Question

If μs is greater than some critical value, the woman cannot start the crate moving no matter how hard she pushes. calculate this critical value of μs.

Answer 1

weight = mg acts downwards 
normal force = N acts upwards. 
and force F acts at an angle θ below the horizontal. 
(Let us assume that the woman pushes from the left, so F is acted towards the right, which is below the horizontal) 
so that, Frictional force, f=us*N acts towards the left 

Now we balance the forces along x and y directions: 
y direction: N = mg + F sinΘ 
x direction: us * N = F cosΘ 

We let the value of µs be equal to a value such that any F will not be able to move the crate. Then, if we increase F by an amount F', then the force pushing the crate towards the right also increases by F' cosΘ. Additionally, the frictional force f must raise by exactly this amount. 
Since f can’t exceed us*N, so the normal force must increase by F' cosΘ/us. 
Also, from the y direction equation, the normal force exceeds by F' sin Θ. 

These two values must be the same, therefore:
us = cot θ