If rotational kinetic energy is `50%` of total kinetic energy then the body will be

Options

A. ring
B. cylinder
C. hollow sphere
D. solid sphere

Answer

To solve the problem, we need to analyze the relationship between the rotational kinetic energy and the total kinetic energy of a body in motion. The key is to find out which body has its rotational kinetic energy equal to 50% of the total kinetic energy.

### Step-by-Step Solution:

1. **Understanding Kinetic Energy**:
- The total kinetic energy (TKE) of a rolling body is the sum of its translational kinetic energy (TKE_trans) and rotational kinetic energy (TKE_rot).
- The formulas are:
- TKE_rot = (1/2) I ω²
- TKE_trans = (1/2) m v²
- Therefore, TKE = TKE_rot + TKE_trans = (1/2) I ω² + (1/2) m v²

2. **Given Condition**:
- We are given that the rotational kinetic energy is 50% of the total kinetic energy:
\[ \frac{TKE_{rot}}{TKE} = \frac{1}{2} \]
- This implies:
\[ TKE_{rot} = \frac{1}{2} TKE \]

3. **Substituting the Formulas**:
- We can substitute the expressions for kinetic energies:
\[ \frac{\frac{1}{2} I ω²}{\frac{1}{2} I ω² + \frac{1}{2} m v²} = \frac{1}{2} \]
- Simplifying this gives:
\[ \frac{I ω²}{I ω² + m v²} = \frac{1}{2} \]

4. **Cross-Multiplying**:
- Cross-multiplying gives:
\[ 2 I ω² = I ω² + m v² \]
- Rearranging this leads to:
\[ I ω² = m v² \]

5. **Relating v and ω**:
- For a rolling object without slipping, the relationship between linear velocity (v) and angular velocity (ω) is:
\[ v = r ω \]
- Substituting this into the equation gives:
\[ I ω² = m (r ω)² \]
- Simplifying leads to:
\[ I = m r² \]

6. **Finding Moment of Inertia**:
- Now we will check the moment of inertia for different shapes:
- Ring: \( I = m r² \)
- Cylinder: \( I = \frac{1}{2} m r² \)
- Hollow Sphere: \( I = \frac{2}{3} m r² \)
- Solid Sphere: \( I = \frac{2}{5} m r² \)

7. **Checking Each Body**:
- For the ring:
\[ \frac{I}{mr²} = 1 \quad \text{(This satisfies the condition)} \]
- For the cylinder:
\[ \frac{I}{mr²} = \frac{1}{2} \quad \text{(This does not satisfy)} \]
- For the hollow sphere:
\[ \frac{I}{mr²} = \frac{2}{3} \quad \text{(This does not satisfy)} \]
- For the solid sphere:
\[ \frac{I}{mr²} = \frac{2}{5} \quad \text{(This does not satisfy)} \]

8. **Conclusion**:
- The only body for which the rotational kinetic energy is 50% of the total kinetic energy is the **ring**.

### Final Answer:
The body will be a **ring**.