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 regarding the two rings A and B rolling down an inclined plane, we will analyze the motion of both rings based on their diameters and the physics of rolling motion.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two rings, A and B, with a diameter ratio of 1:2. This means that if the diameter of ring A is \(d\), then the diameter of ring B is \(2d\).
- The radius of ring A, \(r_A = \frac{d}{2}\) and the radius of ring B, \(r_B = d\).

2. **Moment of Inertia**:
- The moment of inertia \(I\) for a ring about its central axis is given by:
\[ I = m r^2 \]
- For ring A, \(I_A = m_A \left(\frac{d}{2}\right)^2 = \frac{m_A d^2}{4}\).
- For ring B, \(I_B = m_B (d)^2 = m_B d^2\).

3. **Acceleration of the Center of Mass**:
- The formula for the acceleration \(a\) of the center of mass of a rolling object down an incline is given by:
\[ a = \frac{g \sin \theta}{1 + \frac{I}{m r^2}} \]
- For ring A:
\[ a_A = \frac{g \sin \theta}{1 + \frac{I_A}{m_A r_A^2}} = \frac{g \sin \theta}{1 + \frac{\frac{m_A d^2}{4}}{m_A \left(\frac{d}{2}\right)^2}} = \frac{g \sin \theta}{1 + \frac{\frac{m_A d^2}{4}}{m_A \frac{d^2}{4}}} = \frac{g \sin \theta}{1 + 1} = \frac{g \sin \theta}{2} \]

- For ring B:
\[ a_B = \frac{g \sin \theta}{1 + \frac{I_B}{m_B r_B^2}} = \frac{g \sin \theta}{1 + \frac{m_B d^2}{m_B d^2}} = \frac{g \sin \theta}{1 + 1} = \frac{g \sin \theta}{2} \]

4. **Comparison of Accelerations**:
- From the calculations, we see that:
\[ a_A = \frac{g \sin \theta}{2} \quad \text{and} \quad a_B = \frac{g \sin \theta}{2} \]
- Both rings have the same acceleration down the incline.

5. **Conclusion**:
- Since both rings A and B have the same acceleration and are released simultaneously from the same height on the incline, they will reach the bottom at the same time.
- Therefore, the correct answer is that both rings will reach the bottom simultaneously.

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

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