A `1 kg` solid sphere rolls without slipping on a rough horizontal surface under the influence of a `7 N` force. The force acts tangentially at the highest point of the sphere. Which of the following statements are correct? (`7 N` force acts towards right).

Options

A. The frictional force on the sphere acts towards right.
B. The value of the frictional force is `3 N`
C. The acceleration of the centre of the sphere is `9.8 m//s^(2)`
D. The acceleration of the centre of the sphere is `10 m//s^(2)`

Explanation

To solve the problem, we need to analyze the forces acting on the solid sphere and apply Newton's laws of motion. Let's break it down step by step.

### Step 1: Identify the Forces Acting on the Sphere
The forces acting on the sphere are:
- A tangential force \( F = 7 \, \text{N} \) acting to the right at the highest point of the sphere.
- A frictional force \( f \) acting to the left at the point of contact with the ground.

### Step 2: Write the Equations of Motion
Since the sphere rolls without slipping, we can apply Newton's second law for translation and rotation.

1. **Translational Motion:**
\[ F - f = m a \]
where \( m = 1 \, \text{kg} \) and \( a \) is the linear acceleration of the center of mass.

2. **Rotational Motion:**
The torque \( \tau \) about the center of mass due to the forces is given by:
\[ \tau = F \cdot r - f \cdot r = I \alpha \]
where \( I \) is the moment of inertia of the sphere and \( \alpha \) is the angular acceleration. For a solid sphere, the moment of inertia is:
\[ I = \frac{2}{5} m r^2 \]
Since \( \alpha = \frac{a}{r} \) (due to rolling without slipping), we can substitute \( \alpha \) into the torque equation.

### Step 3: Substitute Values into the Equations
Substituting the values into the equations:
1. From the translational motion:
\[ 7 - f = 1 \cdot a \quad \text{(1)} \]

2. From the rotational motion:
\[ 7 \cdot r - f \cdot r = \frac{2}{5} m r^2 \cdot \frac{a}{r} \]
Simplifying gives:
\[ 7 - f = \frac{2}{5} a \quad \text{(2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( 7 - f = a \) (from equation 1)
2. \( 7 - f = \frac{2}{5} a \) (from equation 2)

From equation (1), we can express \( a \) in terms of \( f \):
\[ a = 7 - f \]

Substituting this into equation (2):
\[ 7 - f = \frac{2}{5}(7 - f) \]

### Step 5: Solve for \( f \)
Multiplying through by 5 to eliminate the fraction:
\[ 5(7 - f) = 2(7 - f) \]
Expanding gives:
\[ 35 - 5f = 14 - 2f \]
Rearranging terms:
\[ 35 - 14 = 5f - 2f \]
\[ 21 = 3f \]
Thus, we find:
\[ f = 7 \, \text{N} \]

### Step 6: Calculate the Acceleration
Substituting \( f \) back into equation (1):
\[ 7 - 7 = a \implies a = 0 \, \text{m/s}^2 \]

### Conclusion
The sphere does not accelerate, and the friction force acts to the left. The correct statements based on our calculations are:
- The acceleration of the sphere is \( 0 \, \text{m/s}^2 \).
- The friction force acts to the left.