# Moving a box with a force that is less than gravity

1. Oct 3, 2012

### kotchenski

1. The problem statement, all variables and given/known data
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=30o. The magnitude of Fo 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 fo that is greater than the gravity on the box

2. Relevant equations
A drawing of the situation:

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

3. The attempt at a solution

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

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

2. Oct 3, 2012

### 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 < Fg

3. Oct 4, 2012

### kotchenski

So if I use that Fnet(x)=Fo*cos(θ)-μn

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

4. Oct 5, 2012

### Simon Bridge

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) ...