A solid sphere is in rolling motion. In rolling motion a body prosseses translational kinetic energy `(K_(t))` as well as rotational kinetic energy `(K_(r))` simutaneously. The ratio
`K_(t) : (K_(t) + K_(r))` for the sphere is

Options

A. `10:7`
B. `5:7`
C. `7:10`
D. `2:5`

Answer

To solve the problem, we need to find the ratio of translational kinetic energy \( K_t \) to the total kinetic energy, which is the sum of translational kinetic energy \( K_t \) and rotational kinetic energy \( K_r \) for a solid sphere in rolling motion.

### Step-by-Step Solution:

1. **Identify the Kinetic Energies**:
- The translational kinetic energy \( K_t \) is given by:
\[ K_t = \frac{1}{2} m v_0^2 \]
- The rotational kinetic energy \( K_r \) is given by:
\[ K_r = \frac{1}{2} I \omega^2 \]
- For a solid sphere, the moment of inertia \( I \) is:
\[ I = \frac{2}{5} m r^2 \]

2. **Relate Angular Velocity to Translational Velocity**:
- In pure rolling motion, the relationship between translational velocity \( v_0 \) and angular velocity \( \omega \) is:
\[ v_0 = r \omega \quad \Rightarrow \quad \omega = \frac{v_0}{r} \]

3. **Substitute \( \omega \) in the Rotational Kinetic Energy**:
- Substitute \( \omega \) into the equation for \( K_r \):
\[ K_r = \frac{1}{2} I \left(\frac{v_0}{r}\right)^2 = \frac{1}{2} \left(\frac{2}{5} m r^2\right) \left(\frac{v_0^2}{r^2}\right) \]
- Simplifying this gives:
\[ K_r = \frac{1}{2} \cdot \frac{2}{5} m v_0^2 = \frac{1}{5} m v_0^2 \]

4. **Calculate the Total Kinetic Energy**:
- The total kinetic energy \( K_t + K_r \) is:
\[ K_t + K_r = \frac{1}{2} m v_0^2 + \frac{1}{5} m v_0^2 \]
- To combine these, find a common denominator:
\[ K_t + K_r = \frac{5}{10} m v_0^2 + \frac{2}{10} m v_0^2 = \frac{7}{10} m v_0^2 \]

5. **Find the Ratio**:
- Now, we can find the ratio \( \frac{K_t}{K_t + K_r} \):
\[ \frac{K_t}{K_t + K_r} = \frac{\frac{1}{2} m v_0^2}{\frac{7}{10} m v_0^2} \]
- The \( m v_0^2 \) cancels out:
\[ \frac{K_t}{K_t + K_r} = \frac{\frac{1}{2}}{\frac{7}{10}} = \frac{1}{2} \cdot \frac{10}{7} = \frac{5}{7} \]

### Final Answer:
The ratio \( K_t : (K_t + K_r) \) for the sphere is \( 5 : 7 \).