A ring is rolling without slipping. Its energy of translation is E. Its total kinetic energy will be :-

Options

A. `E`
B. `2E`
C. `3E`
D. `4E`

Answer

To solve the problem of finding the total kinetic energy of a ring rolling without slipping, we can break it down into a few steps.

### Step-by-Step Solution:

1. **Understanding the Motion of the Ring**:
A ring rolling without slipping has both translational and rotational motion. The translational motion is due to the center of mass moving, while the rotational motion is due to the ring spinning around its center.

2. **Translational Kinetic Energy**:
The translational kinetic energy (TKE) of the ring is given by the formula:
\[ TKE = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the ring and \( v \) is the linear velocity of the center of mass. According to the problem, this energy is given as \( E \):
\[ E = \frac{1}{2} mv^2 \]

3. **Rotational Kinetic Energy**:
The rotational kinetic energy (RKE) of the ring can be calculated using the formula:
\[ RKE = \frac{1}{2} I \omega^2 \]
where \( I \) is the moment of inertia and \( \omega \) is the angular velocity. For a ring, the moment of inertia \( I \) is:
\[ I = mr^2 \]
where \( r \) is the radius of the ring. The angular velocity \( \omega \) can be related to the linear velocity \( v \) by the equation:
\[ \omega = \frac{v}{r} \]

4. **Substituting Values**:
Now, substituting \( I \) and \( \omega \) into the RKE formula:
\[ RKE = \frac{1}{2} (mr^2) \left(\frac{v}{r}\right)^2 \]
Simplifying this gives:
\[ RKE = \frac{1}{2} (mr^2) \left(\frac{v^2}{r^2}\right) = \frac{1}{2} mv^2 \]

5. **Total Kinetic Energy**:
The total kinetic energy (TKE) of the rolling ring is the sum of the translational and rotational kinetic energies:
\[ TKE = TKE + RKE = \frac{1}{2} mv^2 + \frac{1}{2} mv^2 = mv^2 \]
Since we know \( E = \frac{1}{2} mv^2 \), we can express the total kinetic energy in terms of \( E \):
\[ TKE = 2E \]

### Final Answer:
The total kinetic energy of the ring rolling without slipping is:
\[ \text{Total Kinetic Energy} = 2E \]