Introduction:
When a force is applied to a disc at its center, it can either roll or slide on a surface depending on the coefficient of friction between the disc and the surface. In this case, we need to determine the minimum value of the coefficient of friction required for the disc to roll instead of sliding.

Explanation:
To analyze this situation, we need to consider the forces acting on the disc and the conditions for rolling without sliding.

Forces Acting on the Disc:
1. Applied Force (F): This force is applied at the center of the disc.
2. Normal Force (N): The reaction force exerted by the surface on the disc in the upward direction.
3. Frictional Force (f): The force opposing the relative motion between the disc and the surface.

Conditions for Rolling without Sliding:
For the disc to roll without sliding, the following conditions must be met:
1. The frictional force (f) should be equal to or greater than zero.
2. The torque produced by the applied force (F) should be equal to the torque produced by the frictional force (f).

Calculating the Minimum Coefficient of Friction:
Let's assume the radius of the disc is R and the coefficient of friction is μ.

1. Torque produced by the applied force (F):
The torque produced by an applied force at the center of the disc is zero since the distance from the center is zero.

2. Torque produced by the frictional force (f):
The torque produced by the frictional force is given by τ = f * R, where R is the radius of the disc.

Equating the torques:
f * R = 0
f = 0

From the condition for rolling without sliding, we know that the frictional force should be greater than or equal to zero. However, in this case, the frictional force is zero, which means the disc will slide rather than roll.

Therefore, the minimum value of the coefficient of friction required for the disc to roll is zero.