Moving a box with a force that is less than gravity

[kotchenski]

Homework Statement

The object is a box with a given mass m. Our person has the choice between pushing the box with a horizontal force, or pulling the box with a wire with an angle of θ=30^o. The magnitude of F_o is the force vector he affects the box with in both cases.

Is it possible (If you can freely choose a value for θ) to keep the box moving without using a force f_o that is greater than the gravity on the box

Homework Equations

A drawing of the situation:
http://myupload.dk/handleupload/64d68oBb8KHy8

I believe I need an equation that describes the force F_o of θ, which I've found is given as:
F_o(θ)=\frac{\mu_k mg}{cos(θ)-μ_k sin(θ)}

The Attempt at a Solution

I could choose to solve F'_o(θ)=0 for theta which gives θ=arctan(μ_k)

This would describe the minimum force required but I don't know how to relate that to F_g

 

[Simon Bridge]

What is the relationship between friction and the weight of the box?

But what would the turning point tell you?
Don't you want to know where F₀ < F_g

 

[kotchenski]

What is the relationship between friction and the weight of the box?

But what would the turning point tell you?
Don't you want to know where F₀ < F_g

So if I use that F_net(x)=F_o*cos(θ)-μn

And that F_net(x)<0, then the value of θ<arctan(μ), and therefor I'm applying a force F_o that is less than gravity? Wouldn't that technically mean the box is not moving and is being held back by the friction?

 

[Simon Bridge]

Wouldn't that technically mean the box is not moving and is being held back by the friction?

Why would that be? Gravity is mg... and force less than mg would be less than gravity. To overcome friction it just has to overcome μmg.cos(θ) ...