A circular disc rolls down on an inclined plane. The fraction of its total rolling energy associated with its rotational energy is,

Options

A. 1
B. `1//3`
C. `1//2`
D. `1//4`

Explanation

To solve the problem, we need to determine the fraction of the total rolling energy of a circular disc that is associated with its rotational energy as it rolls down an inclined plane.

### Step-by-Step Solution:

1. **Understand the Types of Kinetic Energy**:
- The total kinetic energy (KE_total) of the rolling disc consists of two components: translational kinetic energy (KE_trans) and rotational kinetic energy (KE_rot).
- The formulas are:
- KE_trans = (1/2) * M * V^2
- KE_rot = (1/2) * I * ω^2

2. **Identify the Moment of Inertia (I) of the Disc**:
- For a solid circular disc, the moment of inertia (I) about its center is given by:
- I = (1/2) * M * R^2

3. **Relate Angular Velocity (ω) to Linear Velocity (V)**:
- When the disc rolls without slipping, the relationship between linear velocity (V) and angular velocity (ω) is:
- ω = V / R

4. **Substitute ω in the Rotational Kinetic Energy Formula**:
- Substitute ω in the expression for KE_rot:
- KE_rot = (1/2) * I * ω^2
- KE_rot = (1/2) * (1/2) * M * R^2 * (V/R)^2
- KE_rot = (1/2) * (1/2) * M * R^2 * (V^2/R^2)
- KE_rot = (1/4) * M * V^2

5. **Calculate Total Kinetic Energy**:
- Now, substitute the expression for KE_rot into the total kinetic energy:
- KE_total = KE_rot + KE_trans
- KE_total = (1/4) * M * V^2 + (1/2) * M * V^2
- To combine these, find a common denominator:
- KE_total = (1/4) * M * V^2 + (2/4) * M * V^2
- KE_total = (3/4) * M * V^2

6. **Find the Fraction of Rotational Energy**:
- The fraction of the total kinetic energy that is rotational is given by:
- Fraction = KE_rot / KE_total
- Fraction = [(1/4) * M * V^2] / [(3/4) * M * V^2]
- The M * V^2 cancels out:
- Fraction = (1/4) / (3/4) = 1/3

7. **Conclusion**:
- Therefore, the fraction of the total rolling energy associated with its rotational energy is **1/3**.