What is the ratio of the rolling kinetic energy and rotational kinetic energy in the motion of a disc ?

Options

A. `1:1`
B. `2:7`
C. `1:2`
D. `3:1`

Explanation

To find the ratio of the rolling kinetic energy to the rotational kinetic energy of a disc, we can follow these steps:

### Step 1: Define the parameters
Let the mass of the disc be \( M \) and the radius be \( R \).

### Step 2: Calculate the moment of inertia
The moment of inertia \( I \) of a solid disc about its central axis is given by:
\[ I = \frac{1}{2} M R^2 \]

### Step 3: Calculate the rotational kinetic energy
The rotational kinetic energy \( K_r \) is given by the formula:
\[ K_r = \frac{1}{2} I \omega^2 \]
Substituting the expression for \( I \):
\[ K_r = \frac{1}{2} \left(\frac{1}{2} M R^2\right) \omega^2 = \frac{1}{4} M R^2 \omega^2 \]

### Step 4: Calculate the rolling kinetic energy
The rolling kinetic energy \( K_{rolling} \) consists of both translational and rotational kinetic energy. The translational kinetic energy \( K_t \) is given by:
\[ K_t = \frac{1}{2} M V^2 \]
In rolling motion, the relationship between linear velocity \( V \) and angular velocity \( \omega \) is:
\[ V = \omega R \]
Substituting \( V \) in the translational kinetic energy:
\[ K_t = \frac{1}{2} M (\omega R)^2 = \frac{1}{2} M \omega^2 R^2 \]
Now, the total rolling kinetic energy is:
\[ K_{rolling} = K_t + K_r = \frac{1}{2} M \omega^2 R^2 + \frac{1}{4} M R^2 \omega^2 \]
To combine these, we can express \( K_t \) with a common denominator:
\[ K_{rolling} = \frac{2}{4} M R^2 \omega^2 + \frac{1}{4} M R^2 \omega^2 = \frac{3}{4} M R^2 \omega^2 \]

### Step 5: Find the ratio of rolling kinetic energy to rotational kinetic energy
Now we can find the ratio:
\[ \text{Ratio} = \frac{K_{rolling}}{K_r} = \frac{\frac{3}{4} M R^2 \omega^2}{\frac{1}{4} M R^2 \omega^2} \]
The \( M R^2 \omega^2 \) terms cancel out:
\[ \text{Ratio} = \frac{3/4}{1/4} = \frac{3}{1} \]

### Conclusion
Thus, the ratio of the rolling kinetic energy to the rotational kinetic energy in the motion of a disc is:
\[ \boxed{3:1} \]