What is the Force of Friction on a Painter's Ladder?

[Jbreezy]

Homework Statement

A house painter stands 3 m above the ground on a 5.0 m long ladder that leans against the wall at a point 4.7 m above the ground. The painter weighs 651 N and the ladder weighs 140 N. Assuming no friction between the house and the upper end of the ladder, find the force of friction that the driveway exerts on the bottom of the ladder.

Homework Equations

Ʃ F = 0
Ʃ τ = 0 , where T is torque we use tau in my book I can't find the symbol.

The Attempt at a Solution

I have been at this for some time. I'm stuck I looked at my solution after 2 hours. I understand that the normal force of the wall is equal to the force of friction when you sum the forces in the x direction.
Now you can find the torque
τ = Nw(4.7) - W1(2.5)cos(theta) - Wp(3.0/4.7)(5.0)cos(theta)
I understand this what they are doing in general I can see you will solve for the Nw (normal force of wall). I really am having a hard time with the part I have in bold. I don't know what this ratio is. I also think that I'm having issues with the line of action. If someone could please explain this to me.
Thanks

 

[collinsmark]

Homework Statement

A house painter stands 3 m above the ground on a 5.0 m long ladder that leans against the wall at a point 4.7 m above the ground. The painter weighs 651 N and the ladder weighs 140 N. Assuming no friction between the house and the upper end of the ladder, find the force of friction that the driveway exerts on the bottom of the ladder.

Homework Equations

Ʃ F = 0
Ʃ τ = 0 , where T is torque we use tau in my book I can't find the symbol.

The Attempt at a Solution

I have been at this for some time. I'm stuck I looked at my solution after 2 hours. I understand that the normal force of the wall is equal to the force of friction when you sum the forces in the x direction.
Now you can find the torque
τ = Nw(4.7) - W1(2.5)cos(theta) - Wp(3.0/4.7)(5.0)cos(theta)
I understand this what they are doing in general I can see you will solve for the Nw (normal force of wall). I really am having a hard time with the part I have in bold. I don't know what this ratio is. I also think that I'm having issues with the line of action. If someone could please explain this to me.
Thanks

It's about similar triangles.

Let's start with the big triangle.
Hypotenuse: 5 m.
Opposite: 4.7 m.
Adjacent: Well, we haven't calculated that. But we could if we wanted to.

The triangle that being worked with regarding the part you bold faced, is a similar triangle. It has the same shape as the big triangle, but it's smaller. It starts at the same corner of the ladder meets the ground, but it ends up at where the painter is standing (not all the way to the wall).

You know the painter stands 3 m above the ground. The key phrase here is above the ground. That's not 3 m along the ladder, so you know that that 3 m is not the hypotenuse. It's the Opposite of the smaller triangle. But if you want to be consistent with using the cosine function (to find the adjacent), you'll need to calculate the hypotenuse of the smaller triangle. You can do that using similar triangles, and that's where the (3.0)(5.0/4.7)* comes from.

*(I took the liberty of rearranging the terms a little here)

 

[Jbreezy]

Thanks. I appreciate it. Honestly I never had similar triangles. Which is horrible I need to learn it. It is a skill that cannot be ignored.