The ratio of the accelerations for a solid sphere (mass `m, and radius R`) rolling down an incline of angle `theta` without slipping, and slipping down the incline without rolling is

Options

A. `5 : 7`
B. `2 : 3`
C. `2 : 5`
D. `7 : 5`

Answer

To find the ratio of the accelerations for a solid sphere rolling down an incline without slipping and slipping down the incline without rolling, we can follow these steps:

### Step 1: Understand the accelerations
- When a solid sphere rolls down an incline without slipping, its acceleration can be derived from the forces acting on it and the rotational motion.
- When it slips down the incline, it only experiences translational motion.

### Step 2: Acceleration for slipping
- For a solid sphere slipping down the incline, the acceleration \( a_S \) is given by:
\[ a_S = g \sin \theta \]
where \( g \) is the acceleration due to gravity and \( \theta \) is the angle of the incline.

### Step 3: Acceleration for rolling
- For a solid sphere rolling down the incline without slipping, the acceleration \( a_R \) can be derived using the moment of inertia. The formula for the acceleration of a rolling object is:
\[ a_R = \frac{g \sin \theta}{1 + \frac{k^2}{r^2}} \]
where \( k \) is the radius of gyration and \( r \) is the radius of the sphere.

### Step 4: Determine the radius of gyration for a solid sphere
- For a solid sphere, the radius of gyration \( k \) is given by:
\[ k^2 = \frac{2}{5} r^2 \]
Thus, substituting \( k^2 \) into the equation for \( a_R \):
\[ a_R = \frac{g \sin \theta}{1 + \frac{2/5 \cdot r^2}{r^2}} = \frac{g \sin \theta}{1 + \frac{2}{5}} = \frac{g \sin \theta}{\frac{7}{5}} = \frac{5}{7} g \sin \theta \]

### Step 5: Calculate the ratio of accelerations
- Now, we can find the ratio of the accelerations:
\[ \frac{a_R}{a_S} = \frac{\frac{5}{7} g \sin \theta}{g \sin \theta} \]
- Simplifying this gives:
\[ \frac{a_R}{a_S} = \frac{5}{7} \]

### Conclusion
- Therefore, the ratio of the accelerations for a solid sphere rolling down the incline without slipping to slipping down the incline without rolling is:
\[ \frac{a_R}{a_S} = \frac{5}{7} \]