What is the minimum angle at which the ladder will not slip?

[formulajoe]

a ladder of mass M and length L leans at an angle theta against a frictionless wall. coefficient of static friction between ladder and ground is MU, what is the minimum angle at which the ladder will not slip? this problem sounds fairly simple, but I've been looking at it for the past 20 minutes and have no idea where to begin.

 

[gnome]

The approach is the same as the boom problem.

Begin by drawing a force diagram.

Then figure out the horizontal & vertical components.

Then pick an axis & figure out the torques.

Hint: Even though it may not be obvious, you do have to consider rotation to solve this.

Another hint: there are 4 forces acting on the ladder. Can you identify them all?

 

[formulajoe]

the forces are gravity, friction, normal force from wall and normal force from ground right?
Fn from ground = mg sin 90-theta correct?
but how would i found the normal force from the wall?

 

[gnome]

how would i found the normal force from the wall?

How many horizontal forces are there? What are they? What must their sum be, remembering that the ladder (hopefully) isn't moving?

Fn from ground = mg sin 90-theta correct?

Before I answer that, what is the magnitude of the normal force? How many vertical forces are there? What must their sum be?

 

[formulajoe]

2 horizontal forces and two vertical forces. each is a component of the normal force.

normal force = m * g, sum of vertical forces must be zero.

 

[gnome]

normal force = m * g, sum of vertical forces must be zero.

So do you want to change your statement about the magnitude of the friction force?

2 horizontal forces and two vertical forces. each is a component of the normal force.

I'm not sure I understand what you're saying there. There are TWO normal forces in this problem, aren't there?

 

[formulajoe]

there is a normal force from the wall and a normal force from the ground. there is a horizontal force and a vertical force component vector of each.
Friction = m*g*tan theta * mu right?

what kind of problems are these anyways? I've never seen anything this complicated.

 

[gnome]

there is a normal force from the wall and a normal force from the ground. there is a horizontal force and a vertical force component vector of each.

REALLY think about this. What is the direction of the normal force from the ground? Does it really have a horizontal component? What is the direction of ANY Normal force relative to the surface that applies it?

Friction = m*g*tan theta * mu right?

Now you're just guessing, right? Why tan theta? Why anything theta?

ive never seen anything this complicated.

It all depends on your point of view. Wait till you see problems on electrical and magnetic fields. Then these statics problems will seem simple.

 

[formulajoe]

is this right? i did sum of torques to find normal force from wall, which ended up being d*m*g/2*h.
than i figured theta = arctan mg/(dmg/2h)

am i getting close?

 

[gnome]

Where did d and h come from? Don't introduce new variables. You already have enough of them.

 

[formulajoe]

d is distance from base of ladder to wall and h is height. don't i need these for the sum of torques?

 

[gnome]

No. You should express any distance that you need in terms of L and theta.
i.e. some fraction of L times some function of θ

Also, I just realized, for this problem you can't use the point where the ladder touches the floor as the pivot for your torque equation,
[[because then μ drops out, and you need μ for the solution.]]
edit: the preceding statement should be:
because then θ drops out, and you need θ for the solution.

So figure the torques about a pivot either where the ladder touches the wall, or at the center of mass. Either way, there will be 3 torques in the equation.

 

[gnome]

Unless I blew it, you should end up ultimately solving for θ as a function of μ.

All the other factors -- m, g and L -- drop out.

 

[formulajoe]

sum of torques = 1/2L * m + cos theta * L * m + sin theat * L * m = 0
thats center of mass, point on wall and base of ladder.
how do i figure in mu?

 

[gnome]

Take the point where ladder meets wall as the pivot.

Then (assuming the wall is on the left), the clockwise torques are
due to gravity:
(mgL/2)sinθ

and due to the friction:
μmgLcosθ

The counterclockwise torque due to the normal force from the floor is
mgLsinθ

Now, that's really all you need. Make an equation from that & solve for θ.

 

[formulajoe]

so theta = arcsin(sin theta/2 + mu cos theta)

 

[gnome]

(Sigh)

(mgL/2)sinθ + μmgLcosθ = mgLsinθ

divide by mgL:

(1/2)sinθ + μcosθ = sinθ

μcosθ = (1/2)sinθ

divide by cosθ, multiply by 2:

tanθ = 2μ

θ = arctan(2μ)

If θ is any bigger than that, (1/2)sinθ will be greater than μcosθ, and the ladder falls.

 

[formulajoe]

sorry dude, i don't mean to look like a total idiot.
after i posted the answer last night i realized it was wrong. i figured it out to be the arctan statement you gave. thanks for confirmation.

 

[gnome]

No problem. If I didn't enjoy helping I wouldn't be here.