Solve Friction Problem: Find Force P on 20kg Block on 30° Incline

[Patdon10]

Homework Statement

a 20-kg block rests on a 30 degree incline. knowing that the coefficient of static friction is 0.25, determine the magnitude and direction of the smallest force P required to cause the block to start sliding up the incline.

Homework Equations

F = ma

The Attempt at a Solution

I'm trying to solve it using x and y components. For this example, I consider the X direction to be straight up the ramp, and the y to be in the direction of the normal force. The problem is I have 2 equations and ?3? unknowns. How can I solve for Normal force if the normal force is being affected by the pull in the y direction. I've tried to solve this problem many times with no luck. Can someone point me in the right direction?
My teacher told me it would be much easier to solve this using a force Triangle, but I would prefer to use this method. If there were some rule, saying the angle is so-so, because that leads to the least pull this would be a very simple problem.

F_x = Pcos(θ) - mgsin(30) - F_s
F_y = Psin(θ) - mgcos(30) + N
F_s = μN

 

[tiny-tim]

Hi Patdon10!

The problem is I have 2 equations and ?3? unknowns.

That's not a problem … it's an opportunity!

For any value of θ, those equations give you definite values of P and N.

Now choose the minimum value of P. ​

 

[Patdon10]

so I can literally plug in any value for θ and get the values of P and N? If I'm understanding what you're saying...any value of θ will always give the same value of P and N or will the numbers vary and I just need to play with it until I get the lowest value of P?

Is it fair to assume that θ has to be At Least 30 degrees because it's not being pulled into the ramp?

 

[tiny-tim]

Hi Patdon10!

(just got up :zzz: …)

so I can literally plug in any value for θ and get the values of P and N? If I'm understanding what you're saying...any value of θ will always give the same value of P and N or will the numbers vary and I just need to play with it until I get the lowest value of P?

They'll vary …

it'll be something like P = cosθ + sinθ, which you'll have to minimise.

Is it fair to assume that θ has to be At Least 30 degrees because it's not being pulled into the ramp?

Yeah, I suppose so … but does it save any time?