What is the Relationship Between Friction and Horizontal Force?

[caduceus]

Homework Statement

There is a heavy box across the floor, and you apply a horizontal force just sufficient to get the box moving. As the box starts moving, you continue to apply the same force. Show that the acceleration of the box, once it gets started, is

a=(us-uk)*g

Homework Equations

uk <= us
Fk=uk*Fn

The Attempt at a Solution

I thought that at the initial condition there will be Fs,max=F where Fs,max=us*Fn=us*mg

When the body starts moving the disaster comes!

Since Fk=uk*Fn=uk*mg

Well, I could not combine these equations since something is missing...

What method shall I apply for this question?

Any helps will be appreciated.

 

[PiratePhysicist]

Well when the box first starts moving the force applied will be:
$$F_{applied}={\mu}_s F_N$$
And it stays at that value. While moving the frictional force is:
$$F_k=-{\mu}_k F_N$$
So the acceleration will be equal to the net force divided by mass.
$$F_{net}=F_{applied}+F_k={\mu}_s F_N-{\mu}_k F_N=( {\mu}_s -{\mu}_k )F_N$$

 

[caduceus]

Thanks for your assistance.