Finding the direction of the force knowing the mu static

[Mafalda]

1. A crate resting on a rough horizontal floor is to be moved horizontally. The coefficient of static friction is 0.40. To start the crate moving with the least possible applied force, in what direction should the force be applied?

a. Horizontal

b. 24 degrees below horizontal

c. 22 degrees above the horizontal

d. 24 degrees above the horizontal

e. 66 degrees below the horizontal

Almost 1 hour on this and I really cannot find how to do it! This is the work I’ve done

Fx-fs = 0 <=> Fcosθ = µs . N

Fsenθ + N - W = 0 <=> N = W - Fsenθ

So:
Fcosθ = µs . (W - Fsenθ)

And then I can’t continue! I really need to understand! Thanks!

 

[berkeman]

1. A crate resting on a rough horizontal floor is to be moved horizontally. The coefficient of static friction is 0.40. To start the crate moving with the least possible applied force, in what direction should the force be applied?

a. Horizontal

b. 24 degrees below horizontal

c. 22 degrees above the horizontal

d. 24 degrees above the horizontal

e. 66 degrees below the horizontal

Almost 1 hour on this and I really cannot find how to do it! This is the work I’ve done

Fx-fs = 0 <=> Fcosθ = µs . N

Fsenθ + N - W = 0 <=> N = W - Fsenθ

So:
Fcosθ = µs . (W - Fsenθ)

And then I can’t continue! I really need to understand! Thanks!

Welcome to the PF.

I think I know the answer, but I'd like to see your Free Body Diagram (FBD). Could you please Upload a JPEG or PDF of your FBD for this problem, and show how you did the math to minimize the total force vecror needed? Thanks.

 

[Delta2]

It will help to gather all the terms containing F and $\theta$ in one side of the equation. Then you ll have the equation in the form

$F\cdot f(\theta)=...$ where $f(\theta)$ is a function of theta containing sine and cosine of theta.. In order to minimize F, maximize $f(\theta)$. It will not be too hard to find the value of theta where f is at maximum.

 

[scottdave]

As @Delta² stated, get F on one side, then if you have F multiplied by a function of Θ equal to some constant, you can maximize the function. You can try plotting it vs Θ if you don't know how to find the maximum.