Ladder leaning against a vertical wall.

[Reid]

I have encountered some problems with this exercise:

A ladder is leaning against a wall, it will slide due to its weight down the wall and along the floor.

I am supposed to determine the time at which the ladder looses contact with the wall, but how? :S

 

[HallsofIvy]

Calculate the horizontal force against the wall as the ladder slides down. The ladder "loses contact" with the wall when that horizontal force is 0.

 

[pervect]

Easier said than done. I suppose what one could do is first find the Lagrangian assuming the ladder end touches the wall, solve the equations of motion, find the horizontal component of the momentum. The force will be zero when the horizontal momentum is an extremum (maximum). Such a maximum should exist - if we consider the case where the ladder cannot leave the wall the horizontal component of the momentum starts out as zero, and winds up as zero, but is non-zero in between.

 

[GBart]

How could you set this up, though? I had a similar problem in my mechanics class this week and I really want to figure it out. The prof said he used energy to solve it (we'll find out how in 2 days) but there has to be a way to solve it with moments and forces, right? I want to figure it out.

The book says the angle at which the ladder loses contact is $$sin^{-1}(sin(2/3\theta_{0}))$$, where $$\theta_{0}$$ is the initial angle between the ladder and the ground (I think, I left my book in my car, I'll double check later). Working backward has been so far fruitless.