Solving Static Equilibrum Homework: Can a Man Reach the Top?

[abramsay]

I need an assessment on the homework before I submit it.

Homework Statement

A Ladder of length L and negligible mass leans again a vertical wall, making an angle θ with the horizontal. A work man of mass M climbs the ladder a distance x from the bottom along the length of the ladder. Assuming the wall is completely frictionless, but the ground possesses a coefficient of static friction µ,
(a ) how far up the ladder can the man climb before it slips along the ground?
(b) Is it possible for the man to climb to the top of the ladder without slippage occurring?

Homework Equations

The Attempt at a Solution

Let N1, N2 and Fs be the normal reaction from the wall, the ground and the frictional force respectively.
ƩFx=Fs+(-N1)= 0 ⇒Fs=N1
ƩFy=N2+(-Mg)= 0 ⇒N2=Mg
Taking moment about B,
Ʃԏ=N1(Lsinθ)+(-MgLcosθ)x/L=0
N1Lsinθ=xMgcosθ
N1= xMgcosθ/Lsinθ
⇒Fs=xMgcosθ/Lsinθ= µ_s N2
µ_s=Fs/N2= xMgcosθ/MgLsinθ=xcosθ/Lsinθ
Let y be the fraction of the ladder the man climbs before slipping occurs
Fs=N1 and N2=Mg
N1Lsinθ-MgyLcosθ=0
N1= MgyLcosθ/Lsinθ
But Fs= µ_s N2
xcosθMg/Lsinθ=MgyLcosθ/Lsinθ
y=x/L

 

[kuruman]

Your analysis is correct so far. Is there a question you wish to ask?

 

[abramsay]

Your analysis is correct so far. Is there a question you wish to ask?

Ok.

The second part:
(b) Is it possible for the man to climb to the top of the ladder without slippage occurring?
Do you think I have to show that with my workings or say it in a statement.

 

[kuruman]

Ok.

The second part:
(b) Is it possible for the man to climb to the top of the ladder without slippage occurring?
Do you think I have to show that with my workings or say it in a statement.

Both. Use your equations to find what the minimum coefficient of static friction ought to be so that the ladder does not slip no matter how far up the ladder the man climbs.