Statics and CM - Ladder Problem

[topgun08]

Homework Statement

A uniform ladder length L stands upon the ground; the coefficient of friction there is μ .
It leans safely against a smooth (friction-free) wall with the base of the ladder at an angle
! to the horizontal. Jim is the same weight mg as the ladder, and so is Jim's friend Bill. Jim
finds by experience that if he attempts to climb up it, the ladder falls before he reachs the top.
So Jim asks Bill to stand on the bottom rung of the ladder. Will this work?

The Attempt at a Solution

The ladder doesn't fall when no one is on it because the forces and torques are balanced. The CM of the ladder is in the middle of the ladder (as it is uniform) and as long as the CM stays at or below the the center of the ladder the ladder will not tip. When Jim climbs the latter and goes up past the half way mark the CM of the ladder (and now Jim) changes and is past the half way mark, causing the ladder to fall. So if Bill stands on the bottom rung, even when Jim is on the top rung the CM will never exceed the half way up the ladder.

I believe this is the reason, But why does the the ladder not fall when Jim is below the half way mark of the ladder. Can someone explain the reasoning I gave above?

 

[ehild]

Draw the forces acting opon the ladder and write up the resultant force components and the resultant torque and see if equilibrium can maintain.

ehild