Coefficient of static friction Problem?

Question

Romeo takes a uniform 12 m ladder and leans it against the smooth wall of the Capulet residence. The ladder's mass is 21.0 kg, and the bottom rests on the ground 2.8 m from the wall. When Romeo, whose mass is 72 kg, gets 90% of the way to the top, the ladder begins to slip. What is the coefficient of static friction between the ground and the ladder?

Answer 1

Take moments about the center of the ladder

R2 f = R1 F1
f=R1 F1/R2

R1 - horizontal arm from center of the ladder projected on the x axis to the projected Romeo's position
F1 - Romeo's weight = m1g
R2 vertical distance from the ground to the center of the ladders mass.
f - force of friction and it is directed towards the wall
f= u (Reaction of F1 + reaction of F2)
f=u g cos(A) (m1+m2)
u- coef of friction
u=f/(g cos(A) (m1+m2))
u=[R1 F1/R2]/(g cos(A) (m1+m2))

A - the angle the ladder makes with the wall
A= arcSin(2.8/12)=13.5 degrees
finally

u=[R1 m1 g /R2]/(g cos(A) (m1+m2))
u=[R1 m1 /( cos(A) (m1+m2) R2)

R1= 0.9 x 2.8 - 0.5 x 2.8)=1.12 m
R2= .5 x 12 x cos (A)=5.8 m


u=[1.12 x 75 /( cos(13.5) (75.0+21.0) 5.8)
u=0.196 This is the coefficient if static friction. A bit slippery I would say.

Answer 2

Ok, here we need to apply torques, relative to the point where the ladder touches de wall, the vertical wall, not the ground.

The angle that the ladder makes with the ground :

cos(angle) = 2.8 / 12

sin(angle) = 0.94

Now, we have the weight of Romeo = 72 kg, and he will be at 90/100*12 meters from the wall, in other words :

Distance of Romeo = 10.8 meters

The weight of the ladder = 21 kg

And the reaction from the ground = R

Let's decompose the for ces, and applying torques :

21*9.8*6*sin(angle) + 72*9.8*1.2*sin(angle) = R*12*cos(angle)

1160.7 + 795.9 = 2.8R

R = 698.78 Newtons

Now, let's find the force that the wall exerts to the ladder :

Force = F

F*sin(angle)*12 = 21*9.8*6*sin(angle) + 72*9.8*1.2*sin(angle)

11.28*F = 1956.6

F = 173.45

Then, that force, is equal to the force that is making the ladder to slip :

173.45 = 698.78*u

u = 0.25

That's it

Hope that helps