Assertion : The total kinetic energy of a rolling solid sphere is the sum of translational and rotational kinetic energies.
Reason : For all solid bodies total kinetic energy is always twice the translational kinetic energy.

Options

A. If both assertion and reason are true and the reason is the correct explanation of the assertion.
B. If both assertion and reason are true but reason is not the correct explanation of the assertion.
C. If assertion is true but reason is false.
D. If the assertion and reason both are false.

Explanation

To solve the given question, we need to analyze both the assertion and the reason provided.

### Step-by-Step Solution:

1. **Understanding the Assertion**:
- The assertion states that the total kinetic energy of a rolling solid sphere is the sum of its translational and rotational kinetic energies.
- For a solid sphere rolling without slipping, the total kinetic energy (TKE) can be expressed as:
\[ \text{TKE} = \text{Translational KE} + \text{Rotational KE} \]
- Translational kinetic energy (TKE_trans) is given by:
\[ \text{TKE}_{\text{trans}} = \frac{1}{2} m v^2 \]
- Rotational kinetic energy (TKE_rot) is given by:
\[ \text{TKE}_{\text{rot}} = \frac{1}{2} I \omega^2 \]
- For a solid sphere, the moment of inertia \( I \) about its center of mass is:
\[ I = \frac{2}{5} m r^2 \]
- When the sphere rolls, the relationship between linear velocity \( v \) and angular velocity \( \omega \) is:
\[ v = r \omega \]
- Therefore, we can express the total kinetic energy as:
\[ \text{TKE} = \frac{1}{2} m v^2 + \frac{1}{2} \left(\frac{2}{5} m r^2\right) \left(\frac{v^2}{r^2}\right) \]
- Simplifying this gives:
\[ \text{TKE} = \frac{1}{2} m v^2 + \frac{1}{5} m v^2 = \frac{7}{10} m v^2 \]
- Thus, the assertion is **true**.

2. **Understanding the Reason**:
- The reason states that for all solid bodies, the total kinetic energy is always twice the translational kinetic energy.
- This statement is not universally true. For example, in the case of a solid sphere, the total kinetic energy is not twice the translational kinetic energy; it is a combination of both translational and rotational energies.
- For certain shapes, like a ring or hollow cylinder, the relationship may differ, and thus the reason is **false**.

3. **Conclusion**:
- The assertion is true, while the reason is false. Therefore, the correct option would be that the assertion is true, and the reason is false.

### Final Answer:
- Assertion: True
- Reason: False

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