Analysis:
When a force is applied to a solid sphere placed on a rough surface, two types of friction forces come into play: static friction and rolling friction. In order for the sphere to roll, the friction force must be equal to or greater than the force applied.

Static Friction:
Static friction is the force that opposes the relative motion between two surfaces that are in contact and not moving with respect to each other. The maximum static friction force can be calculated using the equation:
Fs(max) = μs * N
where Fs(max) is the maximum static friction force, μs is the coefficient of static friction, and N is the normal force.

Rolling Friction:
Rolling friction is the force that opposes the rolling motion of a sphere. The rolling friction force can be calculated using the equation:
Fr = μr * N
where Fr is the rolling friction force, μr is the coefficient of rolling friction, and N is the normal force.

Equilibrium Condition:
For the sphere to roll without slipping, the friction force must be equal to or greater than the applied force. Mathematically, this can be expressed as:
Fr ≥ F
where Fr is the rolling friction force and F is the applied force.

Calculation:
Given that the applied force F is 14 N, we need to find the friction force so that the sphere rolls. Since the question does not provide the coefficient of rolling friction, we can assume that the sphere rolls without slipping, which means the coefficient of rolling friction is equal to the coefficient of static friction.

To find the static friction force, we need to know the coefficient of static friction and the normal force. Since the sphere is initially at rest, the normal force is equal to the weight of the sphere, which can be calculated using the equation:
N = mg
where m is the mass of the sphere and g is the acceleration due to gravity.

Since the sphere is solid, its mass can be calculated using the equation:
m = (4/3) * π * r^3 * ρ
where r is the radius of the sphere and ρ is the density of the material.

By assuming the values of the radius of the sphere, the density of the material, and the coefficient of static friction, we can calculate the normal force and the static friction force. If the static friction force is equal to or greater than the applied force, the sphere will roll.

Conclusion:
To summarize, the friction force required for the sphere to roll can be calculated by finding the maximum static friction force and comparing it to the applied force. By assuming values for the radius, density, and coefficient of static friction, the normal force and static friction force can be calculated. If the static friction force is equal to or greater than the applied force, the sphere will roll without slipping.