How to Solve a Friction Problem with Statics

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The discussion focuses on determining the minimum horizontal force required to prevent a crate from sliding down an inclined plane, given its mass and the coefficient of static friction. The initial calculations yield an incorrect force of 51.6 N, while the correct answer is 49.5 N. Participants highlight the importance of solving the equations symbolically before substituting numerical values to avoid errors. A key mistake identified is misinterpreting the algebraic manipulation of terms involving P, leading to confusion in the final result. The conversation emphasizes careful algebraic handling to achieve the correct solution.
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Homework Statement



Transtutors001_0f7b69b2-1e48-4ca4-8d27-89c487b0202a.PNG


Determine the minimum horizontal force P required to hold the crate from sliding down the plane. The crate has a mass of 35kg and the coefficient of static friction between the crate and the plane is μs = 0.40

2. The attempt at a solution

Fx = P cos 30 - 343.35 sin 30 + F = 0
Fy = -P sin 30 - 343.35 cos 30 + N = 0

F = (μs)(lNl) = 0.40 (N)

so,

Fx = P cos 30 - 343.35 sin 30 + 0.40(N)

therefore,

Fy = -P sin 30 - 343.35 cos 30 = -N

so, F = 0.40 (P sin 30 + 343.35 cos 30)Sub into Fx,

P cos 30 - 343.35 sin 30 + 0.40 (P sin 30 + 343.35 cos 30) = 0Now solving for P I get 51.6 N, which is incorrect.
The answer is given. P = 49.5 N

Can anyone give me an indication of where I have gone wrong please.

Thank You.
 
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You went wrong when you plugged numbers in your equations very early. Work out the result symbolically, then plug in the numbers.
 
P cos θ - mg sin θ + μs ( P sin θ + mg cos θ) = 0

Still yields the same answer?
 
This is not the result, this is just a step toward it. The result would have the form P = ... where the right hand side would not contain P.
 
P ( cos θ - mg sin θ + μs (sin θ + mg cos θ)) = 0

I'm having major trouble transposing.

P(-mg sin θ + μs (sin θ + mg cos θ)) = -cos θ

P(μs(sin θ + mg cos θ)) = -cos θ + mg sin θ

P(sin θ + mg cos θ) = -cos θ + mg sin θ / μs

P (sin θ) = (-cos θ + mg sin θ / μs) - mg cos θ

P = (-cos θ + mg sin θ / μs sin θ) - mg cos θ)
 
Jud said:
P cos θ - mg sin θ + μs ( P sin θ + mg cos θ) = 0

Open the brackets.

P cos θ - mg sin θ + μs P sin θ + μs mg cos θ = 0

Then collect like terms.

P (cos θ + μs sin θ) - mg (sin θ - μs cos θ) = 0

And this is just one step away from P = ...
 
voko said:
Open the brackets.

P cos θ - mg sin θ + μs P sin θ + μs mg cos θ = 0

Then collect like terms.

P (cos θ + μs sin θ) - mg (sin θ - μs cos θ) = 0

And this is just one step away from P = ...

Sorry but I need bludgeoning in the head because I get 51.6N again, and I cannot see only one step available.

-P = (cos 30 + 0.4 sin 30) - 343.35 (sin 30 - 0.4 cos 30)
-P = -51.669N
P = 51.669N
 
Jud said:

Homework Statement



[ IMG]http://questions.transtutors.com/Transtutors001/Images/Transtutors001_0f7b69b2-1e48-4ca4-8d27-89c487b0202a.PNG[/PLAIN]

Determine the minimum horizontal force P required to hold the crate from sliding down the plane. The crate has a mass of 35kg and the coefficient of static friction between the crate and the plane is μs = 0.40

2. The attempt at a solution
...

P cos 30 - 343.35 sin 30 + 0.40 (P sin 30 + 343.35 cos 30) = 0Now solving for P I get 51.6 N, which is incorrect.
The answer is given. P = 49.5 N

Can anyone give me an indication of where I have gone wrong please.

Thank You.
Solving

P cos 30° - 343.35 sin 30°+ 0.40 (P sin 30° + 343.35 cos 30°) = 0

for P gives 49.4689 .

Check your algebra.
 
Jud said:
Sorry but I need bludgeoning in the head because I get 51.6N again, and I cannot see only one step available.

-P = (cos 30 + 0.4 sin 30) - 343.35 (sin 30 - 0.4 cos 30)

P (cos θ + μs sin θ) - mg (sin θ - μs cos θ) = 0 does not lead to that. P (cos θ + μs sin θ) means P multiplied by (cos θ + μs sin θ), not P plus (cos θ + μs sin θ).
 
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