The moment of inertia of a solid cylinder about its axis is given by `(1//2)MR^(2)`. If this cylinder rolls without slipping the ratio of its rotational kinetic energy to its translational kinetic energy is -

Options

A. `1:1`
B. `2:2`
C. `1:2`
D. `2:3`

Answer

To solve the problem of finding the ratio of the rotational kinetic energy to the translational kinetic energy for a solid cylinder rolling without slipping, we can follow these steps:

### Step 1: Understand the formulas for kinetic energy
The rotational kinetic energy (Re) and translational kinetic energy (Te) are given by the following formulas:
- Rotational Kinetic Energy:
\[ Re = \frac{1}{2} I \omega^2 \]
- Translational Kinetic Energy:
\[ Te = \frac{1}{2} mv^2 \]

### Step 2: Substitute the moment of inertia
The moment of inertia (I) for a solid cylinder about its axis is given by:
\[ I = \frac{1}{2} m R^2 \]
where \(m\) is the mass of the cylinder and \(R\) is its radius.

### Step 3: Relate angular velocity and linear velocity
For a cylinder rolling without slipping, the relationship between angular velocity (\(\omega\)) and linear velocity (\(v\)) is:
\[ \omega = \frac{v}{R} \]

### Step 4: Substitute I and \(\omega\) into the kinetic energy formulas
Now, substituting \(I\) and \(\omega\) into the equation for rotational kinetic energy:
\[ Re = \frac{1}{2} \left(\frac{1}{2} m R^2\right) \left(\frac{v}{R}\right)^2 \]
This simplifies to:
\[ Re = \frac{1}{2} \cdot \frac{1}{2} m R^2 \cdot \frac{v^2}{R^2} = \frac{1}{4} mv^2 \]

### Step 5: Write the ratio of Re to Te
Now, we can write the ratio of the rotational kinetic energy to the translational kinetic energy:
\[ \frac{Re}{Te} = \frac{\frac{1}{4} mv^2}{\frac{1}{2} mv^2} \]

### Step 6: Simplify the ratio
The \(mv^2\) terms cancel out:
\[ \frac{Re}{Te} = \frac{\frac{1}{4}}{\frac{1}{2}} = \frac{1}{4} \cdot \frac{2}{1} = \frac{1}{2} \]

### Step 7: Conclusion
Thus, the ratio of the rotational kinetic energy to the translational kinetic energy is:
\[ \frac{Re}{Te} = \frac{1}{2} \]
This can be expressed as a ratio of 1:2.