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?

 

[rwisz]

To solve this problem, we can use the concept of equilibrium. This means that the net force in both the x and y directions must be zero for the block to remain at rest.

In the x direction, we have:
Pcos(30) - mgsin(30) - μN = 0

In the y direction, we have:
Psin(30) - mgcos(30) + N = 0

We also know that the normal force and the force of friction are related by:
N = μN

Substituting this into the equations above, we get:
Pcos(30) - mgsin(30) - μμN = 0
Psin(30) - mgcos(30) + μN = 0

Now, we have two equations with two unknowns (P and N). We can solve this system of equations using algebra or a graphing calculator to find the values of P and N.

Once we have the value of N, we can use it to find the force of friction, F_s, using the equation F_s = μN.

Therefore, the smallest force P required to cause the block to start sliding up the incline is:
P = μN + mgsin(30)

We can also find the direction of this force by looking at the direction of Pcos(30) and Psin(30). Since the block is on an incline, Pcos(30) is pointing up the ramp and Psin(30) is pointing perpendicular to the ramp. Therefore, the direction of the force P will be slightly up the ramp.

In summary, the magnitude of the smallest force P required to cause the block to start sliding up the incline is given by the equation P = μN + mgsin(30), and the direction of this force will be slightly up the ramp.