How can I find the critical mu of friction?

[ehild]

I see; cosθ ≥ μ*sinθ. But that is so beyond any kind of critical thinking that the textbook has ever expected in any other questions. This is very humbling.

You just have to read a problem text carefully.

The problem said

If μ(static) is greater than some critical value, the woman cannot start the crate moving no matter how hard she pushes. Calculate this critical value of μ(static)

Assume that μ is given. The woman can just set the cart moving if her force is

F = (μ*m*g)/(cosθ - μ*sinθ).

If μ=1/tan(θ) the required force becomes infinite, the women can not exert infinite force no matter how hard she pushes. If μ is greater than that the equation can not be fulfilled by a positive F.

ehild

 

[PhysicsLord]

The answer is as follows: as the woman pushes harder and harder, the effect of gravity becomes less important overall, in proportion to her pushing, in deciding what happens to the box. As we approach this extreme point, or the limit, since mg <<< Fsinø, You can essentially take mg+Fsinø = Fsinø. Now reducing becomes simple algebra.

In other words, in the case that she is pushing infinitely hard, or somewhere near there, if mu_s is cotø, the box will not budge.

 

[haruspex]

mg+Fsinø = Fsinø

I assume that is not what you meant.
Anyway, the student seems to have arrived at the answer in post #27, three years ago.