A large crate with mass m rests on a horizontal floor. The static and kinetic coefficients of friction between the crate and the floor are μs and μk, respectively. A woman pushes downward on the crate at an angle θ below the horizontal with a force F⃗ .
a)What is the magnitude of the force vector F⃗ required to keep the crate moving at constant velocity?
Express your answer in terms of m, g, θ, and μk.
b)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.
Express your answer in terms of θ.

[tex]F cos \theta - \mu_k (Fsin \theta +mg) = 0\\F cos \theta -\mu _k F sin \theta = \mu_k mg\\F(cos \theta - \mu_k sin \theta) = \mu_k mg\\F=\frac{\mu_k mg}{cos \theta - \mu_k sin \theta}[/tex]

b) [tex]\mu_s=cot(\theta)[/tex]

In this second case, the crate is still at rest, so we have to consider the static force of friction, not the kinetic one.

The equations of the forces will be:

[tex]R-Fsin \theta - mg = 0[/tex] (1)

[tex]F cos \theta - \mu_s R = 0[/tex] (2)

In this second case, we want to find the critical value of [tex]\mu_s[/tex] such that the woman cannot start the crate: this means that the force of friction must be at least equal to the component of the force pushing on the horizontal direction, [tex]F cos \theta[/tex].

Therefore, using the same procedure as before,

[tex]R=Fsin \theta +mg[/tex]

[tex]F cos \theta - \mu_s (Fsin \theta +mg) = 0[/tex]

And solving for [tex]\mu_s[/tex],

[tex]F cos \theta = \mu_s (Fsin \theta +mg) \\\mu_s = \frac{F cos \theta}{F sin \theta + mg}[/tex]

Now we analyze the expression that we found. We notice that if the force applied F is very large, [tex]F sin \theta >> mg[/tex], therefore we can rewrite the expression as

[tex]\mu_s \sim \frac{F cos \theta}{F sin \theta}\\\mu_s=cot(\theta)[/tex]

So, this is the critical value of the coefficient of static friction.