Static Equilibrium - Ladder problem with a twist.

[Emilyneedshelp]

Homework Statement

A board that is of length L = 6.0 m and weight W = 335.7 N rests on the ground and against a frictionless contact at the top of a wall of height h = 2 m (see figure). The board does not move for any value of

greater than or equal to 65 degrees but slides along the floor if

<65 degrees. Find the coefficient of static friction between the board and the floor.

Homework Equations

Σt=0
F(roller)=Fs
Fn=W
Fs=Fn(μ)

Horizontal Forces.
F(roller) is the horizontal force are the roller point.
Fs is the force of friction at the point of contact with the ground and direction towards the wall
Vertical Forces.
Fn is the normal force from the point of contact.
W is the force of the rod.

The Attempt at a Solution

Fn=335.7
Fs=335.7μ
F(roller)=335.7μ

Σt=0=Fs*d+Fn*d+W*d1-F(rol)*d2

d=0,[Fs*d+Fn*d]=0

Σt=0=W*d1-F(rol)*d2

F(rol)*d2, d2=h

Fr(2)=W*d1

W*d1,
d1= L/2cos65

[335.7(6/2)(cos65)]/2=Fr

Fr=335.7μ

( [335.7(6/2)(cos65)]/2 )/335.7 =μ

Am I making a huge mistake with my thought process?

 

[TSny]

One of your forces listed as horizontal is not actually horizontal.

 

[Emilyneedshelp]

One of your forces listed as horizontal is not actually horizontal.

The force of the roller is perpendicular to the to the board, using this I got it to F(rol)cos(25)

Fn=W*d - F(rol)cos(25)

But now I am confused what to do.

 

[TSny]

The force of the roller is perpendicular to the to the board,

OK. Good

using this I got it to F(rol)cos(25)

Is this the magnitude of the horizontal component of the force of the roller? If so, that looks right.

Fn=W*d - F(rol)cos(25)

This looks strange. On the left side you have a vertical force. The first term on the right side looks like a torque. The second term on the right looks like a horizontal force.

But now I am confused what to do.

You know all the forces acting on the ladder and I'm sure you've carefully drawn a free body diagram. Use the diagram to set up the conditions for static equilibrium.

 

[Emilyneedshelp]

F(rol)cos25=Fs=Fnμ
Fn=W

Σt=0=w*d1-F(rol)cos(25)d2

d1=(L/2)sin(65)
d2=h
w*d1=F(rol)cos(25)d2
w*d1/d2=F(rol)cos(25)
F(rol)cos(25)=Fnμ
(W(d1))/2=Wμ

Getting 1.5 out for μ which is ridiculous.

I feel like my issues lies with the d1 selection. I tried it with cos(65) too and still a wrong answer of .7=μ

EDIT: I also just noticed my final answer in the first post is the same as this one. Seems like I need to do some more review.

 

[TSny]

You've left out one of the torques. Note that F(rol) also has a vertical component.
[EDIT: You might find it easier to write the expression for the torque due to F(rol) by finding the lever arm for F(rol), rather than finding separate torques for each component of F(rol). But either approach will work.]

d1 is not quite correct. Make sure you have identified the lever arm for W correctly.

Otherwise, you are close.

 

[Emilyneedshelp]

3cos(65)W-2tan(25)sin(25)Fr-2cos(25)Fr=0

335.7(3)cos65=Fr(2tan(25)sin(25)+2cos(25))
Fr = [335.7(3)cos65]/(2tan(25)sin(25)+2cos(25))
Fs = Fr*cos25 = Fnμ
Fn = W-Fr*sin25

μ=(Fr*cos25)/(W-Fr*sin25)
μ=.688 ANS

Thank you for the help it was the correct answer.

 

[TSny]

OK, good work!