What is the minimum horizontal force needed to make the box start moving? the coefficients of kinetic and static friction between the box and the floor are .24 and .41

minimum horizontal force required to start the block motion is F = 0.41 mg

Explanation:

As we know that box is placed on the horizontal floor

so here the normal force on the box is counter balanced by the weight of the box

so we will have

now in order to displace the box from its position we need to apply the external force which will just exceed the limiting static friction force

So here we will have

here we know that

so minimum horizontal force required to start the block motion is F = 0.41 mg