A man is pushing a box of mass m along the ground with a force F. The coefficient of kinetic friction between the box and the ground is μk. The box is accelerating, but then the person stops pushing and the box slides to a halt. The person then starts pushing on the box again with a force F, but the box doesn't budge. The maximum coefficient of static friction between the box and the ground is μs. Identify which of the following statements is true? a. F > μk/μs
b. μk > μs
c. μkmg < F <= μsmg
d. μsmg= F

Option C.

Explanation:

When the box is moving, he pushes with a force F.

in this case, the coefficient is μk, and the box is accelerating, this means that the net force is not zero, so F is larger than the friction force.

When the box is still, the man pushes again with a force F, but now the box does not move, so there is no acceleration, which means that the net force is zero, then F is not greater than the maximum static friction force.

Now, the friction force between an object of mass M, and a surface with a coefficient of friction  μ is:  μ*m*g

where g is the gravitational acceleration.

Then, from the first part, we can conclude that:

μk*m*g < F

(the force F is larger than the kinetic friction force)

and from the second part, we know that:

F ≤  μs*m*g

(The force F is not greater than the static friction force)

If we write those two together, we have:

μk*m*g < F ≤  μs*m*g

Then the correct option is c.