The ratio of diameters of two rings A and B is `1:2`. On rolling down an inclined plane simultaneously:

Options

A. A will reach the bottom first
B. B will reach the bottom first
C. both with reach simultaneously
D. nothing can be said in the absence of mass

Answer

To solve the problem of two rings A and B rolling down an inclined plane simultaneously, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- The ratio of the diameters of the two rings A and B is \(1:2\).
- This implies that the ratio of their radii \(r_A : r_B = 1 : 2\).

2. **Understand the Forces Acting on the Rings:**
- Both rings experience gravitational force \(mg\) acting downwards.
- The component of the gravitational force acting along the incline is \(mg \sin \theta\).
- There is also a frictional force \(F\) acting up the incline.

3. **Set Up the Equations of Motion:**
- For ring A:
\[ mg \sin \theta - F = ma_A \]
- For ring B:
\[ mg \sin \theta - F = ma_B \]

4. **Torque Equation:**
- The torque about the center of mass due to friction is given by:
\[ \tau = F \cdot r \]
- For a ring, the moment of inertia \(I\) is given by \(I = m r^2\).
- The angular acceleration \(\alpha\) is related to the linear acceleration \(a\) by:
\[ \alpha = \frac{a}{r} \]

5. **Relate Linear and Angular Acceleration:**
- The torque equation can be expressed as:
\[ F \cdot r = I \cdot \alpha \]
- Substituting \(I\) and \(\alpha\):
\[ F \cdot r = (m r^2) \cdot \left(\frac{a}{r}\right) \]
- Simplifying gives:
\[ F = \frac{m a}{2} \]

6. **Substituting Back into the Motion Equations:**
- Substitute \(F\) into the equations of motion:
\[ mg \sin \theta - \frac{m a}{2} = ma \]
- Rearranging gives:
\[ mg \sin \theta = ma + \frac{m a}{2} = \frac{3m a}{2} \]
- Thus, we find:
\[ a = \frac{2g \sin \theta}{3} \]

7. **Conclusion on Time to Reach the Bottom:**
- Since both rings experience the same acceleration \(a\) which is independent of their mass and radius, they will reach the bottom of the incline simultaneously.

### Final Answer:
Both rings A and B will reach the bottom of the incline at the same time.