:
The angular speed of the cylinder \( \omega_c \) is given as twice that of the sphere:
\[ \omega_c = 2 \omega_s \]
The kinetic energy of the cylinder \( E_c \) is:
\[ E_c = \frac{1}{2} I_c \omega_c^2 = \frac{1}{2} \left(\frac{1}{2} m R^2\right) (2 \omega_s)^2 \]
Simplifying this gives:
\[ E_c = \frac{1}{2} \left(\frac{1}{2} m R^2\right) (4 \omega_s^2) = \frac{1}{4} m R^2 \cdot 4 \omega_s^2 = m R^2 \omega_s^2 \]

6. **Find the Ratio of Kinetic Energies**:
Now, we need to find the ratio \( \frac{E_s}{E_c} \):
\[ \frac{E_s}{E_c} = \frac{\frac{1}{5} m R^2 \omega_s^2}{m R^2 \omega_s^2} \]
Cancelling \( m R^2 \omega_s^2 \) from both the numerator and the denominator gives:
\[ \frac{E_s}{E_c} = \frac{1}{5} \]

### Final Answer:
The ratio of the kinetic energies of rotation of the sphere to the cylinder is:
\[ \frac{E_{\text{sphere}}}{E_{\text{cylinder}}} = \frac{1}{5} \]