Box on a Board: Finding Minimum Pulling Force

[hvthvt]

Homework Statement

A small box of mass m1 is sitting on a board of mass m2 and length L. The board rests on a frictionless horizontal surface. The coefficient of static friction between the board and the box is, as usual, less than μ_s. The board is pulled with a force F, as shown. We wish to know the minimum force necessary to pull the board out from under the box. Express your answers in terms of any of the variables F, L, μ_s,g, m1 and m2.

Find the acceleration of the board when the force of static friction reaches its maximum possible value.

What is the requirement for the board starting to slide under the box?

Homework Equations

F=ma

The Attempt at a Solution

I already solved the largest possible acceleration of the box, which is
F=(m1+m2)a up until m1 falls. I do not actually understand how I can find the maximum possible value. This is probably when the acceleration is the largest I guess?.. Can anybody help me out with these two questions please! Thank you in advance

 

[Pythagorean]

Maybe this will help you understand the maximum value of static friction (where the "threshold of motion" occurs).

http://hyperphysics.phy-astr.gsu.edu/hbase/frict2.html

Yes, it's when the acceleration is largest: when the block goes from not moving to moving.

 

[haruspex]

The coefficient of static friction between the board and the box is, as usual, less than μ_s.

No, that coefficient is usually written μ_s. It would not be less than μ_s. Do you mean the coefficient of kinetic friction?

The board is pulled with a force F, as shown. We wish to know the minimum force necessary to pull the board out from under the box.

What did you get for that?

I already solved the largest possible acceleration of the box, which is F=(m1+m2)a

That is not right. F is unlimited, so it would make a unlimited.
Draw a FBD diagram for the box. What are the forces on it?