If rotational kinetic energy is `50% ` of translational kinetic energy, then the body is

Options

A. Ring
B. Cylinder
C. Hollow sphere
D. Solid sphere

Explanation

To solve the problem, we need to analyze the relationship between rotational kinetic energy (RKE) and translational kinetic energy (TKE) for different bodies. We are given that the rotational kinetic energy is 50% of the translational kinetic energy. We will examine four different shapes: ring, solid cylinder, hollow sphere, and solid sphere.

### Step-by-Step Solution:

1. **Understanding Kinetic Energy**:
- The translational kinetic energy (TKE) of a body is given by the formula:
\[ TKE = \frac{1}{2} mv^2 \]
- The rotational kinetic energy (RKE) is given by:
\[ RKE = \frac{1}{2} I \omega^2 \]
- Here, \(m\) is the mass, \(v\) is the linear velocity, \(I\) is the moment of inertia, and \(\omega\) is the angular velocity.

2. **Assuming Pure Rolling Motion**:
- For pure rolling motion, the relationship between linear velocity \(v\) and angular velocity \(\omega\) is:
\[ v = R \omega \]
- Thus, we can express \(\omega\) as:
\[ \omega = \frac{v}{R} \]

3. **Calculating for Each Body**:
- **Ring**:
- Moment of inertia \(I = mR^2\)
- RKE:
\[ RKE = \frac{1}{2} (mR^2) \left(\frac{v}{R}\right)^2 = \frac{1}{2} mv^2 \]
- Here, RKE = TKE, so it cannot be 50% of TKE.

- **Solid Cylinder**:
- Moment of inertia \(I = \frac{1}{2} mR^2\)
- RKE:
\[ RKE = \frac{1}{2} \left(\frac{1}{2} mR^2\right) \left(\frac{v}{R}\right)^2 = \frac{1}{4} mv^2 \]
- Here, RKE = 0.5 TKE, which satisfies the condition.

- **Hollow Sphere**:
- Moment of inertia \(I = \frac{2}{3} mR^2\)
- RKE:
\[ RKE = \frac{1}{2} \left(\frac{2}{3} mR^2\right) \left(\frac{v}{R}\right)^2 = \frac{1}{3} mv^2 \]
- Here, RKE is not equal to 0.5 TKE.

- **Solid Sphere**:
- Moment of inertia \(I = \frac{2}{5} mR^2\)
- RKE:
\[ RKE = \frac{1}{2} \left(\frac{2}{5} mR^2\right) \left(\frac{v}{R}\right)^2 = \frac{1}{5} mv^2 \]
- Here, RKE is not equal to 0.5 TKE.

4. **Conclusion**:
- The only body for which the rotational kinetic energy is 50% of the translational kinetic energy is the **solid cylinder**.

### Final Answer:
The body is a **solid cylinder**.