The Toppling Stack: Finding Equilibrium and Minimum Values

[mrknowknow]

Homework Statement

The Toppling Stack

There is a stack of N boxes (rectangular solids) on a level frictionless table. Each box has the same uniform density, has the same mass M, and has the same dimensions L x W x H. The bottom box is pulled with a constant force F parallel to the table. The dimension H is perpendicular to the table and perpendicular to F; the dimension L is parallel to the table and is parallel to F; and the dimension W is parallel to the table and perpendicular to F. The coefficient of friction between the boxes is mu_s. Assume the acceleration due to gravity, g, is constant for the entire stack. The bottom box is Box 1, the box directly on top of Box 1 is Box 2, ..., the top box is Box N.

If F is large enough, some boxes will fall by toppling over (not sliding off).

a) Show mathematically that the separation should occur between the Box 1 and Box 2.
b) What is the minimum value of F which will cause the boxes to fall. Answer in terms of g, H, L, M, N, and / or W.
c) Find the minimum value of mu_s which will make falling over impossible without sliding first. Answer in terms of g, H, L, M, N, and / or W.

The three parts can be done separately.

Homework Equations

F=μ_kN

The Attempt at a Solution

I drew a picture of 4 boxes and labeled the forces but I don't know how to proceed further...

 

[haruspex]

Consider the stack as a set of N-n blocks on top of a set of n blocks.
Taking moments about the point where the upper set might topple from the lower set, what inequality do you get for equilibrium? (Put in an unknown 'a' for the linear acceleration.)