Introduction:
In order for the block to move with constant velocity, the net force acting on it must be zero. The tension in the string will counteract the force of friction, allowing the block to move without accelerating. To find the minimum value of F and the tension in the string, we need to consider the forces acting on the block and apply Newton's second law.

Forces acting on the block:
1. Weight (W): The force due to gravity acting vertically downward. Its magnitude is given by W = mg, where m is the mass of the block and g is the acceleration due to gravity.
2. Tension (T): The force exerted by the string on the block, directed horizontally.
3. Friction (F_f): The force of friction opposing the motion of the block. Its magnitude is given by F_f = μN, where μ is the coefficient of friction and N is the normal force exerted on the block.

Equation of motion:
Using Newton's second law, we can write the equation of motion for the block:
ΣF = ma,
where ΣF is the sum of all forces acting on the block, m is the mass of the block, and a is its acceleration. Since the block is moving with constant velocity, the acceleration is zero, and the equation becomes ΣF = 0.

Analysis:
1. The net force acting on the block in the horizontal direction is given by ΣF_horizontal = T - F_f.
2. The normal force N is equal to the weight of the block, N = W = mg.
3. The frictional force F_f is equal to μN = μmg.
4. Substituting the values into the equation of motion, we have T - μmg = 0.

Finding the minimum value of F and the tension:
To find the minimum value of F and the tension in the string, we need to consider the case where the block is on the verge of moving. This occurs when the force of friction reaches its maximum value, which is given by F_f,max = μN = μmg.

1. When the block is on the verge of moving, the net force acting on it is zero, so T - μmg = 0.
2. Rearranging the equation, we find T = μmg.

Conclusion:
In order for the block to move with constant velocity, the minimum value of F is equal to the maximum frictional force, given by F_f,max = μmg. The tension in the string is also equal to μmg. By setting the tension equal to the maximum frictional force, the block will move with constant velocity and not accelerate.

Great thanks