How high up on a ladder can he go without the ladder slipping?

[rachie9]

The ladder is 5m leaning on a wall, and it is 2.5 m away from the bottom of the wall, so the angle between the ground and the ladder is 60 degrees. The contact interaction between the ladder and the ground has a static friction force of no more than .4 mg, and mg is the weight of the person on the ladder. How high can he climb w/o the ladder slipping?



The relevant equations are acceleration = 0 and torque = 0 because the ladder isn't moving.



I know that the static friction and normal force from the wall on the ladder must equal 0, and that the normal force from the ground plus the force of gravity must equal 0. Since the torque is 0, I think the total force x length of ladder x cos 60 must equal 0, but I'm not sure how to break up the individual forces and fit them into this equation.

 

[Hootenanny]

The most important point to realize here is that the net torque on the ladder must be zero, as you said.

To start, try taking the torques about the upper end of the ladder.

 

[physics girl phd]

I think the total force x length of ladder x cos 60 must equal 0

Note: This statement is definitely incorrect -- it doesn't say net torque is zero. In fact, I don't think it is mathematically identified with anything physical about the problem.

but I'm not sure how to break up the individual forces and fit them into this equation.

Heres an outline of what you need to do in ANY statics problems (even when something like a distance is unknown, and variables have to be carried along):

1) Find all the bits of force and locate them along the length of the object with an orientation. some of these are related to gravitational force, some of them are related to other forces (like you say: friction, normal forces).
i.e. DRAW A FREE BODY DIAGRAM of the ladder!

2) Make lists of your forces in some coordinate system that has perpendicular axes: You can list either of:
a) Upwards forces vs. downwards forces
b) Forces parallel to the ladder versus forces perpendicular to the ladder
Note: you may have to break forces into components and use these components in your chosen listing.

You will use the above information to make a net force equals zero equation.

3) Chose a pivot point for referencing torques an identify the forces that cause torques when the system is oriented from the pivot.
edited to add: Hootenanny gives a wise choice for this pivot... why is the choice smart?

4) Find the components of those forces that are parallel to and perpendicular to the ladder.

5) Find all the moments of inertia/torques using ONLY the components of forces perpendicular to the length of the ladder -- using your CHOSEN pivot.
(I make separate lists of clockwise torques and counterclockwise torques).

You will use the step 3-5 information to make a net torque equals zero equation.

Can you start to fill in these steps?

 

[rachie9]

Thanks for your help.

I'm not sure how many components are in the net torque equation. I know there is a torque from the force of gravity, is there one for all the normal forces (acting on the ladder from the person, from the wall, and from the ground) as well? What about from friction?

Once I have my net torque = 0, how do I relate this to my net force = 0 equation?