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`

Explanation

To solve the problem of finding the ratio of the accelerations for a solid sphere rolling down an incline without slipping and slipping down the incline without rolling, we will analyze both scenarios step by step.

### Step 1: Analyze the case of rolling without slipping

1. **Identify Forces**: For a solid sphere rolling down an incline, the forces acting on it are:
- Gravitational force component along the incline: \( F_{\text{gravity}} = mg \sin \theta \)
- Normal force: \( N = mg \cos \theta \)
- Frictional force \( f \) (which is necessary for rolling).

2. **Apply Newton's Second Law**: The net force acting on the sphere along the incline can be expressed as:
\[ mg \sin \theta - f = ma \]
where \( a \) is the linear acceleration of the center of mass.

3. **Relate Linear and Angular Acceleration**: For rolling without slipping, the condition is:
\[ a = \alpha R \]
where \( \alpha \) is the angular acceleration and \( R \) is the radius of the sphere.

4. **Torque Equation**: The torque due to friction about the center of mass is:
\[ \tau = fR = I \alpha \]
where \( I \) is the moment of inertia of the solid sphere, given by \( I = \frac{2}{5} m R^2 \).

5. **Substituting \( \alpha \)**: We can express \( \alpha \) in terms of \( a \):
\[ fR = \frac{2}{5} m R^2 \cdot \frac{a}{R} \implies f = \frac{2}{5} ma \]

6. **Substituting \( f \) back into the force equation**:
\[ mg \sin \theta - \frac{2}{5} ma = ma \]
\[ mg \sin \theta = ma + \frac{2}{5} ma = \frac{7}{5} ma \]
\[ a = \frac{5}{7} g \sin \theta \]

### Step 2: Analyze the case of slipping without rolling

1. **Identify Forces**: In the case of slipping, the only forces acting along the incline are:
- Gravitational force component along the incline: \( mg \sin \theta \)
- Normal force: \( N = mg \cos \theta \)

2. **Apply Newton's Second Law**: Since there is no friction (the sphere is slipping), the equation simplifies to:
\[ mg \sin \theta = ma \]
\[ a = g \sin \theta \]

### Step 3: Calculate the ratio of accelerations

Now we have the accelerations for both cases:
- For rolling without slipping: \( a_1 = \frac{5}{7} g \sin \theta \)
- For slipping without rolling: \( a_2 = g \sin \theta \)

To find the ratio of the accelerations:
\[ \text{Ratio} = \frac{a_1}{a_2} = \frac{\frac{5}{7} g \sin \theta}{g \sin \theta} = \frac{5}{7} \]

### Final Answer
The ratio of the accelerations for a solid sphere rolling down an incline without slipping to slipping down the incline without rolling is \( \frac{5}{7} \).

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