What is the Force Required to Move a Crate on an Incline Plane with Friction?

[dolpho]

Homework Statement

Find the force necessary to start the crate moving, given that the mass of the crate is 32 and the coefficient of static friction between the crate and the floor is 0.58. The angle is 21 degrees below horizontal.

Homework Equations

The Attempt at a Solution

I don't really understand where to go from here, I've found out the answer but really don't understand why we use the equations we do. Would appreciate any help regarding the logic behind the methods!

My attempt:

Weight = 32kg x 9.81 = 313.9 Newtons
Normal Force = 313.9 cos(21) = 293 Newtons
The force parallel to the plane = 313.9(sin21) = 112.5N

The Force of Friction = 293 x .57 = 167N

Now this is pretty much the farthest I can get without understanding what I'm doing. I think I did the above steps right I'm not completely sure. Would appreciate any help.

 

[SHISHKABOB]

so the crate is sitting on some slope, but is not moving because the frictional force acting on it is greater than the force due to gravity pushing it down the slope (which is the weight parallel to the surface)

so right now F_fric > F_weight

let's say you apply some force F in the same direction as F_weight

how big would F need to be in order to make F_weight + F > F_fric?

 

[dolpho]

so the crate is sitting on some slope, but is not moving because the frictional force acting on it is greater than the force due to gravity pushing it down the slope (which is the weight parallel to the surface)

so right now F_fric > F_weight

let's say you apply some force F in the same direction as F_weight

how big would F need to be in order to make F_weight + F > F_fric?

I guess any number that you can add to make the force greater than the frictional force? So in our case the force parallel is is 112.5 and the Frictional force is 167. So we need an extra 54.5 Newtons to make the crate move.

 

[SHISHKABOB]

pretty much

 

[dolpho]

pretty much

So how would that help us in getting the answer of 250 Newtons?

In the solution they put,

(.57) (313) / cos21 - .57sin21 = 250 Newtons

I don't really get how we end up there from what we have so far.